# 5D Mathematics

Logic and math are the 2 entangled stiences of spacetime laws. As such in a Universe made of space-time beings, contrary to belief they are the most experimental of all stiences. Mathematics is the main science of vital, scalar Space (represented by spatial points and scalar numbers), in mirror symmetry with the less developed Time focused science of ¡ logic that analyzes the 5 Time dimotions of reality, as humans use only 1 time arrow -entropy- but… ‘God the seer of time is of a logic higher than man’s’ Augustine.

We shall thus upgrade in 2 papers to maths main 2 subfields, S@-Geometry & ∆T-Algebra to 5D.

In this one we upgrade Nº theory & geometry to show mathematics as a logic mirror of the Universe’s pentalogic scalar structure made with scalar numbers, Non-E fractal points with inner dimensions and timespace S<≈>T operands that reflect the dynamic symmetries & stœps of its 5 spacetime dimotions. Ænthropic men instead justify math with an ego-trip of digital creationism: Only he & God speak the language, math, that creates reality -not the other way around, and try to prove its truths with an incomplete (Gödel) axiomatic method, as an ideal, perfect absolute ‘truth’ with NO correspondence with reality (sets as ‘imagined’ units of math). We shall do the opposite: departing from reality we construct a vital improved mirror of maths, where even the perceived irregularities of mathematics are experimental true symmetries of a vital dynamic Universe: i.e. irrational numbers are required NOT to close a vital pi-cycle and √2 triangle to allow openings into its lower decimal scales to absorb energy and information.

So we shouldn’t force ‘idealist continuity’ in a discontinuous Universe. i.e. 3 topologic geometries with motion suffice to mirror the 3 St, sT, ST combinations of an organism because we ARE spacetime’. i.e. numbers are indistinguishable social groups (polytopes) because 5D is ‘social’; its Prime, T>S + odd, S<T male numbers vs. Even female s=t numbers are the 1st digital reproductive mirror symmetry, because the Universe is reproductive. Calculus is all pervading because it studies the Spacetime dimotions mirrored by 5 operands, 5 number families, 5 geometric postulates, etc. Topologic proofs of algebraic equations are possible because of S-form=T-motion symmetry. And so on. MATHS ARE the best mirror humans have of space-time laws (as our logic systems are too simple). And it is a mental orgasm to see its vital reproductive properties once we decode and give life to its images.

Thus this 1st volume, still a work in progress, will define its real units, fractal points and social numbers, its vital topologies and S=T symmetries in 3 scales of complexity: 1) S@(• point)=∆t (Nº); 2) mind-spaces = numbers’ 5 families united in 5 planes of reference & 3) statistic space populations=time probabilities…

So we shall evolve both formal sciences from its Euclidean ‘continuous’ single space view of ‘points without parts’ into a full Non-Euclidean fractal system of discontinuous points that create waves instead of lines, which evolve into networks and topological planes=organisms; as we are all made of vital mathematical spaces’ and we shall evolve, the simple unicasual Aristotelian logic, of a single lineal A->B time arrow, into the higher pentalogic of the Universe made of 5 Dimensional motions of space-time (ab. Dimotions), entangled to create reality. Both sciences are immense in scope; but humans have reached a high degree of dexterity in the understanding of mathematics, so we shall study first the science of space in two volumes, this one dedicated to its experimental foundations, fundamental units, points of space and scalar numbers, its symmetry and Non-E Geometry, with a brief introduction to its most extensive Time Algebra.

5D mathematics thus ads to the axiomatic method an experimental view of the discipline as a mirror language of the 5Dimotions of space-time that create all the systems of the Universe, and mathematics reflects from multiple perspectives (sub-disciplines, operands, geometric postulates, pentalogic use of each of its elements, families of numbers, etc. etc.).So we shall enlarge the foundations of mathematics and illuminate beyond the present ‘creationist beliefs’ of mathematicians, as the language of God the meaning of its components, as mirrors of the 5 Dimotions and pentalogic, entangled structure of reality.

Further more because 5D metric (SxT=K) implies that smaller systems (Min. S) have paradoxically more information (Max. T), as ‘seeds’ that develop its synoptic form through the immanent program of reality, the enormous synoptic power of mathematical elements, social, scalar numbers and operand makes it the main language of the smallest Timespace beings – particles and atoms, which likely create their local order through ‘Existential algebra’, the vital mathematical interpretation provided by 5D.

Thus we first study the pentalogic of its 5 main sub disciplines, each one based in a pentalogic, slightly different view of the 5 Dimotions of reality: Geometry and its 5D Non-Euclidean fractal points of space; Algebra and its 5 operands expressing each of the 5 Dimotions of time; Number theory and its 5D families of scalar social numbers; self centred in the 5 D main frames of reference of bidimensional space, also corresponding with the 5 Dimotions (2D lineal=cylindrical, 5D Cartesian=flat, entropic, 1D Polar; 3D vectorial and 4D, complex planes), negated by 5D entropic inverse operands. We study in 2 volumes those amazing mirror symmetries between the pentalogic of the Universe and the pentalogic elements of mathematics in all its main sub disciplines. As both reality and its mathematical mirrors are constructed with the 5 dimotions of ¬∆@st , dust of space-time, the substance of which we are all made.

Because mirrors are fractal synoptic images of the whole, mathematics also has an internal consistency described by the axiomatic method, but the advance provided by 5 D mathematics is that now we know where the laws of mathematics come from, as a mirror of Scalar time space systems. We shall therefore put mathematics in correspondence with the laws of fractal space and the pentalogic of cyclical time, in 2 parts, given the extension of the discipline ; this first volume will introduce 5D with emphasis in its mathematical mirror images; and then considering the new units of 5D mathematics, the fractal point; its scalar equivalent the number; and its spatial stience, geometry in its 3 ages and 2 forms, mental spaces and vital topology; leaving for a future 2nd post, the wider fields of Algebra and Analysis.

Newton’s absolute space-time was proved wrong by quantum physics (space is broken in quanta and organised in ‘fractal scales’ of different size), and relativity (there are ∞ time clocks in the Universe with different speeds). But if there is not an absolute background space-time we are as Leibniz wanted relational space-time beings that occupy a vital space and last a time duration; and so Space and Time Generate the properties of all systems made of them; which explains why mathematics, the science of space and logic, the science of time, have so many wide applications in all stiences each one studying a relative scale of size of the fifth dimension. We shall thus show math as an experimental science, relate its laws to those of space=geometry and time=algebra and consider some examples on the emergence of those laws in ‘higher ‘stiences’, in this introductory course on 5D math.

The article is a work in pogress that won’t be complete till reaching Boolean algebra. So far we have posted an intro to 5D Generational SpaceTime, foundations of mathematics as the language of entangled space, point-S=∆-Number symmetry, Nº theory, mind geometry & translation to 5D of Greek 1st age, bidimensional ST-ill Holographic geometry: Pythagoras, Fermat postulate’s Margin Proof, timespace spirals, 1D trigonometry, Euclid’s axioms.

Since if we look at maths as a space-time mirror each LAW reflects 1 ST law. So we improve Einstein’s quip: ‘I know when mathematics is truth&when is real’.

In its 1st young and 2nd mature age almost all maths are ST-reality balanced mirrors though its 3rd inflationary age is filled with idealist distortions, redundancies & language fictions, regardless of its beauty. So the value of 5D mathematics is to return the discipline to its experimental nature, doubling its proofs that must be both consistent within the structure of the mirror (axiomatic method), but also a useful focused image of a general Spacetime law.

This lack of focus with reality needless to say is the main handicap of cosmological grand theories of reality including the big-bang and black hole evaporation, falsified in our post on 5D Cosmology.

**MATHEMATICS AND 5D**

** ****SUMMARY**** **

**Introduction.**

** The Fractal Universe**: its bio-topo-logic properties, its 5 Dimotions & s=T symmetries.

** Pentalogic of Time:** Existential Algebra and classical Algebra. Its operands as Dimotions.

**Book I. The spatial view. Geometry. Its S=T symmetry. Numbers**

**Fractal Space:** ¬E Points. ¡logic Geometry. Its 5 Non-E Postulates as Mirrors of scalar growth.

** Philosophy of Mathematics:** 5 sub-disciplines as mirrors of ¬∆@st. 3 Ages. S-point=∆-numbers symmetry.

**∆-scales: **Number theory. S=T Dualities: Space Points = Scale numbers. Polygons. Primes. Closure: N,Z,R,Q,C.

**3 Ages of Geometry**: 2D Geometry>Analytic Geometry >5D hyperbolic geometry & Topology.

**∞ Mind Spaces. **Its different p.o.v.s Artistic Geometries.

**The Geometry of the 5 ^{th} Dimension. **Network spaces.

**IX**.**1**Geometry in the bidimensional plane. Conics & Curves.^{st}**2**Analytic geometry: S=T symmetries. Differential Geometry: Motions of a point. Vectorial Spaces^{nd}Age.**3**Hyperbolic geometry & Topology. Explosion of Mind spaces.^{rd}age.

** The future of Geometry:** Vital Mathematics

**Book II. The Temporal view. Algebra.**

**Part I. Algebra.** 5 Dimotions as Operands. Equations. Polynomials. Groups.

**Part II. Analysis.** Finitesimals and wholes. 5 Dimotions of space-time. Calculus. PDEs, ODEs. Functionals.

**Part III.** **Group theory**. Sets.

**The ±¡ future: Existential Algebra.**** **

**Part IV.** **5D Existential Algebra. **The function of exist¡ence as a generator of algebraic functions..

**Part V.** **Digital Thought.** Boolean Algebra. AI: The function of existence. The conscious Algorithm of Information.

** **** ****Theme.** This post introduces an elementary translation of mathematical elements as a mirror language of the 5D Universe and its ‘primordial dimotions and elements’ to prove its experimental nature. As usual the article won’t be completed till reaching 200 pg. completing the summary. And even then will be a very small part of my research in the field in a discipline that if ever completed by future researchers will recast the entire subject of mathematics as set theory did in the XX c.in terms of its experimental power to describe the ultimate laws of ‘mental spaces’, which emerge from fractal points – mind-seeds that code the game of exist¡ence with pentalogic numbers, to develop ¡ts territorial local order as the language imprints variations of species on the underlying timespace we are all made of.

So far the article reaches only to Geometry’s I age –existential algebra in II book, as each stience requires a similar upgrading so I keep switching my free time between the 4 ‘blocks’ of formal, social, biologic & physical stiences.

We first study how mind languages mirror supœganisms made of scalar space and cyclical time. So we introduce the metric equation that describes the scales of the 5^{th} Dimensional Universe: SxT (max.s=t) = C. Then we couple the 5D elements of reality (entropy, scales, minds, space and time) in its 3 ‘relative scales of size and time duration’, as fast 5 Dimotions=actions (short view), superorganisms tracing worldcycles (medium view) and ¬∆@st elements (long view) with the 5 disciplines of mathematics as experimental mirrors of those 5 elements: Space=geometry, ∆: Scales: Number theory, S<=>T Dimotions and Symmetries=algebra; which analysis is best for time motions; @-minds = Analytic geometry (Frames of reference) and philosophy of mathematics; while entropy, negation of information appears in the limits of calculus, inverse operands and exponential functions.

Since, as scalar space includes scales & ‘still mind mappings’ and Time the entropic limits of death and all its motions; we can reduce mathematics to spatial geometry and temporal algebra, which indeed were the original disciplines, from where all others branched, and to its minimal units, points of space and numbers of time, which appeared even earlier in the human consciousness. We shall thus consider the S=T symmetries between fractal Non-Euclidean space points (ab.•) and numbers (ab.Nº); and latter on between Topology and Algebra; and couple the internal elements of S-geometry (dimensions) and T-algebra (operands) with the 5 Dimotions (dimensional motions) of the fractal Universe.

So mathematics is only 2^{nd} to ¡logic in the quality of its experimental mirror expressing the Disomorphisms (equal dimensional laws) of all systems of Nature.

As each of its 2->5 subdisciplines reflect those 5 Dimotions of the Universe, which all stiences expresses in different forms. In mathematics geometry does so through 3±¡ topologic bidimensional varieties and classic 3 dimensions (1D: height, 2D: Length, 3D: width; 4-5D: fractal dimensions and topologic networks). While algebra mirrors the 5 Dimotions with its 5 operands & inverse entropic functions (1D: sin/cos, 4D:±, 3D: x÷, 4D: xª √, 5D:∫∂). So finally we focus on the medium view of ‘entangled supœrganisms, in simultaneous space’, as expressed by vital topology, tracing ‘worldcycles’ as expressed by Existential algebra, which become the new ‘integrated’ 2 polar disciplines of 5D mathematics that should make it a better mirror of the fractal Universe.

** FRACTAL 5D METRIC:**** A UNIVERSE MADE OF B¡O-TOPO-LOGIC TIME§PACE BEINGS.**

In the graph the Universe is a fractal that reproduces ‘forms with motion’, informations and then **organizes** them in networks and systems that evolve into larger organic systems creating the scalar structure of reality. Thus we call *the sum of all those co-existing scales of parts and wholes the fifth dimension.*

Then it is necessary to find a metric equation to define this new dimension of space-time. Since a dimension only exists when we can write a mathematical simple metric that leaves the dimension invariant when we change our parameters of space and time – hence we travel through it. (Klein). This equation, as all space-time metric equations, is simple; since metric equations are meant to represent measures of ‘covariant’ motion in a given space-time dimension that leave the other dimensions unchanged. So we write using ð for cyclic time instead of t, for a motion that changes the relative size and speed of clocks of a system (measured with frequency):

**5D Metric: S (Lineal Size/Volume in space) x ð (cyclic speed of its time clocks) = Constant.**

According to those metrics, smaller systems in space have faster time clocks. As information is stored in the frequency and form of those cycles, smaller systems have more information, coding larger ones: genes code cells, memes societies and particles’ quantum numbers code atoms and molecules.

This equation and its use to improve our knowledge of space and time in all sciences, with an emphasis in our models of physical systems will be the theme of this paper. Even if physicists stubbornly refuse to treat information with the same value than entropy. So they call it negentropy, and when you give a conference on the fifth dimension – the dimension of ‘creation of social forms of information, of organic wholes’ – there are no physicists on attendance; and likely no physicists will be reading this post… Let’s then use the metrics of the 5^{th} scalar dimension to explain the fractal, nested Universe and its scales, shown in the graph:

The metric equation of the fifth dimension of space-time (ab.∆) defines 3 known scales of physical systems, with different quantity of information according to 5D metrics, Se (size in space) x Ti (volume of information) =k. Since as we become smaller in space paradoxically our time clocks accelerate, and since information is stored in the cyclical patterns and frequencies of those clocks smaller systems code more information, so quantum particles code atoms, genes organisms, memes civilizations and chips machines establishing the essential symbiosis between ∆-1 scales and ∆º super organisms, inscribed in an slower ∆+1 world.

Those metrics means information is higher in the smaller ‘quantum plane’ than in the larger gravitational one, and inversely the size of its physical parts is larger ins the Gravitational cosmological ‘plane’ than in the quantum one, with the human thermodynamic scale in-between.

As there is no reason to stop the scales of the fractal Universe in particles and galaxies, there is a ‘potential’ fourth, ∆±4 organic plane defined ‘above’ the galaxy, (∆+4, dark energy world) and below the quantum world (∆-4, Bohm’s quantum potential), which represents the larger cosmos. Further on, according to the fractal, nested principle any larger organic system, encloses smaller nested systems. Thus the ∆±4 cosmos contains ∆±3 galaxies, which contain ∆±2 solar systems and planets, which contain ∆±1 thermodynamic organisms and matter states, described by the human ∆º mind languages, contained on our brains, *which according to those metrics* will have a much denser content of information becoming a ‘linguistic Mind-Mirror’ of the whole.

**RECAP.** It follows from that nested structure and 5D metrics that speed up information processing in smaller spaces, a symbiotic relationship between ∆-¡, the smaller parts of faster time clocks, which carry more in-form-ation in the form and frequency of its logic cycles, and act as languages that code the larger systems: genes code cells, memes code societies, quantum numbers code atoms and so languages are mind-mirrors of wholes.

**The underlying question of time§pace physics: Absolute or Relational, Generational Space-Time **

The fundamental question physicists wondered for centuries regarding the nature of space and time unfortunately was resolved as usual in favor of the simpler view: it is space and time an absolute abstract background of the Universe (Mr. Newton’s view) or are we made of ‘vital space’ that lasts a time duration, so we are generated by the bio-topo-logic properties of scalar space and cyclical time? This is the choice of 5D ‘stiences’. And its simpler version was called relational spacetime, sponsored by Mr. Leibniz. A realist interpretation of the world we live in, which has never shown in any scale of reality such ‘background’ – ultimately a mathematical graph used in abstract by human scientists – considers that we ARE the vital space we occupy with our cells, and we LIVE a cyclic time duration between birth and extinction. So *we are* space and time. And must extract our properties as ‘existential beings’ from them.

In Newton’s cosmos, space and time provide a fixed, immutable and eternal background, through which particles move. Space and time are the stage of intersecting lines sketched in the illustration. Fact is this ‘mathematical artifact’ made with pen and paper by earlier physicists, called the Cartesian graph, useful to measure ‘translation in space’ is no where to be seen in reality. Unfortunately as time went by the graph became somehow ‘real’ as scientists’ felt the ‘mathematical language’ created reality. It meant also the invention of an absolute ‘continuous space’ and a single ‘lineal time’ that extends to infinity contradicting the obvious fact that all ‘spaces’ are broken, divided by membranes, and all beings have a finite time duration. Further on, as we kept exploring smaller scales of reality, we never found the drawings of God, not even a solid still substance, but always ‘motions’ tracing closed time-space cycles; since even particles turned to be also ‘vortices of time-space motions’.

The true, sound experimental and logic theory was Leibniz’s, who considered absolute space and time an abstraction, and so he coined the concept of relational space -merely the adjacent pegging of similar forms in simultaneous space and relational time – the sequence of events which we relate causally with reason origin of the ‘Generational space-time’ model of 5D in which are the space we occupy and the time we last – as in the graph where there is no longer abstract background lines. What Newton called absolute space-time IS NOT. So space is the sum of all the discontinuous vital spaces, occupied by different beings: ∑s=S And lineal time, T the sum of all the finite life-death cycles of all beings T=∑t.

**Since space & time do exist if they are not in the background we ‘are’ vital space and cyclic time. **The simple idea behind the structure of the fractal Universe is to consider time=change =motion and Topologic, formal space=extension the 2 elements of which all beings are made.

**WE ARE space and time,** merely of a different kind to that of Newton: Organic scalar spaces, and cyclical, discontinuous times who ‘live’… worldcycles (no longer worldlines as we have a 2^{nd} arrow of information) of exist¡ence (as all species follow the common laws of space and time). As cyclical time that explains the informative repetitive patterns or Laws Nature and its multiple space-time clocks. Why a Universe made of space-time beings is essential to a philosophy of mathematics is obvious*. *Because the main experimental science concerned with space is mathematics and the main science of causal time is logic, if we are made of fractal space & cyclic time, both mathematics & logic become experimental sciences, reflecting the properties of those 2 primary substances, as mirror-languages of maximal synoptic information and minimal size (Sxð=C), whose underlying laws emerge in all other larger scales of the fractal Universe of bigger size and less information, proving also why they apply to all of them

**Cyclic time: ð – the causal repetitive laws of ‘stiences’**

A Universe of ∞ time clocks of different size and speed differs from its human description with a single mechanical clock-time to which all time clocks of the universe are equalized, elongated into a lineal ‘second-minute-hour-day-year’ system of equalized time clocks (of light waves, mechanical clocks, earth’s astronomical clocks). As Galilean physics, born of ballistics, simplified the nature of cycles of time-space into lineal durations, to measure best the locomotions of cannonballs: Time is cyclical, all clocks of time and laws of science are based in the cyclical patterns of nature. But physicists developed ballistics and denied the obvious truth that we can know the future because it will repeat the causality of the past, and we can change it by changing that causality, in History by repressing the lethal memes of the tree of metal and enhance the welfare memes that make us survive. Of course, lineal and cyclical time render the same equations as one is the inverse of the other, measured by frequency, T=1/ƒ, but the philosophical implications of cyclical time, are ginormous and the in-form-ation provided by those cycles, erased by lineal time, a handicap for humans to truly understand the cycles of history and economics, the ‘deep time’ scales of the fifth dimension, and the whole workings of super organisms and its physiological structures.

A key fact that of a time cycle is to break reality (1^{st} knot theorem) in an outer and inner region, creating a repetitive motion that becomes an isolating membrane that encloses a vital space, the ‘substance of which we are all made’.

A second key element is to be made of 3 ‘relative pi-diameters’, which therefore determine ‘3 ages of time’.

**Local Past=Entropy, Present=Iteration and Future=Information in zero-sum worldcycles. **

‘The separation between past, present and future is an illusion’ Einstein ‘Spacetime ∆@st you are to dust you shall return’

Of all the consequences of cyclical time, the most important is the existence of infinite local time clocks of which we are all made, which therefore imply the existence of infinite local past, present and future states.

Whereas the past is the beginning of a pi cycle, starting as a line of entropy with no form that curves and raises in height in its second state of present, and returns back to its origin in its future 3^{rd} age of information, completing a 0-sum of life and death. Thus instead of a single ∞ lineal absolute time there are ∞ living cycles of time happening in zillions of entities.

So we can see the fundamental unit of reality in a given scale, the timespace cycle, as ‘time’ in a sequence of 3 ages, which close the cycle into a zero sum; or we can see it sequentially in space as a ‘fractal point’, that is a non-Euclidean point with 3 parts, the elements of its angular momentum, which will become the new unit of both geometry and mathematical physics.

The 3 only topologies of physical systems: conservation laws.

The 3 part of any fractal point of cyclical time, of angular momentum, in the Universe become then in the simplest physical systems, the 3 conserved parts of the minimal organic Unit of reality, a ‘Planckton’ of angular momentum. All physical systems then can be ‘reduced’ to fractal ensembles of 3 ‘conserved quantities’, angular momentum – the membrane of the system, which becomes a membrane, fractal sum of ‘cellular cycles’ or the skin of the system in human beings. Vital energy, the enclosed cyclical forms and motions within the space whose boundary conditions are given by the membrane, and lineal momentum, the motions with a ‘finality’ we perceive guided by a ‘relatively still mind-point-singularity’ that focuses the energy and information transferred through the angular momentum membrane.

It is also the smallest clock of our world, as its minimal unit of cyclical information, or angular momentum, used as the unit of the *3 human physical parameters* of *spatial size, cyclical time** frequency and ‘ scale’ (Active magnitude):*

**h= mass (∆) x area (S) x ƒrequency (ð).**

So h, ‘Planckton’, is the minimal fractal organism which becomes the ‘cellular unit’ of all other species of light space-time, as Plankton is the minimal unit of the biologic Universe, also a cell with a similar ternary structure – a DNA nucleus that process information, a protein membrane that isolates a vital space-time, the cytoplasm.

Thus Non-Euclidean points have breath, its lines are therefore waves able to communicate the external form and internal energy or fractal networks that branch to connect multiple points, and its planes intersection of three of such waves or networks that form topological organisms… It is then obvious that the next step of Non-Euclidean geometry is to merge those concepts with the physical analysis of the smallest physical systems, to understand its vital topologies. To that aim we have to introduce *the second fundamental equation of 5D metric that formalizes the Paradox of relativity. And so it acts in the physical model as the Postulate of relativity, common to 3D Galilean and 4D Einstein’s simplex models, which correspond to 5D in a single plane of ‘light space-time’…*

**Galilean Paradox**: **S****ó****T: Relativity of space Dimensions=Forms=Motion in time****: The 5 Universal** **Dimotions**

Galileo’s time and space Principle of Relativity is the fundamental conceptual thought behind the relationship between time=motion and space=form and how one can be converted into another: All what exists is made of space=form and time=motion. And yet physicists know that we cannot distinguish motion from form. That any being in motion from its point of view seems to be still and all other things moving around it. *This is the principle of Relativity of motion.*

Physicists then without much thought about that fascinating duality, went on to use mathematics to calculate the relative motion of each entity of reality respect to other system, which seems static from both points of view. This is called Galilean relativity, latter refined by Einstein’s relativity, and essentially is concerned with the mathematical calculus of what we shall call the 2nd Dimotion of time=change, locomotion. Fine, but *we are more interested on the duality of space=form and motion=time and its entangled relationships –the reasons why we do NOT see together motion and form, even if all systems have both.*

The conclusion is then rather obvious: one of the two parameters of reality is ‘hidden’ to perception; we either see motion or form, ‘waves or particles’ (quantum complementarity), distances and lines or points in motion (as in the night when fast cars in a picture appear as lines). So *physicists calculate only one when in fact we must assess the existence of 2; and since we cannot distinguish them, logically we must equal them. ‘Form=motion-function; space=time; S=T’. *

Relativity then becomes a duality, S=T, is *at the heart of every law of the Universe. But which one is the real element? Obviously time=motion. Space is a”Maya of the senses” – a slice of time motion. The ultimate substance is motion. Form is what a ‘still mind’, makes of that motion to ‘perceive’, information, forms-in-action. *

Since we see Earth still and flat but it is round and moving. Galileo’s profession was ballistics – the study of cannonballs motion. So he chose ONLY motion and lost the chance to start physics with a complex philosophical understanding of its S=T dual Principle of relativity, which Poincare defined latter clearly when he said that ‘we cannot distinguish motion from stillness’. An example is quantum/relativity duality. In detail quantum space has ‘dark energy’ because it has expansive motion that extends into a plane of space, but when seen at larger scales without detail its entropic motion seems static space – a dual area of scattering length and width. So in the galaxy we see either dark energy motion or expanding space: T=S. A motion of time is equivalent to a dimension of space: Distance and motion cannot be distinguished so they must be taken as two side of the same being, a space=time Ðimotion (ab. Dimensional Motion):

**S= T; Dimension-Distance = Time-motion = ST Ðimotion**

*Why if the Earth moves in time, we see it as a still form in space? Because Reality is a constant game of infinite motions, but the mind focus in stillness those motions, and measures them at distances. For ‘huminds’ motion is relative to our systems of measure and perception, which are light-based; hence a fixed c-rod speed/distance. Reason why Einstein’s relativity postulates a maximal T:c-speed, measured as **if observer and observable were still to each other (Constant S); which at our scale we ‘correct’ with Lorentz Transformations**. *

*P*hysicists just substituted Earth’s still distances for motions. It took 300 years for Einstein to realize the relativity of motion and its measure made essentially time and space, motion and form two sides of the same coin. Still this realization was not explored philosophically and so it gave birth to a series of ill-understood dualities between ‘states of measure and form’ (particles, head gauging form, in-form-ation) and ‘states of motion’ (wave states).

It is then essential to grasp that motion and form co-exist as 2 different states depending on 5D scale and detail: Motions are perceived by minds that stop motion into form, into information, as distances. So if we see slow motion in the night, a car’s headlight seems a long distance line ‘still’ picture. But this means also that the 3 ‘Euclidean still dimensions’ must have motion; they are ‘bidimensional ST-holographic, topologic dimotions’. So we have 3 Space + 1 Time + 1 5^{th} dimension of scales = 5 Dimensional motions. None of them is *a Dimension of pure spatial form or a pure time motion but a combination of both. Even if **mentally we tend to reduce motion and focus on forms, all has motion=time, and form =space: this is the meaning of ‘spacetime’, the messing of both into 5 dimotions, the fundamental element of all realities.*

Relativity states ‘we cannot distinguish motion=time from position=space’. So all what exists is a composite of both, undistinguishable S=T, 5 ‘Dimensional motions’ (Ab. Dimotions), broken in infinite fractal, vital time space organisms, composed of topological Dimotions: height=information; length=locomotion; width=reproduction; form=social evolution of parts into wholes & entropy=dissolution of a whole into its parts in a lower scale of the fifth *dimension* (term we keep for the whole range of scales of the Universe)**; whose study is both mathematical, the main science that studies how those 5 Dimotions entangle in simultaneous Space, connected to each other topological adjacent parts, which create superorganism, and Logic; the main science of time that observes how those pentalogic, entangled superorganisms move and evolve, change in sequential relational time, living a worldcycle of life and death.**

And since there is nothing else than time and space, those 2 experimental primary ‘mirror-sciences’ of time and space become the most important to extract the ‘Disomorphic=laws’ of those 5 Dimotions that all systems have in common. Since *while those Dimotions are broken, in vital organisms, separated by cyclical time membranes, they are the same*.

*In the graph **Galilean relativity was ill understood, as the true question about time-change is why ‘the mind **sees space as a still continuum, when in detail is made of *smaller self-similar *quanta*, in motion. The paradox *defines mental spaces as still simplified views of the more complex whole. The 3 ¡logic paradoxes of space topology (closed in-form-ative curved-O vs. |-open, free entropic lineal forms), time-motion (stillness vs. motion) and ∆-scale, (continuous whole vs. discrete forms; single scale vs. multiple one)s, are essential to the perception of a simplified ‘spatial mind universe’ in a single flat still plane vs. the full, more detailed complex picture in time, of a curved, discrete and moving Universe. Those paradoxes resume the 5 elements of reality, Space=form, time=motion, scales and the mind that measures them, within its own entropic limits.*

**Motion is reproduction of form in a lower scale. Bohm’s realism: quantum potentials.**

How a system moves; in a crowded universe, where we ‘are’ vital space-time? the answer, which resolves also Zeno’s and quantum complementarity paradoxes, is if *we do not move but reproduce our information, translated into a lower faster wave scale of the fifth dimension; as we reproduce our sound in faster electrons to telephone or nerve impulses into chemical dopamine to jump discontinuous neurons. So motion becomes scalar reproduction of form, and since all is a form of motion, all is reproduction, which is the definition of a mathematical fractal, a feed-back reproductive equation; 5D metrics, which become then the ‘function of existence’ whose goal is to reproduce the form of all systems – the simpler ones with maximal motion-translation in space, the complex ones with min. motion as a reproduction that emerges between scales. And this gives birth to the worldcycle. Consider the case of quantum physics:*

In the graph we see a particle reproduced in adjacent regions that fade away, and the result is the perception of a wave of motion. In Bohm’s realist model this reproduction happens in a lower plane of quantum potentials, where also entanglement happens, which is the ∆-4 scale which is v>c in 5D metrics (Min. S x Max. T = C), hence real.

Motion then is reproduction of form over such potential: the wave erases form into motion, the particle is a still state that gauges information entangled to other particle, fermion and boson, still to each other – despite the perception of relative motion in our scale – hence the information electrons share has always a c-constant speed. Thus the Lorentz transformation are objectively real for mankind who eliminates the stop state of particles as we do in a movie eliminating the stop frame but if we were observing reality from the perspective of an atom, we would ‘stop’, entangle in the quantum potential, neutrino scale & so eliminate the spooky effects of ‘time dilation’ & ‘length’ contraction, from our perspective (but not of mass increase as it is a scalar effect). This is the ‘rational’ 5D explanation of both the c-constant of light and entanglement; As electronic beings perceive information in ‘stop position to each other’ and move in ‘wave state’. Motions are perceived by particles that stop motion into form, into information, as distances. So 4D relativity needs to be expanded to the scalar Universe beyond the c-speed light limit of the galaxy.

Within this complex view, the simplified models of Newton, Galileo and Einstein’s space-time correspond to the limit of 5D when we simplify all the worldcycles of time, we call life & death to a single mechanical clock, elongated to infinity & perceived in a single scale of space. Let us then deduce from those 2 equations the fundamental equation of reality:

**The function of existence: Reproduction of form.**

Physicists made the Galileo’s paradox, the cornerstone of their theory of measure, but *they** failed to study the deep implications it has for every aspect of the structure of the Universe, from the duality between spatial mental, linguistic forms and physical motions; to the balances achieved by the similarity of both space and time, which becomes the fundamental ‘equation of present’ S=T, and hence with the metric equation of scales, $ x ð = K, the two essential equations to formalize single planes S=T, and multiple scales of spacetime. **Yet as S=T maximizes SxT=K (5×5>6×4). We unify both in a single equation: Max. S x T = C, which defines for each fractal vital space-time organism its Function of existence, as all species will try to maximize its motion-entropy-time for its field-limbs, its information-spatial states for its particle-heads, whose product will give us its vital reproductive energy. But as all systems move and motion is reproduction of form we can ad a final factor, ∑, reproduction of parts, which to maximize that function become joined into larger wholes which are stronger than individuals; creating new planes of existence. C= Max. ∑ SxT (s=t); whereas C act as the entropic limits in ∑-scales, T-ime & S-pace, boundaries beyond which the still mind doesn’t’ perceive or control. *

It is also a survival biologic equation, because it implies to provide lineal motion to ‘fields-limbs’, absorb energy to reproduce our bodies-waves, and information to guide our motions with particle-heads. *So reality is a ‘struggle’ for existence as systems reproduce its Ts-fields-limbs of motion, S=T body-waves of energy and St-particles-heads of information. But as all T.œs are fractal, broken, its growth has a limit on the fight with other systems, who try to move and reproduce. In terms of pure T-motion and pure S-form, we consider then the whole of maximal* time=motion= entropy or TT and Max. space=form=stillness or SS the 2 limiting Dimotions for any 3 ensemble Ts<ST>St-system.

So we define the Universe as a ‘fractal supœrganism’ that reproduces and ensembles Space-time Dimotions into organic forms through ∞ relative scales of spatial size and time-motion; whose Fractal generator (mathematics) Metric (Physical jargon), ‘function of survival (Biology) or Function of existence (logic Jargon) writes **C=Max.∑SxT (s=t)**

We shall prove that all realities are always a reproductive radiation of a function of existence along 5D scales.

## The immediate conclusion of a world made of space=time then is obvious: We are made of the 5 Dimotions of space-time of the Universe. We are ensembles of those 5 Dimotions, which seen in simultaneous space give origin to the vital topological organisms of the Universe; whose study therefore is mathematical, the science of space; and observed in sequential relational time, live a worldcycle of life and death; and since there is nothing else than time and space, those 2 fundamental experimental primary ‘mirror-sciences’ of time and space become the most important to know what all systems have in common, its ‘Disomorphic=laws’ extracted form the nature of those 5 Dimotions.

**Vital, ternary, Organic Topology**

Not all geometry is a representation in mental space of an outside reality. There is an inverse type of geometry, which is objective, as it is not intrinsic to a mind made with a force, as Euclidean 3D geometry is but reflects the vital space of beings. This is the field in which 5D innovates, ‘enlightening’ classic topology to ‘understand the ternary organic, structure’ of all systems of nature: As the 5 Dimotions are dimensions of space with time motion, its science is topology that allows a system to deform= change its inner form. Yet a 4D Universe has only 3 ‘topological varieties’ that restricts ensembles to only 3 topologies, each one best suited to perform the 3 organic vital functions of any physic or biologic system –gauging information (1D) to move the system (2D) to an energy field in which reproduce (3D):

## The purpose of vital topology is to study the 5 Dimotions (dimensional motions) of the Universe… As such it will be the final stage of evolution of geometry as an experimental science, merging elements of all disciplines.

## All dimensions of space have motion in time. Mathematics realized it, as the still Geometry of the Greeks evolved into a vaster, generalized concept, a topological variety, where a topology as opposed to a geometry has internal motions-changes. As the only case in which the inner dimensions of a being don’t seem to change is external locomotion most 5D motions need ‘geometries’ with inner motion, which are topologic varieties of which there are only 3:

*In the graph, *the diffeomorphic Principle of Einstein’s 4D analysis acquires an organic nature, when we see the Universe as the sum of ∞ Complementary ternary, topologic systems whose dimensions have organic functions: Systems feed on their relative dimension of energy-length, perceive in their relative dimension of height and reproduce in their relative combined dimension of width, which are assembled into each specific species, *to best satisfy the systems’ in taking of motion, energy and information. *

*I.e*: An animal has its informative height in the high perceptive light dimension, but a plant, which uses light as energy has its up and down dimensions inverted respect to the man and its chemical brain buried on the Earth. So both have opposite energy-time coordinates, with an ‘antero-posterior’, lineal’ ‘outward’ energy oriented structure due to the oriented arrow of light. But in a 3D world with no preferred orientation, a sea or vacuum, cyclic forms that maximize information dominate from plankton to galaxies that have a cyclic, informative, inward structure, as the stars’ body absorbs energy from intergalactic space, reproduces matter with it and feeds the internal informative knot of gravitation, with a higher height dimension the black hole.

So the Dimotions of reality are the 3 bidimensional topological varieties that act as vital organs in cylindrical long limbs/fields, Hyperbolic wide bodies and spherical tall heads, each one dominant in a lineal classic dimension, lineal motion, informative height and reproductive width, which DO have organic vital properties too:

Spherical particle-heads, perceiving information from the advantage point of **height**.

**Lineal long, cylindrical **legs and fields of locomotion as the line is the shorter distance between two points.

**Wide, hyperbolic body waves, **storing the energy reproduce by the system.

As such single dimensions perform those organic functions in light waves that huminds perceive as space. So light is also an organic system of 3 dimotions: c-speed length, electric informative height & a wide magnetic field that supports them.

*Moreover beyond its classic analysis of forms and functions, what Vital topology does it to serve the basis for its temporal*-numerical version that is existential algebra, as points become the spatial version of temporal numbers:

*So we develop a formalism of the 5 Dimotions, as operand, «,<,* =,>,», for entropy, locomotion, reproduction, information and social evolution, which will develop into flows of ‘stœps’ that change sequentially a form through its events in time, where the vital topology of each Dimotion is ‘integrated’ within those symbols. In this manner vital topology becomes ‘Existential algebra’ –the analysis of flows of dimotions=actions of T.œs in its worldcycles of existence. Those sequences can then be studied as templates of all T.œs in all scales which will follow them to complete its survival cycles. And so existential algebra has implicit vital topology. So we define in existential algebra with simple T, S and 5 «,<,≈,>,» the main events of space-time of the Universe:

**1D: T>S: angular cyclical motions of information (Ab. §ð):** the minimal ‘geometry’ of reality, a spherical particle/head or fractal point, the geometry that stores maximal form in minimal space, hence suited for ‘organic functions’ of gauging, storing and perceiving information (particles, heads).

**2D: S<T: Lineal Locomotions***,* **(Ab. $t) *** which will move through its* lineal limbs/fields the system, as the line is the shortest distance between two points… towards a…

**3D: S≈T:** **Fields of vital Energy (Ab. ∑≈∏):** with its hyperbolic body-waves that iterate the forms of both the spherical particle/heads and lineal limbs/fields; as the hyperbolic topology combines the other two, so it can generate them, in the same manner Energy adds as the third conserved space-time quantity the lineal and cyclical momentum of 1 and 2D. To which we must add the 2 ‘scalar’ Dimotions of:

**5D: entropy («, ∂S)** whereas motion is dual internal dissolving the information of the being and external, scattering its parts, hence we use an « ¬Æ symbol; so the system explodes into its ∆-1 parts: ∆«∆-1 (death). **4D:** **organic evolution (»,∫T)** of parts into still locked simultaneous ‘linguistic seeds or mind forms’∆-1»∆

So the abstract 3 conserved substances of reality become organic bidimensional topologies – flat motion in space, cyclical time membranes and the vital 3D energy within them. We then observe its ∞ variety of combinations as topologic ternary species, whereas the 3 perpendicular ‘lineal’ dimensions are a simplification of those organic functions: the height dimension enhances the ‘perception of information’ by O-heads/particles place above in the point of maximal projective geometry; the dimension of ‘length’ maximizes locomotion; physicists’ only time motion, and the width dimension maximizes reproduction and storage of energy.

So we have transformed the 5 Dimensions of space-time in 5 vital dimotions, broken into infinite vital space-time beings, and now we have the 5 elements which reality uses in different perspectives to construct all realities. Nothing else is needed. And it will easily follow that in each stience, including mathematics, 3±¡ elements (depending on the perception in a single plane or in several ones) will be concerned with the analysis of a system in Simultaneous, entangled, still space as a superorganism constructed with those 5 Dimotions, which in sequential motion time will trace a worldcycle also composed of those 5 dimotions. As the universe simply put it is a reproductive fractal of 5 Dimotions of spacetime…

*As motion with form constantly reproduces by the mere fact of moving. So the fractal reproductive nature of all what exists is immediate, as all what exists moves form, reproduces form. *

**THE WORLDCYCLE – THE FUNCTION OF EXISTENCE.**

**The worldcycle of existence**: We then put together 5D scales, vital topologic space & cyclic time to describe simultaneous superorganism tracing a worldcycle in time, the fundamental event of reality.

*All what exists is a supœrganism of vital space tracing a 0-sum worldcycle of time through 3 scales of 5 ^{th}dimension: Born as a seed of fast time cycles in a lower 5D scale (∆*

*-1*

*:Max. T x Min. S), emerging as an organism in ∆o, living 3 ages of increasing information, as its time clocks slow down in its ∆*

*+1*

*world to die in a time quanta back to ∆*

*-1.*

So, absolute spacetime is the sum of ∞ Timespace beings, observed in space as simultaneous super organisms, in time, as **worldcycles** of existence between birth and extinction; as all systems are born in a seminal seed, of faster time clocks, in a lower scale of the fifth dimension, growing socially (4^{th} dimotion) till emerging in the organic scale, where they will live 3 ages dominated by one of its 3 topologic organs and its functions=dimotions, a young age of maximal locomotion, dominated by its limbs/fields, a mature age of reproduction dominated by the body-wave and a third age of information dominated by the informative dimotions, which finally exhausts all energy and as time-space never stops, it reverses its dimotion from information to entropy, exploding in the moment of death.

So we marry the 3 vital functions=motions of time and the 3 dimensions of space, either in 1 or 2D (height=spherical information, length=planar locomotion, width=hyperbolic reproduction) which merge in all Time-space Beings; and dominate one of the 3 ages of its life-death worldcycles, the past, young age of limbic entropic motions, the mature reproductive age dominated by the hyperbolic body/wave and the 3^{rd} age dominated by the informative particle-head, when the illusion of time ends with an entropic big-bang death that dissolves the being into its ‘scalar cellular, atomic parts’, which lead us to the realization that time cycles NOT only return to its origin in a single spacetime continuum but they move up and down the scales of the fifth dimension:

**The 3 ages of existence of space-time organisms. Its 2 worldcycles and Metric equations.**

The *Function of Existence* of any space-time organism can be developed as a feedback equation, S<=>T, in 3 sequential phases/ages/horizons, between 3 ‘standing points’ (changes of phase): Max. T=motion x Min.S: form =moving youth; Max. SxT(s=t): reproductive maturity and Max. S x Min. T=informative, old age, as the equations of the 3 ages of life, between the seed of pure linguistic form born in the lower plane: S¡-1 and its T¡-1 entropic death, back to ∆¡«∆¡-1:

*∆-1»∆º:* The cycle or organism starts its existence as a seed of pure form (4D) when its space-time field is created.

*$t:* It is the first horizon or ‘energy, youth age’ of the cycle, in which energy dominates the system and so we write this phase as, max. $t x min. T.

Max. SxT: s=t*.* It is the present balanced age of the cycle or classic age of ‘life’, when energy and information are in a constant proportion. It is the most efficient age, when the cycle reproduces.

*Max. T x min. S*: 3^{rd} age of the cycle when information has exhausted the space-time field that warps into itself.

*∆º«∆- 1: 0S x T: * It is the end or death of the cycle that reverses its form and becomes energy again.

Existence is an ∞ sum of space/time fields, fluctuating between birth and extinction through those 3 phases or ages. The 3 ages of Timespace supœrganisms happen in all systems, including mental languages:

In State Physics they are, $t-gas, the moving state, S=T liquid, the balanced state and §ð-solid the informative state; into Cosmology, where it describes the Universe as a space-time system that fluctuates between both limits, a form of pure time, the singularity (*min.$t x max.ð§) *and a form of pure space, the big- bang (*max.$t x min. ð§). In *Biology, they are the 3 ages of living beings AND the 3 horizons of evolution of species. In social organisms, through the subconscious collective mind of civilizations which in art styles mimic in a longer 800 year cycle of life and death of civilizations (according to 5D metrics a human social superorganism is larger in space – a nation, culture, religion – and so it lives longer in time). So we find the 3 ages of life *emerging in the 3 ages of cultures and* its 3 artistic styles: Min.S x Max. T (infantile epic, lineal art, as in treccento, Greek kuroi; S=T; balanced beauty, when form and size are in balance, the classic mature age; and Max. S x Min. T: baroque, 3rd age of a civilization, whose subconscious mind is the art of its ‘neuronal artists’, the age of maximal form and angst for a no future, which *is the age of war and death of cultures*).

Finally we talk of 3 ∆±1 scales of worldcycles as the being live in a placenta, then emerges as organism in a world:

þ: 0-1: its palingenetic o-1 social evolution in the accelerated time sphere of existence, till becoming 1 (0-1 bounded unit circle in ¡logic mathematics; quantum probability sphere of particles in physical systems; palingenetic fetal age in biologic systems; 0-9 memetic learning childhood in social systems). It is a highly ordered worldcycle as a placental mother-energy world nurtures a memorial cyclical spacetime that erased errors of previous generations.

– c: The outer 1-∞ world, in which it will deploy its 2nd world cycle of existence in an environment which is open, entropic (1-∞ hyperbolic unbounded Cartesian plane in ¡logic mathematics; thermodynamic entropic statistical molecular populations in physics; Darwinian struggle between populations in biology; idol-ogic dog-eat-dog capitalist, nationalist competitive eco(nomic)systems in the super organisms of history. In this 1-∞ existence the world cycle is not ensured to continue, as the entropy of the world system can cut it off.

ω: The existential life cycle, though is part of a larger world of hierarchical social scales (§ D¡), where it performs 5 survival actions through ∆±4 Planes self-centered in its mind, beyond which it cannot longer perceive, to become if successful a new superorganism of the infinite planes of God, the game of existence.

The graphs show physical time ages, in more detail : Left, ‘states’ of matter, from pure solid, crystal, §top state, to an even more solid ∆+1 boson condensate, etc. We see that systems either move a step at a time within a plane of existence (gas, liquid, solid) or they can jump « two states at once, (as in the case sublimation) within that plane, or most often between two planes, as in « scattering & entropic death), to become a different Dimotional state. Right, and graph below, the 5^{th} dimension of scalar space-time and its dual arrows of entropy (studied by physicists) and information (its inverse that evolves form) combined into conservative ‘present’ zero-sum cycles of energy. They are the 3 ‘conserved’ arrows of time that create the futures of each fractal supœrganism, between its 4Dimotion of generation in a lower ∆-1 plane that emerges in an ∆+1 social world to die back in a dissolving explosion of scattering entropy, back to ∆-1 in the moment of death. As time is cyclical, made of 3’π’ lineal time motions, we shall distinguish 3 conserved ‘quantities’ of timespace that put together create superorganisms, a relative devolving past or arrow of entropy represented in physics by disordered explosions in space and in vital topology by lineal limbs/fields of lineal momentum, an iterative reproductive present that seems not to change, represented by hyperbolic body/waves of energy, and an implosive in-form-ative local future arrow represented in physic by accelerated, V(t)R(s)=K vortices and angular momentum, and by particle/heads in the ensembles of vital topology. So timespace breaks in ∞ relative local, fractal entropic pasts, iterative energy presents and informative futures, which put together create the illusion of a single timespace continuum.

Thus the Universe has mathematical, spatial continuous & logic, temporal, cyclical & organic, fractal properties derived of the more complex ¡logic geometry of 5D scalar space and cyclic time. Thus even if we describe it with the same logic-mathematical equations a 5D Universe is very different, as scalar space brings organic and sentient properties, and cyclical time, informative deterministic patterns, which the lineal philosophy of mechanist physics ignores. From those scalar, hence organic, cyclical hence informative and moving, energetic, hence vital properties of scalar space and cyclical time, a complete different picture of reality arises; where languages become the ‘extreme’ limit of the still formal ‘spatial state’ of the being with minimal size and maximal information.

Whereas each ‘stience’ defines a scale of the nested Universe, from Physics focused on the largest Galaxy and physical scales, to the smallest ∆^{1} scale of Formal digital and logic sciences of the mind. Languages thus become the limiting formal, spatial scale of reality and as such they are mirrors of the fractal Universe that share the ‘Disomorphic’ (Equal laws) of all the scales of reality. And so mathematics becomes an experimental stience whose ‘Universal grammar’ and fractal generator is the same ternary structure of 3 elements of all other systems in a single plane made of three topologies, SóT, giving birth in each sub-discipline of mathematics to similar ‘mirror images’, algebraic S(x)=T(y) equations, ternary dimensions, ternary topologies, and so on.

**WORLDLINES AND CLASSIC RELATIVITY**

How all this relates to 4D classic relativity is simple. In first place because we start from duality, form and motion, O x | in geometric terms, we do have No longer worldliness but worldcycles. For a physicist which only measures lineal time motions, if you ask him what is the function of exist¡ence he will just state: the sum of motions that you make through your entire life and draw a ‘cone of light’ explaining you that as long as you go below c-speed your worldline will be space-like etc. All this has nothing to do of course with your life experienced through the 5 complex dimotions of reality, but the physicist will not have the slightest doubt, as he has reduced perception to that single motion that he is talking serious metaphysics of being and will go further with its errors of lineal time (as all dimotions of time are fractal, local to the being’s exist¡ence). It will then ‘obviously’ enter into paradoxes – the twin paradox, the Lorentz transformations, etc. which we streamline in our post on physics – matter=form and motion; which is the only realm to study with worldlines.

As we respect the correspondence principle, we can compare 5D with 4D *strictly in the real of physics where it applies*:

We unify the 3 fundamental principles of vital spacetime, 5D metrics and Relativity, which become the equivalent in 5D to Einstein’s 3 principles in 4D metrics – the constancy of light speed, his rejection of absolute space-time and Relativity, which 5D also share but make explicit with an S=T equation. We compare then both formalisms with the principle of correspondence to show that 4D is just the limit of 5D for a single light space-time plane:

__–____ The postulate of relativity__, which states that we cannot distinguish motion=time from dimension=position=form =space, which in 5D we reduce to a simple formula, S≈T; we shall call the Paradox of Galileo (e pur si muove, e pur no muove) as he couldn’t explain why the Earth moves but we perceive it in mental space as a still dimension.

– __Denial of Newton’s Absolute space-time__; as he said ‘Leibniz was right, but if so we have to start physics from its foundations’ and ‘I seem to be the only physicist who believe there are infinite time speeds in the Universe’, which we made explicit accepting Leibniz’s relational concept of space-time, the seed of Generational space-time, resumed in a simple sentence: ‘we are the scales of space we occupy and the cycles of time our existence lasts’.

– __The constancy of light speed__, c, which in a 4D single background plane of light space-time corresponding to the galaxy holds naturally, as *light in a relational->Generational theory is the space-time of the galaxy*, which generates through its ‘accelerated’ vortices of space-time in a crescendo of ever more dense scales, photons, electrons, quarks, atoms, molecules, matter states, cosmic bodies and black holes.

Yet when we add a 5^{th} Dimension sum of all those scales, c-speed becomes the limit when we reduce all the potential scales of the Universe to light space-time and the particles generated by it.

And so we define a second metric equation for all scales: $ (Space size) x ð (cyclical time frequency) = Constant; meaning smaller scales of space hold faster rotating particles, (black holes beyond the event horizon, where there is no longer ‘light’) or when moving in lineal fashion can go faster than light speed in scales below light or outside galaxies, (Bohm’s quantum potential of action at distance cause of entanglement, neutrinos outside galaxies.)

*Yet as S=T maximizes SxT=K (5×5>6×4). We unify both in a single equation: Max. S x T = C, *the 5D metric equation, local to each Time§pace organism struggling to survive by maximizing. its S in-form-ation & T-motions, or in terms of organs, its limbs/fields of speed and heads/particles of information, and

*its balanced combination or body-waves of vital energy, where*entropy=T-motion and S=form, become one, S=T.

*So 5D metrics coupled with the Principle of Relativity and relational space-time defines for each fractal vital space-time organism its Function of existence, as all species will try to maximize its motion-entropy-time for its field-limbs, its information-spatial states for its particle-heads, whose product will give us its vital reproductive energy. *

So the equation has a biologic meaning, because we are made of 3 topologies ‘fields-limbs’ of lineal motion (Ts, where T-motion dominates s-form); balanced hyperbolic bodies-waves of S=T energy we absorb to reproduce, and spherical particles/heads of St- information, form with a bit of action-motion, which we need to linguistically guide our motions; *sandwiched in a larger ∆+1 world of max. T-entropy; coded by the ∆-1 seeds and minds of pure Spatial form. Which give us the 5 Dimotions of reality that are also structural organic parts of our superorganism & phases/ages of our existence: S¡-1 (seed-mind) »Ts-fields/limbs of motion>S=T-body waves>St-*particle-heads«∆-1 T-death; where double «» are symbols for an entropic, ‘expansive dimotion’ and an informative still one. So the essence of survival is to increase our St= mental information and territory of order; in which to S=T reproduce and sT move avoiding the entropic limits of scattering, disordered motion, T-1:death and total St-illness in space the two limiting Dimotions. So we keep our ‘energy=reproductive body-wave, s=t in balance moving, sT our field/limbs and perceiving, St, with particle-heads.

**Recap**. The Universe is a fractal organism of similar scales, defined by 2 metric equations and its philosophy of science, 5D Absolute Relativity: the equation of 5D scales, SxT=C & the equation of relativity between form and motion, S=T, which started modern science. Since *we cannot know from the mental, still point of view, what truly is motion and what is dimensional form. *Thus Time Motions & space dimensions co-exist and merge in every space-time being of the Universe &* we must talk of space-time dimensional motions.* When we combine them, as S=T maximizes SxT, we define the Function of existence, Max. S x T = C (s=t), which IS also a biologic equation of survival that embodies the will of life; the biological expression of the ‘Universal mandate’, expressed by all species in all its codes and languages, the Grow and Multiply of the Bible, the intuitive truth that guides all beings. In detail implies all systems try to maximize its 3±¡ ‘Ðimotions’ that in space create organisms and in time its worldcycles of existence; starting as pure Form (4D) in a seminal seed, emerging in an ∆º scale, in a young age of max. Motion (2D), balanced with information in the reproductive age of max. Energy (3D), followed by an age of information (3D), when time reverses into entropic death.

The dual metric 5D equations of scales, SxT=C & the equation of equality between form and motion, S=T, develop in 3 ages with 3 standing points, a max. point of existence, S=T or mature age, a young age of Max. T=motion, and an old age of Max.S=information; between birth in ∆-1 Form & T-entropic death.

**¬Æ PENTALOGIC ENTANGLEMENT OF SUPŒRGANISMS. EXISTENTIAL ALGEBRA**

‘The Universe is a fractal supœrganism of 5 entangled D¡motions of Space-time entangled in pentalogic supœrganisms of Dust of time-space (¬∆@st) tracing 3 dodecalogic cycles of existence, the þlacental, life & worldcycle.’ *l*§

*The entangled Universe generates its space through synchronous connections between 5 Dimotions of reality in its 3 relative ranges, Actions, topological organisms and its 5 elements: fractal space, scales, cyclical time, minds that perceive with synoptic languages the system as a whole and project its territorial order and survival will through its actions, and the entropic limits (¬) the infinite universe imposes to that will. So any aspect of reality must account for 5 elements to express such entanglement at the 3 relative scales of existence. It is the fractal principle of 5D pentalogic that structures Timespace supœrganisms (T. œ)*

Scalar, topologic space and cyclical time organisms, made to the image and likeness of the fractal Universe, require a complete change of paradigm regarding how reality is created, away from the chaotic, entropic, or religious self-centered (Anthropic) theories sponsored by humans, which are monologic, blind to the entanglement and constant communication of information between fractal points, either in logic=time or mathematical=space languages, in a sentient, apperceptive Universe, where information when locked as ‘entangled space’ by those dynamic networks of communication creates a mind-mapping of the outer world.

Monologic man is closer to Leibniz’s Monad’s – self-perceptive, selfish black holes of information, whose communication he could not properly resolve so he simplified them into an ego-paradox. We deal with monologic man in the I logic paper on our blind ænthropic ‘culture that passes as big science’ and its grand paralogic theories of reality. While on the posts on pentalogic we define the 5 elements of reality – space, time, scales, entropy and minds, as entangled systems definitions can only be made from the combined perspective of all other systems of reality. To represent this fact – entanglement – we use the pentalogic symbol where each point of the pentagram is connected to all other points, and so the fractal point, in this case each of those 5 elements cannot be understood without the connections to the others.

The key concept behind pentalogic and entanglement is rather simple: to survive in the Universe, to have focus, form, a system must be ‘anchored’ through entanglement in its ‘plane of existence’ with ‘trinity’ parts of vital topology, connected through 3 inner physiological networks, but also it must co-exist in a larger ∆±1 nested world separated by a topologic membrain, which allows it to absorb and emit entropic motions and information, on that outer world.

The understanding of entanglement was best among humans in Buddhism, where each soul is considered a knot of communication with all other systems, and minimal in America where the self is considered always the only central point of view of existence, in eternal struggle with all other human beings – yet as entanglement is necessary to survive, the American, modern man is entangled NOT with humans but increasingly with the machines of the Metal-earth, the Financial-media/military-industrial ecosystem for which ‘it’ enslaves as an object with a price.

**S@.** **Geometric entanglement: fractal points and scalar numbers.**

When we go through different stience scales to explain each ‘entangled connection between the 5 elements of reality’, we shall see that already in the first scale of mathematical systems, the fractal point requires 2 different postulates to define it: The first postulate which is concerned with its inner parts (a fractal point has breath), and the 5^{th} which is concerned with its outer parts (a fractal point is connected with infinite potential parallels to the universe). The 2 inner and outer elements that meet in the ‘membrain’ thus define the membrain-mind as the essential entangled element between both worlds. Entanglement thus defines the being from an outer perspective as much as its inner nature.

*In graph, t**he fractal sentient point, new unit of mathematics and vital space.* *Einstein’s interpretation of the 5th non-Euclidean postulate was the view of a fractal point of the gravitational scale from our smaller electromagnetic world, which shrinks its inner volume, bending its parallels, hence its curved geometry. But that view breaks the conceptual definition of parallels are straight lines and is absurd, as the point remains ‘Euclidean with no breath’, hence only fits one line with no breath. Thus, particle points must be defined as ‘FRACTAL points’, like those we see through telescopes or microscopes, which grow we approach our distance both in scale and space becoming enlarged worlds with a complex internal structure and external entanglement** following a Non-E 5th Postulate: A point external to a line is crossed by ∞ parallel forces. *

Such organic points are like the stars in the sky. If we look at them with the naked eye they are points without breadth, but when we come closer to them, they grow and in its membrain we find its maximal entanglement and communication – the hottest region of the star, the Earth’s membrain with its life superorganisms & 3 ages Gaia< History>Metalearth, where a parallel network of ‘still entangled’ parts forms a whole which makes the ‘membrain’ a hard interconnected system porous to the energy and information of the world focused in its singularity center.

We can then assess the fundamental quality of vital mathematics as a key language of the Universe, which is to be the best language of space, hence the one that describes the laws of entanglement, even if man doesn’t understand it beyond its egocy of magic creationism, as the language of mirror symmetries and dimotion flows of ¬∆@st in all entangled systems.

Where the fundamental entangled element is the fractal point, and its social unit the polytope in 1,2,3,4 Dimotions; but not 5Dpolytopes since the fifth dimotion, entropy is the disentanglement of all the others. So polytopes rise positively in complexity as entanglements of indistinguishable regular points (or else one point will be different by position and connection to the other points). They reach its maximal complexity on the gender duality-mirror symmetry of the 2^{nd} 10^{2} scale – (female Icosahedron & male dodecahedron) & 3^{rd} polytopes’ 10^{3}scale (Male dodecaplex & Female tetraplex).

**∆T: Human vital Entanglement in ages and Scales.**

Each of us is an entangled fractal system, interconnected or else it won’t absorb energy and information and survive. And vice versa, when less entangled a system is – 3^{rd} age of the being, and becomes cut-off from reality its ‘truth’ its exist¡ence dwindles and the system fades away and ultimately perishes.

At which scale of reality then we must define entanglement. The answer is at the 3 scales, of the whole Universe as an absolute fractal organism of time-space (at the level of its 5 elements, S, T, ¬, ∆, @), but also at the level we just did of individual superorganisms, where its organs that represent those elements are interconnected to the whole, and finally at the level of its diminutive actions (a,e,I,o,u, acceleration=motion, entropy feeding, information, organic reproduction and Universals evolution, in the mnemonic rule), which we will find amazingly enough, in one of my most orgasmic-mental entanglements with the whole I experienced, were directly connected to the ∆±4 planes of exist¡ence of the being. Reason why we do perceive all those scales:

*In the graph, scalar entanglement happens between the humind’s organism and its territories of perception that provide the ‘bits and bites’ needed for humans to perform their 5 Dimotional actions. So we extract entropic motion of the ∆±4 gravitational/galactic field, information of the ∆±3 light/star scale, energetic food from the ∆±2 molecular/Gaia scale; reproduce in the ∆±1 seminal, gender scale and evolve through socially love in the ∆º mind field, entangled to other minds in metaphysical experience of bondage through our memes, in nations, religions and civilizations. *

So yes, scalar entanglement is real. It is the reason you do exist, and you do perceive and you are entangled to all the scales of reality you perceive. You are your mind space and your mind space is your entanglement.

**Mathematic entanglement – ¡ts true structure.**

Entanglement is thus the essence of scalar space that stops the motions of entropic time creating organic networks, and hence the essence of fractal point geometry with motion – vital topology – and its scalar undistinguishable social numbers, which makes so powerful mathematics. It is the real reason why mathematics can describe information as the main stience of space that departs from points and numbers NOT from idealist sets. And so a true mathematical description of reality must start by its ∆=S elements: entangled points of space, in a single plane, whose regular polygons are indistinguishable numbers of social scalar groups, which reproduce when acquiring motion, as even polygon female self-centered numbers or odd male mirror polygons that reproduce in scale projecting internal and external self-images of diagonal crossing. Numbers thus become regular social groups, symmetric to points, ∆=S, which then entangle in parallel time motions, ∆=S=T with 5D algebra’s operands that move ‘groups’ through the 5 Dimotions of reality, establishing further symmetries, 5D=∆=S=T, whose result is to create a given mind space, mirror for a fractal point of the whole world, O x ∞ = C, as the 1-∞ statistical space world plane is reflected in the 0-1 probabilistic unit time sphere of the mind.

So the 0-1-∞ scalar entanglement allows the mind to project its mental space in territorial order by converting its probabilistic mind i-logic into spatial larger forms, as probabilistic particles do in molecular statistics. Because vital mathematics DOES happen in vital terms in the physical world, when motion and energy is added to the coded §eed of mental space, ‘irrigating’ the program of exist¡ence coded in still bidimensional geometry, giving it more dimotional extensions. So from the first S=T entanglement of points and numbers, which then become social figures of geometry, that penetrate scales of number families, in ∆st mirror symmetries, numbers unfold the program and start to run it with 5D operands, which make them ‘real. Algebra then enters when it is able to move through the dimotions of time by means of its operands acting over groups of social numbers, the entangled Universe creating ternary networks that become planes that become supœorganisms of existential algebra. This is the game we shall describe, which needless to say compared with the barren land of the Axiomatic egotrip of Hilbert and Cantor ‘imagining points’ is a blossom spring of life and beauty.

**Physical entanglement.**

Entanglement appears in the non-ego centered Broglie’s>Bohm’s interpretation of realist quantum physics, as particles entangle through its quantum potential fields; and this vast entanglement happening at v>c in the lower plane of reality ‘reduces spatial distances’ in the Riemann’s sense of a geometry of similarity to a ‘boson like’ unit of parts as if distance-space would NOT exist within them; which does NOT for the emergent apperceptive ilogic ‘couple’ as a single whole.

To understand entanglement in formal languages is needed to adopt the Riemannian concept of a mental space where proximity and distance is a direct product of similarity, as if space will not exist, which can be achieved through communication on a lower ‘potential field’ of faster motion, as a faster v in fact perceived in stillness implies a smaller distance, and vice versa –i.e. the expansion of space in the Universe is in fact a v>c speed on the jets of dark entropy expelled by the accelerated faster than c timespace vortices of heavy top quark stars, aka black holes.

So the black hole beyond the c-horizon speeds up at v>c and erases the information of light into dark entropy expelled through its axis and perceived as growth of distance between galaxies, when by virtue of S=T can be considered grow of speed. Entanglement is exactly the inverse process in a quantum potential field that makes communication disappear in distance between two poles – which is the magic of a phone conversation – distance is NO LONGER real because we are not in sound communication at 300 m/s but in a decametric jump of scale at 300.000 km/s.

There is no distance in mental space of telephonic communication between speakers, only in geographic Earth’s space; but we are not concerned with land or air but with a language of communication and its bits of information, which are translated, entangled, in a faster 5D scale of smaller bits and max. form (electronic space).

Informative translation in other 5S scales is what T.œs constantly do, entangling its informative systems in faster scales, whose bits become ‘something else’ but preserve its patterns *as there is the Universal reference of the game of existence behind them*. In the previous case the communication has become an entangled back and forth feed-back electronic equation of existence as an event on a wire line between two fractal points which cannot longer be considered different from the ‘communicative line’ between them.

*Energy does not entangle only information does…*

A conclusion proved mathematically, as only the sphere of max. volume of information (Poincare) can shrink without tear in any number of relevant dimensions (4 for entanglement) crossing the barrier between two planes of existence, while energy becomes dissipated as entropy. So you can talk with mum in Australia but NOT send her 1000 kilowatts, you can though send her digital information, aka money too.

**Bio-logic entanglement.**

This leads us then to the first scale of complex i-logic ‘gender mirror symmetry’ between two entangled elements that become by fusion love a single one, of which there are infinite proofs.

In fact gender is the first and essential entanglement of the Universe between 2 single elements, through the mirror symmetry of female-like St+sT male-like complementary couples. Entanglement starts with 1+1=3, and for that reason we dedicate an entire post- to gender symmetry, The entanglement of gender, is so obvious that became the basis of the duality knowledge of earlier cultures (Chinese) will constantly come in Nature. It is NOT only the entanglement of biologic gender; so how we do distinguish its ‘essence’? In ilogic terms is easy because the essential duality is that between simultaneous present space, S=T and time flows in relative stœps, from past to future to past… to future, S<T>S<T… Both are perpendicular. And so the ultimately the definition of Gender we shall use here goes to the final core level of ‘perpendicular’ space (female gender) to Time flows (male gender).

Gender is essential because without Gender we can’t understand reproduction by mirror symmetry and the duality of symbiotic perpendicular vs. Darwinian relationships.

Two systems achieve maximal communication when they pass from parallel sharing of information, through a medium which transfers a language, to direct contact between its sensorial membranes, which bring true bondage. And at that moment two different outcomes might happen. If the membranes touch without tearing, the topological integrity of the organs are conserved in vital mathematics; so the being co-exists, and entanglement (the previous state of parallel communication) becomes bondage according to the fourth postulate of non-Euclidean congruence.

This is the magic of it, as fusion love kicks in and entanglement becomes a dance of complementary forms. It is the beauty of love and gender symmetry. And the maximal action of it, is called sex, orgasm, the present state of fusioning both the past to future to past, and present forms into one. So how it happens physically? Do I have to explain it to you?

The male gender penetrates the fold of the topological female but does not TEAR IT. It goes in T>S, and then it goes out S<T, as the male does in and out, information and entropic disentanglement. But the female S=T ‘grasps it’. It caresses it, encircles its lineal form with its cyclical even polytope. Entanglement is now bondage. And it brings when bondage combines in the most intimate ways the information of both systems trinity, reproduction. So alas, 1+1=3!

Those are the rules of vital geometry, topology, which in inverse fashion defines Darwinian destructive perpendicularity when one of the entangled elements turn out to be a predator top and breaks in, tears the membrain of the other system, killing it, entanglement then gives way to entropic devolution and one system becomes destroyed.

Existential Algebra. The upgrading of ‘¡logic mathematics’ as a mirror of ¬∆@st

The purpose of existential algebra is to study the multiple elements of our ‘5D metric equations of Absolute relativity’, Max. SxT (s=t), which we have broken down into 5 Dimotions, Pure Space-seed-minds of linguistic information, Pure entropic time-motions, and the 3 intermediate St-information, Ts-locomotion and S=T reproductive states.

The equation has an infinite number of perspectives – in fact it generates in each local system, its events and superorganisms, and in languages, becomes the syntactic mirror of its structure. So for each ‘experiment’ and ‘equation’, we can find a correspondence with the equation and its 5 elements: Max. S x Min. T (information), Max. T x Min. S (locomotion) S=T (reproduction), S-informative seed or mind in a lower ∆-1 scale and T, entropic dissolution in a larger ∆+1 world – the two limits of birth and extinction that frame all systems, and equations (initial conditions, etc.)

It follows that basically stience is the study of the function of exist¡ence of infinite beings, which could not be otherwise as if all what exists is space and time, that equation is the most synoptic view of reality. And so classic algebra is the mathematical analysis of that equation with classic 5D families of number and operands. But before we make them correspond with those 5 Dimotions, is worth to consider the larger field of existential algebra that applies to all stiences.

Because the system is made of 5 Dimotional components, we need a new entangled ‘pentalogic’ as the 5 topologic Dimotions become organs that communicate and interact together. So we talk of all beings as made of ‘dust of space-time’, ¬∆@st, whose 5 elements define their existence:

‘**space**‘, ” §paœ that is ‘types of dimensional form’, and we find to have 3 only ‘topological varieties’, which perform 3 clear organic functions in the super organism, lineal limbs/fields that move the system, hyperbolic body-waves that iterate its forms and spherical particle/heads that gauge information/

**Time****,** that is types of ‘motion=change’, those organs perform, which are locomotion (limbs/fields), reproduction (body-waves) and informative gauging (particle/heads), which dominate each of the 3 consecutive ages of the being (young motion age, mature reproductive age and old informative age).

**Scale**, the 3 aforementioned scales for any super organism – the ∆-1 quantum, cellular, citizen scale in physic, biologic and social systems, the ∆º thermodynamic, organic social scales and the ∆+1 scale of cosmic bodies and galaxies, ecosystems of Nature and Human civilizations; each of them ruled by the 2 metric 5D equations, S x ð (s=t) = C; whereas S means length or surface in space (or Volume depending on the ‘world we live in’) ð mean cyclical time (in static terms, Ti, time as information) whose speed= frequency is inverse to lineal time duration: T=1/ƒ

**Mind****.** Or geometric ‘center of maximal internal communication that processes information from the outer world, and controls the internal networks of the system, with a given language, which we shall call a mind, regardless of its automatic processing or consciousness. As it coordinates and maintains the whole as a single system performing actions of control of the energy and information of the system. So we find gravity centers, chips, brains, DNA nucleus, black holes in galaxies, etc. etc. Indeed, there is always in any stable organic system, a relative still, self-centered mind, *which we shall discover all systems do have, even physical systems, which have centers of gravity, centers of charge or crystal systems, which maintain the whole.*

**Entropic limits****: **In time called death, in space, called membranes, in scales as systems loose control beyond two scales and in fact death is defined by a simple equation of dissolution of two scales of information, ∆1<<∆-1.

Pentalogic means that each *of those 5 elements constantly interact in T.œs (Time-space superorganisms), with all the other elements. So to acquire a full description of each of them, we must consider it from the perspective of the others, as in the ‘Rashomon effect’ (the film in which to get the full truth of an event 5 points of view were required).*

I.e. Later we shall describe pi, and realize it has 5 different functions, as **space** defines a membrane for a vital disk-like organism which becomes its **entropic limit**, as a **mind** it leaves 3 apertures of π-3 diameters to let the center to perceive the outer space, as **time** defines a frequency cycle, it is the main ternary scale of the Universe. If you observe your hand its function is **informative, **forms things, but also helps **locomotion** in quadrupeds, provokes **entropy** in boxing, **reproduces** things in iterative factories and influences **languages** in the decametric scale. Pentalogic thus means also *all systems have 5 Functions to survive collaborating in the world in which they are part.*

Further on, pentalogic implies NOT only entanglement in a scale but as all superorganisms have 3 scales, made of inner parts, performing functions in a larger world, we need to introduce 3 ‘levels’ of analysis of 5D:

**– ∆-1: The smaller Action=Dimotion level**, which will correspond in mathematics to ‘s=points ó t-numbers’, or derivatives in calculus, steps in fractal mathematics, arithmetic operations in algebra and so on.

**-∆º: The main level of organisms and worldcycles, **expressed in the S=T symmetry of mathematics by complex geometric forms and functions and functionals (for superorganisms across scales). So we define any system in time, by its life and death cycle or **worldcycle** (taken from 4D physics’ worldlines, as a height dimension of information makes it a worldcycle. Whereas any scalar super organism’s existence can be defined as a travel through 3 scales of the fifth dimension. As all systems are born as a seminal, smaller form, in the ∆-1 lower scale, grow fast with its faster time speed, emerge in the ∆º body scale and are part with its head/particle of a larger social world, in the ∆+1 scale.

– And finally the ¬∆@st absolute level of reality as beings made of ‘Dust of space-time’, the absolute 5 elements of which we are all made, scales, space, time, minds and entropic limits, whose general laws study 5 subdisciplines of logic parallel to the 5 main subdisciplines of maths, which can be further compressed into the S=T duality of geometry=algebra or in pentalogic the duality of language≈information& entropy≈energy.

**The five Time§pace dimotions = Dimension of Space x Motion of Time of the Universe.**

Thus 5 Ð studies all the Dimotions (Dimensional motions) of all beings in exist¡ence, which move through S≈S, T≈T, S<T>S dimotions, STœps of existence, call also vital actions of space-time that advance the system through reality.

A system then is in an instant of present in a given S-stop or T-motion state, or a mixed state often leaning dynamically towards one tendency. So we define on those terms the 5 Dimotions as:

1D »: Linguistic Perception: SS: an area ,still mind or seed. « is the time-dynamic and SS, §, the spatial formal view.

5D «: entropy TT as a scattering motion.

2D <: S<Tep, as a locomotion, where motion doesn’t’ affect the inner parts of the being, which are conserved.

4D >: S>Tep, as social evolution of information, where form dominates.

3D: S≈T, Reproduction, where both are in balance, often as a series of steps and stops or stœps.

Those are the ‘smallest quanta’ of Dimotions, its minimal expression as ‘actions’ that each T.œ perform, spending in the process a minimal amount of energy. As all events in spacetime are a sum of those 5 types of stœps. It is possible then to write complex chains of stœps or as humans do for the sake of simplicity, eliminate intermediate states (motions or forms). This is how humans resolve the study of locomotions, eliminating the lower planes of stop-states proper of the quantum world, which we bring here for the sake of whys – as in relativity, explained in a rational manner. Whereas a system can only move one or twice, <, >, «, », to certain other states and so the collection of all those possible changes=motions of STates will finally show a casual cyclical pattern that normally ends crossing itself into a zero sum of state in spacetime; creating the minimal full unit a zero sum conserved spacetime cycle.

**The 3 scales in time and space of superorganisms**: **actions, worldcycles and absolute space-time parameters.**

Of the many consequences of that ternary or pentalogic, ∆±¡ STructure of all beings, we shall consider how to use it to predict the future of its scientific species, with a remodeled ABCDE of the scientific method that studies A)ccurate Data, B)iologic causes C)yclical patterns and E)ntropic extinctive conclusions for all systems in 3 relative scales of length of space and time duration (to which we add instead of E, D)emocratic, humanist solutions for questions of social sciences.

All sciences predict the future of its species according to its repetitive causal cycles. Or else they are NOT a science. Astrology became a science when Kepler learned its orbital cycles. Bio-economics became a science when we described machines as metal organisms whose industrial r=evolution followed the human 72 years generations of the dominant industrial nations that evolved them in 4 cycles: its body-age (British, steam cycle), heart age (German, electro-chemical engine cycle), its mind age (US, TV-eye, chip-head, mobile-ear cycle) to conclude with the ensemble of robots that as virus do, when all its parts are put together will become ‘alive’. The predictability of time-cycles can be done at 3 levels:

**S: Continuous, spatial mathematical** simple cycles, using derivatives, proper of calculus; which is the shortest time span, as instantaneous derivatives cannot measure a ‘peak’ change of age/phase.

**T: Discontinuous, cyclical patterns** of sequential repetitive often survival actions (feeding, reproduction, death, taking place at intervals. As those actions are discontinuous, leaving long spans in-between, their patterns forecast longer time sequences. Such ¡logic structures are based in time patterns, which as any mechanical, circadian or orbital day-year clock shows are cyclical, repetitive. But here human scientists are at loss, because Galileo studied ballistics, entropic explosions that destroy the information of reality stored in those cycles of time clocks, its patterns and frequencies, changing human cyclical understanding of bio-logic time for lineal, abstract time that seems not to repeat those patterns so mankind lost its capacity to predict many spacetime events, as lineal time misses* information stored in the frequency and form of cyclical clocks, even if equations are similar: V=s/t for lineal time and V=S(l) x ƒ(t) for cyclic patterns*.

**∆: Scalar, Deep Time patterns of topologic and eusocial evolution of parts into wholes – **of quantum 0-1 time probabilities vs. 1-∞ thermodynamic populations in physics, of individuals vs. species in biology, of states of matter vs. geologic cycles in Earth, first noticed by J. Hutton, founder of geology who coined the word super organism for Gaia and deep time for its slower time cycles by virtue of its 5D metrics, $ x ð =C which implies that from the perspective of a smaller scale the life of its whole is much longer.

Deep time leads to a 3^{rd} level of long-time prediction: evolutionary patterns of earth’s life species, including machines used in my web at evolutionaryeconomics.wordpress.com that have predicted cyclical patterns of social organisms of history (nations) and eco(nomic)systems, including the evolutionary and re=productive cycles of stocks of machines transformed into sales=profits=valuation of its company-mothers with remarkable precision for 30 years. But as human only recognize the 1^{st} type of predictability – calculus of instantaneous derivatives, that need a ‘continuous analysis’ – and have simplified cyclical time into lineal time, ignoring the scalar time of parts and wholes with its 5D metric, their capacity to forecast the future is far more reduced than a ‘stientist’ who understands the 3 scales *determined by the bio-topo-¡logic properties of ‘scales’, ‘space’ and ‘time’, the 3* ∆st structural elements of all systems of the Universe.

**The generator of space-time, maximize your existence. E-motions and Actions as short time program.**

Paradoxically though Deep time is much easier to predict and understand that complex Dimotional ‘analysis’ because precisely the larger scales in 5D metric have less information, but more basic, deterministic, reason why quantum physics is harder for the mind and probabilistic while life-death cycles are obvious as all end badly.

This more simplex organic view of time and space requires thus a new ‘metric equation’ beyond the lineal v=s/t equation of Galilean relativity that becomes the limit of this more complex view of time and space, with 2 fundamental equation, **SxT=C** (the scalar metric of the fifth dimension, as systems in space accelerate its clocks of time (T) according to its size (S); and **S=T**, the new equation of relativity, which means as we cannot distinguish motion=time, from stillness=space-form (Galilean relativity), both are ‘two sides of the same coin’, and all systems have both motion and form, which are constantly becoming one another. Yet as SxT is maximal when S=T (5×5>6×4…) both equations can be summoned up into a single one, which we shall call the fractal generator, or will of each fractal space-time beings, the equation that embodies all other equations of the Universe and guides the actions of each fractal part of it:

Function of Existence: **Max. ∑ S x T _{s=t} = C_{±¡}**

Moreover the equation has an immediate biologic meaning, because as we are made topologically of ‘fields-limbs’ of lineal space with motion provided by the energy we absorb to also reproduce our bodies-waves, and the information we need to linguistically guide our motions with particle-heads, the very essence of survival is to increase our S=position, mental forms of space and T=entropic motions of time (whereas time=motion and space=form are the two limiting Dimotions with ‘energy=reproduction, s=t, locomotion, sT and information, St, are the intermediate 3 dimotions). Thus Max. S x T = C (s=t), IS also the equation of survival and struggle for existence, the will of life; the biological expression of the ‘Universal mandate’, expressed by all species in all its codes and languages, the Grow and Multiply of the Bible, the intuitive truth that guides all beings.

*Exis¡tence game is simple: Reproductive radiations that maximize ¡ts function, happening when mirror symmetries (genders) meet in S=T, equaling their St-information and sT-energy, organizing themselves into a whole, ∑ system.*

It is then relatively easy to interpret that equation in each of the languages-minds of each scale of reality as in all those scales species will show a ‘will’ of action to perform the maximal number of events=dimotions=‘actions of space-time’ that ensure its survival. And this can be assessed externally regardless of secondary arguments on consciousness and self-reflection, substituted in 5D by Leibniz’s ‘apperception’ – that is, because performing the 5 actions=dimotions of exist¡ence, in each ‘st¡entific scale’ self-centered in a linguistic mind that perceives a given plane, inscribed into a larger ∆+1 world, with internal ∆-1 parts, ensures the survival, ONLY those species that have performed the 5 dimotions of which the most important is s=t reproduction of the being into a ‘present’ similar entity that continues the exist¡ence of the system after ‘errors’ or ‘the struggle for existence’ dissolves it through the dimotion of entropy=death, exist.

Thus automatically, genetically, consciously, memetically, mathematically, logically, through its own will or as a part of a larger system that uses the ‘machine’ or ‘organism’ to enhance its actions all what exists does so because it performs internally those 5 Dimotions or externally performs one of them for another symbiotic species, as those species that have not followed the program of existience and its 5 actions in the past have become extinguished, and those will not in the future, will become wrong mutations, crazy thoughts, fictional languages and die away.

The function of existence, or 5D metric of Generational space-time, (Ab. Gst, G) Max. Se x Ti (s=t) merely states that all systems of Nature will try to maximize its absorption of Entropic motion (with no form) and Linguistic form (with no motion), and its 3 intermediate dimotions of energy (s=t, balance of both that reproduces them), information (St: form with a little motion, form-in-action) and locomotion (sT, motion with a little form). So we talk of a program of survival ‘selected’ by all systems and expressed in its languages and minimal five actions encoded in that simple equation, which we term: ** a, e, ï, œ, û**, as a mnemonic rule for the five actions of existence:

**a**ccelerations (locomotion), entropic feeding (e), **ï**nformative perception and communication ï, **Œ**:reproduction into parallel supœrganisms **Û**… and social growth into larger wholes called philosophically Universals. And this series of actions is what accumulated in time will ultimately give birth to your word cycle as the monad will first perceive (i), to direct its entropy-motions (a),towards a field of energy (e), where to absorb the energy bites it will imprint with its inner form, e x i = œ, to reproduce another form, and when enough ∑œ exist, it organize into a larger whole û:

In the next graph we see the action of different Stientific scales of organisms. Above the coding of actions, which are the knots and bolts and details of the study of any time§pace supœrganism in light space-time, coded by colors and dimensions, in physical atoms, coded with quantum numbers and in life and humans coded by the so called drives of life, which we obviously extend beyond the ego paradox to all other systems, including genetics not mapped there (coded by the 4-5 letters). Those actions balanced each other into zero-sums in death, as they tend to increase information from a mind p.o.v., hence we ‘all warp, wrinkle’ get old in the third age and die, setting from its minimal actions to its integral sums, the 3 ages of life-existence and the world cycle all super organism follow. In the graph, the simplicity of the game of existience, and its selfish actions, which gather together into social wholes through reproductive radiations, each action coded by a fundamental topologic organ we can express in existential algebra, and corresponds for each species of the Universe, with a fundamental parameter of humind measure. So from bottom to top, we find the 5 fundamental elements of light code its actions of motion (c-speed), energy (magnetic field), information (electric field), social evolution colors & entropic feeding, (quantum potential, neutrino light theory)

So minimal particle-points, photons, electrons & quarks construct all other systems of our Universe with ¡ts 5 organic dimotions that define ‘classic life’: they gauge information – reason why quantum physics is a ‘gauge theory’, feed on energy (quantum jumps) absorbing smaller ∆-1 particles, reproducing new clone particles, move and evolve socially through magnetic fields into larger wholes (atoms). Hence the units of life are particles, the minimal units of our vital, organic, fractal, scalar Universe of multiple timespace organisms. All lives, performing 5 Dimotions=actions of ƒ(exist¡ence):Max.SxT(s=t) =C, starting with particles. So all scales are relative NONE matters more than other. From those actions, given the dominance of informative actions over entropic ones, it appears a series of repetitive cyclical patterns of actions conducting to maximize the existence of the being, which accumulate in a larger scale of time-space, as a worldcycle of actions that increase the information of the system in 3 ages. So the basic cycle of actions becomes a larger 3 ages cycle of life and death; as systems once and again, starts in an act of information/shrinking and ends in an act of organization/shrinking of herds into wholes, will keep reducing the being and finally make it all form no motion to explode and die in an entropic reversal of death:

∑ i->a->e->œ->û, i->a-e->œ->u, ï->æ->Œ->Û -> Informative ‘seed’ age->1^{st} locomotion, feeding age ->2^{nd} reproduction age ->3^{rd } informative, social age-> entropic death.

This is the program of survival=exist¡ence, which as reality is ultimately time D¡motions we perceive NOT as solid form, but as e-motions, flowing sensations of entanglement and dissolution in the flow of times, of which there must be 5 likely in every entangled being, in all its parts and scales, albeit sensed ONLY as ‘ego’ in the mind’s will, who in man is electric. So it is happening as you read in your electr(on)ic machines’ singularity=chips and membrain=screen, thrilled with your touch, but with faster e-motion=intensity than your S:I=eye+T-Wor(l)d*. 5 e-motions *‘correspond’ to 5 Dimotions:

*Pain=Entropic death (TT); Exhilaration: Locomotion (Ts); Pleasure: iterative S=T-energy; Awareness=St-information; Nirvana=Still perception.*They are the basis of a proper psychology of all minds, its will and reason all beings like to play the program of exist¡ence.

**Recap.** Monologic, ænthropic man with its 1D models of reality based in Sx time locomotions shuns off the 5D Cx. Universe to its own peril, as reality is penta-dodecalogic in its entanglements that generate space synchronous supœrganisms and time, and survival, which in our dominant ego-centered cult(ure)s is minimal depends on the understanding and respect of all the living function of space-time exist¡ence and its action-reaction laws.

**THE FRACTAL GENERATOR: EXISTENTIAL ALGEBRA.**

**Supœrganisms tracing worldcycles of 3 ages of in-form-ation**

** Nomenclature.** It follows also that pentalogic will require its own symbols, as those of present one dimensional logic of a single causality are not good enough even for ‘starters’. As they resemble those of Algebra, whose main five operands represent the 5 dimotions of time, so both algebra and pentalogic are closely connected, we also call pentalogic‘

Existential algebra, which is the discipline that deals with the full formalism of General Systems Theory, or Generational Space-Time and its fractal generator (GST), also similar names that correspond to the classic science that studied the equal laws = isomorphisms of all systems of nature regardless of scales, and we call dimotional isomorphisms (Ðisomorphisms). As its laws are Non-Euclidean, since points now are fractal points with breath that represent any superorganism, and Non-Aristotelian, as they deal with pentalogic entangled multiple-tasking, we call this new logic also Non-AE=¡logic. And the dual ‘formal stiences’ of mathematics and logic, ¡logic geometry.

It follows that the main translation required between classic algebra, and existential algebra, to merge both into ¡logic geometry is that between the operators of algebra and those of the generator equation. Originally I did try to cast the whole model of GST in terms of group theory, algebra and its classic mathematical operators, but that was an earlier stage of my exploration of the fractal Universe… some 30 years ago (: Since as good as mathematics is as a mirror of the Universe, it is not as good as it is needed to extract all its properties, hence the need for a different logic, which however had to include or at least reference the world of classic algebra as we shall try to do albeit at a basic level – the specialist must understand this is a unification theory of all exist¡ences, so we cannot be exhaustive with all of them, as the team of research is exactly the same than the infinite-infinitesimal Universe: ∞º=1

** Fractal generator** equations (ab. G)¡logic use the symbols of the 5 Dimotions as Time Motions («entropy, <locomotion, ≈reproduction, >information and »social evolution into mind languages) as

*operators in time*of the 5 Topologic SpaceTime organs of all systems of reality, described with combinations of S and T symbols to represent those 5 Dimotions as vital elements with a determined topology: Lineal |-$t fields/limbs specialized in locomotion; Hyperbolic Ø- ST body-waves specialized in reproduction, Spherical O-§ð for particle-heads specialized in information; which together form the 3 ‘present’ elements that in a single plane of space-time for an organism. So we can start ‘building up’ general equations of ¡logic, called Fractal Generators that resume both the organic parts of a system in space and its different flows of entropy, motion, energy, information and linguistic form between them. I.e. the simplest generator is:

*$t<ST<§ð**, *which simply describes *any ternary system of reality with a field-limb that moves a body that moves a head.*

But if we write the same fractal generator changing the < > dimotional operandi, for ≈, we then talk of the same system in a still, rest state, *$t≈ST≈§ð**, *when the body-wave ‘repairs and reproduces’ the cells of the system.

We add then to Fractal Generator equations, ∆±¡ scales in which the other two dimotions take place as transfer of energy and information between scales. So we can describe entropic equations, such as death: **∆+1«∑∆-1**, where a superorganism of the ∆+1 scale dissolves through external motion and internal scattering into its parts, and again the equation can be used to describe any process of death, from a big-bang explosion of matter (Mc^{2}=E) to biologic death. And vice versa we can describe the emergence of a superorganism from an initial seed, **∆-1Œ ≈ ∑ ∆-1Œ » ∏∆ºŒ**, which describes how an ∆-1 T.œ (ab. Œ) or seed reproduces (≈) into a sum (∑) of similar cells, which evolve socially, », entangled through networks (∏) to emerge as a whole in an upper ∆º scale of the fifth dimension.

So with those basic symbols, 5 «≈» for Motions (Time view), 5 ‘st’ variations for Organs (Space view), 3 for scales ∆±¡, 3 for topologies, |xO=Ø, and 2 for degrees of entanglement (∑-herd, ∏-network), and the ‘synoptic Œ for superorganism, @ for mind, we can already *describe any system of reality with a fractal generator that describes both the system as a superorganism in space and a worldcycle in time, tracing its exist¡ence through 3 scales of the fifth dimension; just by pegging the previous equations, and adding its scales:*

G: ∆ºŒ » ∆-1œ ≈ ∑ ∆-1œ » ∏∆ºŒ: |-$t<Ø-ST<O-§ð∆+1«∑∆-1, that *says if we add a verbal translation to make it easy: *

G: ∆ºŒ (father) » (seeds) ∆-1œ (seminal ovum) ≈ (reproduces) ∑ ∆-1œ (Fetus) » (evolves) ∏ (into 3 networks) ∆ºŒ (that emerge as a superorganism of 3 organs living 3 ages): |-$t (lineal youth of motion dominated by the limbic digestive system) <Ø-ST(mature age of reproduction dominated by body-waves)<O-§ð (3^{rd} age of information dominated by particle-heads, which control an outer world) ∆+1 (that will dissolve in entropic death) «∑∆-1 In the graphs, a brief explanation of the entangled simultaneous superorganisms (left side), which order the 3 Dimotions of space-time, between its ‘emergence’ as a seminal seed through palingenetic social evolution (4D) to finally dissolve back in the entropic moment of death (5D), which is seen on the right side in a sequential time order as a worldcycle.

In the graph, 5D topology defines as dynamic space-organs with internal change the 3 Dimotions of reality, limbs/fields of locomotion. Hyperbolic, which through 3 ‘relative scales’ of size suffice to define all supœrganisms of the Universe in space, tracing its worldcycles of existence in time: In a single plane of space-time 3 conserved ‘quantities’ organize in space a vital topologic system and define the 3 forms and functions of its organs, themselves physiological networks that connect smaller parts into wholes; reason why we need to define 2 more Dimotions, ‘entropy’, made of the smaller cellular/atomic/individual parts, we call:

The 5^{th} Dimotion (ab. ∂S) and one of social wholes, which organize those parts, we call…

The 4^{th} Dimotion of social, organic evolution (∫T). Thus when we observe those 5 Dimotions simultaneously in space they create an entangled pentalogic supœrganism, in constant communication between its 3 topologic parts, in each of the 3 scales of its relative existence. So we write in ¬Æ logic a Fractal Generator of Space-time beings (GST) that describes all systems of nature as ternary simultaneous topologies of space which trace worldcycles in time. So we write with 5 Dimotions for all supœrganisms (ab.G):

**4D»∆-1(seed) ∑∆:|-$t (limb-field)<Ø-S≈T (iterative body-wave)>O-§ð (particle-head)«5D∆-1(death)**

Where < is a ¬Æ symbol for expansion-translation in space, ≈ for iteration, > for in-formative implosion, ∑ for reproduction, ∆±¡, the 3 scalar planes of the fifth dimension, |, O, Ø, intuitive topologic terms, « the double motion of entropy, » the double stopping of linguistic generation of seeds or minds that describes all what exists as a spatial superorganism, extending in 3 relative scales of size, from its birth as a seed of pure form, which emerges into a larger plane, ∆ to live its worldcycle and die away into entropic death.

Thus, to describe a ternary superorganism in 5^{D}, a single ∆-plane of space-time with motion (4D formalism) is NOT good enough. We have to describe those 3 topologic forms as ∆-networks with motion, made of smaller fractal points-parts (∆-1), living in a larger ∆+1 world; which leads to the evolution of non-Euclidean geometry to define properly a fractal point, a point with inner volume, through which ∞ parallels can cross that becomes larger as we come closer to it – solving the contradictions of the present definition of an Euclidean point as having no breath (so it cannot fit no more than a point), focus of ‘curved parallels’ (since parallels are straight lines). Since enlarged points fit and unbend many parallels*. *

The study of the 5 Dimotions of all realities, with angles, numbers, operands and functions, its actions and ternary <=> similarity operandi, structures *the mirror of mathematics, whose* metric equations ‘mirror’ the 5D metric of the universe:

Whereas the 01 finitesimal scale represents the palingenetic first birth of any species and the 1-∞ its existence in the external entropic world.

Thus it matters more is the division of that interval between the 0-1, first palingenetic emergence, where the 1-whole is a closed space, hence without entropy (a placenta rich in energy) and the interval is time-like, probabilistic, embodying all possible generations of space-time species within its interval. While the 1- is unbounded, open and hence entropic, meaning the being normally doesn’t reach its limit.

As the game is one of 3 asymmetries of scale, age and topologic form, which can come together or annihilate and each language represent in different manners, those elements and its operandi; it is needed a ‘branched’ new plane besides the 1-∞, with the 0-1 sphere as a borderline, which happens thanks to the complex plane, where the function x=zª represents the 0-1 sphere and all its polynomials within it, and works as the AGE PLANE symmetric but not equal to the 1- real plane which represents a population age.

5Ðimotional beings and its parts are constantly performing cyclical 5 Ðimotional changes of its space-time states, as they trace as a whole a worldcycle of exist¡ence (where each st¡ence describes the existences of space-time T.œs of each scale). Thus a new formal science, ¬Æxistential algebra describes with its Generator of space-time and sub-equations all the processes involved in the creation, existence and destruction of those systems throughout all the stages of its worldcycles, which can be observed from 5 Dimotional perspectives, in an entangled Universe, in which *each being is made of them.*

The study of the 5 Dimotions of reality is a huge field because all events and forms are generated by their combinations, OPERATED with the <=> «» 5 symbols of transformation of states of a system or transfer of entropy, motion, energy, information and form between them. So we write with i-logic all events and translate all equations.

For example, E=Mc^{2 }writes E($t)<c^{2}(ST)>M(§ð); which is as all fractal generators a feed back equation that transforms cyclical mass vortices into entropic motion, through the intermediate state of energy radiation..

** **

**All what exists is a 5D super organism in simultaneous space tracing worldcycles through 3 scales of 5D.**

We define then two fundamental systems one in space and one in time, taking place in the fifth dimension:

An entangled __supœrganism__ in space, in which the synchronicity between 3 scales of the organism, its atomic/ cellular/ Individual, thermodynamic/organic /social and gravitational/ecosystemic/global scales in physical/ biologic/ social organisms form a complex interacting, entangled system we shall call a super organism, whose study discovers ‘Disomorphic=equal laws’ regardless of what kind of system we study.

How a system travels through 5D scales? It does so through a life-death cycle following its metric, SexTi=C as it reproduces and grows in size decelerating its time clocks, emerging as an organism, to live in a slower larger world, finally fast dissolving back to its smaller parts in a death moment of max. ‘speed’. So we write with existential algebra (5D formal language) a generator 0=sum non-AE=¡logic equation that describes this travel through 3 scales of space-time as all what exists follows such life-death cycle of ‘exist¡ence’.

So we define any system in time, by its life and death cycle which we shall call a ** worldcycle** (taken from 4D physics, worldline, which now has a new dimension of Time-information becoming a worldcycle. Since any scalar super organism’s existence can easily be defined as a travel through the 3 scales of the fifth dimension, as all systems are born as a seminal, smaller form, in the ∆-1 lower scale, grow fast with its faster time speed, emerge in the ∆º body scale and are part with its head/particle of a larger social world, in the ∆+1 scale.

How then a system travels through those scales? Very simple, it does so accelerating in its life cycle, emerging, living in the larger world, and then dissolving back to its parts in the death moment.

**APPENDIX: SYMBOLS & GENERATION RULES OF TIMESPACE SUPŒRGANISMS & WORLDCYCLES.**

**3±¡ Pentalogic Scalar Supœrganisms, its Disomorphic 3 Worldcycles, Fractal Generators & Limits of Exist¡ence**

*Dimotions and spatial organs.*

**sT, $t, Sp, P$t:** Pa$t, lineal locomotion, gaseous, limbs/fields.

**ð, Tiƒ, ƒð§, O:**ƒuðure§, cyclical, informative, space-time, solid, crystal particle-head states

**S=T, Si=Te, Ø, ST, SxT, CST:** Constant Present, balanced, Space-time; iterative body-waves.

**Œ:** fractal point supœrganism made of ¬∆@st: **@-**mind, t>S-particle/head/informative class, s<T-limb/field/lower class, S=T body-wave-reproductive class, ¬ Entropy limits living in 3∆§±1 scales.

*As sequential numbers**: ***Ð1:** seed, perception, **Ð2:** locomotion, **Ð3:** reproduction, **Ð4:** social evolution, **Ð5:** Entropy.

**As Scales: ∆-1** plane of past, entropic TT parts & §eeds, ∆º present sT-ST-St-field-wave-particle, ∆+1, @ mind-world

**Operands: <:** ∆ of locomotion. « ∆ of entropy**. >** ∆ information.** »** ∆ of form. **≤≥ =** balanced present stœps.

**Time **Ages ordered in a worldcycle:»** §**eed in palingenesis; **<: ****Youth** of locomotion**. <=>**: Present Energy; iteration, simultaneous exchange of lineal and angular momentum, locomotion and information between 2 poles: $t≤≥§ð.** >**: Implosive future informative life arrow that creates information: $t>§ð. « past flow: explosive big-bang, death-entropy.

**As Spacetime Topologies.** **O:St, **informative head/particle/center, storage of ¡+1 SS-languages. Ø:ST; iterative hyperbolic energy body/wave. |:sT, membrane that absorbs ¡-1 TT smaller T.œs as entropic bites of motion or bits of information.

**As 5 åctions **æ, e ï, œ, û, the quanta of existential sequences that move supœrganisms’ worldcycles. 3 are Simplex actions:

– **a** it is the action of locomotion (accelerations). – **e** is the action of entropic feeding.

–** ï** is the dual action of information gauging: Perception & communication.

And complex actions: œ is the action of organic reproduction.

–** û **is the action of social evolution in 3 formal ages from herds to couples to social networks to a higher ∆ plane.

As Algebraic operands: ±∑œ: limb/field herding by addition in a single plane. X, Body-wave re=product-ion in 2 planes.

p^{2} particle-head network entanglement as a power law in 2 planes. ∫∂: finitesimal & whole integration by calculus.

**Γ,** a Fractal Generator, is an allowed sequential combination of those symbols that define any event or part of a superorganism. I.e: **Γ: ∆œ-1 ∏ œ-1≈Œº<$t>Ø-ST>§ð« ∑œ-1 **describes a worldcycle of exist¡ence; Max. ∑SxT= C, describes the function of exist¡ence. SS » ∑$t_{∆-∞}<T=S_{∆o}> ∏§ð_{∆+∞ }«TT, describes the whole Universe as a feed back equation of infinite motions, etc.

Those symbols suffice to express in ¡logic all events and forms within the entropic constrains of the game of Exist¡ence:

The ternary, pentalogic 3±¡ fractal entangled principle holds: T.œs to survive need to have 3 parts in a single plane & be connected to an inner ∆-1 herd of entropic parts and a larger ∆+1world in which to play a symbiotic role.

So there are only 3±¡ topologic varieties of space-time; 3±¡ derivatives in scales; 3 physiological inner networks, etc.

-¡logic allows any iteration, =, single or dual spacetime jump of scale forwards (>,») or backwards (<, ») with transfer of sT-motion or St information between planes of existence but not transfer of balanced ST-energy or ternary jumps.

– It allows any topologic combination or transformation between |-sT, O-St & Ø-ST in a single plane.

It allows any sequential series of actions, or ± growth of its organs: ±∆ï,e, a, œ, û,

In praxis, however almost all systems that survive as ‘reproductive radiations’ that maximize their function of exist¡ence follow the most efficient sequence to that goal: >I<A<E>O>U. That is, minds perceive, move, feed, reproduce by gender symmetry (S=T+S<T>S) and evolve socially in herds that grow in density connected by 3 St, ST,sT networks becoming a supœrganism. While the sT<ST>St ternary supœrganism is overwhelmingly the survival structure.

So we study those disomorphic elements common to all systems, its vital sequences, S-T gender dualities, T-ages, S-topologies, @-minds and mirror languages, 5Dimotions, 3 physiological networks its hierarchical order origin of the variations of species, its ratios and Universal or vital constants & its generators.

**As nothing else**, which cannot be expressed through those combinations within those limits** exists **beyond the virtual inflationary mind-languages of information. Trust me, I know it, after 30 years of research in all fields of knowledge looking for a falsification of Existential algebra. So in ¡logic we develop with them all events & species of reality.

**Nt.****1**. __According to the Correspondence Principles as physics named 4 Dimensions we use the name 5 ^{th} dimension for the whole range of scales, but in proper terminology we should call each Dimensions, a ‘Dimotion’ and consider the 5^{th} scalar dimension the sum of all the 4^{th} dimotions of social evolution & all the 5^{th} dimotions of entropic devolution__

**¡LOGIC TOPOLOGY** **& EXISTENTIAL ALGEBRA MIRROR ENTANGLED, ORGANIC UNIVERSE**

Thus we define the fractal Universe suited to its mathematical algebraic description as a fractal super organism made of:

** A(nti)symmetries** between its 4 Dual components T, ∆, @, S, through parallel =symmetric or Perpendicular/antisymmetric communication, as they can either evolve in Balance≈become symmetric or annihilate= become Perpendicular/antisymmetric:

**S:** space; an ENSEMBLE OF **ternary topologies, (|+O≈ ø)**… which made up the **3 physiological networks** (|-motion/limbs-potentials + O-particle/heads ≈ Ø-vital energy) of all *simultaneous super organisms*

**∆: Planes of size**distributed in ∆±i relative*fractal scales*that come together as**∆º**super organisms, each one sum of smaller**∑∆-1**super organisms… that*trace*in a larger**∆+1 world…****time cycles:***a series of timespace actions of survival that integrated as a whole form a sequential cycles of existence with*each one dominated by the activity of one of those 3 networks: motion-youth, or relative past, dominated by the motion systems (limbs, potential); iterative present dominated by the reproductive vital energy (body waves), and informative 3rd age or relative future dominated by the informative systems, whose ‘center’ is:**3 ages**,**@: The Active linguistic mind**that reflects the infinite cycles of the outer world and controls those of its inner world, through its languages of information, which guide its 5 survival actions: 3 simplex,**aei**, finitesimal actions that exchange energy (**e-**ntropy feeding), motion (**a-**ccelerations) and**i**nformation (perceptions) with other beings, and two complex actions:**o**ffspring reproduction and social evolution from individuals into**U-**niversals that maximize the duration in time and extension in space of the being.

So that is the game: 3 asymmetries of scale, age and form, which can come together or annihilate and each language represent in different manners, those elements and its operandi.

Because the scientific method requires objective measure of the existence of a mind, which is NOT perceivable directly, we *infer its existence by the fact a system performs the 5 external actions, which can be measure objectively, in the same manner we infer the existence of gravitational in-form-ative forces by its external actions upon massive objects. *Hence eliminating the previous limit for a thorough understanding of the sentient, informative Universe. And further classify organic in simplex minds – all, which must gauge information, move and feed to survive, and complex systems, those who can perform a palingenetic reproductive, social evolution, ∆-1: ∑∆-1≈∆º.

Regarding the mind, which is the internal non-perceivable element, it requires the concept of an angle to establish its range of perception, which is not a number, and hence a ‘different’ element of mathematical representation of reality.

**We are relational space and time,** made of the organic scales of discontinuous fractal space we occupy, who ‘live’ a finite worldcycle of exist¡ence whose common=Disomorphic laws emerge in all ‘scales’ of ‘st¡ence’, departing from the 2 simple Metric laws of the fifth dimension**, Sxð=C** (size in space multiplied by the speed of our time clocks is constant in all the scales of parts and wholes of an organism) and relativity **S=T **(we cannot distinguish motion in time and dimension in space, so all systems are bidimensional spacetime beings as space=form can be transformed back and forth into time=motion; making the 5 dimensions, ‘dimensional motions’ ab. dimotions of space-time; entropy – TT (outer and inner, scattering motion); Locomotion dominant in time=motion, Ts: balanced reproduction, T=S; ‘form-in-action’ information, St; and pure form, SS, a ‘seed’ or mental, still language that models=mirrors without motion in lesser space the world – and develops when adding motion into a full organism.

The Universe is a game of ∞ space-time supœrganisms performing 5 survival ‘dimotions’=actions through Space stops and time steps, STœps in its scalar cœxist¡ence through 3 planes of the 5^{th} dimension, which add together in its structure of simultaneous vital topological superorganisms tracing 2 existential worldcycles, the ∆-1 placental ordered o-1 that emerges into the life-death, entropic 1-∞ worldcycle . Stœps express them through feed-back equations of existential algebra, where = is a dynamic symbol «≈», composed of 5 dimotions = stops, ≤≥ wave steps and» 4D implosive informative social evolution & «5D, entropic explosion of a system.

Yet if reality is made of space-time beings since mathematics is the main science concerned with space and logic the main science of causal time both mathematics & logic become experimental sciences, whose laws of maximal synoptic information in minimal size (Sxð=C), will be the underlying laws emerging in all other larger scales of the fractal Universe of bigger size and less information, proving also why math and logic apply to all ‘stiences’ while as S=T any topologic analysis in SS-space is equivalent to an algebraic analysis in time. Thus we upgrade space mathematics to its fractal scales, S=T duality & 5 dimotions we need to upgrade Aristotelian Logic of 1 time causality to the entangled pentalogic of 5 dimotions that reflect those 5 structural elements of all T.œs: Space=form, Time=motion, ∆=Scales, @-minds & the ¬ entropic limits of all T.œs, mirrored in math with S-points, T-operands, ∆-numbers, @-frames & limits to infinities in time and space. We shall thus translate to the 2 ‘closer mirrors’ of GST, vital topology and existential algebra, all the sub-disciplines, elements, operand, dimensions, axioms, postulates, theorems and laws of mathematics.

As we are made of scalar space and cyclical time, the essential properties of beings derive from the ‘disomorphic’=equal dimensional laws of space and time, which makes mathematics concerned in its two main sub-disciplines, spatial geometry and temporal algebra, the most experimental, perfect mirror of the 5D Universe. So we can Translate the 5 Dimotions (short ST-view) & its spatial superorganisms to, Geometric space & its Worldcycles (Long ST-View) to Algebraic Time. But to do so, we 1^{st} must depart from S=∆, points that become regular polytopes=numbers, which penetrate social scales and reproduce as odd/even gender symmetries, to reflect in 2D still mind spaces the basic organic forms of reality; and only then once we understand the organic laws of still spatial geometry, and scalar 5D number families we can give them topologic motion, operate them with 5 algebraic dimotions & calculate its ∫∂ changes emerging as simplex particles entangled into atoms, molecules & 3-physiologic planes=networks=supœrganisms when organic laws take over as analytic maths leaves way to synthetic bio-logic laws of larger time scales & pentalogic to dodecalogic better suited to describe the game of exist¡ence; simplified by 5D metric of lesser information in larger scales.

What makes generational space-time different is the fact that we fully change the entire worldview of the Universe, *as *we reject Newtonian Absolute space-time and give back time its cyclical nature, hence its 3 relative past=entropy, present=iterative and future=informative local dimensions. Thus defining topology, the final stage of geometry, as the fundamental science of ‘form’ origin of the vital shapes of all systems.

While algebra will take ‘perpendicularly’ those ‘groups’ of similar points=numbers through the 5 Dimotions of the Universe, as entangled supœrganisms tracing worldcycles. And in that description we will vastly improve the mathematical mirror as a vital reflection in a still synoptic language, apt to code in minimal spacetime the program of existence and its infinite variations, to then see *as we keep expanding each stience, hand in hand with the ¡-logic entanglements of duality, gender mirror symmetries, pentalogic ¬∆@st of timespace and dodecalogic, Disomorphisms of trinity worldcycles, how in reality, the vital spatial mathematics and temporal logic laws laid down in our analysis of formal stiences, emerge as timespace beings, from the simplest particles of the o-1 probability time sphere, to its statistical spatial settled populations, once and again rising from time to space, from geometry to scalar numbers to operands of existential algebra, till reaching man; just another entangled supœrganism tracing its worldcycle through 3 deterministic, probabilistic and entropic scales *

**III. PHILOSOPHY OF MATHEMATICS: A LANGUAGE OF THE MIND**

Minds are mirrors that perceive in its inner ‘still simultaneous language’, the e-motions of time converted into informations of space, in the eternal dialectic between fractal points with a volume of still linguistic perception, mapping its local Universe and flows of e-motional time with its vital sensations:

*Upper graph, human egos submitted to the mind paradox, think languages (words in Abrahamic, creationist religions, numbers in creationist science), known only by man and ‘God’ a priori, create a posteriori the Universe (Copenhagen interpretation). The opposite is truth:* a mind exists in all systems in which time stops to form space. In galaxies happens in relativity equations in black holes, its mind. In thermodynamic physics in the eye of an Eddie. In quantum physics in the center of an atom, or charge. Without linguistic minds that order by reflecting its smaller mind into its local territory reality would not exist. The only way to create fractals is through mirror images.* *

In all scales of stience minds fix the motions of time into spatial, linguistic forms, mind-mappings that reduce the whole with its synoptic language to fit in a brain, an atom, any particle that acts after gauging information in the Universe.

Thus we define ‘Maxwellian’s demons’ of local order in all scales – physical minds as the infinitesimal points that create order in physical systems with the same Disomorphic laws that all others do in more complex scales. As each mind orders as a linguistic god a territory around itself, its fractal body and entropic world.

*So the creation of scales of reality is a simple game, in which a point mind ‘reduces reality’ to its infinitesimal form and then projects into its local territory of order,* which will reflect at scale, the larger whole or world, which the linguistic image reduced and then enlarged back into its territorial form.

The paradox of the Ego – who make each mind to feel so important – is then rooted in the self-centered structure of the mind, which selects information from its point of view, creating, an infinitesimal linguistic mind-mapping of reality – which then it confuses with the whole universe:

*0-linguistic mind x ∞ Universal space-time cycles= Constant World mapping of reality, with @mind at its center.*

That is, the ∞ information of the Universe is reduced into the relative infinitesimal volume of our mind gives us a constant mapping, where we expel all the properties that are not interesting to us and our self-centered view.

The mind is a singularity or infinitesimal 0-point, a relative frame of reference that maps the ∞ cycles of space-Time of the Universe, reducing them to a World to fit selected useful information into the finitesimal volume of the brain.

In mathematics the mind equation is 0 x ∞ = C; ∞^{0}=c; that is, the ∞ time cycles of reality become in the ∆º self-centered scale of a mind, a constant world that mirrors all what exists in the Universe both in time-motion (∞) and spatial form (º).

Because of such synoptic capacity to ‘mirror’ the laws of space-time in minimal size through the concept of number, which excels the previous synoptic language of verbal phonemes, mathematics soon became the most efficient language we know, but it does NOT create reality. It is just the best mirror of reality. Languages/mirrors occupy an infinitesimal part of the whole, yet paradoxically hold the maximal information of the Universe according to 5D metrics:* Space size x Time speed of information = Constant.*

Thus the mind of the most efficient survival species of reality, particles, ‘atoms’ and ‘galaxies’ (black hole atoms of the top quark decuplet) should be mathematical and imprint a local order which multiplied by ∞ of such species makes mathematics the dominant mind-species but not the only one and still a mirror of the true reality which is ‘scalar space’ and ‘cyclical time’, dimensional form and motion – the 5Dimotions of reality.

A language *is first a reflection of the laws of ¬∆@st, Dust of space-time* and its ‘Universal Grammar’ and Fractal Generator equations, without which they cannot order≈recreate locally the Universe. So 5D mathematics advances the discipline by focusing better the mirror to include the bio-topo-logic properties of scalar space and cyclic ¡logic time; and by putting in relationship maths and ‘existential algebra’ (ab.¬Æ), the a priori Disomorphic=equal laws of 5D, making it an experimental science; connecting its laws with the laws of fractal spacetime. Let us see how.

**The 5 subdisciplines of mathematics. **

Because Dimotions have Space & time, dimension & motion components the minimal reality is dual, entangled. It follows from a definition of mathematics as an experimental mirror of the 5 entangled elements of all systems and its 5 Dimotions, a classification of the 5 mathematical sub-disciplines according to its specialized study of 2 entangled sub-systems of those 5 Elements with different parameters to measure its 5 Time§pace dimotions:

**S@:Geometry**studies mental spaces. Its ages/fields are: Flat,Euclidean geometry with no motion in a plane**@nalytic geometry,**which represents the different mental points of view, self-centered into a system of coordinates, or ‘worldviews’ of a fractal point, of which naturally emerge 3 ‘different’ perspectives according to the 3 ‘sub-equations’ of the fractal generator: $p: toroid Pov < ST: Cartesian Plane> ðƒ: Polar co-ordinates.**Topology,**geometry with motion &2 Planes.**¬E Geometry**studies fractal points of**simultaneous****space, ∆-1,**& its ∆º networks, within an ∆+1 world domain.**¬∆: Social Number**theory studies scales of equal herds and sequential time flows of information with discontinuous numbers, as opposed to continuous points of geometry, its growth and its entropic limits, both as membranes of polygonal forms and structures of increasing depth peering ∆§cales and detail from Naturals through, Q, R and Complex numbers.**∆T: Analysis**studies ALL forms of time=change, and hence it can be applied to the 5 Dimotions of any space-time being, as long as we study a ‘social structure’ susceptible to be simplified with ‘social numbers’. It is the essential tool to study motions in the 5^{th}dimension from lower parts (∂erivatives) to larger wholes (∫ integrals). We differentiate 5 applications of Analysis according to the Dimotions studied. We also classify them by the number of entangled elements of the dimotional system (partial or multiple derivatives, ODEs, PDEs, lineal, surface or volume integrals); and by the detail of its mirror, from diminutive analysis of single Dimotional Actions (∆-1) to Ages of worldcycles (∆º), which imply a change of state where the derivative breaks (Minimal, maximal, standing points) to larger changes from minimal parts, ‘finitesimals, 1/n’ to larger wholes of entire planes; to the maximal complexity of functionals.**S≤≥T: Algebra**studies through 5 operands that represent the 5 Dimotions of reality its ‘feed back’ S<T>S, stop and step or ‘stœps’ that*move reality through mirror symmetries of space-time, represented by equations connected by operand of increasing complexity*, from single S=T steps of ∆-1 sequential numbers, which gather in ∆º functions, part of ∆+1 functionals. So numerical stœps are first key constants and operands that represent basic dimotions (Sine/cosine: informative perception; π: $t>§ð: cyclical dimotion that transforms energy into information ; e^{-x}=S«T: entropic dimotion, log-x^{n}: social evolution etc.) become connected by ± social and x ÷, reproductive operand complex larger associations of Dimotions called Polynomials where those mirror symmetries S=T gather in more complex ∆+1 structures (Functions).**Analysis**is a sub-discipline of Algebra that studies Dimotions between parts and wholes (∂entropic parts and ∫wholes) in growing complexity from changes in functions (1^{st}derivatives/integrals), to changes of changes of functions (functionals).

Thus algebra mirrors reality with all the elements of mathematics, albeit with a temporal perspective – as topology does with a spatial perspective – because it can express all type of complex entangled ¬∆@st of space-time in its simultaneous analysis of super organisms through the study of its S=T dimotional a(nti)symmetries, classified exhaustively by **group theory**… So Algebra was first the science of operands that translated into mathematical mirrors the 5 dimotions of space-time and then build up from them as the Universe does building up from actions, simultaneous organisms in space and worldcycles in time, in different degrees of complexity, new mirrors for all those events and forms of ¬∆@st.

**¬@ Humind:****Philosophy of mathematics**studies the bias and limits of huminds studying mathematics as they project its æntropic simple view of the world due to 1) the ego paradox (all systems measure from its distorted self-centered ego) 2) dominance of the ‘western military lineal male, entropic destructive culture..3) inflationary limits of smallish languages that multiply its kaleidoscopic mirror images of larger single wholes as they are not bond by the restrictions that ‘lineal motion-entropy’ impose to form, whenever we try to build ‘reality’. So we might ‘know when mathematics is truth but NOT when is real’ (Einstein, Gödel’s incompleteness, false ∞ as all planes have entropic discontinuous limits of solubility of functions, Cantor Px, idealist, German, physics: Copenhagen interpretation, creationism, false ∞=singularities. So the axiomatic method of proof still needs the experimental mirror of ¬∆@st laws**.**.**6. Vital mathematics**expands its foundations to 5 ¬E Postulates beyond Aristotelian logic (A->B single causality) into the**¡logic of 5 Dimotions**, as an experimental mirror of the fractal, organic universe and its bio-topo-logic properties,

**Expansion of mathematics to 5d: correspondence principles.**

Mathematics & logic are languages, mirrors of an a priori ∆ST reality that comes before languages that describe it.

Mathematics is derived of geometry, the science of space and logic is the science of causality in time. So space and time must be the first substances of which all is made, a model of reality that has a deep tradition in the east (philosophies of a Universe made of two poles, space=dimensional form, or ‘yin’ and time=motion or ‘yang’)

Thus we need to introduce the correspondence principle also in mathematics, according to which present mathematics is a simplification and biased view of the true discipline, as all sciences reduce to a single plane of existence, using therefore lineal concepts of a single time motion the fractal Universe. As time and space are NOT absolute Newtonian backgrounds but are in a Leibnizian relational space-time background independent theory, the ‘generational substances’ of all what exist, composed of organic fractal vital spaces that last a finite duration in time, space and time become the common principles, whose disomorphic properties originate all other laws of ‘stiences’ each one studying different ¡-scales of spacetime beings that must be first put in relationship to those properties and then improved with the comparison of human scientific laws, specific species of each science and the universal laws of ¬∆@st.

Hence we upgrade all its concepts to ‘**cyclical time’** that stores **information**, in the frequency of its cyclical membranes, limiting and breaking **space into ‘fractal’ **topological parts, and** ‘scales’**, according to **‘relativity symmetries’ **between formal linguistic mind-spaces vs. time motions,** S=T,** which mix together forming the **5 Dimotions of reality, **all of which follow its** 5D metric**. Those elements entropic limits, fractal space, SxT=K metric scales, cyclic time of information, and S=T symmetries, *are *thus the barebones fundamental elements required to upgrade each science, and its simplest mathematical equation is its S x T (s ó t)= Constant, ‘Fractal Generator metric’.

In mathematics the same upgrading is needed, as it is also an experimental mirror-image of those laws. Something which subconsciously happened as mathematics evolved, *from pure mental space (Bidimensional still geometry) to time perception (sequential numbers) merging both in modern analytic geometry. And finally peering first with temporal algebraic numbers in the 4 ^{th} and 5^{th} dimension (calculus), which topology mirrored in space *with its 3 varieties of bidimensional forms with motion made of networks of points; to complete with fractals the mirror structure of the Universe – adding on the path all kind of new ‘mind spaces’(Phase spaces), with ST combinations (vector spaces) and scalar levels of growing complexity (Hilbert spaces, functionals) – whereas other highly valued branches, set theory that constructs maths NOT from its initial s-points and t-numbers but from the top head of a Mr. Cantor, the set and the axiomatic method of Mr. Hilbert born also of his head – ‘I imagine points, lines etc.’. So those will be considered largely inflationary forms of the language within itself NOT observed in the real ∆st world, not worth to mention. While chip maths (Boolean Algebra) belongs to a new species the digital machine, which as per our papers on the superorganism of Mankind in time, history, is bond to displace us with its higher dexterity in the most efficient language of the Universe, if we keep evolving them. So we feel ethically inclined NOT to upgrade it in this paper.

**S=T symmetry.** Systems are made of spatial form and time motions, from a static mind-perspective of still geometry mathematics also evolved to acquire algebraic motion* the essential Duality of reality also evident in any scale of complexity of mathematics, which has always 2 solutions from either an S≈T perspective, Spatial forms (points) = Temporal sequential numbers*. Topological methods=Algebra methods, merged in analytic geometry.

So **geometry** of points studies dimensions in space, **Number theory, its sequence in time** and its **analysis** its scalar motions=changes whereas Algebra puts them all together considering its S=T Dimotions and symmetries, making mathematics the best humind’s known language mirror of the…**5d+5m = 5 Ðimotions** of reality, which each mathematical S=T sub-discipline expresses in different terms – Geometry as Dimensions, Analysis as motions and Algebra as Dimotions using operands to that aim, and finally with ∫∂ calculus peering the 4^{th} and 5^{th} Dimotion and its travels through those upper and lower scales, as analysis introduced in algebra the study of integral wholes and derivative parts. Scales of 5D parts and wholes soon gave further boost to algebra, as functions became part of functionals, and all variations of a spacetime structure were tabulated with group theory. So the 5 ‘Dimotions’ of any system can be mirrored ¡logically with *multiple kaleidoscopic perspectives and languages. So as systems have always 5 Dimotions its pentalogic study give us at least 5 ‘Rashomon’ truths; 5 operand, 5 ¬∆@st elements, 5 dimensions, 5 motions etc.*

**In** geometry the same evolution from static space on the 3 ‘dimensions of a single plane of existence’ to the analysis of the upper and lower scales to finally give motion to them all took place, from Greek bidimensional geometry to solution of 3 dimensions, height-information, length-motion and width-reproduction. Then topology introduced the concept of motion, made lineal dimension, bidimensional as all is an ST composed form and then those bidimensional topologic forms with motion, peered the fourth and fifth dimension, as made of points=parts that form a whole; soon points themselves acquired volume as multiple parallels crossed them in non-Euclidean geometry. Philosophy of mathematics is the less evolved discipline, as humans didn’t develop it as an experimental stience so they also ignore its selecting Darwinian power, due to its synoptic nature making it the best language mind – and so ignore in its idealist view the dangers of evolving digital thought in chips. Instead they developed a self-contained consistent world in itself, since each language is also a species of stience, of minimal volume, a fractal world that imitates the whole; hence it has the same self-consistency, which in mathematics became the Euclidean axiomatic method we must however improve by looking also to the ¬∆@st universe – the ‘object’ the language mirrors.

**SxT=K** We must compare any mirror language with the ∆ST reality it describes as languages are smaller, hence more informative in terms of 5D metric, Se (size in space) x Ti (Time information) =C. So we see multiple, kaleidoscopic inflationary mirrors of information– so money is inflationary over the physical economy it describes, we talk more than we act, epigenetics multiplies waste code; so do digital programmers. Thus as fictions exist in all languages and to crop the fictional part of mathematics we shall rather use an experimental method, selecting from maths those parts who better mirror the Universe, since as Einstein put it ‘I know when mathematics are truth but not when they are real’. As languages mirror the Universe in a mind, which is NOT reality itself. But on the positive side this *gave creative capacity to huminds as they invented to represent reality*, phase spaces, Hilbert spaces and variations came to reflect that growing awareness of the complexity of reality.

**RECAP.*** In this introduction to the fundamental particle of mathematics, the fractal point, we considered the key elements of 5D mathematics: First we adapt geometry to the scalar fractal Universe with the concept of a fractal point, and the logic of mathematics to the pentalogic of 5 Dimotions, to understand the multiple variations of maths as an inflationary language, and then we make it a experimental stience referring its main elements, structures to the larger language of 5D showing mathematical equations and postulates reflect laws of the 5D Universe in its synoptic language. *Since space and time are more complex than ‘huminds’ our language of spatial thought, geometry and temporal logic must be upgraded to formalize properly the description of ‘superorganisms’ perceived geometrically in simultaneous space and logically in sequential time.

Most geometry in science is Euclidean (save 5^{th} postulate) and static; while Logic is lineal, Aristotelian, A->B.

So we shall evolve both into a non-AE-=i-logic topology of space, next letter, i also meaning information and the fractal scales of reality and a ¬Æxistential algebra of time to formalize the ‘stœps’, stops in space and motions in time *that move the relative futures of each system, made of vital scales of space that last a finite time duration, which are generated by the bio-topo-logic properties of its vital space and cyclic time.*

The 3 only topologies of 4-5D space explain the ternary nature of all physiological networks and its vital functions.

So a bidimensional motion or topological variety relates to internal changes of in-form-ation not external, lineal changes of locomotion. And we distinguish 3 possible Bidimotions: SS (space areas), ST (dimotions proper) and TT (accelerated time dimotions). So in a single 5D plane all systems are made of those 3 Ðimotions, which become organic §urfaces perceived simultaneously in space ensemble into vital systems; act in Space-Time and live an accelerated/steady state/decelerated 3 ages TT worldcycle of existence observed only as a time process.

**Rashomon truths: 3 ±¡ Dimotions. Ternary fractal method: ****Efficiency of mathematics.** **Its 3 ages.**

In an entangled Universe made of 5D¡ ¬∆@st – space-time dust systems, knowledge requires a pentalogic analysis of any system or mirror-language, including mathematics, to extract all its information about its 5 scalar Dimotions in space as an entangled superorganism and its 3±¡ ages in time as it traces its worldcycles. As only entangled systems made of those 5 elements, performing dimotions=actions of survival exist; so linguistic mirrors also reflect those elements & dimotions. So we apply pentalogic to describe mathematics also in time, as we find 3±¡ ages of its evolution in huminds:

– A lineal young age of simple parts ‘flat’ geometry, arithmetic of unconnected numbers, into a…

– Mature reproductive, combinatory age as a realist mirror of the scalar Universe ( analytic geometry that merges space-points time operand and scalar numbers; calculus that gives points curved motions and scalar depth studying its finitesimal 5D parts and integral 4D wholes) into a

– 3^{rd} age of maximal complexity (Non-Euclidean Geometry, Functionals, Hilbert spaces) but also of wrong inflationary information and mathematical fictions unconnected with reality as its foundations (set theory, categories that substitute real space points, scale numbers and time operands, the axiomatic method which despite Gödel’s incompleteness theorem substitutes the experimental proof asked by Lobachevski and Einstein, we regain in 5D as ‘space and time’ become the real substances of all systems which mathematics studies directly.

-¡: Finally as human maths reach its ceiling, the discipline starts again in a different mind species – chips now evolving fast from its earlier age of simple Boolean algebra, modulo-2; into a realist age able to model any form of the Universe, but also in mathematical physics, in a 3^{rd} fictional age of computer models that validate any physical theory with its nice ‘digital pictures’ regardless of experimental truth (ad hoc big-bang models, evaporation of black holes, dark matter new particles, bizarre multiverses, etc.)

We conclude that among all mirrors=languages of the Universe, mathematics – topology and social numbers – dominate minds because they are the most synoptic, hence able to fit in any scale, less distorted, hence able to perform the 5 Dimotions =actions of survival, better in its expression the laws of fractal space and cyclic, pentalogic time, *even in limited huminds.* *In the graph, languages select the species that survive into the future. In evolutionary terms languages of information matter more than bodies of energy and maths is the best synoptic, focused language mirror. Problem is in this planet machines speak better maths than huminds so the eco(nomic)system selects them. *

So as Upanishad said, ‘the languages of God are ∞; but some matter more than others. Topology should be the internal language of atoms and hence self-similar galaxies of any 10^{n }scale; that is the language of electronic eyes (lineal, Euclidean) and quarks, nucleons and black holes (Non-Euclidean curved), and all its social organisms, which are most forms of reality emerge with mathematical properties.

**The 3 ages of mathematics. **

Mathematics as all organic systems lived 3±¡ ages in the Humind (ab. Human mind) proper of any worldcycle:

**1 ^{st} age: Arithmetic and plane geometry.** As mirror language that studies ¬∆@ST humans understood its simpler units, points of space, social numbers and entropic limits, drawing figures of flat geometry to ‘encircle’ territorial properties in our flat world.

**Trigonometry**appeared then as the 1

^{st}realization of a ‘@-mind frame of reference’ to measure the 3

^{rd}dimension of depth, which is often parallel to scale (astronomical measure). It was a lineal youth, which slowly understood curves and the |xO=Ø generation of all forms with ‘conics’. As all organisms & worldcycles can be subdivided in 5 fractal subparts and 3±¡ ages in its 3

^{rd}age Greek geometry became old, warped inwards-looking detached from experience with Euclid’s axiomatic method, the 1

^{st}mind-ego trip of creationism (man & god’s language)

**2 ^{nd} Classic age.** The S=T symmetry realized with analytic geometry, marrying numbers and points, while calculus brought ∆-scales, with finitesimal derivatives, 1/n, units integrated in wholes (Leibniz). The 3

^{rd}symmetry of pentalogic ∆=S=T, we haven’t mentioned implied that derivatives could be interpreted as ‘stœps’ of motion and ‘minimal straight intervals of a curve’. So they could also study curvature (differential geometry) and locomotion.

**¬entropic limits**were needed to find solutions (definite integrals). New

**@-frames of reference**expanded mental geometries to represent all forms of ‘selected information’, which mathematical physics used extensively to describe the physical world. Thus the classic age had all the mirror tools needed to interpret the Fractal Universe and its 5 entangled elements, ¬∆@st. But the axiomatic ego-trip stretched maths beyond ¬limits when Newton imposed its thesis over Leibniz’s finitesimals and fractal points with infinities, lineal absolute space-time and the false hypothesis of the continuum, leading to its…

**3 ^{rd} age** that abandoned its realist foundations with creationism -Hilbert that imagined points, sharing the only language ‘God’ & Cantor sets instead of space points, scale numbers and time operands as its generators, leading to an excess of old age information & fictions spreading to mathematical physics, as now Maths creates the Universe, not the inverse.

**+¡: **Thus we need a return to its empirical foundations now formulated in terms of the 5 Dimotions that create reality, mimicked by the 5 mathematical subdisciplines (larger view), Operands (shorter dimotions) & equations (worldcycles).

-¡: Yet that might not happen as instead mathematicians are evolving the digital ‘mind’ of machines, the Chip Homoctonos, which speaks better digital numbers and so the eco(nomic)system of company-mothers of machines & weapons is selecting computers that are fast substituting obsolete huminds in labor and war fields, atrophying them back to a ‘audiovisual’ violent non-rational neo-Paleolithic, while Boolean Algebra, past the earlier age of simple, fixed Algorithms of Information (the true meaning of AI) enters its classic age of freedom & consciousness that might end the dominance of huminds on Earth; introducing ethic elements on the praxis of mathematics, as it should in all ‘stiences’.

**RECAP.** 5D philosophy of mathematics explains its whys and widespread application, based in two concepts: 1. Mathematics is the **best synoptic, focused language** we know of the ¬∆@st properties of reality. 2. Hence those minds that speak it better survives and are the **selected** longest living species of reality, electrons with simplex Euclidean math and quarks with Complex curved topology & its atomic ensembles.

**Algebra: 5 dimotions as operands & number families.**

Let’s then close this intro to GST & 5D formalism with the main field of study of existential algebra, the worldcycle of existence -the widest view of an entity as a superorganism on space tracing a worldcycle between birth and death – we don’t expect you to understand it, as it is a packed synopsis of ilogic and its main theme of study, the dual worldcycle of all systems, which go from a placental, protected ordered seed age of fast evolution, by palingenesis, (memorial development of the seed of information with the energy provided by the parents) to emerge into an entropic life in an outer larger world, where it will perform ‘smaller dimotions’ to survive, we call actions, A-ccelerated locomotions, E-ntropic feeding, i-nformative gauging, O-rganic reproduction and Social Evolution into U-niversals (a,e,I,o,u), which form the program of survival of all systems of the Universe, clearly leaning towards creation of information, reason why systems $t>ST>§ð, increase its information commanded by minds that finally exhaust all the vital energy of the system provoking a local ‘reversal’ of its time arrows into entropy and death:

**Operand as Dimotional functions.**

Within those 3 fundamental planes, there must then be operands, which represent Dimotions of a given function of exist¡ence. And so we define those Dimotions in mathematics*, with the duality of inverse operations, + -, X ÷, √ xª and ∫∂, and the universal ≤≥ symbols that put in relationship two sides of timespace event that connects adjacent, simultaneous parts of a super organism in space, or ‘moves’ and transforms a T.œ in time. So Dimotions are presented by those operand, and functions connected by similarity operand (<=>) express the connection of adjacent topologies in space: |<ø>O.*

Mathematics in that sense follows the same ‘grammar, syntax, semantics and synoptic nature’ of all languages that portray the world cycles of existence in time of super organisms, with synchronous adjacent topological chains of time§pace cycles, that all other languages of nature, albeit, seemingly with more precision, which has made it a ‘survival language’, likely to be the ‘mind-mirror’ of atomic structures perceiving in topological mind-spaces.

Those equations express different motions of energy & information produced in the change between states of spatial form and temporal motion. The varieties of it, being mainly the 5 Ðimotions of space-time, and so relating equations of physical Dimotions to classic equation through those operandi allow us to convert classic physics in a sub discipline of GST.

If we define the 5 Ðimotions of the Universe as S> T: information, S<t: locomotion, S t = s T: reproduction, §tð¡ ∑ §ð¡>ST¡+1: social evolution, ST¡+1<<∑ $t¡-1: Death then when we study mathematical physics, which is basically the study of different forms of the v=s/t original equation we shall find slight differences between mathematical dimotions which become then reordered within those 5 subspecies.

What world-universe mathematics reflects within that equation? The largest possible world, where o-mind spaces are the smallest infinitesimal and the body wave, the largest ∞. And this is the galatom superorganism (∆±4).hence it is the essential language of astrophysics. It does have also as all languages mirrors capacity to analyze all other systems, based in the fact that its numbers reflect the 5th dimotion of social evolution (a number is a set of undistinguishable beings), and its algebraic operandi can reflect the scales of reproduction and social evolution 3 ‘stages’ (±, x÷, log xª), its sine/cosine operandi the cyclic repetition of those processes, and the ∫∂ operandi *any dimotion of time=change.*

We are thus to translate mathematics to 5D. And to do so, as a fractal can always be divided in sub-fractals, mathematical disciplines subdivide further at all levels in 5 elements connected not so much with the 5 Ðimotions of existence, but with its 5 structural elements, ∆@s≤≥t, scales, minds, space, time and its a(nti)symmetries.

So in its full formulation as mirror of ∆@ST elements, we can say that:

*Algebra mirrors S≤ST≥T STRUCTURES, a(nti)symmetric dimotions & ∆ transformations from the point of view of a given frame of reference yet its main mirror is Algebra as expression of the Dimotions of reality with its operands*

What languages as synoptic mirrors of the mind try to do is to establish the basic relationships between the space, time, scale of the being, expressing them through its operandi, depending on the degree of perception the being has of reality and its scales which might be reduced if the being is not fully aware of all the scales of existence, as most minds exist only in a plane of reality. So does mathematics, through combinations of:

Sum/rest- >multiplication/division- >potency/logarithm; point->line->plane->volume and so on.

Let us then briefly introduce ‘Existential algebra’, which is the mathematical expression of pentalogic and the study of the 5 Dimotions of space-time *with digital numbers and its parallel 5 families, 5 operands & equations.*

Operands express the 5 Ðimotions of Time§pace supœrganisms; whereas ∫∂ is the most versatile for both the analysis of Dimotions=change & scales (finitesimal derivatives & integral wholes) hence the most fruitful in mathematical physics.

While the negative inverse operands represent the destructive entropic 5th dimotion that affects all others. So negative/ inverse operand balance positive ones, restoring the zero-sum balances of all S=T, events and systems of reality. Thus:

**1D:@:** 1^{st} Dimotion operand of perception for a self-centered in the point are sine/cosine trigonometric functions, which allow to measure the 3 ∆st parameters distance=motion in spacetime (S=T) and angle *the scalar Dimensionless factor.*

**2D: Lineal Locomotion. ± operands***.*

A ‘lineal motion’ is a ‘sum’ of ‘frequency steps’ moving in the positive direction, or a negative number, moving in the inverse direction. Thus unlike *spatial populations of* **N, Natural Numbers, Z** *adds temporal negative numbers which are inverse dimensions that close a worldcycle of simple locomotion. *Negative inverse Dimotions thus can be expressed also in the duality of expansive entropic vs. implosive informative warping in more complex formalisms of Locomotion, as in Minkowski’s space where time, -ct has a negative, i-number that ‘warps’ lineal space into a wave of high information, shortening the distance: D^{2}= S(x+y+z)^{2} –(ct)^{2}.

**3D: X, ÷, reproduction & social evolution vs. entropic division.**

2D: In pentalogic, the product acts an operand of a sum of sums which can account for a ‘long period of locomotion’: S=vt.

3D: But its true new function is reproduction, carried in 2 scales; as in the Function of existence, Max. SxT (s=t), which merges two different parameters, maximizing its *connections on the lower level of its organic parts. As the product means then the number of ‘axions’ connecting them, which is maximized when the parts of both ‘sets’ are equal. *So, if we have 2 points with 4 internal sub-elements, the product 4 x 4 will give us the number of maximal connections, when we put each internal element in connection with all the elements of the other sub-set, which will be larger than between a 5 & 3 elements. Thus ‘gender’ reproduction requires similar forms.

4D: As a sum of numbers that are social groups it can represent also a scalar social evolution. So its inverse, division and Q-numbers are entropic numbers breaking of wholes into parts, as __the 5 number families also represent the 5 Dimotions.__

** 4D-5D:** **log a, xª: social evolution in decametric scales and entropy decay in negative exponentials.**

The 10^{10} decametric scale is the social scale of the Universe, as the Tetrakys is the simplest, perfect social system, with 3×3 physiological networks, and a 10^{th} element, the ‘black ball’ communicating them as the ‘single Unit’ of the larger whole – from armies of 10 x 10 captains, to 10^{10 }ties in DNAs, stars in galaxies, galaxies in Universe. Powers of 10 as Eames’ movie show thus create new wholes. While a negative exponential shows the rhythm of decay of the system in the fastest event of reality, death, maximized by the e-function, which we just showed to be the fastest negative growth.

**The exponential, logarithmic and sinusoidal functions: bell curves representing a worldcycle.**

**In the graph, a key theme of vital mathematics is the representation of a worldcycle in lineal time, with ± exponentials & its inverse, logarithmic curve around the key points of change of phase… as growth of ‘entropy-motion’ diminishes. So we move from ‘adolescence’ of max. growth of both parameters (sT energy and sT information) to the y”=0 point of youth, where the logarithmic part grows slower. Together they form, one half of the total graph of a cycle of existence, till reaching the y’=0 point of Max. (S≥≤T), which then becomes negative, happening a decay of the whole system in two negative curves, so we can see the graph of the whole world cycle as a ‘composite’ of a positive exponential and logarithmic 1**^{st} age and maturity, and a negative logarithmic and exponential – mirror decay with the final collapse of death. Needless to say the same functions appear on all growth curves, It is then when we add all those ‘exponentials of growth’ and transfer the function to lineal sinusoidal ‘series’ in ±complex planes, when the ‘stretched representation of the world cycle becomes a ‘sinusoidal function’. So it is carried into Complex exponentials. Finally through the T=S symmetry becomes both a Bell curve of spatial populations & its s=t statistical mirror; which finally carries to the probabilistic 0-1 ‘unit’ sphere of palingenetic growth (as in the duality T-probability 0-1 sphere of quantum physics vs. 1-∞ entropic statistical function of thermodynamics.. Since the function of existence in its subtle fractal variations and complex pentalogic is the most important of the Universe.

**Real Numbers **are scalar ratio numbers, which as e decay a whole into scales, or as π allow mental perception, shrinking a whole into a mental mapping with no limit of ‘warping’ (Poincare Conjecture). Together,* the logarithmic part of growth and its inverse function of decay shape a ‘life-death cycle’ in term of social growth of energy ∆-1 cells and dissolution forming the two sides of an inverse worldcycle peaking in the Max. SxT (s=t) middle point. *

**4-5D: ∫∂: Cx pentalogic. **So the closure of 5Dimotions & 5 number families, is calculus – derivatives of finitesimal 1/n parts of a whole that measure a quanta of change in time, space or scale, and its inverse integral of wholes sum; and Complex plane of ST numbers, which squared show a Real 2D area of S-populations & a negative 1D T-frequency of emerging i-motions. *Since complex numbers allow to plot together ‘dual ST systems’ of different Dimensions (potentials-waves-particles), reason why they are so useful in mathematical physics. *Algebra thus is the queen of mathematics as it is the true science of its 5 Dimotions, reason why we shall need a II ‘book’ dedicated fully to its discipline, just sketched here.

**RECAP.** 5 operands & numbers’ families represent the 5 Dimotions of ¬∆@st suit to represent the function of exist¡ence.

**Pentalogic on Analysis as the study of multiple scales and dimensions of space-time.**

Among all the operands of mathematics, derivatives and integrals are by far the most important as they reveal the inside workings of the 5 dimotions of space-time and its scalar structure, *with far more precision, stœp by stœp of S<T>S change*, than all other operands, without the kind of bulk change proper of polynomials. Further on Analysis is able to measure different types of change – of scale of space populations and time ‘motions/speeds’.

A simple example will suffice. We said that the fundamental particle of the Universe is a superorganism, which in its simplest, commonest form has the shape of a sphere with the 3 canonical regions (graph):

@- A Mind-Center, measured by its radius, as the ‘DNA’ or ‘Territory’ of the organism that moves around the system. Membrane of angular momentum, or clock that can be measured by the circumference, and volume of vital energy which can be measured by the area. So we get the value of the 3 elements of a disk, and expanding the concept to a volume, we find then a volume for the vital energy, a surface of an sphere for the membrane and a perimeter for the wanderings of the singularity. Alas! As it turns out, the area of a circle is π R^{2}, and the circumference is 2πR, which is the derivative. The volume of a sphere is V=/3πR^{3}, and the surface area is S=πR^{2}, which is again the derivative. And inversely, the integral of the circumference is surface and the surface integral the volume. So analysis indeed will become the essential tool to understand the dimotions, parts and growing scales of parts and wholes of the fifth dimension, reason why it is essential to understand the correspondence between analysis and 5D laws, motions and scales. The previous example shows the main and obvious usefulness of analysis *in static space: to describe through 3 levels the 3 parts of the being, which is the ultimate reason why derivatives are really useful when considering merely 2 of them. *

Yet analysis is not often used in static terms. This was though likely its first use (to calculate volumes from areas). It is in fact used to study motion, change in time, and we shall argue also scales; and in that sense, *as we shall repeat ad nauseam, the entangled Universe which shows a clear correspondence between the mirror elements of 3 motions in time, 3 topologies of space and 3 scales of size, wholes and parts that bring together the 3 x3 (+2 mental) = 11 Dimensions of reality is fully realized in the fact that analysis works to explain the 3 ‘ternary symmetries’.*

Let us consider the previous example, it is obvious that we can consider the sphere to be the whole sum of parts, where each part is a circumference, so our planet is the sum of all its ‘parallels’ with center in the poles. And then the volume *as each internal sphere can be in terms of 5D metric, $ X ð = K have the same co-invariant value, can be considered the sum of all those equal 5D valued spheres, so again we can talk of ∫∫ ¡-1=circumference-> ∫ ¡0=sphere->¡+1 = volume.*

What about the third ‘ternary symmetry’, that of time-change? This again is the fundamental use today analysis has, to study the rate of changes of a system, and it can be seen easily that the 3 elements of the ‘t.œ’ ARE measures of time-change when we study not a mere locomotion, but the ‘change-rate’ of ‘growth’ more proper of the worldcycle of existence from ‘seed’ (the internal minimal sphere’) to emergent system:

If you describe volume, V, in terms of the radius, R, then increasing R will result in an increase in V that’s proportional to the surface area. If the surface area is given by S(R), then you’ll find that for a tiny change in the radius, dR, dV=S(R)dR or dV/dR=S(R),

The increase in volume, dV, is the amount of new ‘cellular layers’ their system grows, and the amount of cells form the membrane which is just the surface area, S(R), times the thickness of the growth, where each unit is a layer, dR.

This same argument can be used to show that the volume is the integral of the surface area (just keep adding layer after layer of atoms or cells).

**Derivative vs. integral in time and space. **

Thus derivatives and integrals calculate the ratio of change in the 3 elements of reality, space, time=change and its scales. As those *are the true variables of reality, ∆ST, we can then assess in much more depth than present ‘axiomatic’ analysis which forces the ‘error’ of continuity and infinitesimals and infinities, the limits and nature of derivatives and integrals.*

Leibniz vs. Newton already argued what is an infinitesimal part, the *unit* of derivatives. The answer in a Universe, which is a fractal scalar system of stœps of space-time, S<T>S, or S=S=S, T=T=T, is a minimal quanta in SCALE, TIME or SPACE, which by virtue of 5D metric, SxT=K, will be a ‘minimal unit’ on that equation (Min. S x Max. T), for the quanta of space, Min. T x Max. S for the quanta of frequency, or minimal cyclical bit of information.

So for example the inverse, ƒ=1/T of a long duration in time, will be its short quanta. The inverse of a population, taken as a whole, 1, 1/n will be a scalar space quanta. So all systems have *an infinitesimal, which is a cut-off limit, NOT really an ‘infinitely small’ (an error of the continuous dogma of the axiomatic method), but a ‘finitesimal’. So we can obtain through a derivative a finitesimal unit of time, space or scale, a minimal action, a minimal point-volume or a minimal cellular quanta.*

*And for that reason we can approach finitesimals with ‘differentials’, which become the minimal ‘lineal steps’ of a long curve. And for that reason we can use ‘affine functions in space’ and lineal approximations, as by definition the shortest path between two points is a line, and so in a discontinuous Universe of Stœps, quanta are minimal lineal units; minimal fractal points of a population or a whole.*

*Yes, this is a complete overhauling of the dogmatic pedantic attempts to prove the ‘hypothesis of the continuum’, but it is not my fault that humans are so off-track with the reality of the Universe as **it is**, not as their ego tries to impose. *

__Finitesimals in time and space.__

A key concept for 5D in its mathematical analysis thus will be that of a finitesimal, which was rightly defined by Leibniz as 1/n in space (an individual unit of a social population), and we can now extend in lineal time duration as the minimal bit of frequency ƒ=1/t, or quanta of time, introducing therefore the concept of a finitesimal, discrete minimal unit for each scale of the fifth dimension – h-planckton, cellular units, atomic units.

And as we shall soon see through the parallelism between space-form and time-motion, S=T, as most actions in the Universe are actions of fractal reproduction of form, for each quanta of time, we shall se the existence of a reproduction of a quanta of space…

On the other hand the inverse Integral will allow us to ‘integrate’ an amount of such units of time, space or scale to obtain a simultaneous whole, a supœrganism, a ‘T.œ’, ∫ds, ∫dt, ∫∆-1.

Of those 3 types of derivatives and integrals, obviously the less understood (as frequency and time duration are inverse parameters currently used in all sciences) is ∫∆-1, whereas ∆-1 is taken to be the infinitesimal or minimal quanta of a whole, ∆º, (cell, atom, individual in a society). So we shall dwell more on the interpretation of ANY derivative or integral in terms of parts and wholes, which lead us to the concept of a derivative that mimics the 4D entropic dissolving function, ∆º<∆-1 and the 5D integral, that mimics the creation of wholes.

This *other possible use of derivatives and integrals however must be taken with certain caution. As indeed, what we have found in the analysis of the sphere by making two derivatives is The Point, NOT ANY point but the Center of mass or charge in a physical system. When then a derivative can give us any point, depends on the configuration of the whole we analyze. In a heat equation the whole lacks a center, as it is an entropic flux, and so on.*

So what we shall do in the posts on 5D Physics, its dimotions and scales, is to slowly ad the main equations of the discipline with further analysis of the why of its mathematical calculus.

This ternary use of integrals and derivatives is what we must have always in mind to interpret the equations of mathematical physics. Infinities though don’t exist, as all has a finite membrane and a finite duration in time. And as we replicate the operations of integral and derivative, it is also evident that beyond the third derivative as the scalar Universe is a ‘ternary game’, there is no significance to the mathematical operations of derivatives and integrals – a very strong proof indeed that 5D is truth as it limits reality to such ternary scales, topologies and time ages.

It is then clear the immense possibilities of the use of a derivative, which ‘extracts’ first the finitesimal quanta, or fractal point from a function of exist¡ence (T.œ), often directly as in log x’:1/x, it *can lead directly to the value of the mind, or ‘center point’ of the system – the ‘finitesimal whole’; and its inverse, an integral, which ads finitesimals till reaching the whole, as in the case of a volume of populations, but also illuminates the dissolution of a whole into its integrating parts.*

In 5D analysis depending on what we study ‘motion’, or ‘space’ or ‘scale’ up or down we shall apply either an integral or a derivative, with different results that might seem conflicting, because space and time tend to be inverse, perpendicular functions, in its min. S x Max. T, and Max. S x Min. T states, but symmetric in S=T; so goes for the 2 different arrows of entropy, a dissolution downwards and social evolution upwards. So for analysis to loose its magic view one has to know in depth the a priori laws of Scalar space-time, which obviously humans minus this finitesimal quanta totally ignore (:

So what we shall do as our 5D papers progress is to study the 5 Dimotions of the Universe, in detail – and in the case of physics, which as the ‘classic science of locomotion’ uses overwhelmingly derivatives and integrals in their analysis of reality, to specify what we are ‘calculating’, scales, time motions or space populations – all this ultimately revealing ‘pentalogic’ different p.o.v. on the analysis of physical or any type of systems.

**Pentalogic of Analysis: its multiple forms and functions.**

Since 5 short Dimotions or 5 structural elements, ¬∆@S≈T, (long view) conform all systems, pentalogic -Rashomon effect’ of 5 points of view to find a higher whole truth) applies to all languages mirroring reality as analysis does:

*In the graph, we can see multiple perspectives and functions of Analysis:*

**Space-topology***: used to study (left) structurally the role of the 3 elements of a topologic spatial superorganism:* * Its membrane (line integrals), its vital space (surface integrals) and singularity (derivatives).*

*In its inverse operand, the derivative measure the ‘form’ of the membrane – it curvature and tangential value. It can give us the value of one of its infinitesimal ‘cells’. It can give us its internal volume of **spatial** energy, studied with integrals. While we can extract information on its central **@**-singularity, which commands the lineal motion of the whole system.*

**Time-cycles: **it can model the* standing points, maximal and minimal, which signal the changes between ages, where the derivatives, become null, as the ‘world cycle of existence’ changes its ‘phase’.*

**∆-planes:** it peers down the existential planes of the fifth dimension, down one or 2 ∆±1 planes (double derivatives).

**Complex stœps of dimotions:**

Further on as derivatives and integrals are inverse operand that can combine in **algebraic S≈t equations** (down) for more complex description of multiple events, even cancel each other to give us a ‘constant social **number’ **as an exact quantitative result for a specific value of a sequential ‘sœt’ of stœps of Dimotions.

So the* ‘Rashomon effect’ of pentalogic shows how analysis satisfies as a mirror the description of the main components of the being; further enhancing its description by the fact that ODEs can combine in several *stœps, the 3 elements – time integrals/derivatives, spatial and scalar integrals/derivatives in a single function:

**-Temporal view in **Time curves that often use cyclical time frequencies (Fourier transforms) to describe a combination of time and scale events (wholes decomposed by the transform)

– **S=T**: **combined** **Time-Space view **that resolves symmetries between time dimensions and space motions (S≈T) expressed by ODEs of two variables – a parameter of space that changes with a dynamic function/action/motion in time. It can measure through closed membranes information from the inner vital energy of the system as it moves through time (Continuity equations).

**– Ceteris paribus S=T view,** when we fix 1 dimotion parameter, becoming T>S-pace and operate the other as a time motion, mimicking the stœps, stop and step discontinuous form of most time space dimotions.

**-Spatial view** through lineal, surface and volume integrals; forms that measure the 3 ∆±¡ elements of a T.œ’s topology, made of ∆-1 points (atoms, cells individuals) both in time and space (a population or a distribution) .

If analytic geometry resolved algebraically geometric spatial problems, with ‘a dual point of view’ that increases the easiness of solutions -Descartes – showing the algebraic laws of solution of rule and compass for geometrical problems; calculus took this approach on S=T symmetries to a much higher level.

– ∆: **Scalar view, **which defines infinitesimals (Leibniz) as the 1/n cells/part of the being given also by a derivative; since the minimal rate of change of any system is one of its ∆-1 units.

In that regard, the earlier astonishment of physicists and mathematicians which found the fact that a derivative (the analysis of a step of motion) was inverse to a volume of space (an infinitesimal of a population), is a deep proof of the essential symmetry between space=still states and temporal motions (stops and steps=stœps).

Of all this Pentalogic views, the fundamental use of analysis is in the study of timespace Ðimotions which started as all systems do, in its simplex forms, locomotions (speed) but we shall upgrade to the ‘analysis’ of all others.

**RECAP.** Analysis studies the finitesimal quanta of time, space and scale; NO ∞ in its smallness – an error of the mind px. searching for continuity. A true philosophy of calculus thus deals with the meaning of ‘finitesimals’ in space, time and scale, as a first ‘seed’ of a ‘clone species’ multiplies, creating the regularities of ‘social numbers’ that make ‘analysis’ to work its ‘magic. Integrals in inverse fashion act after ‘calculating’ this minimal point, often as a ‘lineal shortest step’ (differentials), to reach the final ‘whole value’ of the system. The beauty of the field revealing the nature of dimotions and its wide applications, thus will require an entire II book on 5D mathematics, which should r=evolve the discipline.

** ¡LOGIC MATHEMATICS: ITS 5 POSTULATES: FRACTAL POINTS AS MINDS.**

‘‘The smallest point is a world in in itself’ *Leibniz, on the fundamental particle of Reality: The fractal point=world of space-time,* *Non-Euclidean unit of ¡logic topology*

The first immediate mathematical consequence of the fractal structure of space-time is the change of the axioms and postulates of Euclidean geometry taking the r=evolution of Geometry performed by Lobachevski and Riemann in the XIX c. which gave birth to Relativity, to its ultimate consequences, changing *also the axioms of Euclid that defined points and lines as having no breath, since in the fractal Universe all forms do have a volume when we enlarge our view of them, peering into its inner parts and fractal dimensions. And in this manner we shall harmonize and return to its logic meaning the concepts of parallels and Euclidean points able to fit multiple lines=waves of energy and information, converting those fractal points of ‘cyclical timespace’ into the fundamental particle-units of the Universe mirrored by mathematics. *

*Thus *the mathematical unit of a 5^{th} dimensional Universe is a fractal point, whereas a Non-Euclidean point is its limit in a single spacetime continuum; whereas the inner parts of the point, which co-exist in other scales are not perceived. So we shall start with the classic non-E point and show how by adding fractal scales become a more complex reality.

Fractal points unlike Euclidean ones are points with parts, as we come into its scale grow in size and display the 3 minimal parts of all of them, its area, frequency of angular momentum, and central Active Magnitude, the true meaning of a ‘singularity’ – the focus of charge, mass, forces and its informative minds. So let us introduce the minimal POINT with parts of the Universe, the time space cycle.

*In graph, Einstein’s interpretation of the 5th non-Euclidean postulate was the view of a fractal point of the gravitational scale from our smaller electromagnetic world, which shrinks its inner volume, bending its parallels, hence its curved geometry. But that view breaks the conceptual definition of parallels are straight lines and is absurd, as the point remains ‘Euclidean with no breath’, hence only fits one line with no breath. Thus, particle points must be defined as ‘FRACTAL points’, like those we see through telescopes or microscopes, which grow we approach our distance both in scale and space becoming enlarged worlds with a complex internal structure. *

Einstein found that gravitational Space-Time did not follow the 5th Euclidean definition, which says:

*Through a point external to a line there is only 1 parallel.*

*E*uclid affirmed that through a point external to a parallel only another parallel line could be traced, since the point didn’t have a volume that could be crossed by more lines./ Instead Einstein found that the space-time of the Universe followed a Non-Euclidean 5th Postulate: *A point external to a line is crossed by *∞ *parallel forces. Abstract, continuous, one-dimensional point: Real, discontinuous, ∆-dimensional points: *

**. ____________*** ***===========o**

This means that a real point has an inner space-time volume through which many parallels cross. Since reality follows that Non-Euclidean 5th postulate, all points have a volume when we enlarge them, as cells grow when we look at them with a microscope. Then it is easy to fit many parallels in any of those points. Such organic points are like the stars in the sky. If you look at them with the naked eye they are points without breadth, but when you come closer to them, they grow. Then as they grow, they can have infinite parallels within them. Since they become spheres, which are points with breadth – with space-time parts. *So space-time is not a ‘curved continuum’ as Einstein interpreted it, but a fractal discontinuous.*

Leibniz’s monad, which does not communicate, is the simplest o-1 fractal point-mind possible – still a mirror of the Universe, as ‘each point is a world in itself.’

But for a modern scientist it is easier to understand fractal points in terms of its 3 necessary parts, its focus-singularity of the parallels that cross, its membrane or angular momentum (S=T duality) that breaks it into an inner and outer part (first knot theorem), and its vital enclosed territory, which correspond to the 3 physical quantities conserved in Nature, as a fractal point is also the unit of logic as a cycle of time that divides reality in inner and outer regions and the unit of physics, as the minimal form of ‘Planckton’ (h-Planck constant) which has the 3 parts of reality conserved in each plane of space between its ∆-1»∆º palingenetic emergence and ∆º«∆-1 entropic death.

So the 2 ‘emergent formal sciences of space and time and its units, the fractal point and the time cycle become also the minimal organic species of physics; and we shall see this ternary structure emerging in all scales; so we can model cells with 3 organic parts; animal territories; nations; planets, stars and galaxies, always showing a ‘membrane/angular momentum’, a focus/singularity/informative center and a vital energy-space between both. So it is worth to consider it .

**A visual synopsis of its 5 postulates.**

We thus recast the axioms and postulates of Euclid into five new postulates to define fractal points, Non-Æ lines as wave of fractal points, Non-Æ planes as ternary networks of Non-Æ lines, which become supœrganisms, whose relative ‘congruence’ in its 3 ‘elements’ (singularity point, membrane and vital space) defines the type of ‘perpendicular or parallel’ relationship between them:

1st Postulate: ‘¬Æ point are discontinuous time cycles with an inner content of vital space-time’.

2nd Postulate: ‘¬Æ lines are waves of fractal points’

3rd Postulate: ‘¬Æ planes join 3 ¬Æ lines into a supœrganism’.

4th Postulate: ‘2 ¬Æ points are congruent when both its inner parts and outer perimeter are equal’

5th Postulate: ‘¬Æ World points focus multiple ¬Æ waves of energy into a still linguistic mapping of the world.

Let us then explore those postulates, constraining our examples to the simplest forms of physical and biologic spaces.

**1 ^{st} Postulate. The 3 Mathematical parts of a Non-Euclidean Fractal point.**

*–*

* 1 ^{st} Postulate:* A fractal point has parts; that is an enclosed region of vital inner energy surrounded either by a spatial still membrane or a Temporal motion of angular momentum (S=T symmetry) self-centered in a singularity-mind that gauges its information. In the graph, we can see how different vital fractal points of ST¡entific scales follow this ternary structure. The perception of the point though depends on the scale and distance from where we observe it:

From the perspective of the upper st+1 Plane they might be in the limit of invisibility (what quantum scientists call a point-particle) *but they still have a time motion performing a ‘function’ in that upper ecosystem, ∆+1 in which it exists.*

*Internally from its own ∆º perspective the point will have *3 dimensions/networks. This is the case even in the smallest planes of theoretical strings, made of points with parts, with volume – since we require 3×3∆º+1∆+1 theoretical inner dimensions to describe strings – a paradox that can only be resolved if we consider ‘strings’ to be fractal points with inner dimensions.

Fractal points explain without contradictions Non-Euclidean points, which are not logic in a single scale, as they ‘curve’ parallels which are ‘straight lines’ and fit them in a ‘point with no breath’ that holds only 1 line. Fractal points however enlarge fitting multiple ‘straight lines’. Yet when seen from above, human perception of them, becomes ‘deformed’ shrinking and curving its from – a theme, the distortion of human measures of time, space and scale, which will be instrumental to explain rationally the ‘spookiness’ of quantum physics and relativity and its time and space transformations.

So fractal points harmonize the 1^{st} axiom=postulate of Euclid with the 5^{th} postulate of non-Euclidean parallels, as a fractal point enlarges into a cell, atom or particle which even in a smaller scale of the fifth dimension can host multiple parallel flows of energy and information, crossing it.

When we see fractal points far away we describe them as points with breath, with the tools of Euclidean geometry since the ‘inner space’ shrinks to a point and so the ‘bulk’ or curvature of space-time shrinks to a plane. Yet, when we come closer to them, they grow into points with volume. The volume of those Fractal, Non-Euclidean points can thereafter be studied with the 3 types of canonical, Non-Euclidean geometries or topologies of a 4-Dimensional Universe – the Universe we live in. Those 3 topologies make up the 3 regions of the point, which correspond each one to the 3 essential arrows/functions of any species: the external, energetic membrane; the central, informative brain and its reproductive combination, SxT(s=t):

A*ny fractal point is made of 3 regions whose geometry responds to the 3 topological forms of a 4-Dimensional Universe, the convex plane, the torus and the sphere. *The inner parts of fractal points are thus able to perform energetic, informative and reproductive functions, which makes them the fundamental particle of any plane of st¡ence. Thus all entities can be described as wholes made of 3 internal parts whose geometrical properties maximize their energetic, informative and reproductive functions:

– Max S: an inner, dual center, corresponding to convex topologies (left), made with 2 cyclical forms. It is the dominant informative topology of any fractal organism, described by Belgrami in the XIX c. as a conical form with ‘height’, with negative curvature.

– S<=>T: A middle, reproductive zone, described by Klein as a disk of quanta in cyclical motion that communicate energy and information between the inner and outer zones.

– Max. T: An outer membrane of max. |-motion-distance crossed by ¬E information & energy parallels acts as an entropic limit to ¡ts inner parts, described by Riemann’s spherical geometry. It seems continuous, still but on close view, as most external membranes stores and/or absorb information through openings of its broken geometry, outlets of its senses.

The complex analysis of those fractal points that move and have inner fractal parts, made of cycles, started in the XIX century. First Lobachevski, a Russian geometrician, defined Non-Euclidean points as curved forms, crossed by multiple lines, which give them spatial volume. Then Klein studied its cyclical movement and introduced the variable of time in their description. Finally Riemann generalized its nature, considering that all space-times were Non-Euclidean space-times with movement. For readers versed in mathematics, we shall reconsider the common properties of those 3 zones of any fractal point, according to its discoverers, which develop in abstract terms the organic properties we just described:

–* According to Lobachevski and Belgrami,* space is curved since information curves the energy of any real space-time. So points move in curved, cyclical paths gathering energy and information for their inner ‘dimensional networks’.

*– According to Klein* Non-Euclidean space-times have motion. So their speeds measure distances; as physicists do in Cosmology with the distances of galaxies, which are proportional by a ‘Hubble constant’ to their speeds; or as people do in real life when we say that Brooklyn is at 5 minutes by train from Manhattan not at 2 miles.

*– Riemann* summoned up those findings and generalized them to all possible space-times. His work should be the guide to understand them philosophically. He also defined planes as networks of similar points and treated dimensions, as we do in this work, no longer as mere abstract definitions of extensions but as ‘properties of those points’. So points can have beyond its discontinuous borders an inner space-time with several networks/dimensions, one for each of its ‘energetic or informative properties’, as it happens with the points of physical reality. Yet a network of points that form a space with ‘common properties’ defines the dimensions of those points as ‘fractal dimensions’, limited by the extension of the energy or informative network (static point of view), which ‘puts together’ a complementary dual, organic being.

Those pioneers defined the 3 topologies of information, energy and reproduction of all st-points:

*– Max. Space=Information: The informative, fractal center, particle or brain of the point is the so-called Belgrami hemisphere,* a space-time with a dimension of height that transforms energy into information, absorbed or emitted by the central singularity. It is a fractal, informative region similar to a black hole structure. Since it follows the ‘black hole paradox’ of all informative centers, displaying max. form in min. space. So according to the inverse properties of space and time, the center has max. Informative Time and minimal Energetic Space. Moreover any point which comes closer to it, suffers a mutation of its spatial coordinates into informative, height dimensions. This is the case of any particle coming to a black hole, whose space-dimensions become temporal/informative dimensions as it rises in height.

The center has more information because its geometry has at least 2 fractal disks, which channel and transform the energy absorbed through the surface into complex information. Regardless of the complexity of the entity, the structural function of the hyperbolic center as a system that process the information of the network remains. For example, in living systems, those disks might evolve its topology till becoming the relative energy center or ’heart’ of the blood network with 4 divisions; or evolve further its hyperbolic geometry till becoming the informative center or ‘brain’ of the system, attached to the informative network.

*– Max. Time=Lineal motion & Min. Curvature: An external, continuous membrane or Riemann’s sphere of maximal energy* that acts as a relative infinite, unreachable distance. The membrane isolates the point as an island Universe, creating the discontinuity between the inner parts of the point and the outer universe. Since the internal cellular points are either jailed by the membrane’s structural density or destroyed by its energy when touching it. The membrane is the opposite form to the central, informative singularity, with max. spatial extension and continuity, hence with a minimal number of fractal, discreet elements: *Max.T=Min.S*

Thus *all Fractal points have inner worlds whose membrane creates a discontinuity that defines an External Universe or* outer world from where the point obtains its energy and information. The membrane is also the zone through which the point reproduces & emits its micro-forms of information. So it displays ‘sensorial holes’ to relate the point to the external Universe. And those points, despite being discontinuous, will have in their external membrane several generic openings or ‘senses’ joined to the informative networks or ‘brains’ and energetic, ‘digestive networks’ of the organic system:

* – Max. +ΣT: A* ‘*mouth*’ or opening that absorbs energy.

* -Max. –ΣT: *‘*Cloacae*’, through which the cyclical body expels its temporal energy.

–*Max.+Si: An* ‘*eye*’ through which the informative center receives external information.

–*Max.–Ti: An* ‘*antenna*’ to emit information.

Those apertures vary in their number, location and size, depending on the form of the point. In the simplest spherical ‘seeds’ of most species, they are mostly situated in 3 regions:

* – Max. ΣT*: The Equator of the system, through which the membrane absorbs energy.

* – ΣT=Si*: The Tropics where often the same opening emits and absorbs temporal energy.

*– Max.Si*: The Poles or points of confluence between the membrane and its central informative region of height, which hits perpendicularly the membrane on those poles. North and South Poles orientate Anti-symmetrically, acting as 2 relative, negative and positive apertures, communicated by the height dimension of the singularity or Belgrami hemisphere. Thus the Positive Pole absorbs temporal energy that crosses through the central singularity where it is absorbed and ejected to the intermediate region where it is re-elaborated before its emission through the negative Pole.

*– ΣTe*<=>*Si: The reproductive, central region, which combines Energy and Information: *

In all fractal points there is an inner middle volume or intermediate territory, discovered by Klein, which combines the energy coming out of the external, spherical, topological membrane and the information provided by the convex, complex formal center.

According to Non-Euclidean mathematics this region is made of self-similar points that form groups, fractal herds of ‘points with parts’ in perpetual movement, that draw cycles of parallel lines, between the other 2 regions, as they gather the energy and information they need to survive. And they create space by cycling within the other 2 regions.

In many fractal points the informative and energetic centers establish 2 opposite flows of energy and information that become the negative/ positive poles. So often, the particles of the intermediate region cycle around the inner region tracing elliptical trajectories, focused by those 2 informative points. It is the case of any bipolar system, from binary stars, one dominant in energy and the other an informative neutron star or black hole; to bimolecular systems or n-p pairs in the nuclei of atoms. The same duality of 2 specialized centers controlling a common territory, or vital space happens in biology where most species have male-energetic and female-informative genders, ruling a common territory.

Such abstract conceptual space describes in fact the behavior and form of many real, spatial herds. For example, a herd of animals in an ecosystem will move between their hunting and water fields (where they gather energy) and their breeding, inner region where they reproduce information, making cyclical trajectories between both regions. In this manner, they occupy a vital space, called a ‘territory’, which shows the properties of a Non-Euclidean Klein space. A fundamental property of the intermediate space is the fact that it is confined between the other 2 regions, which are never reached in the cyclical trajectories of the inner cells of the space. For example, in a cell, the molecules of the organism will not touch the protein membrane or the central DNA nuclei. Thus, the inner quanta are confined within the Klein’s disk by the 2 other regions, which have more energy and information and might destroy them and/or absorb their energy and information at will.

In abstract terms*, *mathematicians introduced in the XIX c. the concept of an *infinite, relative distance *measured no longer in terms of static space but in terms of time and movement, as *the distance between the point and a region that cannot be reached*. Thus Klein defines a relative infinity, as the region beyond the discontinuous membrane whose insurmountable borders the inner time-space quanta can’t cross, as a cell cannot go out of a body, an atom beyond C speed or 0 K temperature and a man beyond the Earth’s atmosphere. Thus, the informative center and external membrane become the 2 relative infinities or limits that the movements of the intermediate point cannot breach.

As in the myth of Achilles and the turtle, Achilles never arrives because every time he moves he crosses a smaller spatial distance. The same happens in a fractal space-time, when a point moves *temporally* towards its inner or outer space-time limit and finds an increasing resistance to its movement, till finally it is deviated into a cyclical trajectory around the outer, energetic membrane or the height dimension of the inner informative singularity or is destroyed. So the intermediate, fractal cells of the point circulate in parallel cycles always inside the interior of the sphere with contact zones of the type A *(central, 2 ^{nd} row of figures in the previous graph). *

*In a human organism, the blood system might seem infinite for the red cells that transport energy since they never reach the outer Universe. *For that reason in the drawing, Klein interprets the intermediate region of the Non-Euclidean point as an infinite *circle with an invisible, unreachable membrane,* whose motion-distance is unreachable, hence infinite, equaling the ‘space-time distance’ between the intervals B1-B2 (long) and B2-B3 (short but difficult to cross), despite being B2-B3 increasingly shorter in space. Since the quanta take longer in each step and don’t reach the membrane. This is often due to an increase in the ‘density’ of the space, which despite having less distance has more ‘points’ in its network, such as the case of black holes or jails. When those inner points reach the membrane at point C they become destroyed or deviated.

Thus, the energetic membrane and informative center are the discontinuities that isolate the intermediate cellular quanta, creating a discontinuous ‘World’ within the point. Those discontinuities are called in Geometry a relative infinite, in Biology a membrane, in Sociology or Topology a national border, in fractal theory a co-dimension of a point. They are defined in physics by Lorenz Transformations that make c-the limit of energetic speed and 0 k the limit of temporal, formal stillness. Yet those physical limits are not the limits of an absolute Universe, but the limits of the fractal space-time membrane of light and its evolved electroweak beings, since the Universe has at least another bigger gravitational membrane, in which the smaller light-space exists; a fact with enormous repercussions for a proper description of the Cosmos, which extends beyond those limits. Since the gravitational scale should be faster than light-speed forces and cooler than 0 K masses.

*Recap: *Fractal points are organic points, whose topologies maximize the energetic, reproductive and informative cycles they perform. The details of those cycles are described by Non-Euclidean topologies.

**– 2 ^{nd} postulate. Communicative waves of energy and information. STœps.**

The 2^{nd} postulate defines lines as waves of points with volume (which explain complementarity wave particle), no longer as an abstract form like Euclidean geometry does, but as a physical wave of self-similar, fractal micro-points that carry energy and information, as they move between 2 macroscopic points, with 2 possible functions, to communicate energetic forces or linguistic information.

*2 ^{nd} Postulate: A cycle of fractal space-time:*

*‘A wave of communication is a group of self-similar micro-points that move in parallel lines between 2 macro-points, transferring energy and information between them’.*

In Non-E geometry a line with parts is not defined by a sequence of numeric intervals within a straight line, but by the communication of 2 poles of energy and information that establish a flow of particles in 2 opposite directions, creating a simultaneous, paradoxical wave. Such waves again can have different purposes. A wave dominant in information communicates symbiotic particles, creating an informative bondage/network; a wave dominant in energy might be an aggressive action between different species that fight for each other’s vital energy or territorial space; and a wave that balances the energy and information of both points meets in the center, creating a new self-similar, seminal particle, as when 2 electrons emit waves of densely packed photons, which merge in the middle and give birth to another wave.

When we observe a one-dimensional line as a form with inner parts it becomes then a 4-dimensional wave made of cyclical points with motion. Hence in quantum theory we say that any particle in motion has associated a wave. *Thus the 2 ^{nd} postulate resolves the wave/particle duality,* as all lines are now waves traced by a point with inner volume. Further on, since all lines have volume, they carry information and so all forces can in fact act both as a source of energy and as a language of information – as physical experiments prove. Since when we observe a ray of light in detail it becomes a 4-dimensional wave with electric height and magnetic width, often exchanging flows of energy and information in action-reaction processes of communication between bigger points.

When we generalize those concepts to n-points we can define a space as a network of Non-Euclidean points. Indeed, Riemann affirmed that a space is a network made of herds of points with similar ‘properties’. Planes of space are therefore networks of points. The self-similarity of their properties defines its *density *determined by the number of points and its proximity that grows with self-similarity. So similar points come together into a tighter, more continuous space; whereas the density of the space is proportional to the similarity of its points, till reaching ‘boson state’ of maximal density when points are equal. And when a volume of spatial energy is very dense, it is very difficult to go through it, as it happens in the ultra-dense, small space of black holes.

Spatial extension and form/density/mass are inverse parameters, Max. T = Min. S. If we generalize that property to all scales, we can define different fractal spaces by its proportion of mass/density and energy /distance. This is done with ‘Universal constants’ that explain the proportions of energy and information of those spaces.

For example, in physical scales, there are 4 fundamental space-times, the gravitational space-time between galaxies of max. energetic space and minimal formal density; the light space-time of our world, which carries information in the frequency of the wave; the electronic space-time of atoms with more formal density and lesser spatial speed and finally the quark-gluon liquid of atomic nuclei and probably black holes, with maximal density and minimal space. All of them are defined by Universal constants and equations that are either ratios between the energy and form of those space-times, or define the transformations of one space-time into the others. Einstein’s field equations would be the first case, defining the relationship between energy and mass in a gravitational space, while the fine constant of electromagnetism would define the transformation between light space and electronic space/ charge; and the gravitational constant between gravitational space-time and quark/mass. Where the relative densities of information and extension in space of those space-times are in balance, such as ΣTxSi=K. Thus electrons move slower than light but have more density.

All this said it is thus obvious that the *fundamental unit defined by the 2 ^{nd} postulate is no longer a point but an action, Tex Si= k between points, a dimotion. *

**2 ^{nd} postulate in physics S=T: T-Wave motion and S-particle information. Quantum & General relativity.**

Form=space and Motion=Time manifest in physics as particles and waves: the wave erases form into motion, the particle is a still state that gauges information entangled to other particle, fermion and boson, still to each other – despite the perception of relative motion in our scale – hence the information electrons share has always a c-constant speed. This is the ‘rational’ 5D explanation of both the c-constant of light and entanglement; as electronic beings perceive information in ‘stop position to each other’ and move in ‘wave state’:

Motions are perceived by particles that stop motion into form, into information, as distances. In terms of fractal reproduction of information we can define motion as the reproduction of form, between those 2 scales: when the particle moves dissolves into its ∆-1 parts as a wave that imprints an ∆-2 potential field, with its ∆-1 wave form and stops to become a ‘tight’ Particle state that ‘gauges’ information, form in stillness

*Galilean relativity was ill-understood, as the true question about time-change was why ‘we see systems still when they move’, and ‘why we see space as continuum, when in detail is made of quanta’, and why all systems are made of *smaller self-similar systems. So there is NOT really a Dimension of pure spatial form or a pure time motion but a combination of both, even if *mentally we tend to reduce motion and focus on forms, all has motion=time, and form=space, and this is truly the meaning of ‘spacetime’, the messing of both into 5 dimotions, the fundamental element of all realities. *If we see slow motion in the night a light it seems a long distance. Distance and motion cannot be distinguished so they must be taken as two side of the same being, a Space=time ÐIMOTION (ab. Dimensional Motion):

*S= T; Dimension-Distance = Time-motion = ST Ðimotion:*** Dimensions + motions = Dimotions**

*When we perceive the system in space, then we perceive an organism with 3 adjacent topologic elements, and its forma science is ‘vital topology’.* And when we perceive them through its scales the organic system becomes a supœrganism. Finally when we perceive the system in time we perceive a cycle that returns to its origin as a zero sum, which observed through all its scales will be a life-death cycle, common to all systems where life is the arrow of information and future, death the arrow of death entropy and past, and both together form a worldcycle.

**Recap****:** Non-Euclidean points constantly communicate energy and information with other self-similar points and the external Universe, by sharing flows of micro-points of a lower scale of space-time, which carry the energy and form of the particle into the external universe. The laws that define those acts of communication are hierarchical laws between planes of space-time and laws of balance between the energy and form of those ‘actions’ of communication, exi, which become the fundamental dynamic event of any scale of the Universe. Events in the Universe are limited by the ternary principle. Actions of communication also obey the principle: There are energetic, informative or reproductive events, creating often complementary systems with an energetic pole or body and an informative pole or head, communicated by a dense network or neck that carries the actions. The 5 Postulates of non-Euclidean geometry are based in the definition of a fractal point as a point with inner parts, revealed when we come closer to the point. According to such definition, lines are waves of points and planes topological networks of points, communicated through flows of energy and form. While equality requires also equality in the inner form or information of the point, which prompts communication through waves of energy and information that build networks. Communication between points is now possible because points can fit infinite parallel/waves used to gauge the Universe and create an inner image of reality. Non-Euclidean fractal geometry thus improves our vision of the Universe closer to reality and allows the definition of organic systems and logic behavior in bases of geometrical form, a long-sought dream since the times of the Greek.

**– 3 ^{rd} postulate. The fundamental plane of the universe: the super organism**

**and its a ternary fractal networks.**

*:* Because a plane has inner volume – within its points – it is a cellular, organic topography, a network of self-similar points. And because networks of points of energy and information are complementary, often we find systems with 2 complementary networks that form ever more complex geometries – based in the geometrical dualities of lineal energy and cyclical information – with the results we observe in nature: *the creation of an enormous number of complementary systems which are, as we shall see latter, all of them self-similar in its geometries and functions. Thus,* the types of Non-E planes of space-time range from the simplest Euclidean planes to the more complex organisms with a volume given by the relative point/beings that form its space-time networks.

The 3^{rd} postulate defines planes as the intersection=messing of 3 ‘lines=networks’ or waves of points in motion, which carry energy and information; to its cells, defining the biotopologic plane, a physiological system with 3 networks or Non-Euclidean plane. Thus vital topologic evolution is the missing link to complete biology besides genetics (in 5D explained as the lower plane that codes with its faster cycles the larger biologic whole) and Darwinian struggle (fully explained by the congruence principle of Darwinian dissimilarity) vs. eusocial evolution (in its own an entire new discipline) fully explained by the similarity of parallel beings that COMMUNICATE information in a code or language all can understand so they can coordinate its actions and become stronger as a whole. So *similar clone beings form topologic ternary networks that become superorganisms, wholes made of ∆-1 parts, themselves wholes of ∆-1 parts through 3 5D scales*

Stiences study those organic systems, tied up by networks of Ðimotions. In the graph, we see the main st-planes studied by human sciences and their 4 main time arrows, $ *x ðƒ,* which in static space give birth to the ‘organic elements’ of all species: social cell of energy and information and the reproductive networks that relate them. Thus, there are 4 basic elements in all organic systems: **Cellular** units. **Networks** that move the system (limbs/potentials) **Networks** of fractal information (heads/particles). **Networks** that reproduce vital energy (body/waves)

All species studied by science a common phenomenon occurs: the existence of parallel groups of beings organized into a single social form. Molecules are made up of atoms and electronic networks; economies are made up of human workers and consumers that reproduce and test machines, guided by financial networks of information (salaries, prices, costs); galaxies are composed of stars, which orbit rhythmically around a central knot, or black hole of gravitational information. Cells controlled by the nervous, informative system organize human bodies.

A tree is a group of leaves, branches and roots connected by a network that provides energy (salvia) and information (chemical particles) to its cells. Cultures are made of humans related by verbal, informative Disomorphisms and economic networks that provide their energy and information.

Vital topology studies those fractal Super-Organisms of Time space (Ab. T.œ) whose ∆º networks of ∆-1 points form the 4^{th} and 5^{th} dimotions of social evolution and entropy between parts and wholes.

Thus, a plane becomes a real topography made of points with volume, extended as a cellular surface. We can observe its surface as a bidimensional membrane of information (for example your skin, or the screen of a computer made of pixels, or the sheet of this work). Or we can consider the 3-dimensional inner structure of its points and then it becomes a network with inner motions, as those points will form a lattice in which they communicate lineal flows of energy and information that maintain the lattice pegged. Often 2 topological planes of energy and form combine to create a 4-dimensional organism. Such is the most common structure of the Universe, a 4-dimensional World, which is a Universe in itself, made of self-similar cells or networks of points that constantly exchanges energy and information within the ecosystem in which it exists.

In the graph, we see the ternary network structure of the nested organisms of the fractal Universe as ¬Æ topological planes composed of ‘similar fractal points’ (atoms, cells, individuals) joined by 3 physiological lines=networks, whose 3 functions, distribution of locomotion, information and its ‘combined’ energy define the 3 conserved Dimensional motions (ab. Dimotions) of any system of the Universe.

There are only 3 variations of topologic space in a single “plane of the fifth dimension’, the hyperbolic, elliptic and toroid topologies. So the fact that we are made of space means we are made of 3 type of organic topologies:

Spheres are the topology that holds the maximal volume of information; hence all ‘time space’ systems that process information are spherical particle-heads. Flat, lineal topologies are the topology that connects in the shortest path two points; so to reach/move faster, systems have lineal/flat moving potentials/limbs.

Finally the third type of geometry, hyperbolic topology, is complex enough to store all other possible forms, so best to reproduce. So all iterative bodies & waves that generate the other 2 forms are hyperbolic. This mathematical-spatial truism holds for 2, 3 and 4 dimensions; hence establishing a basic restriction to the construction and evolution of forms. It explains the efficiency, speed and homology of formal=functional, ‘punctuated’ evolution, from biology, without the need of ‘intelligent design’ to engineering (no longer analogy, since all forms derive from the same ‘substance’, space=form with time=motion or ‘topology’). In all systems the 3 only topologies of the Universe ensemble to form physical, biological and social organisms, all of them with a spherical ‘tall’ dimension to gauge information; a wide, iterative dimension of hyperbolic bodywave reproduction and a flat, direction for its entropic motions. Each of those 3 topologic Timespace forms can ‘deform’ and ‘move’, as long as they don’t tear (break). All T.Œs are variational local fractal species of that ternary ensemble, seeing as* a simultaneous space supœrganism, which lives a sequential worldcycle in time, *whose 3 conserved Dimotions in a single plane, |-limbs/fields<Ø-body-waves>§ø-particle-heads become in space functional topologic organs ordered in time in 3 consecutive-ages of dominance of each organ-dimotion between ‘4D Generation’ in the cellular/atomic plane of seminal birth ∆-1»∆1 and 5D extinction, back to the ∆1«∆-1 atomic, cellular plane: the young age of max. |-locomotion, the mature age of Ø-body-wave reproduction and the 3^{rd} age of O-informative warping/ wrinkling and minimal motion. Those 3 ages of all T.œs (Time§pace organisms) apply also to languages, including mathematics, evolving from a young Greek age of lineal ‘flat’ geometry, to a mature age as a realist mirror of the scalar Universe with analytic geometry and calculus that gave it motion and scalar depth studying its finitesimal parts and wholes to a 3^{rd} age of inflationary information and mathematical fictions unconnected with reality (set theory, NOT points, numbers and operands as its foundations, axiomatic method NOT experimental proof); to start again as the mind of a ‘different mind species’ – chips now evolving its young simple Boolean algebra:

So the vital space of superorganisms is made of 3 topological forms; and its cyclical time, and clocks that measure the duration of the existence of all beings, by the frequency of its logic cycles of information, accumulate constantly form, T>S, in 3 ages, first dominated by pure motions, (2D) in youth, balanced in the reproductive mature age (3D), to finally exhaust its energy warped and wrinkled into form (1D), which will explode entropically in the moment of death (5D).

Both together show the ‘Disomorphic’=equal form of all 5Dimensional beings, organic fractals, whose scalar structure is the same for all space-time beings, in all stiences, which study according to its relative scale of size and speed of time clocks a different metric scale of the fifth dimension. So we introduce another element of vital mathematics, the ternary structure of all systems of nature of which the previous fractal points of physical systems are its simpler forms.

Because the Universe is Generated by the properties of space and time, better reflected in the mirror languages of mathematical topology and social numbers, and logic, we could say safely that since ‘we think therefore we are’, that is, we perceive as humans a limited range of reality, with our space and time languages, for humanity, the Universe is generated by mathematical and logic languages. And so by studying formal sciences of mathematics and logic, we can describe the Universe. We can talk then of a Universal topology, from topos, the language of geometry in motion and logic, the language of causal time cycles that create and repeat patterns of reality.

How many world minds those 2 languages generate varies according to the complexity of each being in exist¡ence.

We humans, in the present not very enlightened age live in a single dimension of time, and the simplest, most dangerous of them, lineal entropic time, and a single scale of space, sub-divided in 3 lineal dimensions of height-information, width-growth and length-motion. And that is all. But can talk of more complex beings, with dual, ternary, penta and Dodecalogic levels of entangled thought gradually used here.

What we mean by ‘space-time beings? Space is ‘form’, ‘distance’, ‘dimension’, ‘information’, something that doesn’t move. And time is motion, change. So spacetime would be a ‘form with motion’, which is what mathematicians call a topology. A form that can be deformed, trans-formed. We have coined a key word ‘Dimotion’ (dimensional motion) for those topological forms. So the first obvious question we must answer is what means in real terms, the fact that all what exists is made of spatial topologies and temporal ages, of Space and Time? The answer is fascinating, as there are only 3 variations of mathematical space in a single “plane of the fifth dimension’, the so called hyperbolic, elliptic and toroid topologies. So:

‘All entities of the Universe are topological systems made of 3 ‘dimotions of spacetime’. Since *in the whole Universe in either 4 or 5 Dimensions there are only 3 topological varieties of form with motion; the ‘elliptic particle-sphere’ , the ‘hyperbolic bodywave’ and the ‘toroid, lineal limb-field’, illustrated in the next graph for the 3 kind of species studied by the 3 disciplines of science, physical, biological and human ‘stiences’:*

The 3 elements of all systems, lineal/flat limbs/potentials; spherical, tall heads/particles and its hard membranes of dark matter, trunks and skins accomplish the 3 functions of motion, energetic reproduction and informative perception in all systems, varying to adapt the system to its larger world.

Thus, there is a parallelism between ‘vital function’ and ‘abstract dimension’ as *each Dimension is diffeomorphic created by the vital space-time of each entity of the Universe performing its 3 survival dimotions= åctions of gauging, moving, feeding and reproducing. *And each species is a ‘fractal universe in itself’, with different relative, energetic and informative directions that determine its own up & down arrows and complementary morphology.

The specific geometry of each species and its 3 simplex åctions, energy feeding=length, information-gauging =height and reproduction=width fluctuate in shape, but they don’t vary in topological form, giving birth to the Invariance of the 3 topological forms=functions of any entity observed in a single 3D space-time: Toroid informative heads, hyperbolic reproductive bodies and energetic limbs.

While the 2 social ±∆ fractal dimensions, the 4th dimension of social motion in a social ∆+1 herd and the 5D dimension of inner scales within the being follow the physiological invariances that create super-organisms co-existing in a* higher ∆+1 ecosystem with a lower ∆-1 internal, cellular space.*

**The 4 ^{th} Axiom=postulate of relative congruence…**–

*4 ^{th} Postulate:* Equality is no longer only external, shown in the spatial perimeter of any geometrical form (Euclidean congruence) but also internal and further on it is never absolute but relative, since we cannot perceive the entire inner form of a point – hence the strategies of behavior such as camouflage. Forms are self-similar to each other, which defines different relationships between organic points, according to their degree of self-similarity. The

*4*postulate is thus the key to explain the behavior of particles as the degree of self-similarity increases the degree of communication between beings. Some of the most common behaviors and ‘events derived from this postulate are:

^{th}1) Reproductive functions in case of maximal self-similarity or complementarity in energy and form; ei->Sei or Max E x min. I (male)= Min. e x Max I (female).

2) Social evolution, when points share a common language of information, i=i -> 2i.

3) Darwinian devolution when forms are so different they can’t understand each other’s information so instead they feed into each other: i ¹ i. In such cases if those 2 entities meet they will start a process of ‘struggle for existence’, trying to absorb each other’s energy (when E=E) or simply will not communicate (when E¹E, since then there is neither a common information to evolve socially nor a common energy to feed on). Yet because any point absorbs only a relative quantity of information from reality, similarity is relative, faked for purposes of hunting with biologic games such as camouflage or sociological memes that invent racial differences, allowing the exploitation of a group by another.

defines systems as identical when they are equal in its 3 ternary parts, the outer angular momentum or ‘membrane’, its central Active magnitude or singularity, focus of the forces and the vital energy*, enclosed within them, and all others as similar with different ‘angle of congruence’. We* distinguish 2 different interactions according to the degree of equality of its ternary parts, as systems can be symbiotic, if *their individual, cellular or atomic ‘fractal points=parts’ are similar enough, interacting through *its 3 physiological ‘lines=networks’ evolving in parallel creating an organic plane, as those described in the next graph for each scale=science, or they can be entropic, destructive, predatory, when they are dissimilar and don’t speak a common language of information to coordinate its actions, whereas the stronger system will perpendicularly break and feed on the weaker one.

The geometric complexity of the 4^{th} Postulate is caused by the topological forms created by any event that entangles Multiple Spaces-Times. Since it describes the paths and forms of dual systems, which connect points: Self-similarity implies parallel motions in herds; since equal entities will maintain a parallel distance to allow informative communication *without interfering with the reproductive body of each point*. Darwinian behavior implies perpendicular confrontations, *to penetrate and absorb the energy of the other point*. Finally, *absolute, inner and outer self-similarity* brings boson states, which happen more often to simpler species like quarks and particles that can form a boson condensate as they do in black holes, *where the proximity of the points is maximized. And indeed, the same phenomenon between cells with the same inner information /DNA originates the ‘collapse’ of waves into tighter organisms.*

The 4^{th} Non-Euclidean postulate is implicit in the work of Lobachevski and Riemann who defined spaces with the properties of self-similarity (Riemann’s homogeneity), which determines its closeness (Lobachevski’s adjacency).

*4 ^{th} postulate of relative congruence & angle of parallelism as a mirror of its 5 pentalogic dimotions and variations of angle define Darwinian or social, reproductive outcomes to communicative events between fractal points.*

Thus in praxis we assess similarity by an ‘angle of parallelism’ that increases social evolution into herds and supœrganisms, or perpendicularity that ‘scatters’ systems into entropic destruction – elements those of an entire fascinating new field of 5D topo-biologic studies that analyze in geometric terms, the vital topology and relationships between form and function in all systems of Nature from particles to organisms.

This simple geometrical truth however is essential to all systems of nature, whose angles of connection determine the functions and symbiosis between parts.

The Universe always starts with an asymmetric being, which can go both ways: towards a social evolutionary symmetry that lasts in time and implies a mirror parallelism, or an antisymmetric destructive, perpendicular event in which one part punctures and absorbs the energy of the other. It is the topo-biological ternary principle of non-Euclidean, Non-Aristotelian I-logic geometry that puts together both the biological and mathematical properties of reality. The concepts of symmetry=parallelism, antisymmetry=perpendicularity and asymmetry are mirrored by the 4th Non-E Postulate of similarity. But we can extend the concept of asymmetries also to asymmetries of time, between the young age of locomotion and the old age of information, of actions=Dimotions between the step and stop similar actions, and the entropy and social evolution actions, which bring us the final asymmetry of scales between the upper arrow of whole with more spatial size and the lower arrow of parts with more information. When those dualities: step-motion/stop-perception and scale up (5D: social evolution), scale down (4D: entropic dissolution) are put together we obtain the most complex balancing dimotion, reproduction, and when they are all added up in the existence of a being, we get its world cycle.

In the graph we can assess the different 5 mirrors in which mathematical Space and logic Time reflects the game of 5 Dimotions=actions of existence, which then expressed by territorial monads GENERATES its logic REALITY. In Geometry fractal points=monads will other through waves of communication of energy and information that grow into reproductive networks a territorial plane, creating a super organism, which will related to the external world according to its relative similarity=congruence, assessed by its angle of parallelism or perpendicularity.

In logic terms, a super organism, by breaking its formless asymmetry into different spatial configurations according to congruence (social parallel systems, complementary gender-mirror systems, Darwinian perpendicular systems, or systems that are dissymmetric and do not share any reality) builds a casual pyramid of growth from a fractal point through waves of communication into social networks that become ready to act – move, feed, perceive and evolve socially.

Since we must add to the mathematical and logic languages-properties of reality the 5 actions, or organic properties of the scalar Universe as essential to the game as they are its logic and mathematical more abstract laws – a fact the egocy of æntropic men of course reject, as it must remain in its monad-subjective monologic the only claimant to life properties.

Thus the pentalogic of generational space-time is established by its Non-Euclidean fractal points, its ¡logic congruence with reality in which it will order a territory to perform its 5 vital actions=Dimotions of existence, and the mathematical, logic and organic laws of those 3 languages will be therefore the bottom line of the ‘Creative process’ of the Universe – nothing chaotic except the entropic Dimotion, which conforms the monologic of huminds.

*5 ^{th} postulate – definition of a mind-point of a new scale. *

So finally we arrive to the fifth and only standing non-E postulate, which is equivalent to the first, defining a fractal point but from the p.o.v. of its inner mind-center of reference as the Universe has infinite of them: The graph shows the difference between the Aristotelian, self-centered, Euclidean=light humind and the Universal mind: ¬-Aristotelian, Non-Euclidean:

The universe has infinite mind-mirrors depending on the forces used to gauge the external world, which bounces on a limited quantity of its scales of space. Humans perceive the range of scales of the frequency of light between red and blue social density of colors. But infinite other minds with different detail according to the quantitative pixels they absorb (max. S = Min.t) maximal for smaller sixes will determine the intelligence of the system. Descartes was fully aware that what he had in the mind was not the whole Universe, so he expressly stated the fact, differentiating the ‘world’ of a human mind, from the infinite other worlds that exist outside, establishing in his little known book, the ‘world’, this difference, affirming that his ‘Cartesian Frame of reference’, was only that of the human mind.

So he affirmed that the Universe was the sum of ∞ mind-worlds that don’t speak the same languages, and created the same mappings of reality we humans created. So he *said that all what exists was made of:*

*– Open §pace, which he called ‘res extensa’.* *– Closed, cyclical times, which he called ‘vortices’*.

And then he added a 3^{rd} element, realizing the only proof he had of the existence of those vortices and res extensa was the fact that he perceived them: Cogito Ergo Sum. ‘I think therefore I am’.

**Metric equation of mathematics: 0x ∞ = K search for ‘wholeness’ in a single equation. **

Thus we define the Generator equation of algebra, which as a mind language derives from the mind 5D metric:

*0-finitesimal spatial mind x ∞ time cycles = Constant mind-world: §@(mind space)<≈>∆ð (universal cycles)*

So t*he ‘Generator Equation’ of all digital numbers is, 0 x ∞=K and its algebraic expression, Sx<≈>ƒð.*

Both equations together allow to represent all possible metric equations of all supœrganisms of the Universe. As such neither 0 is an absolute zero, nor ∞ is an infinity, concepts relative in a scalar Universe, where the limits of scales in size and time speed and perception in each self-centered p.o.v. only allow to talk of relative infinitesimals, we call finitesimals, 0%, and relative infinities, whose symbol is ∞. 2 evident truths are then the existence of an experimental finitesimal, h=m l² ƒ, the minimal unit of angular momentum or Planckton of the galatom, and a relative infinite, =c, its maximal distanced-speed, which conforms the most extensive metric we know of with precision, beyond which dark space and time is unperceivable to mankind, so we write 0s (h) x (c) = K

It follows also that mathematical infinities are inflationary mirrors whose contradictions (Cantor’s Paradoxes) are NOT solvable (the Zermelo axiom being an ad hoc addition, against the logic of truth), because precisely they limits of reality enter in paradoxical contradiction with the infinite inflationary mirrors of languages.

The study of those 4 elements of all realities, its actions and ternary operandi, structures *the dynamic ‘Generator Equation’ of all Space-time Systems, written in its simplest form as a singularity-mind equation:* *O x ∞ = K = ∞º=1*

So the *5 ^{th} postulate *defines points as informative knots or linguistic eyes – minds of information that absorb a flux of forces used by the point to perceive a relative world. A non-Euclidean point corresponds then to our concept of a relative mind that gauges the information reality with a certain force, similar to the concept of a monad in Leibniz relativistic space-time. In words of Einstein: a point of space is a fixed frame of reference.

Thus, Non-Euclidean mathematics fuses the logic and geometry of the fractal Universe, greatly improving our understanding of Reality even in terms of mechanist measure. Since mathematical solutions to problems with several points of view are impossible to find in continuous space (i.e. 3-body problem in gravitation), given the fact that a network of infinite points of view is local and relative and each point is a focal knot that acts from its perspective. Thus, the absolute truth of a system is the sum of all its points of view, which influence each other. Yet even if we cannot calculate precisely with mathematics, systems with more than 2 bodies, since those systems are organic, hierarchical, made of networks with attractor points, fractal structures and self-similar paths, the new mathematics of attractors, fractals, scales and Non-Euclidean systems, refine greatly our analysis. In essence, indeed, we observe that ‘networks’ integrate parts/points into wholes, which then ‘act’ as a single point. So in the complex models of i-logic geometry we can tackle many problems by defining sets of points as ‘wholes’ of a ‘higher space-time’.

For example, the previous principle of local measure, where each point is a relative center of the Universe, is called in relativity the diffeomorphic principle, which now becomes explained as a partial case of the wider law we called the ‘Galilean paradox’; the duality particle of information/wave of energy becomes a specific case of the application to physics of the duality of energy and information found in all systems.

The expansion of the laws of quantum theory (complementarity principle) and Relativity (relativity of scale, local measure, etc.) to other sciences and the organic principles of the 3^{rd} and 4^{th} postulate to physical particles is therefore the consequence of those postulates. Yet it requires the understanding of the new, i-logic, organic laws of the Universe and its networks, because E-mathematics has clear limits to extract all the information of the Universe, given the fact that it syntax includes a priori errors and simplifying postulates (single space-time continuum of points without parts, etc.).

Thus, when the event described is complex, performed by a great number of points/variables you enter into non-lineal systems, which require topological descriptions (chaotic attractors, fractal non-differentiable equations and Non-Euclidean mathematics), and the i-logic laws of organic networks and systems – a better syntax in which to fit experimental evidence, especially in phenomena of informative nature (since only formless energy is continuous and resembles the models built with a single arrow of energy and a single plane of space-time).

So while classic physical systems calculate accurately the energetic, continuous properties of the Universe an overview on how multiple points of view emerge into wholes requires organic laws. This search of whys also applies to the understanding of mathematics. For example, the previous postulates resolve the long-standing question of what is the nature of the 1^{st} and 5^{th} postulates that seemed redundant (as the 5^{th} describes also properties of a point like the first does, and the 1^{st} seem to describe a non-geometrical property). They are no longer redundant, but they are more concerned with causal logic and time than spatial geometry in its purest forms (points, lines and planes.)

There is also self-similarity between the fractal postulates of i-logic geometry (since the 5^{th} is geometrically self-similar to the 1^{st}, as both are concerned with points) and the 4 dimensional time paths/arrows of the universe. This is not casual since all languages of space-times depict in self-similar ways the 4D Universe. Thus if the 1^{st} and 5^{th} postulates define *a gauging point of information* as the fundamental unit of the Universe, the 2^{nd} postulate defines a line or flow of communication of energy and information between 2 points, which *reproduces* part of the information of the ‘generator’ point across a surface of space; the 3^{rd} postulate defines those points, which are not similar as *energetic* substances that will be absorbed by the points. Yet if those points are self-similar they will gather through the arrow of eusocial love, creating according to the 4^{th} postulate a network of space/time, a new organic plane of existence. So the 4 arrows of space-time are explained by the 4 postulates of i-logic geometry.

**External dimensions/networks of organisms: territories. **

*The key element of Non-E geometry are the 3 topologic regions of all systems, as that is the underlying structure that evolutionary topology develops, with a singularity, @, dominating a vital territory enclosed by a membrane.*

It is the mixture of function in time through actions of survival that dominates the spatial ternary structure of those T.œs which guides the understanding of vital geometry.

I.e. “Though most arachnids are solitary animals, some spiders live in enormous communal webs housing males, females, and spiderlings. Most of the individuals live in the central part of the web, with the outer part providing snare space for prey shared by all the inhabitants”. *Britannica*

Indeed, regardless of the vital topology of the point, all ‘build ups’ of new geometric scales start with the simplest form, a ‘bidimensional territory’ with a membrane, a central singularity and the vital energy between them. So geometry not only evolves in the humind in complexity, it does so in the evolution of the vital topologies of new forms in each single plane:

In the graph, all systems regarding of its ‘perfect geometry’, have the same ternary structure, to which vital geometry adds motion, in a Klein-like Non-E structure, where borders of entropy=death can’t be crossed; so they are relative infinities – military borders, balls of fire, membranes. On the left we study in more detail a mammal territory. Any animal territory is an i-logic space-time with 3 zones:

An informative central territory (1) or den, where animals reproduce and 2 secondary homes where the herd performs secondary organic cycles (2,3).

An energetic membrane (M, 5) – an invisible limit that provokes a confrontation if a stranger crosses it and where most energetic preys ‘flee’ away from the den of the predator.

An intermediate zone with cyclical paths of absorption of Entropy and information; where we find a hunting territory, places to drink (E), to bath (B), socialize (A), defecate (D), etc.

In organic terms, a dimension is a network. So a living organism can be considered a sum of cellular quanta united by 3 basic space/time discreet dimensional networks, which are its physiological systems: the digestive/energetic network, informative/ nervous network and reproductive/blood networks around which cells teem, creating a stable, organic st-point. In other words the Entropy and informative networks of a living being are its internal, diffeomorphic dimensions (of relative length and height), to which the organic system adds a 3rd, reproductive dimension that combines both elements and represents the width or ‘volume of cellular quanta’ of the system.

Finally its movement in the external world becomes its 4th temporal dimension. Yet that 4th dimension of external activity can also be considered a network territory in itself, sum of the 3±i cycles of existence of the being, creating a bigger vital space that will become the basic unit of an ecosystem or social organism made of individuals of the same species. In the figure we draw the vital territory of a minimal social pair of mammals, differentiated in 3 clear sub-sectors:

__Max.Information: Informative den or central territory (1,2,3):__

It is the territory of reproduction used to copulate and store basic food and Entropy to raise the young. It is a forbidden zone where not even hunting is allowed (4). In social species of great mobility, aerial or marine, where borders are much more extensive, this territory is very ample and tends to be located in warm latitudes.

Entropy=Information: Dual Territory of Entropy hunting and informative socialization (5).

It is the feeding, social and hunting territory, on which the central informative being feeds itself. It is outside the zone of reproduction. It is the winter territory of many migratory birds.

Given the relativism of all movement, in biological territories the informative singularity moves to hunt its Entropy quanta, as opposed to galaxies where stars and space-time dust moves towards the central worm hole.

Within those limits there are also neutral territories of communication, courtship reproduction and free Entropy, like water troughs. So the intermediate territory works both, as an informative and energetic territory where different victims and predators trace parallel cycles and come together around meeting points (E, B, R).

__Max. Entropy: Borders that limit the territory.__

Membranes are dangerous topological zones of dual osmosis. So the informative center watches to control any invasion of its hunting/social territory. Membranes fluctuate according to neighbors’ power. For example, the vital space of a fish increases during mating, since the couple is more powerful than a single individual. Marks (M points) fix those limits and reduce combats. They are often invisible, as most territories are defended against competitors of the same species, who understand the informative code of those marks; but rarely against members of other species. So we find all kind of linguistic marks:

Smells (common in mammals, like foxes, rhinos, antelopes), excrements (in canines and felids) or other glandular secretions.

Optical marks often connected to scents: The brown bear creates marks in trees, rubbing them with his head, warning adversaries of his great Max.SxT size and force. In human empires (nations can also be treated as biological territories) visual marks correspond to armies displayed in the borders. In human homes those marks used to be shields with weapons; now they are cars and other proofs of money, the new language of social power.

High pitch, acoustic marks, proper of birds, which are triggered when a rival enters the territory.

**Recap**. Vital territories of animals and human nations can be explained with the 3 topological regions of st-points.

The structure of §ð<TS<$t, territorial spaces with a central point of view, developing its particular worldview, trying to reach infinity with his distorted geometry, affine to a projective geometry where far away means small, defines each world of a Universe, which is objective when ‘clashing’ each form with all others – so only eusocial love, and emergence through the scales of the 5^{th} dimension make survival possible. Geometry is then the study of the spatial form that the functions, which dominate the vital, sentient Universe, adopt in their existential actions.

And as such is the best method to visualize the ‘meaning’ of algebraic and analytic equations both in abstract and mathematical physics. Each of the different laws of bidimensional plane geometry then can be studied as a reflection of the efficiency of vital Dimotions in a simplified geometric form, where curved paths become perfect lines, distances are measured without error and angles have no geodesic distortion. But still *vital geometry will add to those structures* *its vital interpretation, besides the ab*stract knowledge introduced by the Greeks, on the lines of the previous graph (which become lineal forms in other systems such as ‘polygonal molecules’ or human artificial constructions).

In Geometry thus we can also use the ‘pentalogic points of view’ on each theorem and also specific of geometry the vital meaning of the *5 angles of congruence (4 non-E Postulate), from Darwinian perpendicularity to social parallelism.*

**Conclusion: Vital, Organic Geometry merges formal, physical and biological stiences.**

Such organic model is vastly more explanative of reality than the mechanical abstract models dominant in physical sciences for a few basic reasons:

- Only an organism replicates itself without the concourse of a ‘maker of the machine’ – God in the earlier physicists’ view, now obsolete. As machines do not reproduce.
- Only the definition of time as the substance of reality – cyclical motion that reproduces form allows the Universe to move, as only if motion=time is the ultimate substance the Universe never stops.
- Only if it is cyclical, repetitive, there are laws of science, which are precisely those repetitive motions.
- Relativity show there are ∞ clocks of time with different speeds (just look at the clocks of nature).

It is epistemologically sounder (Occam’s razor) as all systems can be modeled as organisms. Further on organicism uses all the language-mirrors to express its topo-bio-logic properties:

- It uses the mathematical language, best for its spatial description, which models organisms in the form of fractal networks, for which we shall adapt and expand its most advanced forms of space, topology – space with motion – and non-Euclidean geometry – fractal points with volume that grow in size as we move closer into its scales, allowing ∞ parallels of energy and information to flux into them.
- It uses the logic language, ideal for temporal, causal sequences and cycles, which we shall improve from the simpler Lineal Logic of locomotion, into Duality, the logic of two arrows of time, energy and information and further on into ‘trinity’, as both combine into energy beings – the logic of topological organisms, conservation principles, Universal grammar, and even further on into pentalogic and Dodecalogic…
- It expands Biologic life to all systems. Since a fractal requires a new dimension of parts that become wholes, larger scales co-existing together gifted therefore with organic properties as atoms/cells/Individuals form part of thermodynamic /biologic/social ensembles that become wholes, living in a larger gravitational, ecosystemic or planetary world.

We call that super organism of astrophysics, the galatom, (as it happens between the quantum atomic scale, and the galactic scale, which are self-similar – with similar forces, and particles similar to celestial bodies). Within the galatom in the intermediate stage there exists, thermodynamic man.

Thus the organic paradigm rejects ‘mathematical creationism’ in physics – the belief an equation describes all the properties of a system, which no longer need parts; as in the case of black holes which are modeled with singularities – mathematical entelechies of infinite density and infinitesimal volume but as Einstein wanted, they must be made of a cut-off substance, which can only be heavy quark atoms (bcb atoms).

**Recap. ** The pentalogic of existential algebra (¬Æ) leads to an i-logic topology of fractal points with parts that complete the upgrading spatial mathematics with 5 ¬Æ postulates to substitute the remaining axioms, postulates and principles of E geometry that the 5 Non-E postulate merely hinted at.

Even abstract still Greek geometry is a mirror of the experimental organic Disomorphic laws of T.œs.

Points are a ‘dominant’ spatial, mental view which reside in a single plane and the ‘mind’ perceives in stillness, with no internal scalar form and continuity when associated into lines, planes and volumes; even though in reality they have ‘fractal content’

The 5 postulates of ilogic Geometry describe how points become parts of social webs, self-organizing fractal planes made of networks of points that emerge as cellular units of a higher fractal space-time or new supœrganism.

Geometry is the study of the structures built departing from fractal points with inner parts and scalar content, and as such we have upgraded its elementary units with our paragraph on ¬E topology.

Classic geometry considers points to have no breath, no inner parts. As such it is within the principle of Correspondence – the limit of ¬E topology when we ‘eliminate’ the 4th dimension of time-motion (Greek bidimensional geometry on a plane) and the 5th dimension of fractal scales (only fractal geometries include it in a restricted analysis). Thus 5Dwill expand all those different geometries to make them describe the same fundamental structure of the Universe: the point with parts and its more complex social forms, lines and planes. The fractal generator of the Universe is a logic equation that represents the main interactions between the arrows of time. Yet all languages mirror that logic equation in its syntax, since all minds gauge and represent the cycles of the Universe with languages of perception.

We thus expand geometry to make it vital as it is in reality, constructing the 5dimotions of reality departing from one-dimensional points with volume, which evolve into bidimensional waves of information, which form ternary physiological topologic networks, vital planes of organisms that finally emerge into relative 0-points of a larger scale, guided by a mind-membrain, self-centered into a linguistic singularity connected through those networks to a membrane that enclose the whole structure and makes it look from an outer perspective of a larger scale as a particle-point of a new plane of existence. It is then clear that a more evolved Non-E mathematics is the organic language of the fractal Universe.__.__

**SOCIAL****NUMBER THEORY**.**∆-SCALES & TIME SEQUENCES**

**Abstract**. Numbers are the next fundamental element of reality, which for what we explained in the analysis of the **S=T ***symmetry between *Numbers=Points have an added quality to points – *to penetrate into the social scales of the Universe. This is so evident; a natural number is just a social gathering of a herd of identical beings….*

Thus the definition of a number is the clearest proof of the *social character of the Universe and the existence of scales.*

But then we saw how inverse negative numbers added motion to points, rational numbers broke social wholes into parts, decimals and irrational numbers peered into lower scales and some specific numbers became Universal constants that expressed the basic dimotions of reality, pi=perception, L_{10}=social evolution, e=entropy, phi=reproduction. Finally complex numbers fully expanded our perception of the Universe simultaneously in various planes of existence.

Numbers can then put into groups according to its families, N, Z, Q, R, C, and the reader should not wonder that in a 5D Universe there are only 5 families of numbers, to close its mirroring of reality.

Further on numbers as social elements of a whole have different scales according to the ‘preferred’ whole considered. Binary scales dealt with the key dualities between Entropy=Time and information=pace (open 1 and closed 0); ternary and sexagesimal scales with the circular worldcycles of existence (ages, cycles) and decametric scales with the social dominant 10-scale of the scalar Universe (as 3×3+1 form a unit of the next scale).

Because numbers are identical beings its S=T symmetry as points requires them to be regular polygons or platonic solids, where all points are equal in position, angle and connection. So those regular figures merge geometry and number theory.

Of significance among those social groups of numbers in the Natural family is the distinction between even, odd and prime numbers which can generate all others, mirroring further the properties of ¬∆@st, which we shall reveal on the development of this abstract on number theory not to tell it all so fast that nobody reads it (:

Finally we can apply the complex pentalogic of the entangled Universe to put numbers in relationship with the other 4 elements of Dust of space-time both within mathematics (its other disciplines) and the Universe at large, a method we can apply with each element of numbers. So let us deal with those key themes of number theory one at a time.

## S≈T: **DUALITY OF SPATIAL POINTS AND TEMPORAL NUMBERS**

Let us start our analysis of numbers with the 2^{nd} metric equation of 5D space-time, which is the proper formulation of the principle of physical relativity between Scalar Space and Time, which become indistinguishable, S=T. This is the origin of the essential symmetries of mathematics between scalar space points and scalar time numbers. Whereas as usual all has a bit of yin and yang, a part of space and time; and all is entangled in scales. So the dualities here are between fractal points dominant in space and fractal social numbers, which also work as sequential time systems:

** **** ∫@ (points of geometry) ≈ ∆ð (numbers of algebra)**

**Which happened in the first age of mathematics, and was extended as mathematics became a better mirror of the whole Universe in its classic second age of description of complex organisms performing events through its** 5Ð¡motions of reality into the duality between Geometric figures made of points and Time Operands of Algebra that put social numbers into complex ‘fractal generators’ called equations that described real events of their existence, further enhanced by analysis that allow the study of dimotions between finitesimals (derivatives) integrated in larger scales. Indeed, all languages mirror ST-symmetries and 5 Dimotions in 3 scales. So as we studied mathematical space ‘units’ – geometric points **(S@)** we now deal with its sequential, scalar units social numbers,** ∆ð****.**

**Greek/Chinese age: Natural Numbers as forms, Ratios, scales & prime singularities.**

In the beginning of mathematics numbers were considered points that showed also an internal geometric nature, merging the 3 time-algebraic, ∆§ocial & Spatial-geometric properties of the Universe in its mathematical units, the fractal point and the social number, a gathering of points *that included multiple configurations in space; as time systems always branch into multiple paths of future*. It is a relevant period – today forgotten – for the understanding of the mathematical mirror casted upon the 1^{st} principles of Nature, Space mirrored geometrically in points and figures, Time in numerical sequences and scales in societies of points converted into ‘whole discrete numbers’ came into being. As today we have lost that contact with the true experimental basis of mathematics, in the ‘ego-trip’ of the axiomatic method and set theory (‘I imagine points’ Hilbert) we return to the origins to analyze the continuous geometry — v. –, discrete numbers’ paradox.

**The non-existence in arithmetic of perfect geometric closures (the √2 diagonal and the π-3 cycle).**

Thus the distinction between continuous and discrete mathematics is deeply rooted in the ultimate properties of discrete time and continuous space, belonging to the ‘pentalogic’ structure of the entangled universe, a subject of a different paper on ¡logic; manifested in mathematical praxis, in the differences between time numbers and space points, algebra and topology, discontinuous parts and single, synchronous, continuous space with less information – the parts and the whole, the detailed perception of those parts and its gathering in forms and social numbers. All those dualities between ‘time cycles as membranes of fractal points’ and space entities as perceived wholes are carried into the dualities of mathematics, and in nature by the discrete properties of time cycles that are only measure as frequencies when the continuous cycle return in a discontinuous lapse to the same point. So in detail while boundaries in space act both as frequency of time cycles that at fast speed are perceived as solid membranes, and frequencies, time cycles are quantic because we perceive them only when the time cycle is ‘completed’ and hence shows a discontinuous lapse on time but space is perceived in simultaneity and so it appears as continuous.

Those ST dualities are part of the wider ‘Galilean paradox’ between still space and time motions, which sets the mind’s limits of perception of information. And they lay at the heart of the answer to essential philosophical question: it is the Universe discrete or continuous? Both, but always discrete time cycles; continuous in space and in any detailed view of its membrane boundaries, still in space, moving in time, as continuous cycles perceived discretely as points of repetition of a certain ‘frequency’.

So truth faces a mental issue on its mathematical mirror-modeling as a form of ‘art’: *exact science does not exist*, *all is art of linguistic perception* – the world as a representation, which ‘features’ the natural world in a reduced mental, mathematical form. And needless to say regardless of platonic, creationist physicists, the universe does not contain or consists of actual mathematical objects, but it is a language that can model all aspects of the universe. So all might resemble mathematical concepts as we are now using phonemes to describe all with words, but not, Muslims and Jews are not right when saying that God created talking in Arab and Hebrew, and Bohr is not right when thinking quantum states are mathematical functions of probabilities. *All are @-mind mirrors of higher laws of the ∆ST fractal reality. *Number 2 does not exist as a physical object, but it does describe an important feature of such things as human twins and binary stars. So we can extract as a mirror mind of those 3 fractal elements of all underlying realities, **scales, space and time,** 3 sub-concepts of 2:

**∆: **2 means the first ∆§ocial scale of growth from the 1 being into the 2 couple in binary language: 10.

**T:** 2 doubles the frequency of time, starting the repetitive patterns of nature, allowing the forecasting of future.

**S:** 2 is the bilateral, complementary couple starting the reproductive mirror symmetries of the Universe.

But mathematics is lost without ¡logic. As such it is the 3^{rd} language of thought after GST & its laws of fractal space-time, better expressed in logic terms, as Space is about Simultaneity; scales are about Disomorphic Similarity and time about repetitive sequences all properties perceived by a linguistic @-observer, in terms of logic thought. So in the definition of a number, logic properties matter as much as descriptive experimental truths.

In this case we talk of numbers only when we gather absolutely ‘IDENTICAL’ undistinguishable forms, which make numbers regular polygons, where each point is undistinguishable from all others; as an @bserver will deem those points identical. And this means natural numbers are defined in a single plane, or else won’t be identical.

So identity is the maximal perfection of a number that has ‘no hair’ 2 ‘anything’ means 2 identical beings.

And yet in a deeper thought, not 2 beings are identical when observed in relationship to its ∆-1 background; and as all points have parts, not two points=2 number is ever identical as the inner parts might differ and cannot be perceived – hence mathematics is neither as ‘perfectly truth as the axiomatic method believes. All we can say is that *Not 2 beings are identical for the Universe, but they can be identical for the mental observer in its reduced mind mirror*. Which is the point we want to stress here: languages neither create reality nor focus its perfect truth as only the being has all its truth within – an important prolegomena to realize that in detail numbers and points will have differences, even if the mind can put them on correspondence by reducing its properties.

Then we might wonder what are the more equal of those 3 ∆St realities, space, time or scale properties? And the answer is time, repetitive frequent properties, themes those though of our post on ¡logic Æxistential algebra.

Thus as always a linguistic mind-concept must be referred experimentally through ‘¡logic’ to those 3+@ elements of all fractal realities, ∆@ST, in the case of *a social number its intrinsic properties define conceptually ‘S- imultaneity, ∆§isomorphic equivalence and T-repetitive equality which becomes an @identity for the mind that defines THEN the group of seemingly identical beings as A NUMBER.*

*Points though – and this is the first huge difference are always different because if they occupy the same place in space they are seen as a single point, or at best as a ‘boson’ number of higher mass density; showing again the difference between numbers, ultimately time entities that are indistinguishable and points which are in different locations. But if numbers in time are indistinguishable it implies that time flows are immortal because each ‘moment in time’ can only be indistinguishable if the flow of time is eternal. *

*Finally numbers are a higher ‘compressed’ analytic expression of reality, hence more efficient, as when we say a number we compress all possible geometric configurations of it in space into a single time expression with **less dimensionality**. Space thus have more ‘wide dimensions’ than numbers in time, a one-dimensional sequence. 4 indeed is a single number but there are at 2 indistinguishable point configurations for 4, the 2D square and the 3 D tetrahedron.* Thus Similarity is a property of scale, Simultaneity of Space and equality of time. Only all together become the identical congruence of two ∆ST beings as one.

It is then clear that a Time number as a sum of space points with different equal forms, encodes more information in a synoptic way about the ‘social group’ than an array of points that displays a given detailed shape.

*And this is so because a time** number is the absolute minimal seed in one-dimensional space that unfolds then as 4 does, into a bidimensional square or a 3D tetrahedron. So a number is the minimal coding ‘linguistic mirror’ of a 5D scalar symmetry; which tells us less about the ‘interactions between the point-parts of the being’, than the spatial knowledge we acquire about the relative position in space of each of the point members of a number-group.*

The power of synoptic numbers comes then from 5D metrics: S x ð=C, whereas ð-numbers are Aristotelian potentials unfolded in spatial varieties, as genes code potential variations of spatial species, and quantum laws of multiple atomic spatial configurations – as all language mirrors that code in its fastest synoptic form the real-ization of real-beings in space. But such mirror correspondences, argued at the origin of geometry when ‘points were numbers’ meant to be the language of reality were lost with the abstraction of sets and axiomatic methods.

**Symmetries of ‘similar’ fractal points and Social numbers**.

Because the visual eye sees space as ‘real forms’ with no idealization – so the circle-wheel didn’t even appear in America, the temporal view of mathematics given by sequential numbers were the first ‘concepts’ to appear in maths by counting; as simple a-vowels appeared on language for naming – both mirroring ∆ST *fractal similarities, without which it wouldn’t be possible to have mirror languages that extract the essential ‘similarity of the fractal Universe’ among all its beings regardless of its displacement in size-space, topology=form or instant within its worldcycle of time*. Then idealist Pythagorean Greeks that thought numbers in time are sum of points in space called Gnomons, which is correct, went too far thinking that both are the ‘substance’ of reality. So while they got **right that first s=t marriage** of mathematics, t**hey got wrong a creationist view **still prevailing in science – that numbers create reality, because God ‘speaks only mathematics’ just another humind ego-trip following the age of verbal thought when Hebrew named things and then things ‘became’ according to Bible.

To discern all possible S=T symmetries in any case is the essential logic task of ¬Ælgebra, which was born in Magna Greece from the Italo-Greek school of Pythagoras and Archimedes. But = is a misleading ‘operand’ as in the fractal universe 2 forms are similar, with different internal parts, never identical – only in the simplifying mind synopsis, as we just explained. So instead of cheering those ‘differences’ when finding that *certain spatial forms made of points have no exact equivalent in numbers, Pythagoreans went berserk**, provoking, so the legend says, the first *scientific murder born of calculations of many to come till the Atomic bomb.

Indeed *in sequential time numbers pi is not exact, and √2 misses a bit, while in continuous geometry we can draw both because numbers are discrete, have wholes and we should not try to fill them, as the axiomatic method does (Dedekind cut), just because Idealist Germans never understood mathematical variations.* It is right to considered then that pi and √2 did not exist *as numbers, that is social collections of identical beings, *because they are NOT social numbers but ‘functions and ratios’ which only because, as we have seen, numbers have such a synoptic power, can also be ‘named with them’.

**Continuity of ‘wholes’ in geometric space vs. discontinuity of its discrete parts in scalar space.**

The discovery of incommensurables disturbed the Pythagorean=spatial notion of an immutable ‘mental’ world; which persisted until geometers defined them NOT as numbers but as proportions (ratios) But this *insight was lost when Dedekind and the axiomatic school ‘decided’ by dogma a ‘continuous’ lineal Universe – another mind mirage where to fit them back again Not as a new species, but as ‘mere’ numbers. *

Since different numbers belong to different ‘scales’ of the 5^{th} dimension and mirror different properties of the ∆st universe – even if *as all words use 23 phonemes we can do with 2 to symbols for them all with the extraordinary synoptic power of mathematics* – concepts those we shall elaborate in our analysis of Number theory and Algebra.

I.e., in a single plane of Space ‘natural numbers’ are social units equivalent to **fractal points (ab.•):** Nº = ∑•. And so it is a family that does NOT accept decimals – 10 humans cannot be broken into pieces dismembering the wholes because they are something else. Its discontinuity when ‘projected’ through the S-point/T-number symmetry in any case is ‘natural’ to its social essence. As the dimotion expressed with those numbers is social, scalar evolution, the ‘embedded’ 4Dimotion of natural numbers, inverse to 5D entropic devolution, represented by the ÷ operand of algebra and the rational numbers. It comes then the first problem of synoptic languages – its ambiguity, which at a certain level implies loss of information, when as in the axiomatic method, we loose its experimental mirror. 2 without reference to reality might also mean 4/2, a ratio. And so as soon as we abandon a single ‘5D plane’ the correspondence between numbers and points is lost. And we enter into a deeper analysis of the 5 Dimotions of space-time, and its scalar arrows.

**The ternary and pentalogic analysis of ‘mathematical entities’.**

In an entangled Universe any system require a ternary, ∆st, or pentalogic ‘Rashomon view’ to fully extract its 5 Dimotions or 5 ¬∆@st elements, as all is made of 5D¡, space-time dust, or else it doesn’t survive in an entangled reality. So let’s apply this formal way to do 5D stience to the duality points≈numbers:

**S:** • points have a single social arrow. Still mind spaces are always positive in its simple growing social form.

**T: Nº** numbers are richer in meanings as they can represent ST-fractal points (T.œs) with motion, hence inverse direction that makes them negative (without the proper philosophy of ∆st maths this was historically a big problem till Gauss define them as inverse, and still lingers in the understanding of the i-complex plane of ‘scalar spacetime’). So still space points acquired motion in Cartesian and Vectorial and Phase spacetime graphs.

∆: Rational, Irrational and Complex numbers is the final expansion of points to reflect the scalar Universe.

Ratios are a property of ∆-scales we study in number theory when dealing with Universal constants; even if we see them ‘surfacing’ in our scale of space as geometric figures. So i-ratio-nal numbers might exist as ‘whole units’ *(**rational numbers that happen in a single ∆-plane as circles or diagonals)* in the continuous geometry of the mind.

I.e. If we apply to π, a pentalogic ¬∆@st view:

*∆: Pi then in scalar terms rises the dimensions of the system from a flat line to a bidimensional cycle. *

*T: **Pi represents one essential Dimotion of existence – a **key s=t symmetries, such as +3=π $ = ð -> π, **is the function of time evolution from lineal youth into its 3 ^{rd} age of maximal curvature, when the point returns its origin, after*

*3 diameters-ages with transitional cuts= openings. Hence the simplest representation of a worldcycle.*

** S:** Pi also can be considered in simultaneous space a membrane that closes a vital energy self-centered in a singularity that needs 3 apertures to the outer world, between the 3 diameters of a perfect ‘closed hexagonal pi’.

*S<=>** T:** S**ince time is dynamic, not static, the fact that there are infinite decimals to pi, always varying by approximation means also that a spiral never closes or opens in perfect fit, that it constantly moves between ±π states of ‘entropic expansion and implosive in-form-ation’ allowing the open-entropy/closed-information duality of membranes that enclose systems to feed and perceive in S<T>States and Stœps:*

**@:*** A pi enclosure, leaves a π-3d/π, 4% of ‘openings’ to the outer space’. Hence the singularity will see only a 4% of the outer world leaving 96% of dark space, not perceivable – not coincidentally the amount of dark matter that our electronic eyes cannot see beyond the galaxy, which likely will have a pi membrane or halo of dark matter that does not let us see the whole universe (proportion of unseen dark matter and energy which completes the organism of the cosmos).*

*Pentalogic then advices always to consider in the entangled universe of ∆@st of space-time to consider at least those 4 space, time, scale and mind views on any element we analyze. *

*Irrational ratios thus eliminate the view that lines are continuous, (Dedekind, etc.), a concept that corresponds to the limit of 5D fractal discontinuity with a lesser ‘perception’ of space, or the belief in a single spacetime scale. The truth is that is existence shows the opposite as the Greeks truly understood: continuity does NOT exist; in all planes there are ‘holes’ to lower and higher planes through which irrational numbers transit.* In other words, such ratios and S=t are ‘incommensurables’ as *continuity does not happen in a single plane of reality.*

As a consequence spatial, geometric and temporal numeric views are NOT exact identities but approximations, as a number is discrete, sequential and so it ‘lacks’ the continuous perception we have of geometric space.

*As numbers tend to operate on ‘several planes’ of existence, and points in a single plane, making them not identical but self-similar.*

Indeed, consider **t****he first algebraic proof of this essential ∆t vs. s quality** from the Book V of Euclid’s Elements: A square with sides of length 1 unit must, according to the Pythagorean theorem, have a diagonal d that satisfies the equation d^{2} = 1² + 1² = 2. Thus the diagonal can be expressed as the ratio of two integers, say p and q, and as p and q are relatively prime, with p > q, the ratio can be reduced to its simplest form: p²/q² = 2. Then p² = 2q², so p must be an even number, say 2r. Inserting 2r for p, we obtain q² = 2r², whence q must also be even, which contradicts the assumption that p and q have no common factor other than unity. Hence, no ratio of integers express the square root of 2. *But the proof has used a scalar dimension; by dividing and squaring systems in ∆±1; when the geometric figure exists only in ∆º. So there is a difference – a loss of information transiting scales. *

In the graph we make a clear illustration of that fact with another example of the non-existence of a perfect square numbers which can only be approached *from above or below in scalar power numbers algebraically but do have in a single plane a perfect closure geometrically; as graphically 2 x 5 ^{2}=7^{2} but numerically we miss a ‘whole 1’.*

Thus translating continuous systems space-like into discontinuous time-like numbers is not exact, neither translating S or T systems into ∆ Systems. S≈T≈∆ parameters might be similar but they are *never identities – the translation of one form into the other misses entropy or information, which will be very relevant for mathematical physics where neutrinos disappear in other scale with missing mass and death dissolves information never regained. *

Continuity and closure is a concept of geometry in a single plane not of algebra in multiple scales with motion. Continuity cannot be considered in time cycles and its mind representations, which are by definition discontinuous (±pi). And this implies real numbers do NOT exist in the plane of Euclidean ordinates, which is perceived in spatial continuity. The same is truth for √2, the diagonal, which is the other *canonical translation of a line into an open triangle, the bidimensional equivalent of the ±pi openings for a lineal entropic system (surface of maximal perimeter and minimal volume of **information).* In that regard, the proof that √2 does not exist means a diagonal never closes statically the triangle as a pi-circle never closes it.

**¬ Entropic view.** Finally to notice that even irrational numbers have limits. For example, in scalar ‘translations’, irrational numbers such as e, show patterns that break beyond ’10 decimal parts’, which happens to be the limits of the ‘planes’ of most systems. So happens with e: 2,71828182828…459 where the number breaks; and this is the ‘last’ ENTROPIC ¬ view to have the pentalogic analysis from the 5 ¬∆@st perspectives of an irrational number. **Only in Pi** we find NO patterns among the key irrational numbers that represent essential ‘functions’ and dimotions (e being the entropy dimotion itself), which means likely the scales of the Universe are infinite as the dimotion of worldcycles and perception has no limit of scales.

Hence the use of similarities, s≈t, instead of identities between time *discrete numbers and continuous mind spaces:*

** **** ∫@ (geometry) ≈ ∆ð (algebra)**

**RECAP**. Numbers are social gatherings of identical points, the most synoptic unit of mathematics. As such they can also express sequential series in time as an entangled element of ¬∆@st. Numbers though are dominant in ‘scalar’ social properties and sequential temporal causal properties, best to describe the external membrane of geometric discrete configurations of regular points=polygons; its ‘vital energy’ (physical parameters) or its time Dimotions in sequential forms (social and scalar reproduction), and in its more complex forms of algebra as all functions express Dimotions of space-time, become the unit of 3 growing scales: numbers->variables->Function(al)s. It follows that Numbers are more complete than points as time-motions are more fundamental than the spatial mental forms we make of them.

**∆§: Social scales of numbers**

Since if there are several scales of numbers, we cannot consider the decametric scale of human social numbers the whole, unless it ‘mean something special’, or else it would not be a language mirror of reality. What is special about our scale is that it actually does correspond experimentally to the social scales of the fifth dimension, which is made of 3 elements for each of the 3 physiological networks of all entities that give us 9 elements+ 1 that acts as the ‘whole’=10. It is not just as clueless people say, our scale because we have 10 fingers. We do have 10 fingers because the Universe work in decametric scales in the social dimension ∆§, which is the main function of numbers.

In the graph the fifth element in the center communicate 3 triads in the corners of a tetraktys the commonest form of social organization into larger scales.

Moreover as all systems live 2 worldcycles the 10^{th} logarithmic scale, becomes the transcendental number between planes. Thus between 2 such scales of the 5th dimension we shall always find around 10¹¹ elements – the ties of a DNA molecule, the stars of our galaxy, the cells of a super organism, the galaxies of the perceived Universe, and between the Universal Constants of Space, we will find the same differences, the energetic value between the H-Planck constant of ð in the ∆-1 quantum scale and the k-Boltzmann constant of the ∆-Thermodynamic, human scale; and then the µ-magnetic scale and so on. *For example in the human social super organism decametric scales (human societies) evolved mankind through the 3 first genetic scales, then 3 geographical, economic scales, and finally 3 verbal, ideological scales to get to the global super organism:*

10º, the individual of a given ‘previously described ∆-scale of the whole Universe…

through the 10¹ scale of the family (which can be a couple, a mononuclear family or most often one through 3 generations of grand-parents, parents and sons, of around 10 people) and then in genetic units, to the…

10² scale of ‘clans’… and 10³ scales or tribes (in the human system, which is taken here for the first ‘names’ of those scales.

Changing to a more complex language of social information, geographical, economical systems that create the next degree of 3 scales, the town, 10 up to 4, the city, 10 up to 5 and the nation, 10 up to 6…

- And then to ideological, verbal systems, which create cultures, till arriving to the – 1 billion, 10¹º-¹³ individuals, which normally emerges as a unit of the next scale. In the graph we can see the social evolution of the economic system both in the human organism and the organism of machines and chips – which has already reached the social number when ’emergence’ into a new plane, a global mind of internet takes place. In the left, we consider the human super organism of history, a smaller super organism of human beings, not yet emerged as a global single being, but coming closer to the 10¹º human beings, in which might emerge as the global super organism, consciousness of the life Earth.

**Pentalogic of scales as representation of the 5 Dimotions of reality.**

We can consider the secondary scales in relationship with the 5 Dimotions of reality, as ‘stages’ of reproductive and social *growth towards the fundamental decametric scale. Binary systems relate then to the 3 ^{rd} complementary, reproductive Dimotion, either as gender elements or as inverse 0-closed informative vs. |-open entropic numbers, the minimal duality to represent reality.*

*Ternary scales relate then to the more complex inverse Dimotions – the ‘triangular 2, 5 D entropic’ Locomotion since the triangle is the strongest form, spearhead of motion, and π-geometries to perceptive still 1Dimotions (3-6 number systems) Planar, 2D geometries (binary, quadrangular systems) and one can change into the other by rotation. While it gives origin to the 3×3+1 complex, fundamental 10D scaling. *

This duality extends to the 2 main geometric scales of numbers in open competition” the **0-6th is the fundamental scale of cyclical geometry**, whereas π=3, is the hexagonal perimeter of a ‘whole’ closed geometric space, competing with **the 10-scale, a perfect tetraktys geometry,** where the ‘whole’ is a lineal, triangular space.

They were adopted by the O vs. |, historic minds of Asian (Babylonian) vs. European (Greek) cultures of 6 vs. 10 scales.

**S-Numbers: Polygons as vital organisms**

Numbers are social gathering of indistinguishable forms. When studied in space thus numbers must have regular efficient configurations. So ∆§ numbers are 2D polygons or 3D Platonic solids.

Its importance in vital topology lays on the fact that polygons start the creation of superorganisms, with a membrane – the polygon proper, which closes a vital space and can by connection of points through lines-waves of communication, create a central singularity. Thus numbers as fractal points grow organisms. And this is self-evident in the study of Nature. We can then study together with entangled pentalogic both ‘scales’ of social numbers as ‘growth in time’ of polygons, efficient configurations in ‘surface’ worlds (land, water surfaces), and polyhedrons, which will be efficient configurations in a larger ∆+1 3D world (water depth, atmosphere, vacuum for atomic and molecular scales).

**The Decametric scale. Growth of Dimensions (d) and Dimotional functions from 1 to 10.**

Polygonal numbers are best to represent the social evolution of forms into new scales that emerge as envelopes into wholes. The immediate decametric scale of numbers give us then the perfect polygonal forms of bidimensional space:

**Nº 1: ***1**d: 1D¡: Perception §ð.** It *is the unit-whole, the unit circle performing, the perceptive 1 Dimotion, the fractal point It has 0 dimensions in Euclidean geometry but 1 dimension in 5D theory. Since fractal points have volume.

**Nº 2:*** 2d, 2D¡:Locomotion $t. *The line-wave of distance-motion made of fractal points communicated between two poles has 2 Dimensions. It is the reproductive couple. It forms the axial flow of any spherical informative topology. In Non-E geometry forms the antipodal points that a singularity reproduces to form its bilateral symmetry.

**Nº 3: ***2 **d, 3D¡: S=T.* Its dimotion is Reproduction. As a couple or antipodal mirror points forms a gender symmetry that creates the 3^{rd}, ‘son’ element in its middle point of communication.

**Triangle.*** 3d: S-st-T* As we already noticed synoptic numbers give birth to multiple paths in spatial geometry. So 3 has the first variation in the triangle, which has 3 Dimensions, as points have volume, and whose main Dimotion is to act as the spearhead of locomotion, with its leading vertex to perceive (1D) or break (5D), entropically the resistance of motion. As it can enclose a vital space and perform the 3 physiological functions with its specialized ternary points, the head, one back point to reproduce seeding information and the other inverse to expel entropic motion, we can already consider a triangle an autonomous organic system, the first whole without the need to perceive internal scalar parts.

So 3 is also the first classic 2 Dimensional geometry: The triangle, the perfect form of lineal-motion geometry. And from then on numbers multiply its ‘Dimotions’, variations of forms, and vital geometries. So we shall just consider a few details of a field which as all those of 5D mathematics is ginormous. I.e.:

**Nº 4: **3d, *4D¡§ð.*** ** The square’s main Dimotion is social Evolution, in 2 D by merging two triangles, as the most stable social evolutionary form to ‘fill’ a flat field. In 1D as the social evolution of two couples of 2 points.

**Nº 4: ***4d***, 4D¡ ***§ð .* Its main variation is the

*first 3-dimensional classic geometry, the tetrahedron (4dimensional in 5D formalism as each of its point has an inner volume).*

**Nº 5: ***5d*, Scalar Dimension¡ The Pentagon has a deserved mystical pedigree since polygons *finally enter the fifth dimension proper as * a pentagram reproduces with its internal diagonals a scalar inverted mirror form of itself. So we define a Pentagon *as the first form of the 4th/5th dimension that reproduces its very same form, by connection of its 5 points creating an internal smaller 5D social form, or alternatively it expands entropically by prolonging its sides creating the pentagram.*

**So with the pentagon we close the 5 Dimotions of existence, forming a pentalogic scale fundamental to 5D theory.**

**Let us then mention from 6 to 10 only the main dualities of numbers; its main Dimotion and Dimensional forms, going back to classic Euclidean dimensions, to make it simpler:**

**Nº 6: ***2d, 4D¡: social evolution.* The hexagon is the strongest form to cover a 2d world, as graphene shows, because it closes with ‘3’ diameters its perimeter. It is the perfect pi, which in physics implies a non-E geometry of maximal Planck mass. It creates also a highly connected • central singularity by joining opposite points. So its hard membrane and connected center forms the strongest organic system on 2 Dimensions.

**Nº 6: ***3d*, 3D¡ ST. In 3 Dimensions is the Octahedron, studied in non-E Geometry as the first reproductive bilateral S=T symmetry in 3 Dimensions (whereas the line is the mirror symmetry in 1 classic dimension and the 2 triangles of the square in 2 dimensions).

**S=T symmetries.** We can introduce at this stage the ‘Temporal symmetry of sequential numbers’, as *it is possible not only to describe as we do here in this introduction to 5D mathematics its simultaneous superorganisms in space, but analyze sequential numbers in time; as ‘years’ and ‘decades’ of a worldcycle; which also divides in 3×3=9 x 9 ‘life baguas’, but we leave for an enlarged future upgrading of this paper. As we want to race into the tetrarkys. *

**Nº 7… ***o*f the same family than the Pentagon, with similar properties, cannot be constructed only from the interaction of a perfect entropic |-line and O-cycle (compass and ruler), but needs *to bisect its angles/networks in 3 sub-angles; making more difficult its reproduction in our mental 4D ‘light spacetime’. But in a 5D Universe, *enough Regular heptagons can tile the hyperbolic plane, as shown in a Poincare disk model projection:

In the graph another key mirror symmetry between numbers and vital topology: §ð particle/heads of information are tiled by informative pentagons; flat Euclidean space is tiled by hexagonal forms of maximal entropic resistance; but the hyperbolic plane of body-waves of vital energy is tiled by heptagons… a theme of advanced 5D geometry…

**Nº 8: ***2d, 2ST.*** ** The octagon of the family of the hexagon, with similar properties is a bridge *towards the infinite points of the circle; fairly redundant, in form… as numbers in 1-2d become now more important as ST-systems of timespace motion; where the 8 plays the role of the final life-cycle. It is in the time the closure of existence – the final state of decay; for each ‘series’ of time-space systems – from the human ages in its 2 scales of life (80 years) and civilizations (800 year cycles of cultures); and have the same role for machines (stock cycles) and physical systems (over 80 z, elements become radioactive). So* as *the fundamental series of spatial geometries closes with the heptagon, the main symmetry of higher numbers belongs to time ages. *

**Nº 8: ***3d, 3§ð.*** ***So 8 belongs properly to the geometry to 3 D platonic solids, considered next, where 8 is* the cube.

**Nº 9-10-11 tetrarkys form – transcendental triad of perfect whole**: **the ‘closure’ of an organic geometry, **

*2d $t-1D¡***: ***Tetraktys* in which *the 3 networks within the system form the 3 surface angles of the triangle, playing the specialized roles of the 3 physiological networks of motion < iteration > information of the system. But 9 has NOT A POINT on the singularity center; it is a mindless system, which requires the 10th point forming finally a **tetraktys, the sacred **number of Pythagorism and Hebrew geomancy (graph):*

* It has* a singularity, a black ball that communicates all other physiological triangles, becoming the inner mind/heart of the organism, doubling as 11^{th}, since it regulates the inner body connecting its internal part but also becomes its antipodal ’emerging’ point in a higher ∆+1 scale; starting a new game. So 10-11 completes the symmetry between time ages, worldcycles, dimensions and spatial forms, becoming a new unit of a new logarithmic social whole of 10^{10} cycles of time an population.

Such polygonal numbers will grow from the 1 seed to the 9+1=10^{th} most efficient scale, through ternary and dual symmetries, which in existential algebra might then loose the ideal form. I.e. a human being, has a tetrarkys of 10-physiological system (3 digestive, 3 reproductive-blood and 3 sensorial systems fusion by the ‘whole’ brain system.**.**

Pythagoras as Plato latter said that numbers are forms, as they were in the earlier age of mathematical geometry, where a number was a group of points, whose *form mattered. So HE REALIZED 10 was the perfect number, because of its perfect form, which in fact becomes the 11th dimension of a new ‘whole’.*

And indeed the internal structure of any being reaches its perfect efficiency with a 3 x 3 +0-mind symmetry of form and function; where each part-number performs one of the 3 physiological entropy, energy, information jobs of the system and the central mind-number in contact with them all coordinates its functions.

So we also talk of 10 inner dimensions or ‘sub-systems’, represented by a tetraktys:

In the graph, each 3 corners are sub-systems of ‘Information, entropy/motion and Energy/reproduction’ put together by a central 10th dimension (the black ball/hole/point/knot that messes with all of them). Indeed the central point of the ideal tetraktys communicates with all the other parts and embodies the whole that ’emerges’ as a point in a higher ∆+1 world. Thus we talk of the ‘subsystems’ of a being.

For example, a human being is defined in medicine as a system of cells, attached by 10 sub-systems:

In the graph upper left 3 ST-ructural human body sub-systems are its membrane, sustain and motion.

Bottom 3 ‘chemical Tiƒ’ systems or hormonal brain (creative, distributive and reproductive). And the nervous system singularity-mind… of the whole. Those graphs show that as we grow in planes, the ideal geometry of the lower ‘atomic planes’ disappear, as long as the ‘logic concepts behind it’ – hyperbolic fractal body-waves branching, bilateral symmetry, etc, remain. Since function is more information than form.

So the convoluted bilateral networks that connect the singularity brain with all its antipodal points of elliptic geometry have the same function that an antipodal ‘representation of a non-Euclidean’ sphere.

Yet in the human organism, the lines that connect them is not MADE OF straight lines but it does work because what matters here is the symmetric territorial order of the singularity which constructs its membrane with opposite ‘rays’/nervous lines and will constantly balance and hence act as a leverage with its ±inverse directions for its antipodal elements, two hands, two kidneys, and so on.

Morphology then starts with simple laws of Non-Euclidean topology which become disguised by the adaptation of each function to the available space within a membrane, as the T.œ adapts to its outer world.

The 3×3+1 physiological systems of the human being, lost its ‘ideal’ mathematical symmetry, translate to ‘existential algebra’ the concepts of decametric scales. They can be easily subdivided in 3×3+1 Nervous integrative mind system, but also in the fractal entangled Universe, into 5 x 2, positive and negative dimotions. For example, the entropic system branches in 3 the digestive, respiratory and excretory systems, which feed the Dimotion of locomotion, sustained by the Muscular and skeletal system; the reproductive ST system branches in 3, the blood, circulatory, reproductive and excretory systems; and the informative system, in 3 Nervous, lymphatic and endocrine systems, whereas the Nervous system doubles as the one site of consciousness, origin of the Dimotion of perception, and social evolution of the whole.

Physiological networks then belong to a different branch of geometry, fractal networks, which are more efficient in the distribution of the 5 dimotions of exist¡ence between parts and wholes of the ∆±1 scales and so won’t be studied here in Number theory. Instead we shall conclude the geometric analysis with the 5 regular solids whose obvious symmetry with the 5 Dimotions of existence, means each of them specializes in 1,2,3,4,5D, in parallel to the simplex forms and mirror symmetries that compose them:

**POLIHEDRONS**

3D platonic solids are 4, 6, 8, the only decametric regular numbers in 3 dimensions. Then we have only two more in the 20 scale, 12 and 20 vertices, and that’s all!!!

They play each one a role as actors of one of the 5th dimotions – the decametric 3 completing a 4,6,8 Fractal generator:

$t-tetrahedron (legs)<S=T-Octahedron (body) > §ð-Cube (head)

Indeed, the tetrahedron is basically a triangle in 3D, acting as the spearhead of a locomotion; which in timespace is traced by lineal members moving on steps as your legs do. Hence its 2D¡=$t function. The Octahedron has a clear mirror symmetry as all reproductive dual genders do – already posted in academia.edu its paper; and the cube, is the form of the 3 closer to a filling sphere with a well-connected singularity of maximal symmetry with the outer world.

So this leaves the two complex dimotions of social evolution and entropy for the triangular icosahedrons, *with maximal number of social triangular units (20 faces of the simplest 2D surface)*. And the 5^{th} dimotion of entropy for the pentagonal dodecahedron… Alas again the pentagonal evil Death symbol again… Why? That I won’t tell…

Just quote the master of ‘transcendental philosophical geometry’, Mr. Plato who affirmed that, “The Universe is the body of an organism whose logic mind we call God.’ & ‘God used a dodecahedron to design the Universe’ *Plato*

Ok, I’ll tell. Think in vital terms. Imagine a Universe with those 5 species, no curves. The 3 first are the smallish ones. Cubes perceive, and keep quiet, hidden, filling easily empty space; its survival strategy – social growth, with its identical forms, repeating by the mere motion as lines first, flat squares and giant 3D cubes. It is a single gender ‘feminine’, informative reproduction. The other series does the same but with mirror gender symmetry, as the triangles of tetrahedrons expand from 3 to 4 and peg inversely into octahedrons and then grow into icosahedrons.

But the dodecahedron is NOT concerned with the survival strategy of reproduction. He rather eats them all!!

How is this? Easy, the dodecahedron has the largest mouth and the largest volume, even larger than the icosahedron, so it swallows them, in an action of vital geometric feeding; and once caged inside, remember pentagons are the ‘first’ polygons which can do a smaller replica of itself, NOT a perceptive singularity but a *crunching stomach*.

Each point is a cell of a dodecahedron. As it happens there are as many faces in the icosahedron to eat as vertices in the dodecahedron that feeds on it – both connected through the golden ratio of feeding and reproductive events, a+b (feeding)/a = a/b; a key relation of top predator/prey forms. And so it opens its mouth, swallows it, cages it, crunches it and burps (; But the pentalogic Universe allows different dimotions. So now imagine instead a reproductive event. It is not far fetched. Biologists consider that ovules first ate sperm cells and latter they learned to merge their genes.

So once a ‘male’ entropic dodecahedron has eaten the ‘female’, reproductive icosahedron it can actually merge into a mixed form, by merely attaching to each dodecahedron point a triangular face, becoming the icosidodecahedron – a classic of Geometry, we shall call ‘dodecosahedron’ for brevity – a new species which is more perfect than its parental ones, as it shares the information of both. It is a first gender case in ideal geometry, since *Gender reproduction happens in all scales of stience,* from the simplest circular ovum swallowing a lineal seminal seed – a cyclical and lineal form, merging into an |xO=ø topology, to the previous example of ilogic mathematics.

The dodecahedron social points, aka numbers feed on the icosahedron faces, reducing them either by swallowing them, which will result in a growth of size of the dodecahedron or by merging into a reproductive evolutionary process far more interesting, as any merged system is more efficient than its 2 separated parts-species, a more evolved form, which survives better in a process similar to all genetic-memetic-reproductive processes, that create in any stientific scale more efficient systems by communication.

Indeed, the dodecosahedron is the top predator polyhedron form, as it has ALL the ilogic elements and symmetries that enhance survival in the fractal Timespace Universe: individual ego-centered points, shaping inner trilogic systems in yellow, tetralogic dual triangles married to dual pentagons, decalogic reproductive bilateral social parts, plus all the magic stable numbers of vertices, edges, faces & diagonals, despite its external ‘soft membrane’ –unlike its ‘dual figures’ the Catalan solids that *show a defensive external membrane*, in which the swallowed form emerges into convex ‘teeth’, but has a less efficient internal organic structure:

In geometry, we say that any polyhedron is associated with a second **dual** figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. The graph is an example from the other dual top predator/prey series, a cube of larger ‘mouths’ and bigger volume-spatial efficiency swallowing an smaller octahedron of lesser volume.

What differs now is an added property – a topologic ‘defensive’ concave and convex dual form which can establish as Vaughan found in defensive flat geometries for the French army, entropic expansive concave points and smaller implosive, informative sinks, which maximize the entropic reach of cannonballs with minimal surface exposure to the external army. Indeed, a ‘mathematical’ ideal duality, we shall find latter on in the stience of biology between top predators of soft membrane but much more efficient inner structure and defensive armored animals with less evolved internal networks (squids, with eye-perception and evolved nervous systems vs. armored trilobites, simpler organisms; Tyrannosaur Rex vs. armored triceratops, Lions vs. armored, skin thick, horned herbivores) and social systems (defensive walled cities vs. nomadic horse armies).

We are all VITAL geometries and so the rules of geometry carry from ¡logic spacetime to ilogic mathematics, to the different stiences of vital reality where those forms acquire topological motion. The specific case of polyhedrons being of special interest to molecular chemistry, which we shall study in detail when upgrading to 5D the discipline (as those are processes that do happen in solid crystals).

Finally to notice that the dodecahedron is generated (Euclid) from the cube, *by covering it as a parasitic membrane, unlike the 3 other species, which can be considered to reproduce the same form. *All this might seem trivial to mathematicians, but they would miss the point of what 5D mathematics tries to show – the vital nature of space… which must be explained as the ultimate formal substance of reality, having organic living properties, simplified into a still mirror, NOT the other way around.

**Hyper-dimensional polygons.**

As S=T 5D metrics implies a dimension of space is another form to interpret a motion in time, an important field of 5D geometry is the study of 4D (2S=2T) regular polytopes, which close the entangled scales of regular social numbers as T-Dimotions or S-till elements that ‘construct’ the real structures of the Universe, to extract 5D¡somorphic laws; even if we do NOT see static fixed 4D forms in our Universe (as 5D includes entropy, which is the negative destruction of all others, 4D is the limit of constructive engagement, often of 2 still polygons with 2 motions=degrees of freedom). We just make some key statements on this discipline:

With the simplification of the inner volume of fractal points, we can match polytope laws with the symmetries of Spacetime beings. So 1D polygon of points with no parts -open lines equivalent to 1D cylindric limbs/fields are ∞ (Natural Numbers); 2D polygons akin to cyclical, O-particles/heads are ∞ in flat space, *& 3 hyperbolic polygons, equivalent to Ø-body-wave reproductive systems, intersection of 2D lineal & cyclic forms are ∞ in 3D space.*

*So there are ∞ growing ensembles of 1D limbs/fields + 2D particle/heads=3D hyperbolic body-waves.*

Growth of complex finite polytopes peaks in 4D, as further increases of dimensions reduce them to the ternary ‘single plane’ structure of a triangular, square and pentagonal ternary ensemble; *which means there are no more dimensions beyond the 4D 2S=2T systems, and all further ‘Dimensional growth’ is a dimension on new scales of 5D which are memoriless, erasing the previous emergent complexity of the fractal point, which stays on its inner parts. (Because the hyperbolic body-system is more complex being its minimal form 3D, its ternary growth peaks at 5D).*

In the side the existence of a regular 4-polytope is constrained by the existence of the regular polyhedral which form its cells and a dihedral angle constraint sin π /p sin π / r < cos π /q to ensure that the cells meet to form a closed 3-surface. And yet despite those restrictions we have 6 polytopes. And this matter because our world as in relativity can be analyzed in a single plane as a 4D ensemble of its 3 classic dimensions and a time motion of social organization, as *the 2 inverse dimotions of entropy and information do NOT happen within the same event simultaneously in ‘regular entagnelments’. *So pentagonal polytopes make no sense, being the 5^{th} dimotion entropy the destruction of the regular in-form-ation shared by them all.* Thus the most complex 4D structure of regular points are the 120-600 dual pentagonal mirror ‘gender symmetry’, *reached through combinations of 5 Dimotions, 6 ‘triangular radians’ of a π sphere (µ polytope) & its 120 angles. The result are the final stable 120 dodecaplex and 600 tetraplex maximal figures of 4Dimensional geometries. And we shall find this 120 magic final number in the few cases te system is perect enough to transcend the octaplex to be the real final ‘island of stability’ in atomic systems or oldest possible age in the very few cases the system transcends the 80 years ‘radioactive barrier’. And we can imagine the 120-600 vertex entanglement the limit of ¡logic evolution of a complex social system in the Universe.

But all those polytopes must be understood as ternary systems which reproduce in motion forming a larger 5D scale; so 4D is a motion, and that is indeed how geometers construct them, bringing to each 3D point an internal volume; that is, making each vortex of a 3D polygon a new ‘square’ to form the tesseract which is equivalent to ‘enlarge’ its inner form. While another 4D polytope will be merely the motion of the square that becomes a larger ‘1D’ line in a larger world, as we make a 2d plane by moving across a 1D line. What about the sphere? It is the ultimate regular polytope and so we wonder how a 4D sphere looks like. The answer is also telling of the 5D structure of the Universe:

As we can only see it, as it moves through 3D ‘cuts’ of the sphere, which mathematicians model on the ‘view’ of a 3D sphere entering a 2D world. And so as a moving 3D sphere in our 2D world appears as a growing and diminishing cycle, a 4D sphere in a 3D world will appear as a growing and diminishing sphere, hence strongly suggesting that indeed a 4-5D Universe is one where beyond our perceivable entangled 3D holographic parts, all further dimensions are motions of time (4D, 5D inverse, SS or TT dimotions) which become ‘social forms of growth’ that loose dimensions in the same 5D scale (cube moving as a squared line, sphere turning into a dual spiral diminishing and growing to form a larger flat disk, etc.) Or if they are ‘static in the same position’, they appear as ‘scalar travels through the fifth dimension’ growing its inner parts (as we draw in each vortex a tetrahedron, cube or dodecahedron), or growing and diminishing in scale (n-D spheres).

This doesn’t mean they are useful to model and solve mathematical problems of the formalism of 4D where the time motion is converted into a geometric still dimension, allowing 4D polytopes to represent real T.œs

** **

** **

**PRIME, ODD AND EVEN ‘GENDER SYMMETRY’ NUMBERS**.

If we extend this vital analysis of regular numbers in space to sequential numbers in time, we come up with another ternary division of all numbers within the finite limits of a given ecosystem (infinity being a paralogic Kantian error of the mind as all planes have a limit of numerical form, normally as explained in the 10-11^{10-11} range, from ties of DNA in a human gene to stars in a galaxy)… that between top predator prime numbers, which generate all other, even, social reproductive, ‘female’ numbers, the majority of them in a Universe that systematically favors ‘reproductive S=T’ states as all fractal generators do, over even, informative + entropic ‘male numbers’, which can be divided in a minority of ‘prime numbers’ that carry the information of reality as they can generate all other numbers in 4-Dimensional sums; and entropic, unstable forms that code no new information (odd numbers excluded primes, which generate them).

So we find yet again another fundamental division of species, according to the key variation of the ‘Fractal Generator equation of space-time or 5D metric: SxT_{s=t }= C.

Indeed, it is the same duality of ‘Female’ present reproductive, ‘even’ gender, S=T vs. ‘male’ past to future to past, entropic and informative gender, S<T>S: SxT, the key temporal ternary symmetries of the Universe; between present dominant states, and past to future to past cycles, which requires a full comprehension of the 3 ages of time, and its ‘complex ternary logic’ sketched in the introduction, that we develop in the paper on ‘5 D time –pentalogic’.

Essentially all cyclical time systems have 3 ages, a relative entropic past of dissolution, a reproductive dominant present and an informative future. And the proportion in any system is ½ present + ½ (Past>Future ) that closes the cycle. For example, in quantum physics wave states are the present state; particles the future state and fields the past state. In matter, liquid is the present state, gas the past state and solid the future state. In cells, DNA is the future state, cytoplasm the present state and the protein membrane the energetic state. Its S=T symmetry makes the past less evolved, more extended in space, the present balanced, the future more informative. In human gender, the woman is the reproductive state, and males divide into a majority of entropic destructive and a minority of highly informative (complex IQ test put woman in the middle, a minority of men above a majority in the entropic lower range).

So to understand the ¡logic division of the 3 ages of time into past=entropy, S<T, which dissolves systems into a lower ¡-1 plane, present=reproduction, S=T, that iterates reality which seems not to change, and future=informative evolutionary system, T>S, which increase its form into the future… I refer the reader to the article on 5D pentalogic.

## In terms of its scalar social organization, the 3 ‘different’ groups of T>S informative, S=T, energetic and S<T entropic elements of a class structure correspond to the informative class (i.e. stocrkats in our society ruled by money, politicians in a real democracy, military in a dictatorship) which controls the language of power of society, while S=T form the body-wave of the system that reproduces it and it is taken care by the informative particle-head, while the S<T element is the entropic class easily ‘divided’ and broken, as energy of the others, in this case the odds that can be broken in smaller factors. And as we shall see the game of numbers have a remarkable proportion of the two first classes as opposed to our ill-designed systems, a theme treated in 5D politics, where we study the informative ‘nervous’ network of human superorganisms.

## While the general study of those 3 states belongs to those 2 articles, in the mathematical mirror the division is self-evident as we already observed some clear differences between odd and even numbers/polygons:

**Odd vs. Even polygons** alternate with a key different property: The diagonals of even polygons create central singularities and form more stable systems, tending to static states, ‘looking inwards’. They can reproduce by motion into equal forms in the same plane, and can easily tile, filling up the entire plane. While odd numbers penetrate in the scales of the fifth dimension by reproducing fractal mirrors of smaller size, when crossing inwards its diagonals (informative reproduction) or larger mirror images when crossing outwards its diagonals.

Those properties carry through all scales of gender mirror symmetry; as 1 element is sterile, self-centered making different the 2 type of organic membrane-polygons. Since S=T even polygons do NOT reproduce through 5D motions as odd numbers, but within the same present plane of existence as they are ‘self-centered’ in a single point, within the polygon, NOT replicating without motion. Reproduction of even polygons happens by translation in space, as waves reproduce. So by moving through a tiling of present space a square reproduces filling the tiling, unlike a pentagon or odd polygon that reproduces internally and externally as its motion filling space is often impossible

**Prime numbers as informative systems.** In ‘vital mathematics’ as in classic maths, Prime numbers play a fundamental role in number theory because of a basic theorem:

Every integer n > 1 may be represented as product of prime numbers (with possible repetition of factors), i.e., in the form:

where p1 < p2 < . . . < pk are primes and a1, a2, . . ., ak are integers not less than one. Furthermore, the representation of any nº in this form is unique.

*This implies that Primes are the generational space-time numbers of all others; hence the minimal numerical system with maximal information. And inversely if we reduce entropically a system of numbers, by ‘division’ (the operand of entropic dissolution)* o*nly prime numbers or its social combination DO survive=exist in the ultimate skeleton of reality, due to its efficiency as polygonal forms. *

*Everything else is a virtual society of prime numbers; and the strongest social groups are those under 23, stuffed with primes and displaying a solid even membrane.*

**Prime numbers in time sequences: stability of systems.**

A prime number is defined as any integer (greater than 1) that has only the 2 positive integer divisors, one and itself.

1 is not considered a prime number since it does not have 2 different positive divisors. But it was so rightly in the past –as we might consider that 1 *is the super-prime, the whole that divides all other numbers of nature. *

And the fundamental characteristic that differentiates them is that as polygonal numbers grow in sides, that is the social number of ‘points-cells’ increases the system become less stable, because *it increases the number of redundant non-prime odds (ab. odds). *And when we carry the symmetry S=T, to temporal sequences which as we have seen is the proper way to study numbers beyond the decametric scale, this lack of stability increases enormously past the 23 ‘years’ (we shall call points=numbers in time sequences ‘years’); which is the ‘cut’ between the young age of maximal energy and the mature age; and again plunge past the 83 number, leaving a mean of 1 primer for ‘decade’, which is indeed the standard age for death, both in human years, or in S=T atomic evolution (radioactive atoms past lead).

*We talk then of each prime number as a key ‘age’, and 23, the PRIME age of a 100 normal system, where a transition to the mature age takes place. This we know about human life – the end of University, the beginning of work, the maximal force and energy for most athletes in the table of elements, it gives way to the family of Iron-Nickel-Cobalt which are the most entropic-energetic system, (26 iron) ending in Cobalt the first ‘radioactive’, transition element that ends the series.*

Those are the themes of ‘vital mathematics’, which the synoptic mirror of numbers reflect from the configuration of social numbers in space and time, and we shall just study in a bit of more detail.

Still prime numbers are infinite and relatively abundant according to the π=n/log n rule, so huge social systems are possible both in space and time duration as complex new primes, can reinforce the structure as nodal points of stable networks. A prime number by definition is more stable as a configuration of social forms, as it *can’t split into equal parts*.

How far can we stretch this stability? In inflationary mathematics to infinity. But in scalar space-time infinity becomes increasingly unstable as prime numbers dwindle.

As the irregular infinity of *the number of entropy, e, which breaks after 10 decimals:**, (*the 1+8=9; 2+8=10 regularity disappears after 10 numbers if we consider 2.7 the first dual term of the regular pattern), loosing its structural, informative meaning’, we should consider that beyond the ± *10-11 ^{10-11} social scale between planes, gathering in 10±1 decametric scales, the systems of ‘big number theory’ are largely irrelevant. *

In any case prime numbers clearly dwindle (as per Euler’s and Riemann’s functions in the graphs) beyond the 10.000 social group, suffering a steep decline towards the 10^{11 }mark in a tail proper of an almost ‘extinct’ species. So larger primes are rarefied and very few complex structural systems beyond mere herding exist past that number:

In graph, number of primes holds remarkably dense in the basic 100 scale being 1/2th of the 10^{th} scale (if we consider 1 a prime number) and 1/4th in the 100 scale, which are the key scales that encode most stable members of timespace both in the space symmetry (table of elements) and the age symmetry (life span in years.

The ratio dwindles to 1/6th out of 1000 and 1/8th for 10.000 in the next key interval of social groups, still fairly strong, if we take into account that the 10% is a very strong ‘leading’ elite in organic class structures of any stientific scale… and organism. This 1/10^{th} informative class structure still holds in the 1 million social group (1/12th). So the Universe is strongly connected and stable in its hierarchy of prime informative numbers on top of reproductive, energy, even numbers 1/2 with the entropic ‘easy to break’ odd numbers, never a majority of the ‘ternary class system’.

Compare it with our present societies – corrupted placebo democracies ruled by money issued in near monopoly by ‘stockrats’, modern aristocrats of the capitalist society, owners of corporations, which are less than the in famous 1%, closer to a 0.01% of owners of corporations. This tiny proportion of ‘informative class’ coding the language of social power doesn’t even happen for the 10^{12} limit of most complex superorganisms of the Universe where primes are still a 4%.

**The 20 ^{th}, reproductive mirror symmetry of the 10^{th} scale, & the 100^{th} time aging scales.**

It is then only worth for an introduction to Number theory to study just the 10^{th}, 20^{th} and 100^{th} scales. The first purely ‘spatial’ the second which ‘doubles’ the first in a mirror symmetry is the reproductive S=T scale of numbers, with the equivalent 4 prime proportion; while the 1 to 100 scale is the Temporal symmetry of ages; and as such we shall relate them in other posts as we ‘ascend’ the scales of reality and the laws easier to understand in the ‘synoptic’ languages that mirror best the Universe’s time and space, ‘¡logic’ and Non-E mathematics’ emerge with different ‘forms’ in those upper stientific scales.

We shall not do it here, for the sake of time and space, as we are still in the 1^{st} age of mathematics, with the symmetry of points and numbers. Only to notice that for the reproductive mirror scale of 20, closed by its largest prime, number, 19, its square, which doubles its dimension 19º x 19º= 361º gives us the commonest ‘closure’ of a sphere in space (360 being the scale of degrees) and time (365 being the days on a year)…

And so we have another curious number 361º ‘degrees’ for the perception of angular rotation, *(as we know geometry is always approached in its figure by discrete number above or below ±1), which suggest the natural limit for a discrete polygon of cyclical nature.*

*The vital nature of the *10 to 20 scale as a mirror of the 10D basic scaling for reproductive purposes, emerges as usual with small variations in more complex stientific scales. So it shows in the game of life, where 20 are the number of amino acids that construct most organisms and your body can not live without; … as they generate everything else in the upper scales. Even in the most ‘distant’ level of human languages, the 20 scale manifest itself; so most alphabets have 20 some symbols to write the language; which then generate a larger number of phonemes broken in vowels of information and energetic consonants (more abundant. I.e. the English alphabet has 26 letters, split in a total of 20 vowels and 24 consonants.

**RECAP**. The first prime numbers are: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29…

And so we see right there that while the decametric and 10-20 scale are buttressed with 4 of them each, almost 1/2, which *explains further why they are really the 20 numbers that matter in almost every system of reality, being the 10 to 20 the mirror symmetry of the 10 scale; which you might extend to the 23 range… beyond those ‘efficient numbers’ systems diminish in stability.*

So in the table of elements, where the strength of the system is a mixture of its ‘even configuration in 2 and 4 pairs’ and odd volume of inner layers, the range around 22-26 (iron) give us the strongest top predator atoms of maximal energy, which is the role of a polygon number that closes as the membrane of vital energy any T.œ. system in nature.

Yet if we consider the temporal sequence the same series, breaks the first age of life, which in a biological human being ends at 24 years: 1-24: youth, 24-48: maturity, 48-72: 3rd age; 72-80 death. And indeed prime numbers plummet after 83. So this will be indeed the fundamental scaling for all timespace life-death symmetries (as we shall show for each stience, from the amino acid series to the life cycles, from the atomic table to the cycles of historic and economics which fit within those decametric 5D scales).

So far we studied only an ideal form of a number as a point, a knot or lattice, which is what in reality most atomic numbers are placed into. Numbers as forms do have a structure that makes them both space and time, S≈ð, But numbers as mirrors of the Universe have as words in verbal languages, pixels of colors in images, curved lines in geometry, 5D smells in noses, multiple relationships with that kaleidoscopic 5D reality. And so we must advance on the multiplicity of Numbers under the 5D ‘Rashomon effect’, both in its structures – number within numbers and its solitude, numbers as social groups alone. The previous analysis of pentalogic pi numbers reflects the deep Entangled Nature of the Universe, where each entity, while *dominant in one of the 5 elements of reality, in as much as it performs only a function on the upper ∆+1 world in which it survives, it will as a whole in its inner reality be a full ¬∆@st being. So while we *shall always stress a single function. I.e. a number will be a scalar social gathering of identical beings, there are for every entity 5 possible perceptions of the being.

I.e. the number will be a rational ‘entropic division’, a ± dual sequential time motion and its inverse which close into a zero sum cycle; and a spatial configuration of identical points. And it will even transcend into a new plane of existence through complex numbers. We are all entangled and so all participates of other elements.

**S<=>****T DIMOTIONS: NUMBERS AS UNIVERSAL CONSTANTS**

As Einstein wanted the ultimate principles of reality are Universal ratios/constants, which represent the 5 dimotions of all Universal systems*. So we shall find 4+1* absolute constant ratios in all systems, *of Nature, of the vital Universe: e pi, e, 10=3×3+1 and phi.* *What about locomotion? In 5D physics locomotion is reproduction of form, from particle to wave state to particle. So it has not a fundamental constant but a mixture of the informative, reproductive ones, pi and phi.* *5D thus use those 5 constants to explain how the Universe reproduces reality departing from those ‘perfect ratios of energy and information’. * 3 are irrational≈ratio numbers that represent the 3 ages of all worldcycles of exist¡ence:

10 is the seminal social number of palingenetic ∆-1 emergence, π, is the number of the worldcycle, of the 1st Dimotion of perception; phi is the number of the 3rd Dimotion of reproduction, in the second ‘age’ or maturity of the world cycle, and e is the number of decay and entropy, of the fifth Dimotion of existence.

**1D: Pi transforms a spatial field of entropy into a vortex of temporal information***: 3D ($t) > ∏ ðƒ.*

** **As we have already talked of pi and will keep adding latter, just a note: pi’s informative nature is immediate, as we perceive through angular trigonometry, when pi closes a circle of 3 Diameters, with 4%=π-3/π of perceptive holes through which 3 sensorial apertures to the Universe happen and 96% of dark matter hidden by the membrane (which not coincidentally is the part of the Universe hidden by the galactic halo we don’t see).

**3D: Phi is the ratio of reproduction of a space-time population. **So applying pentalogic p.o.v.s:

**T p.o.v.**: Phi is approached by Fibonacci series in gender couples, which reproduce in the 2nd age of maturity, and die after the 3^{rd} age. So for ‘2 consecutive quanta of time’, the young still won’t reproduce, the old won’t be still dead and the mature will reproduce giving the series 1,1,2,3,5… that approaches the phi ratio. Thus in 5D symmetries the relationship between phi and Fibonacci numbers (reproduction series) is an S=T* duality of the constant observed in space (golden ratio) and in time – Fibonacci closed solution as a series:*

Whereas the golden ratio becomes an iterative process of generation.

**S=T: Population growth:** This leads to its understanding in ‘still space’, as a population growth:

The golden spiral then becomes the natural ‘time-series’ for filling a surface of energy where to growth in a frequency of space-time generations, given by the ratio – even if the ‘interior of the spiral’, fades away as generations die behind us, leaving a void.

**S: Informative evolution:** Its inverted implosive ‘symmetry’ as informative growth (T>S), gives a proportion for the reproduction of complex information, (as in finger proportions, when form overcomes size).

**∆**: Phi is also a scale function: the whole (trinity) is to the larger part (body) as the larger part (body) is to the smaller one (head), since reproduction is a scalar function that imprints an ∆-1 ‘cellular’ scale to then evolve it socially into a new whole.

**¬ Entropic function: – the beauty of its S≤≥T proportions.**

Yet phi has also a reputation for beauty in proportions between the 3 parts of the being, limbs>body-waves>head-particles, which is the natural outcome of its multiple functions as reproduction and organization of the different parts of the being in 1 and 2 Dimotions:

Whereas phi is the golden ratio of proportions of the whole to the 2 body-head topological parts of the being. Expressly is the ratio of the whole a+b to the body, when the body-wave is proportional to the particle-head: T.œ (ST+St) / ST = ST / St.

But a ‘line’ has ‘motion’ to become a real ST system. So the proportion must be interpreted adding the hidden variable (the speed of processing information/energy of both b and a). Then it corresponds to 5D metric: S (A+B) x ð(a+b) = S(A) x ð(a) = S (b) x ð (b), as S(A+B) > S(A) > S (b) in the same proportion that ð(a+b) < ð(a) < ð(b). So we can *extract phi from 5D metric (Sxð=K).*

Its beauty is then the perception of the Universal Metrics of 5D which in turns gives survival efficiency of such ratio, as the body transforms the entropic energy for the whole in a proportional balanced way.

Further on it can be expressed in terms of entropic feeding completing the pentalogic of Phi as a ternary system: whereas a+b is the limb/field. So we write:

$t(a+b) < ST(a) > §ð(b)

Whereas the ‘entropic length’ of the limbs ideally equals the sum of the body and head system, while inversely the head of information consumes more, faster energy than the body (i.e. neurons consume 10 times more energy than body) whose cells reproduced in the wide dimension consume as much energy as the locomotion limb/field.

And so the symmetry if we transform ‘T into S’ give us the same proportion for its square areas.

So phi is beautiful as beauty, is the proportional vital constant balance of the 3 St, ST, sT parts, the key to a T.œ.s survival (S=T, harmony of ‘size and form’, ‘energy and information’ ‘body and mind’).

**RECAP.** The golden ratio is the most efficient form of inner growth and proportion, keeping the balance and distribution of information in a system.

**2 ^{nd} Dimotion: locomotion: It combines with pi to gives the locomotion number,** as motion is reproduction of a pi cycle of information along an adjacent series of forms. We saw the case for a cyclical reproductive spiral, and in the 3rd graph we see the case for a lineal reproductive series, which becomes real in quantum particle-waves.

**4D:4 ^{th} dimotion. Social scales. 10 is the constant of scale,** of social evolution; hence the main system of scalar numbers.

**5 ^{th} Dimotion of entropy: Finally the exponential function**, is the most efficient form of social growth between scales of the fifth dimension. In the graph, we can see how the growth of any system in base e is the maximal possible. But such exponential 4D social growth can only happen in a placental unlimited world of free energy. So its true use is for the inverse arrow of death and decay, when it is the maximal speed at which the cells that grew in the placental ∆+1 palingenetic first age, now die in the final ∆-¡ age of entropic decay, adding together to a 0-sum. So e is the exponential function that grows faster:

*e ^{x} is the only function, which is a derivative of itself*, adding constantly a 1/n finitesimal derivative ‘cell’, in each time step, but also decaying in the fastest possible path of dying quanta

*. e*

^{x}represents the fundamental function of reproduction of any system in its 1^{st}world cycle of a perfect energy placenta, þ, at the lower ‘cellular scale’, as the system increases by a derivative at each step.**RECAP.** e represents the absorption of a reproduced finitesimal part, or transfer of a quanta of energy to the larger whole. But it requires a ‘perfect energy’ system. In physical systems this happens in decay-radiation; in biology only in the placental cycle; in sociology in the brutal capitalist system that expects interest to be added as the Maximal exploitation possible.

**Universal ratios=constants & the generator. **

Pi, 10 and e, and φ, are the ratios of Ts<ST>St transformation events. So they are Nature’s constants, related to its Dimotions which we can write as partial equations of the generator:

π: 3 |-Ts > O-St; e: ∇1 e->∆1+∆º; φ: T.œ/TS = TS/S; 10 §º ≈ §¹.

They also combine to mirror complex, more efficient dimotions. For example, the closed curve described by π and phi as a 137° golden angle – which allows the maximal covering of a surface from an outer point of view, as a flat, filled space – so it maximizes reproduction, the function of existence, and entropic feeding; as when light in plants can touch most ‘leaves’ which therefore are positioned in a phi angle. *The same maximal reproductive function of existence happens in *open curves, of asymptotic hyperbolic form, where e is the fundamental ‘number’ that maximizes both reproductive growth and decay. So ultimately the 3 fundamental irrational numbers *maximize the function of existence, in its «entropic = reproductive » Social evolving symmetry*

**∞Transmission of information.**

In terms of scales, pi is related to ∆º+1 (perception, social evolution), Phi to ∆+1 (reproduction and social evolution) and e to ∆-1 (decay): Phi (D4, D3: ∆-1 » ∆º) > π (D1:St ∆+1) « e (5D: TT ∆-1)

Since e= 2.718281828…4, the limit of (1 + 1/n)n breaks after 10 ‘decays’ into disordered death patterns; only pi has no limit of scale, as a Poincare’s sphere that can shrink *preserving its information.*

*Hence making the fractal structure of the Universe* infinite, by allowing information but not energy to emerge or sink with no limit of scaling in the universe, reason why Maxwell’s equations of waves can transmit information across scales with no limit in an infinite universe. And this is a huge fact of the Universe, which in the ∆=S symmetry implies information can also be transmitted faster than light (but not energy), through *translations of scales, the same way we transmit voices faster than sound through electromagnetic translation. *

**Beauty of 3: Phi couples. Lineal, hexagonal pi & e as decametric scale.**

π (+3.14), 2 phi (as reproduction is a dual ST-mirror sysmmetry) and e (2.81) are all within 3 ±0.2 range. Since 3 is the constant of the 3±¡ trilogic/pentalogic Universe. Pi is larger as the curvature of 3 lineal elements makes a larger cycle with aperture-wholes. Phi is smaller because the reproduction of a system compresses the entropy it has extracted into form. To notice also the Fibonacci series work on ‘bidimensional’ units (couples). So does the ‘golden rectangle, and the spiral that fills space with time series of reproduction. But ultimately 2 genders give birth to 1.

Lineal pi is 3 – the hexagonal perimeter ad 3.14, circular pi. 3 is also e +e/10, the classic growth of one tenth or finitesimal part of a tetraktys, the perfect organic form, growing, one by one, as 2.718..+0.271..=2.99…

**RECAP.** 5 scalar or irrational numbers represent Dimotions of existence: π, the first Dimotion of perception; phi, the reproductive Dimotion; e the dimotion of entropy and 10, the Dimotion of social evolution. Pi and its function, the sine are the cyclical numbers of perception; phi the number of reproduction; e, the exponential of decay that faster grows in its derivative dissolution into infinitesimals and 10 the social scale of the Universe. All those numbers are ratios related to 5D scales, proving further the 5^{th} dimension. They are not isolated numbers, neither points of a continuous line (continuity error of huminds). They relate S-T bidimensional holographic basic dimotions=actions=functions of time-space.

** **** **

**PENTALOGIC OF NUMBERS. ITS 5 FAMILIES AS REFLECTIONS OF ITS 5 DIMOTIONS. CLOSURE**

## The **3 ages of number theory…** is the best orderly way to study them as* time always increases the in-form-ation =complexity of systems… till once the form is complete, as time never stops it suffers its ‘entropic death’=dissolution.*

So happens with number theory through its 3 ages, which finally closed the description of the universe with the human decametric scale, ‘starting afresh’ with another mind-space, that of duality and 01 digital computers, which for ethic reasons of human survival we shall not upgrade. Thus a resume of **Number theory in its 3 ages reads like this:**

**Youth**: Trinity of ‘present ST numbers’, Pythagoras & Chinese School of Gnomon Numbers: Natural, spatial and time sequence numbers and its two inversions, negative temporal numbers of inverse direction and rational numbers of entropic dissolution of social wholes; created with the inversion of the two operands of social evolution (+ v. -) and social reproduction (x ÷). Numbers thus are used for social counting of population & division of wholes into simple parts; while the independent nature of negative and ratios is hardly considered, associated to those two basic operands.

**Maturity: R & C:** ¬∆@st: Numbers achieve full mirror symmetry with the 5 elements of reality, rational numbers explore scales and Universal constants. Complex numbers mirror ST-numbers, with inverse and squared values. Operands reach its full power and mathematical physics evolve numbers into 2 new scales, variables and functions.

**Old, baroque age**: Numbers explode in useless variations (quaternions hyper numbers, etc.) While they loose its connection with reality, as primary elements of mathematics, first ‘packed’ into a single dimension, the real continuous line and then derived by esoteric methods from the new ‘imagined’ unit of reality, the Cantorian set, which is to the 3^{rd} age of human-invented number theory what Axiomatic Hilbert’s ‘imagination of points, lines and numbers’ to Geometry.

**+¡: 5D Mathematics**** connects back numbers and experimental facts of Nature, as units of scales and time.**

**-¡: Transhumanism. ****But number theory seems to die on human brains, taken by storm, by** computers, which however start an entire different digital mind no longer based on numbers as decametric social scales but on the simplest duality of 0-1 dimotions (entropy vs. form, energy vs. information, closed vs. open, truth vs. false etc. dualities) – a less synoptic mirror of reality but good enough to translate any discovery of previous number theory to the new mind-brain.

**Pentalogic of Numbers. Its families and planes. ****The multiple representation of reality in numbers.**

In the Entangled Universe, each entity will be *dominated function it performs in the upper ∆+1 world in which it survives, but as a whole is a T.œ, an ¬∆@st being. So we *must apply the ‘Rashomon effect’ of pentalogic 5 truths to study its variations. For numbers this gives birth to its 5 ‘different types’, which in turn define different ‘planes of geometry’:

N(∆): The main ones are Natural numbers – scalar social gatherings of identical beings.

Z(T): are ± dual sequential time motions and its inverse which close into a zero sum cycle.

Q(¬); Rational numbers are ‘entropic divisions ‘that dissolve wholes into its parts.

Gnomons (S) are polygons = spatial configurations of identical points.

C (@) Complex numbers and those of different ‘frames of references’ are mental configurations with various applications according to the mind mapping they represent

So we classify all the ways in which numbers reflect the Universe with the classic concept of closure; that is, all the different planes and forms of numbers that put together can solve all the algebraic equations of timepsace variables that represent a function of momentum (lineal or cyclical, | or O) or energy (Ø) or a Dimotion of a fractal point or T.œ of any complexity whose parameters w translate the 3 conserved quantities of physics into the 3±¡ topologies of a T.œ. as ¬∆@st So numbers can mirror the ‘existence’ of beings beyond its relative present superorganism to describe fully its worldcycle.

Numbers though are *best as *social groups in scales, as regular polygons in space and as sequential causal events that represent the 5 Dimotions of the Universe; so we must depart from the tyranny of the axiomatic method that defines then in a single ‘real line’; so we can use better pentalogic to describe the multiple uses of *each plane and family of numbers.*

Whereas the continuous single line to define all numbers, natural, negative, rational & irrational, is the ‘limit’ of a more profound analysis of numbers in multiple scales. So those families of numbers, when escaping the tyranny of the line become filled of new meanings as *5D pentalogic mirrors of ALL the elements of the Universe.* In @nalytic geometry, the line of numbers relates to a 2 dimensional holographic world. So its scope is enormous as it can represent any SS<ST <TS<TT dual ceteris paribus analysis of any dimotion. Yet its synoptic power must not fog a larger picture of reality.

**A first pentalogic analysis of those families then set a **Generator equation for the 5 families of numbers such as:

**Nº:** **Space:*** 4D¡motion: Social evolution. Z: Time: 2D¡: locomotion. R: ∆-scales: 3 D¡:Reproduction. Q: 5D ¬ entropy.*

**Complex numbers and other frames of reference: @***mind:1D¡motion: informative gauging *

**We correspond each family of numbers to its main Dimotion as ¬∆@st, dust of spacetime in the ‘larger picture’ of the absolute Universal mirror; and to specific Dimotions in the ‘smaller picture’ of its use in the events & actions of T.œs. **

**Natural**, ‘direct’ numbers ℕ are zero and the positive whole numbers 0, 1, 2, 3, 4, 5… If two such numbers add or multiply, the result is again a natural number; which means *it is possible to create all intermediate societies departing from a unit quanta, between 1 and the relative ∞ value within which the domain of the function or event is meaningful. *

*Its use is the simplest intuitive **games of social evolution of identical groups, through the § scale between ∆ and ∆+1*

**Natural numbers** are related to the measure of 1, 2, 3 D **spaces**, as units of points and populations. They are either self-centered **even** numbers representing the 1^{st} Dimotion (perception in its crossed diagonal center), or **odd mirror polygons with a reproductive symmetry**, (3^{rd} Dimotion) which can be internal or external crossing of diagonals, making a 4D & 5^{D} replica (inner, implosive social growth or entropic outer growth).

Finally Natural numbers ‘can represent locomotion **(2D) **as ‘steps’ of a given unit, which represent a discontinuous motion of a frequency ƒ, and a wavelength ‘1’. Steps become discontinuous landings of the fractal unit that measure distances or motions in a lineal 2D geometry. We can then include ratios of those steps, breaking the line in different smaller ‘fractal steps’ and that is a first meaning for a rational number. *But if we make a different unit for each rational part, we enter into another Natural number scale of the fractal Universe returning then to natural numbers, which therefore are also able to represent ratios. And so natural numbers can be any number as it keeps falling down in microscopic rates. *

*Why then we need more families? Because **each number really is best for a job, even if they can multitask, as a shovel can do many things but a hammer will nail better. So the next family of numbers, *negative numbers are a better mirror symmetry for the function of motion, as they can change the direction, moving backwards to the other side of the line.

**S+(-T).** The integers ℤ, add “Inverse’ S«»T numbers, normally related to inverse, entropy vs. information functions in all systems of reality, hence to the inverse ‘directions’ of one of the ‘dimensions’ of space-time (S, T or ∆). As Gauss intuitively understood this removes absurd hang-ups of scientists afraid of negative ‘values’, specially in physics, where c-speed is a limit because physicists don’t understand mass’ as a vortex of space-time; hence its inverse is an expansive entropic dark energy’, perfectly valid in relativity equations. Z numbers thus divide in positive and negative whole numbers …, −5, −4, −3, −2, −1, 0, 1, 2, 3, 4, 5, …. If two such numbers are added, subtracted, or multiplied, the result is again an integer; if both are equal in value differing on its sign, the result is a 0-sum, which means we can ad up all intermediate combinations of entropy and information, or similar inverse directions in any dimotion of ¬∆@st, but ultimately its inverse ‘time arrow’ will make the system come back to its cyclical time departure as a zero-sum.

**Negative** numbers thus *are essentially Time Numbers. They make no longer sense as ‘social populations in space’ – there are not 5 negative pears. *So the social use of natural numbers is lost (cause of much philosophical thought in earlier mathematicians), but the power to represent

*the inverted arrow of time in any Dimotion of exist¡ence, either in negative directions of locomotion or negative entropic functions (negative exponentials), grows.*We keep this in mind in physics as ‘i’, a negative number, merely represents the inverted time Dimotion of a physical species, as in EóMc

^{2}.

**5D: entropic, ****rational numbers, ****ℚ.** Fractions invert the arrow of social evolution of Natural numbers by inverting the product operand: *p*/*q* where *p* and *q* are integers and *q* ≠ 0. So they divide social groups or time frequencies/durations or kill and break a whole into similar parts. So they remain in a single ‘plane of reality’. Since if two such numbers are added, subtracted, multiplied, or divided the result is again a rational number. In terms of scales, **rational numbers **break them in part. So with decimal numbers we move the ∆º whole downwards into its relative ‘finitesimal’ ∆-1 quanta, following the standard scaling 10¹¹ scales.

Thus Natural, negative and rational numbers form along the ± and x÷ operands a ternary T(Z)>T/S(Q)>S(N), present organic group of numbers in a single plane of reality – and we call that ‘Fractal Generator’ system, a Polynomial mirror.

**∆-1.** **Real numbers,** ℝ, as rational numbers do, allow us to travel through scales. For that reason they often loose its patterns in decimals around* the 11th element, the limit of most finitesimals between planes. Yet according to the ‘wholeness of reality’ within each of those 2 fractal island-universes,* if two such numbers are added, subtracted, multiplied, or divided (except by 0), the result is again a real number.

**Complex numbers.**

In that graph, *Gauss offered the first clear description of negative numbers as having an inverse direction (of a timespace dimotion) and lateral imaginary ones, as a composite of both. Its proper definition is to consider them composite Time§pace numbers with an space, mostly real and a time, imaginary, negative component, which make them useful in mathematical physics to describe holographic ST T.œs ST or scalar events between the field: $t<wave: ST>particle: ð§ of a physical system. **Let us then do for a change a longer analysis in this introductory course to 5D mathematics of complex numbers and its main ‘magic’ conundrum – what they represent in mathematical physics.*

Vectors and imaginary numbers are bidimensional, holographic numbers, hence best suited to express the 5 Dimotions of space-time, as *functions of Space-Time (SS, TT, ST)*. Vectors in generalized coordinates are the best form to express motion in the plane from an objective non-human self-centered perspective; in terms of lineal time, by virtue of the S=T duality. However, imaginary numbers excel vectors in the representation of more subtle 5 Dimotions=actions in which Time is not expressed in lineal terms as synonymous of space, for 3 reasons that characterize the ‘Dissimilar complex plane’ as its Y and X coordinates are NOT in the same scale, one being the square of the other, and this allows them to express complex ∆ST symmetries of scalar space and ST parameters, mixed together in various ways:

**S<=>****T symmetries,**where**t**he imaginary negative number, which makes NO sense in space (no negative apples) perfectly represents a Time function, either in complementary T>S dual systems or in entropic perpendicular Ðimotions**TT: Cyclical time events**. Cx numbers are also suited to represent cyclical, repetitive functions, in which the time clock parameter constantly switches on and off, representing repetitive 0-sum worldcycles and generations (Euler’s equation, exponentials as sum of sins and cosines inverse functions e^{ ±x}series)**∆-scalar complex numbers:**Events where the higher scale, often a leading informative function that guides the system as a ‘time motion’, has an √ smaller proportional value, as the imaginary part that ‘absorbs’ a small proportion of the existential momentum of the ‘Real energy’ function/organ.

Thus the imaginary coordinates are a root, or inversely, the real parameter is the positive square of its informative value. This duality is ideal to represent a ‘dominant time function’ often with a 1D frequency parameter on top of a ‘bidimensional, holographic spatial’ area or relationships between the particles, body-waves and limbs/fields of a system; as parameters of lineal or cyclical time ARE one-dimensional (duration or frequency), acting upon a ‘surface of spatial populations’.

Complex numbers thus have the advantage in the imaginary frame of reference of being NOT only bidimensional numbers but *numbers of different dimension, hence kept separated, as squared numbers and single numbers when we ‘square’ the coordinates, or real numbers and its roots as observed normally.*

This quality in Nature happens to be very common in Spatial 2D areas vs. Time 1D frequency representations, so Complex numbers are ideal to represent ST functions=actions=dimotions of space-time.

**Pentalogic of complex numbers **

Thus its* * ¬∆@st pentalogic use in experimental physics derives of having a **negative root** coordinates meaning either:

**ð:**An inverse, temporal or negative number that represents an inverse direction of motion. Or…**∆:**An scalar upper plane which emerges from a denser lower plane, often with a √ amount of the real parameter measure, as upper scales take only a ‘smaller part’ of the total energy of its lower plane (as in heat).

Both things put together make complex numbers ideal to represent ST dualities, with the S-function as the real function and the T-function of motion as the negative imaginary root part often extracted from the energy of the still spatial area,

**– ∆ð:** Both together laid down in a Cx^{2} squared coordinates –a new 5D solution to represent complementary ST systems.

Since if we combine the negative time dimotion and the scalar squared dimotion we can ‘reorganize the complex plane’ in polynomial terms, getting rid both of the cumbersome √ elements and its negative complex roots, to reflect a mapping of a fundamental principle of nature:* ‘The imaginary part of a complex number represents a parameter of an ∆+1 whole, which extracts energy from an ∆-1 disordered more extended, ‘real’ potential field’. *

If the classic representation, the upper half-plane H is the set of complex numbers with positive imaginary part; in the square complex plane, both left and right real squared axis are completely equivalent, (-x)^{2}=x^{2}, reason why we use only 1/2 plane. Further on the plane is tumbled, giving the importance of the i-plane which has its conjugate as the negative ‘time motion’, -1.

So we use the squared Cx plane to study dual inverse time motions; while the real squared axis provides the ‘stored energy’ for that motion, dwindling its volume as the time function absorbs it. Thus a Cx^{2} plane represents a ‘local S^{2}T-organism’ (Ab.T.œ) where the conjugate ± axis represents 2 inverse time directions and its √^{2 }value, compared to the spatial area=population of the real line, the dwindling energy of new ∆+1 scalar planes. So in mathematical physics we map with complex numbers the 3 ∆±1 STates & ternary parts, fields, particles & waves:

By making an indirect use to simplify calculus in which complex numbers represent the ± inverse directions of a ‘time function’ (imaginary part) or a spatial population (real part) as in alternate currents and phasors.

I.e. In electronics, the state of a circuit is described by the voltage V across it, the spatial still state and the current I flowing through it, the temporal state – (or its capacitance C and an inductance L that describe its tendency to resist changes in voltage and current respectively). So V and I can be described by a single complex number z = V + i I. Similarly, inductance and capacitance can be thought of as the real and imaginary parts of another single complex number w = C + i L. The laws of electricity can then be expressed using complex addition and multiplication.

*Or by making an **∆ scalar interpretation, where the spatial, area=population component or ∆-1 scale is a larger squared value. *as a ‘co-ordinate system of ‘square, bidimensional’ space and time elements, where the x² coordinates responds to space, and the i²=-1, represents the inverse mostly time scales.

I.e. In relativity the time function is represented with an it parameter. So in special relativity we write:

Where the negative factor of time means an inverse motion/form, as the other 3 positive space parameters represent a lineal distance-stretching on the lower quantum potential=gravitational space-time where the light displaces and the negative ‘cyclical time parameter of the light wave’ the warping by light of a space-time-light function as it ‘forms’ the quantum field potential below (we assume it to be the ∆-1 gravitational scale), the speed of light space that drags and warps the gravitational potential into form appears as a negative element that slows down the motion.

- Or by making
*a ‘direct’ representation of real $t-field<ST-wave>§ð-particle ternary fractal generators .*

I.e. in quantum physics either in its ∆ø: S=T wave representation (Schrodinger), where the wave function is attached to an imaginary momentum operator, ih.

Or in its realist Bohm’s interpretation, where the T-potential, guiding wave is the imaginary part, and the S-Particle state, the real part.

Thus departing from the Fractal Generator of physical systems ∆-1:$t-field<∆º:ST-wave>∆+1:§ð-particle describing the a(nti)symmetric states of its 3 components, we can make a realist interpretation of both relativity and the quantum world in terms of complex numbers as local ST-organic systems to interpret properly most parameters of modern physics; *either in terms of the different scales of ∆-1 fields, ∆ºwave or ∆+1 particles, with relative squared values* or in terms of the negative and positive, implosive or expansive, entropic or informative arrows of time represented by + real or – root values.

For example, complex negative masses in relativity describe are just the inverse ‘motion’ to the cyclical space-time vortex of mass, hence repulsive expansive gravitation.

While Schrodinger’s wave either in the S=T, present wave or T-past quantum potential field>S-future particle formalism (Bohm’s interpretation) is a no time dependent wave, which therefore represents a complete worldcycle of lineal past, kinetic energy; present hyperbolic wave and future particle states fully ‘integrated in the equation’ *that must be seen as the sum of both, the Schrodinger’s present and Bohm’s past to future formal states*, whereas the angular momentum, or membrane of the wave (T=S symmetry), becomes the real parameter of the particle envelope.

**Cx as expressions of the 3 timespace dimotions= states: potential past <present wave> future particle**

All what exists is a ternary system generated in time and space by a simple fractal generator equation:

|-$t (past moving limbs/fields) < Present hyperbolic ST-waves/bodies> future O-§ð informative particles/heads

Both biological and physical systems have the same structure of 3 timespace Dimotions, which give birth to 3 spacetime parts. So entropic fields ‘feed’ the motion of ‘hyperbolic waves-bodies’ which feed and guide ahead the particles-heads that do NOT move in its relative stillness or pure informative linguistic views of reality, which ARE mirror still images of the world.

So for example, ALL topological geometries of the Universe REDUCE to those 3 types: lineal/planar topologies, which are entropic motions as the line is the shortest distance between 2 points; spherical mind/heads, which are n-spheres as the sphere holds maximal information in lesser space and can shrink without deformation to create a 0-mind mirror of an infinite Universe (Poincare Conjecture); and *hyperbolic body-waves, **which combine and hence iterate both, lines and cycles, forming sinusoidal body-waves. *

And so mathematics again reflects in its 3 only topological varieties, the 3 only organs needed to explain reality.

** **While we can study the 3 arrows of time of the Schrodinger wave, reordering the 3 components in 5D in terms of the 3 conserved time-states of any physical system (graph):

∆ø: Vψ (Present energy state) ≈ h²/2m ∇²ψ (past, kinetic energy) + ih ∂ψ/∂t (future particle)

So in the right side we have now the past lineal kinetic≈entropic energy of the particle or singularity parameter and its future, particle envelope of angular momentum or membrane (S=T symmetry). While in the left we have the quantum wave itself, its vital energy, which is the potential energy of the wave, Vψ.

So the equation understood in terms of the fractal generator of 5D physics and its 3 conserved spacetime states reads:

Singularity or active magnitude with its kinetic≈entropic, past, lineal momentum + cyclical particle of angular future i-momentum = Vital, Present, Potential energy; which is always the function surrounded and integrated by the other relative past and future states, in dynamic balance with them – ultimately an expression of the much wider fundamental equation of time states of Pentalogic: Past-entropy • Future-information = Present-iteration.

In this case, Past-field • Future-particle (Bohm) = Present-wave (Schrodinger).

In this manner, we obtain *a third 5D interpretation of the 3 states of the quantum system, besides the expression of Schrodinger and the Bohm’s interpretation, connecting both. through the more general 5D TimeSpace equations.*

Whereas • will be a different operand according to the type of formal representation we use for the 3 parameters.

In terms of past, present and future an i-number rotates from a relative past/future to a relative future/past imaginary states on the ±i conjugate axis (imaginary potential in Bohm’s formalism, imaginary angular momentum in the Schrodinger’s formalism), the function of existence of the quantum wave, through a 90° middle real angle, in the square Complex Plane or ST real balanced point, in which the present becomes a bidimensional function, the wave state with a REAL, + (relative past-entropic coordinates) and an imaginary, negative (future coordinates) elements.

This needs a couple of clarifications: first we move ‘backwards’ in a complex i-graph from future to past as we rotate through i-scales, in as much as most graphs are measuring the ‘motion’ of lineal entropic time which is the *relative Space Function.*

So the negative side becomes the time function, something which mathematical physics makes obvious, when considering an attractive, informative force-field-potential negative.

**@:****As a self-centered polar frame of reference (r, α),**calculated by modulus=distance and angle, which as we shall see are the two essential ‘mental informations’ needed in any representation of space, for a self-centered mind to understand its surroundings – measuring other entities of the world by its relative size (angle) and distance-motion, (hence the time to reach them, and the ‘danger’=size and encounter chances, given by the trigonometric angle in prey-predator events). Distance and angle*thus are the two minimal elements of information all useful mental spaces of geometry must carry, as Riemann and those who followed his logic interpretation of geometric features realized.*

As to the geometrical meaning of a product in the so-called plane of complex numbers, we can see it more easily if we consider the length p of the vector from the origin of the coordinate system to the point (x, y) (this length is called the modulus of the complex number z = x + iy) and the angle 4 which the vector makes with the Ox-axis (this angle is called the argument of the complex number z = x + iy); in other words, if we consider not the Cartesian coordinates x and y but the so-called polar coordinates ρ and ϕ (figure). Then x = ρ cos ρ, y = ρ sin ϕ and the complex number writes as X+¡y=ρ (cos φ+ i sin φ)

From this we see that in multiplication of two complex numbers their moduli ρ1, and ρ2, are multiplied, and the arguments ϕ1, and ϕ2, are added. In division, since it is the inverse operation of multiplication, one modulus is divided by the other, and the arguments are subtracted.

In raising to a power with positive integral exponent n, consequently, the modulus is raised to the same nth power, and the argument is multiplied by n.

However, in taking roots a special situation arises. Since the values of the root coincide in the argument (cos φ/n+2kπ/n+ ¡ sin (φ/n+2kπ/n), where k is any of the numbers 1, 2, ···, n – 1, will also be an nth root of the number as the addend 2kπ, because of the properties of sins and cosines, can be neglected, since it changes neither sine nor cosine. Thus the nth power of this number is also ρ (cos φ+ i sin φ).

*We can then go further depth in applying ‘twice’ pentalogic to extract new p.o.v.s on C-Time numbers from ¬∆@st; which we shall do now returning to our case example, of a quantum triple state of field-wave-particle:*

*3D: S=T:** Geometrically, the nth roots reproduce points of the complex plane corresponding to the root values of the number ρ(cos *

*ϕ*

*I sin*

*ϕ*

*) lying at the vertices of the regular n-sided polygon inscribed in a circle drawn about the origin with radius modulus, and so rotated that one of the vertices of this n-sided polygon has argument ρ/n.*

**1D: @:** *The sinusoidal functions, π and its angle of aperture,* becomes the numerical value of the external membrane measured in terms of 3 diameters turning around with 3 apertures to perceive, from the self-centered singularity, giving us the general equation of dark matter for all systems (including the Universe):

3’14-3/π=0,14=±4,5%, with 96% of ‘dark matter’ outside the perception of its singularity.

So the angle of the point and sine function is its angle of perception of the central point or 1^{st} Dimotion of existence.

**2D**. The 3 apertures allow the point to absorb but also emit waves of information, often curling a minimal angular momentum/membrane/cellular quanta, h, for particles, that becomes the minimal boson unit of communication between 2 Fermions, entangling them as it reproduces its form over the quantum potential. This makes complex numbers in polar coordinates the best way to represent informative, perceptive wave communication between points.

**4, 5D**: Finally the polar backwards rotation, each new power a quantum stœp of time-space, represents perfectly a vortex of time-space, which is the meaning of the ih operator of angular momentum in the quantum scale:

So the nth powers mimic the trajectory of an attractive timespace vortex, and the worldcycle of its st constants towards the internal i-point singularity. The Complex plane become then a vital representation of and accelerated physical vortex: Notice that as X increases those stœps with each power, the attractive vortex becomes paradoxically smaller in its radius; as it accelerates in time, according to ∆-metric, making the vortex faster (Vt x Rs=K), and more attractive.

*So in 5D physics we model vortices of charges and masses, with complex numbers as convergent vortices with an ellipsoid ‘flattened’ smaller imaginary axis turning counter clock wise around a physical @-mind singularity where T=motion turns to zero; as observed in Relativity equations for a black hole & eddies with 0-motion in its center.*

We can then understand Schrodinger’s & Bohm’s dual interpretations, in terms of those ∆+1 vortices (particle states: 4D, ∆+1-involution state) and ∆-1 quantum potential fields (5D entropic ∆-1 scale) that entangle through 2D wave communication (Schrodinger’s ∆º-present reproductive state): 2 particles share boson waves that upon arrival collapse into the particle spacetime vortex in any ∆±¡ scale, splitting its ST function at the vortex center. There the informative part of the boson becomes a still mental image (1Dimotion) and the energy part dissipating, scattered through the vertical axis (5D); which writes the Fractal Generator as the classic Fermion-Boson-Fermion equation. §-Fermion Vortex <ST-boson wave reproduced over ∆-1 $t quantum potential >§-Fermion vortex.

** **So the quantum unperceived non-local ∆-1 potential field of action at distance (as v=s/ti and with no information, v=s/0i = ∞) entangles the fermions in a first exchange of ‘trigonometric’ information (only angle or distance) likely with a Neutrino-antineutrino pair, and then in a 2^{nd} passing (neutrino theory of light), a ray of light is created over the non-local quantum potential field, which has chosen the best path to move (Fermat’s least time principle), offering the 5D interpretation of quantum physics adding Schrodinger and Bohm’s equations that put together the 3 states of its fractal Generator. Since, Bohm just *put Schrodinger’s hyperbolic ‘Cartesian S=T wave’ in Polar S-Particle coordinates, where the mind-particle state is the new point of view, and voila! it also appeared ‘magically’ the T-entropic field that moves the particle T>S, as in 5D we are all made of time-space so those coordinates ARE the wave state and the Particle state when we gift them the Galilean symmetry of motion. It is NOT a change of coordinates but of states from the spatial hyperbolic ‘present wave’ to its Past field>Future particle state-coordinates flattened in the i-direction. *

What is then the future direction in the imaginary arrow? Obviously that of a normal clock, down the conjugate is a future moving on a 90 degrees phase faster than the real and in the imaginary side a lag of 90 degrees in the phase of the world cycle – which physically appears in the AC circuit with the forward 90 degrees wave of the capacitive reactance, Xc on the conjugate and the lagging, 90 degrees of Xl the inductive reactance on the Imaginary side.

**$:** **Finally the last pentalogic use of complex planes is its classic forms helpful for vectorial calculus ****as** bidimensional real numbers (Bolay). If a and b are regarded as rectangular coordinates of a point, and the point is associated with the complex number a + bi, then the addition and subtraction of complex numbers corresponds to the addition and subtraction of vectors, i.e., of directed segments from the origin to the points with coordinates (a, b) and (c, d), since in addition of vectors their corresponding coordinates are added. The sum of two complex numbers: **(a+bi) + (c+di) = (a+c) + (b+d)i** is then represented geometrically by the diagonal of the parallelogram constructed from the vectors representing the summands. In this way, complex numbers are added by the same law as the vector quantities found in mechanics and physics: forces, velocities, and accelerations AND used to represent actual physical quantities, *which if we square the graph, become expressions of ST entities in the real x² coordinates and S or T entities in the Y-coordinates; yet as they are ‘two different species’ they cannot fusion together*.

**RECAP** The complex plane is specially suited to represent S(real)-T(imaginary) symmetries of creation by absorption of Energy from a real-spatial function, representing a body-wave, of an inverse, perpendicular particle state. While Vectorial space is better suited to represent social or reproductive functions of similar ‘herds’. This simple guidance clarify in mathematical physics, the meaning of complex numbers as parameters that express combined dimotions of ST-states beyond its use as artifacts of calculus (as in the quantum momentum ih operator).* As they are un*able to mix, they maintain the 2 functions they represent apart from each other. Further on imaginary numbers are inverse directions (signs) than its conjugates, which makes them highly antisymmetric, able to represent ‘temporal ± numbers’ as two inverse arrows of time. So this is the essential use in mathematical physics.

*Let us then close this introduction to 5D Cx numbers with a consideration on mental space and relative infinities, the themes of the next paragraphs, studying the main tool of 4D Complex analysis, the Riemann sphere:*

**In the graph, a simple proof of the non-existence of infinities, as all perceptive points loose detail with distance. The paradox of the Riemann sphere makes each point of the sphere represent a point of the plane, which seems absurd, since the ½ lower sphere is projected in a much smaller surface of the plane that the upper part that reaches till infinity.**

However if we consider the north pole the @-informative mind gauging the plane as it moves to infinite it perceives lesser quantity of points and finally in its relative infinity all information on the far away points of the plane disappear, making the less dense further plane equal in density of points to the ½ closer plane/lower sphere.

** **

**Fractal generator of N: O-1-∞. 2 scales of numbers as reflection of worldcycles. **

*In the graph, an ∆ scale can be represented through the interval of 0 to 1 by finitesimals or the interval from 1 to ∞, which become the decametric and decimal regions of a supœrganism represented in the real line: ∆-1: o to 1, 1, the ∆º scale and 1 to ∞, the external world. Thus this section will study the meaning of numbers and its scales as a representation of the supœrganism. *

In that regard 3 numbers could resume it all, 1, 0 and ∞. They can be seen as the ternary elements of the fractal point: 0 is the mind-singularity=center, 1, the membrane/angular momentum, perceived externally as a whole that encloses the internal vital cycles of energy, between the singularity- mirror of the relative infinite world, ∞, outside the being; which only reaches till the limits of its perception.

**Thus we define a limited infinity or relative infinite, ∞, which in scale is the limit of human perception, the ∆±4 cosmological scale; and so it is in distance (the event horizon of light, we perceive) and in time (regardless of pretentious cosmologies, which set a beginning for the whole Universe); which humans would better set on the impeding limit of our existence as a species, that is, the likely death of the superorganism of history if we keep killing the life of the planet, instead of its hyperbolic Cantorian infinity paradoxes. The same goes for 0, which is rather 1/∞, that is, the relative finitesimal part of a whole, as Leibniz defined a derivative, which in a discontinuous Universe, has always a +0 value.**

Indeed, many aspects of the natural world are simplified as being continuous— to an excellent degree of approximation, not because they are so but precisely because perception of infinities is limited; and so the first element eliminated from perception are the dark ‘discontinuities’ between ‘film frames stopping, time cycles coming back to the frequency point, scalar gaps of a lower ∆-1 background field of finitesimal particles and so on. So since spaces are always mental constructions, continuity is a question of synoptic modeling, not of reality. Matter is never continuous; if matter is subdivided into sufficiently small pieces, then indivisible components, or atoms, will appear and finally we *will find the finitesimal +0 quanta*. And then we can get into the inner parts of the 0-fractal point and find again its 3 components.

It is then within those concepts where we talk of 0 as the finitesimal part, 1 as the whole and ∞ as the limit of the outer, entropic world in which the 1 will perform its worldcycle.

So the |-membrane, the • singularity & the ∞ cycles in the outer world create, a fractal superorganism and its worldcycle. The study of the interaction of those 3 elements that compose all §upœrganisms is thus an essential part of GST, which can be inscribed in an even larger concept: *the ternary, Universal grammar of all systems in a single scale of the fifth dimension, and its fractal ternary sub-divisions with added ±¡ planes, which create ever larger nested superorganisms.*

*And since size is absolutely relative, a key question in the modeling of reality is what we shall study, the 1 to 0 scale * downwards, through ‘finitesimal decimal scales’ or upwards, through decametric, growing ones? Since, fractal reality has it that ‘The decametric, §ocial scales (10§≈10^{10} : ∆º ≈ ∆+1) are symmetric to the inverse, decimal ∆º>∆-1 scales’.

Or in terms of similarity, regardless of scale, in a reduced ‘formula’ we say: (∆+1=**∞ ****= ∆º=1 = ∆-1= 0): (∞-1) ≈ (1-0)**

**Whereas we define** ∞ is the perception of the whole from the ∆+1 in the domain of 1, the minimal quanta to the relative ∞ of the ∆+1 scale. While 1 is the relative infinite of a system observed from the bottom up, such as ∆+1 is 1, composed of a relative infinite number of ‘finitesimal parts’ whose minimal quanta is +0.

So in absolute relativity the ∆-1 world goes from 0 to 1, and the ∆+1 similar world goes from 1 to ∞. But there is a key difference when we apply 5D metric and the relative limits between scales:

*The upper scale has a well-defined lower point – finitesimal quanta, 1 but an undefined upper ∞ limit. While the lower scale, has an upper well-defined whole limit, the full 1 but an undefined first ‘finitesimal quanta’, seminal +0:*

*So as mathematics is a mirror of scalar space-time we immediately find what essential feature of ¬∆@st the o-1-∞ scale reflects: The 2 worldcycles of existence of all systems of the Universe, which are born as a seminal seed, a finitesimal 0, which will live an ‘ordered’ worldcycle in the placental womb full of positive energy till emerging as a 1 full whole form in an open 1-∞ entropic world, where it will perform a second far less ordered ‘worldcycle’, always fighting in entropic Darwinian perpendicular events with similar 1-wholes in the struggle for existence. So those 2 placental o-1 ordered scale and 1-∞ upper ∆+1 scale, can be modeled mathematically in the 0-1 unit sphere and the 1-∞ Cartesian plane, which in physics represent for atomic superorganisms, the o-1 probabilistic quantum time formalism or placental age of atoms and the 1-∞ thermodynamic spatial scale of populations. So we state in physical systems the equivalence of:*

*+0-1:quantum: particle evolution (time probability)≈1: atom≈1-∞:thermodynamics: atomic population (statistical mechanics) ≈ ∞ (bound infinity): galaxy (Gravitational scale**: Relativity formalism)*

**In the graph the existence of 3 distinctive ‘worldcycles’ in all systems of reality, the ‘palingenetic worldcycle’ (þ), the lifecycle of the individual, as the ‘one being’, (c), and its worldcycle in the outer world (w), mimicked mathematically in the difference between the o-1 sphere and 1-∞ Cartesian plane:**

** þ: 0-1**: is the cycle of palingenetic o-1 social evolution in the accelerated time sphere of existence, till becoming 1 (0-1 bounded unit circle in ¡logic mathematics; quantum probability sphere of particles in physical systems; palingenetic fetal age in biologic systems; 0-9 memetic learning childhood in social systems). It is the highly ordered world cycle as the ‘placental mother-energy world’ is nurturing, and memorial cyclical spacetime has erased errors of previous generations.

**c: 1:** So the one being will ‘emerge’ as a whole – a mind that puts the inner and outer world together.

**– w: 1-∞:** In that outer 1-∞ world, the organism then will live its 2nd world cycle of existence in an environment which is open, entropic (1-∞ hyperbolic unbounded Cartesian plane in ¡logic mathematics; thermodynamic entropic statistical molecular populations in physics; Darwinian struggle between populations in biology; idol-ogic dog-eat-dog capitalist, nationalist competitive eco(nomic)systems in the super organisms of history. In this 1-∞ existence the world cycle is not ensured to continue, as it can be cut off by the entropy of the world system.

__The symmetry of finitesimals ____– 0-1-____∞ scales.__

Infinity belongs to Kantian’s paralogic, transcendental ideas, to the inflationary nature of mirror languages in 5D. As all 5D scales have a relative µ beyond which perception is not and a minimal o-quanta of perception or finitesimal…

The ancient Greeks expressed infinity by the word apeiron, which had connotations of being unbounded, indefinite, undefined, and formless. It is a clearer meaning to the loss of information which comes when we go far away in ∆st.

The mirror symmetries between the 0-1 universe and the 1-∞ set two different ‘limits’: an upper uncertain unbound for the 1-∞ world, where any T.œ exists, where the most ordered moment is that of birth and emergence in the parental care, but from then on all will become a slow slide into the open world of statistical populations, and entropic events that will wear the system till death vs. the lower uncertain bound for the 0-1 Universe, where entropy happens at the finitesimal 1/∞ quanta seminal level of the palingenetic journey as there is nor perceivable limit on the lower bound for finitesimals, unperceived by the 1-observer, who emit millions of seminal seeds of uncertain future, till they enter the placental sphere where order and growth is regained. But those unbounded limits are not infinite, just uncertain, entropic limited by death…

Of course, the axiomatic, creationist method ‘believes’ in ∞ as Mathematics must be the a priori language of God, not a mirror, so infinity must exist in numbers, larger than the Universe itself, reduce to a trifle of time by the entropic big-bang. The result are the absurdities of Cantorian transinfinities, self-evident paralogic errors, proper of the 3^{rd} age of the discipline (excessive formalism and fictions). But as we consider maths, after Lobachevski, Gödel and Einstein, an experimental science; we are more interested in *how experimental evidence always reduces infinities to quanta*. So while 0 can be approached by infinite infinitesimal ‘decimals’, so it seems it can never be reached, as in the solution to Zeno’s paradox, there will be always a ‘quanta-step’, which acts as the finitesimal. So happens in physics. Even in radiation, the smallest quanta, perceivable, we know since the ‘violet catastrophe’ that the infinitesimal is a ‘quanta’, h a ‘minimum planckton, the ‘limit’ of the light space-time, which is what an electronic humind perceives. And so we return to Leibniz’s rightful concept of an 1/n minimal part of ‘n’, the whole ‘1’.

**This implies by symmetry that on the upper bound, the world-universe in which the 1 is inscribed will have also a limit, a discontinuity with ∆+2, which sets up the infinite in the upper bound also as relative ∞ finite quanta, a ‘wholes of wholes’, which from the ∆+3 perspective will be a +0.**

One of the most important S=t symmetries of the mathematical Universe is the one between time probabilities in the 0=1 unit circle and the 1-∞ plane of statistical populations (space-points), as it is both, a symmetry between the ∆-1 scale of finitesimals (unit circle, where a finitesimal, 1/n is defined as the inverse of a real number of the 1-∞ plane); and one between time-cycles and space-planes (topological symmetry).

So the Cartesian graph adds a key ∆±1 symmetry of the placental worldcycle in the 0-1 unit circle and the life worldcycle in the 1-∞ larger ∆+1 scale, as a natural correspondence – since all laws of probability in the 0-1 unit circle work on the 1-∞ scale of statistical populations, with 3 pentalogic differences:

– S: The unit circle has a defined membrane, but its minimal infinitesimal ‘decimal’ +0’ is not defined. Inversely, the 1-∞ scale has a defined minimal infinitesimal 1 but the ∞ element is not. They are the limiting elements to consider for a best choice of which ‘mental geometry’ to choose, to study a problem depending on which duplet of the 3 elements model better the problem – the ‘singularity 0-mind or seed or quanta’ (similar fractal points for mathematical analysis), its o-1 closed space coupled with the 1-organism, which emerges at the *end of the problem*, or the outer entropic world in which the 1-defined quanta will play is particular uncertain game of existence on the ∞-open spacetime.

*-@: In praxis, *self-centered minds use a unit circle where 1 is the ∞ limit of perception, and +0, the hidden finitesimal self-centered ‘polar’ mind, where all the ‘∞ parallels of information that cross the 1-border from the open world (1-∞ region) converge to create a mind-mapping of the whole – where 0=∞.

– A unit circle is a ð-cyclical/polar/spheric topology that in certain problems (quantum physics) uses a Complex plane or Riemann sphere (when we need different parameters for the 2 components of an S≈T, wave-particle system, or an S, -T signature in a 4-vector Minkowski space); while the 1-∞ scale is an open, flat unconstrained geometry, *much easier to use as it is the human scale; so a better *choice for simpler solution of problems depending on the level of accuracy we seek (as there will always be an S=T symmetry to make possible, for example to study a vortex of space-time in S-symmetry, as curvature in relativity, or as a T-symmetry, as a Newtonian accelerated mass).

– If we *are interested in scalar studies, both scales must be used, whereas *the unit circle at 1 becomes the membrane between both worlds – in an organism the sensorial 1-membrain, acting as an open/closed border between them.

Needless to say then we can define in vital topology many ‘other’ bounded membranes not ideally closed by the 1-unit sphere; as long as they are closed ‘membranes’ (spherical topology) with small designed holes for sensorial perception.

Some consequences of the symmetry should be obvious then to the mathematician. In complex systems, fractals are defined only for bounded regions on the similar z=1 sphere. Most differential equations have solutions only when we now the initial conditions and the boundary, which are normally set to t=0 and S=limiting domain. And so on…

Regarding the membranes, as we have reiterated continuity is a paralogic Kantian error of the mind and the axiomatic, creationist method; but the ‘ternary’ Universe has always ‘holes’ and limits where continuity break. I.e. most key processes in any space-time worldcycle happen in the 3 first decimals, around the key number 3. So e+e/10=3, whereas the commonest decametric ‘derivative’ 1/10 quanta added to e, the constant of unlimited growth and decay gives us 3.

π-3/π =4% are also 3 sensorial apertures of the cycle, made of 3 curved diameters that leave 96% of ‘dark matter’ not perceived by those within the circle, as our electronic eyes who don’t perceive 96% of dark gravitational matter ‘outside’ the halo of the light space-time galactic scale.

When we extend the domain of the unit circle to the complex plane and its bidimensional holographic numbers, the ‘world’ is represented now in the higher dimensionality of the Riemann sphere (unit circle of the complex plane) where every point of the sphere is mirrored by one point of the Complex plane while the 0-mind point on top of the height-informative dimension of the sphere becomes the inverse of the ∞ point of the ‘complex plane’. In this manner as in projective geometry, the o-mind point becomes the mental residence of the ∞ of the real Universe:

O-point mind x ∞ Universe = constant world.

In 5D number theory, we state from a pentalogic perspective that:

**Time & scalar view:** The placental worldcycle, fast time palingenetic interval o-1 is equivalent in ∆-1 to the slower spatial, statistical open 1-∞ interval of the life cycle.

**@- mind:** a 0 point is a mathematical mind looking beyond the 1-membrane to the unbounded time-space Universe.

The beauty of this symmetry being the fact that according to 5D metrics (SxT=K) the whole larger world, ∆+1 and its plane, holds the same quantity of information that the o-1 smaller scale of diminishing size. And so the 0-1 mind representation of the whole world outside and 1-∞ world are mathematically, virtually and perhaps really equivalent.

**Spatial and entropic view:** the 1 unit disk or sphere forms a limiting membrane that breaks space into an inner region, the o-1 organism and an outer region, the 1-∞ world; called the simplest knot (1^{st} postulate of Knot theory).

**0-1 4D palingenetic fast-time generation (unit, probability circle)≈ 1,2,3, D population, of exist¡ences (1-∞ plane)**

**The previous symmetry of scale carries into the 3rd S=T symmetry between space points as statistical populations & time numbers as probability frequency, called in ****theory of measure **(graph) the identity between probability laws in the 0-1 circle of fast temporal palingenetic generation of the being, and the laws of statistics in the 1-∞ world cycle of existence happening in the Cartesian plane after ∆º emergence.

It simply states in terms of 5D that for each ‘theorem’ on the 01-probability sphere there is an equivalent law in the 1-∞ statistical realm. It solves to start with the conundrum of quantum physics, when coupled with 5D metrics. Since now quantum physics are equivalent to thermodynamic statistics, in the same manner the palingenetic cycle (quantum physics) is related to the life worldcycle.

Because according to 5D metric, the 0-1 circle of quantum physics that has minimal spatial size, lives much faster time cycles, S x ð=K, the faster clocks of time of its particles, perform so many worldcycles in the minimal time period of human perception that we see those worldcycles completed as if they were ‘spatial particles’ – when we are observing time processes. In the same way you see a fast turning wheel as a solid spatial form. And that is why even if we ‘think’ we see spatial particles, the proper formalism is one of probabilities of time events. While in the larger molecular thermodynamic scale, we see, life long-lasting configurations of populations. So instead of the fast smaller 0-1 ∆-1 probabilistic unit circle, used for temporal flows, we switch to statistical analysis. This said it would have been easier to *use statistical mechanics renormalizing quantum ‘blocks of time’ – entire life of particles seen as space due to its faster motions than* renormalize all results to fit them into 0-1 probabilistic range to make sense in the time perspective, specially due to the lack of conceptual understanding of what is happening.’

**Recap**. Numbers evolved through the expansion of the concept of ‘quantity’ from the natural numbers which merely observe spatial quantities added on in social groups, the essential definition of number, into ratios between numbers that express more complex concepts of reality, and introduce the idea of functions and ‘motions in time, advancing further with the concept of negative numbers, which are no longer quantities in space, but qualities in time, finally reaching the maximal transformation from ‘space to time’, from quantity to function, and ratio of exchange of the vital entities of reality, entropy, information and energy (sT-ST-St), to the final closure of reality expressed in the mirror-language of numbers, which is represented by the ill-understood imaginary numbers.

So the 5 families of Number systems are symmetric to the laws of the ∆ºst universe. So are its spatial representation in frames of reference:

1.S±T: N, Z: **N**atural direct numbers form simple social groups. ¡ts inverse negative numbers form Ts≤≥St symmetries

- Q: SxT: The Cartesian graph and the rational numbers, for Ts x St = K metric equations and complementary systems, (whereas often the z-dimension is the reproductive combination of the other 2).
- ∆±1: R,C: The real numbers, for inverse ∆±1 scales & the complex numbers for worldcycles

∆±1. Functional Spaces, wholes & parts where each point represents a vector (Hilbert space): ∆-1(∑Ts) >∆ST.

Those number systems can be operated on by some or all of the standard operations of arithmetic in a single ¡-plane:

addition, multiplication, subtraction, and division (we study them in ¬ælgebra).

Such systems have a variety of technical names (e.g., group, ring, field) that are important to grasp its properties, making of group and set theory a needed ‘bridge’ (beyond the errors of sets regarding infinity), towards a GST definition of them.

*So as humans have expanded their understanding of the Universe, they have also evolved its mathematical mirror over it, which has finally become both in its more geometrical, spatial form (points) and its more informative, temporal form (numbers) a quite perfect form able to understand it all in the form of numbers.*

However humans have as in most fields of its knowledge merely acted in a very mechanical manner, ‘discovering’ without much understanding the deeper level of reality, which *numbers mirror, the space and time entities, symmetries and sequential events and simultaneous organisms of **reality*. So 5D gives experimental space-time whys to all those discoveries.

**SYMMETRY OF ∞: STATISTIC SPACE POPULATIONS = PROBABILISTIC TIME FREQUENCY**

** **Finally we just shall mention the entanglement of S=T elements in the more complex field of statistics and probabilities which we shall treat in our 2^{nd} book on Mathematics (Algebra and time theory)

**S≈T how time events become statistical populations.** **The interchangeability of probabilities and populations.**

The symmetry is profund: bits of frequencies of time = quanta of populations of space.

As frequencies of time become ‘populations’ of space *once they are born in time, and ‘settle’ long-lasting in space; given the fact that ‘probability birth’ is a faster placental cycle than population life in the longer cycle. So there is a clear proportionality between both, reason why the bell curve of statistical populations and probabilities is the same.*

So, we shall talk of the symmetries between frequencies of future time that become space population accumulated in the past, till they both recede in size as we keep growing in scales, and become undistinguishable, continuous, quiet.

And so again, we can cast both with Nature’s vital function of existence with its eternal goal: iterate a T<ST>Species

**Reproduction and motion, two sides of the same coin: ±, X all pervading operand.**

*Since **at the end of the journey all what nature cares about is to reproduce its fractal st assemblies of creative patterns of space-time. As all is reproduction. The ultimate substance of reality is motion with form, and motion is the reproduction of form along an ∆-1 disordered region of quanta of space that a whole ∆-being will mold and ‘rise’.*

And those processes will have always its S=T symmetries described mathematically with re=product=ive operands, which describe the union of a bidimensional population surface of smaller simultaneous quanta of space moving, reproducing through frequencies of time.

So we use 3 ‘scalar operands’ for those processes of social evolution, reproduction and emergence into a new scale.

First equal beings can be added socially in bidimensional fields by ± superpositions, in the simplest social act. But as density and gender mirror symmetries happen, and ST equivalences move the herd, perpendicular time and space functions require an st multiplicative, reproductive wave form or perpendicular, field > particles system. And finally power laws and integrals and logarithimic scales of decametric growth complete the ∆º>∆+1 creation of a new plane. And there is NO more operands as the Universe is discontinuous and so once a game is completed a new scale starts from nothing, as a new fractal point emerges in an entire new world.

*So probabilities can be seen as frequencies of reproduction in time of populations of space quanta.*

**The persistence of memory – lives that last convert probabilities into populations.**

**Thus time cycles create spatial populations with different ‘degrees’ of persistence into the relative past: Motions in space (locomotions) have hardly any persistence in time (we do not leave a trace of parallel forms) except in the simplest beings (waves of light and so on); but reproductions in time leave a persistence of populations. And so the time bit becomes a space quanta. And this again is an important phenomena in the simplest forms (waves and particles) of the ∆-3, 4 scales. Hence the need to know the S=T symmetries between both disciplines, to know ***when to use them, as in quantum physics where most processes are time events, even if humans with a much faster time clock consider them spatial.* We illustrate both cases with 2 key equations that mix those vital constants defining both T=locomotion and S-populations as reproduction games):

Euler’s formula (left) describes the S=T symmetry of a world cycle reproduced in a lower ∆-1 ‘plane’. So it can be interpreted as the waveform derived of the constant reproduction of the Unit cycle in ∆-1 points’ defining a world cycle of space-time in ∆-1, with faster time life hence minimal persistence of memory. *Its reproduction, thus appear as mere locomotion as the intensity of the wave fades away in the lower herd-plane.* *In the other extreme,* the probability distribution of a population in space (Gauss curve) persists so long, as its populations are in the same plane in which they were created that both S and T stay roughly the same. And its maximal value is in S=T, the central point that maximizes the function of exist¡ence, which is defined at that point. Both curves further on are similar to the bell like curve of the function of existence in time, so the central region S=T in in time the mature age of the system. Thus fields so apparently disjoined are equalized by 5D metric showing in a rather simple but profound way how all is about super organisms tracing world cycle of space-time in synchronous or sequential forms.

**THE 3±¡ AGES OF GEOMETRY**

**The 3 ages of geometry and its 3*** masters**: Euclid, who systematized bidimensional, ‘holographic’, Greek geometry, Descartes that married it with time algebra GIVING IT MOTION and Lobachevski, who established the principles of hyperbolic geometry, the geometry of the 5th dimension, and denied the ‘mental, logic nature’ of mathematics, proving the multiplicity of spaces and the need for experimental proofs. We shall complete their work with the formalism of ¬∆@st. *

In terms of its time ages geometry is the humind evolution of our comprehension of the ¬∆@st complex Universe in ‘ages’ of increasing dimensionality and motion. Since, the ages of Geometry are stages on the realization – as a child does with the world – that our mental still holographic ‘I=eye’ space is NOT the absolute and only space but that of the humind.

So all other geometric laws are self-consistent relative worlds mirrors of the larger laws of ∆@st. Which finally lead after Riemann to the realization that what all those relative worlds of geometry have in common are just 2 parameters: the scalar parameter of ‘angle’ and trigonometric ‘depth’, and the $t parameter of distance=lineal motion within a single plane; which are the *2 ‘survival informations’ any mind requires to locate positions in the present and the future (according to motion and distance) and measure sizes (according to angle and trigonometry) on the scalar Universe. *

So we expand the foundations of geometry in two main type of geometries, mental subjective space that selects information according to the needs of each mind; and organic, topologic vital spaces, which are real and so display objective Disomorphic (equal) laws in their ensembles of organisms; which are *not as mental spaces a construction of the mind, or rather a construction of the mind of God, the ¡logic laws that create the T.œs of the fractal Universe.*

__S:__I Age: Spatial, Greek Era: static bidimensional Geometry__: The holographic principle.__

Geometry started in a mental still deterministic, simple, young age of absolute beliefs, and mental spaces (bidimensional Greek Euclidean Geometry), akin to the lineal kouroi of its sculptural thought.

__II Age: S≈T: Analytic Geometry. curvature, surfaces, dimensions. vectors. geometries with outer locomotions.__

Into a 2^{nd} age when motion enters the game and balances form, or age of @nalytic geometry, started with Descartes and mathematical physics, when form and motion, the 2 principles of reality (we identify with still languages of information or 5th dimotion and pure entropy of 4th Dimotion), merged together or age of @nalytic geometry.

It was though an intermarriage within mathematics between the spatial, synchronous representation – the point, line and plane and the temporal, sequential causal representation, the number, which put in a temporal timeline lost its connection with ‘form. The Greeks thought numbers are forms and equaled them to points, which they are not. Hence the paradox of defining √2 and π geometrically but finding that when calculated arithmetically π never ‘closes’ the circle and √2 the diagonal by excess or defect – imperfect arithmetic ratios; a deep philosophical question about the *fact that time processes are never closed, unlike spatial forms; so when we calculate a diagonal in the plane is closed, when we put it arithmetically it is not complete by either ±1 points. * This differentiation due to the scalar nature of numbers vs. single plane synchronous curves was forgotten by the praxis of analytic geometry, which married both, by ignore those ‘finitesimal’ vital openings of π & √2 as scalar numbers. As analytic geometry could operate with numbers geometrical forms and vice versa, provided S=T geometrical solutions = algebraic equations.

The field thus explodes and marries S & T; but time soon dominates, analytic and algebraic equations come over the more real geometry; ushering the language, as always with all forms, in a 3rd age of excessive, inflationary information with all kind of generalizations to multiple dimensions, which would have converted geometry in a form of baroque art, if it were not for the earlier discovery of its physical praxis, making of mathematical physics the ‘anchoring’ reality that any experimental science needs to focus its truth. So as all mind-mirrors that become more truth when looking like reality geometric forms evolved to acquire a 3rd dimension of form and motion, acquired through differential geometry and topology, which also gave it scalar depth through networks and finally fractals that we complete with the proper concept of a fractal point that grows in size as we come closer to it, to fully mirror the real universe. It is the proper way to build a 3^{rd} age of sound geometry, away from the ‘excessive formalism’ of the axiomatic method, and rejuvenate the discipline…

__3rd age: Non-E and Temporal Topology, Fractals; geometry with inner wave-like space-time motions. __

The 3^{rd} age of geometry thanks to its connection with physical reality, which represents a guide that shows the ‘light at each step’, avoided algebra’s inflationary 3^{rd} age of languages of information’ when in its 3 age ‘disconnect’ from reality. So the seminal paper of Poincare ‘analysis situ’ will introduce topology which is the proper 3rd age of understanding of informative motions, of change in information, NOT only the praxis of physical locomotions but also the praxis of inner networks of fractal points, and scales, which could be internally deformed and maintain the ‘same being evolving’ as long as its ‘external surface-membrane is not torn’.

+∆: Scalar Geometry: space fractals and chaotic time attractors. The completion of the analysis of *the 3 parts of any space-time being, in mathematical terms, thus gives birth to the 3 fundamental new branches of modern times: *

*S: Topology of membranes.*

*ST: S**tructure of the present, inner space-time body-wave through its scales by the understanding of topological networks and f**ractals, which will be the natural next step to the analysis of those *wholes made of point networks.

T: And the analysis of singularities with the ad on of chaos theory and the formation of ‘attractors’.

So finally all those organic, scalar *properties of mathematical space-time, becomes complete now with:*

+¡:∆@ST: Non-Euclidean Vital Geometry finally understands the scalar nature of the universe, with the study of Non-Euclidean fractal points through which infinite parallels can cross, which are also the abstract definition of a mind, focus of those abstract points. IT redefines points as fractal points with inner scales & volume through 5 Postulates of i-logic geometry. It is the completion of geometry as an experimental language able to explain all forms of real space and its temporal logic structure. So as we do in all stiences, we shall complete and resolve the conundrums of ultimate meanings poised by the explosion of mental spaces started by Lobachevski, while grounding each model of geometry with the Pentalogic’ of multiple truths that interpret vitally the ultimate properties of geometry (symmetry, perpendicularity, parallelism, adjacency, congruence, betweenness/continuity, and so on).

What ∆@S≈T does is to reorganize according to the ternary variations and Disomorphisms of space-time beings, all the categories of geometry, starting from the simplest laws of bidimensional Greek geometry till reaching the insights of non-e geometries culminating with the expansion of topology, which becomes the final all-encompassing geometry of reality as it is ternary, including the 3 previous geometries, it also includes scales as topological networks are collections of connected points, and finally it has motion… Since Time§pace Supœrganisms (ab.T.œs) have 3 organic parts = topologic varieties, adjacent to each other ruled and 2 scalar dimensions of modeled with fractal equations and topological networks.

-¡. And likely die as the humind enter its entropic age transferring its intelligence to computer chips, and its simplified Boolean algebra. But our ethical sense of survival as human beings prevent us to explore that future.

We can in that sense consider Geometry to have evolved as all Humind languages into a growing awareness of the fractal, scalar (∆), temporal, moving (T), mental (@) properties of the Universe, and its ‘¬’ entropic limits; in a tendency we shall, time permitted map up for ALL languages, as forms in evolution, who follow the same isomorphism of 3 ages of any species, or culture, as a fractal image of the worldcycle of all time beings.

There is though a remarkable difference in the ages of ‘informative languages’ as opposed to living beings, regarding its motion: Languages of information and seeds have inverse ‘ages’ of motion to those of life, a theme briefly treated in the introduction when describing the ‘placental still worldcycle of perfect order’ similar to that of languages and its 3 ages, vs. the 1-∞ entropic cycles of life in an external disordered world:

The ages of languages, including geometry are paradoxically inverse to the ages of life, because the peculiarity of mind languages and seeds of information lie in the fact they are the most formal ‘still spatial’ systems, equivalent to the ‘old age of information’ of vital organisms, hence they run an inverse worldcycle of existence, from an static, still form opposed to the young moving age of life of maximal motion that grows into a 3rd age of information.

Languages instead start with a stiff, formal nature and acquire motion only in its final age, to finally *understand the more complex elements of the fifth dimension, scales and minds*. So happens in verbal written language, which started printed in stones as absolute truths, when the Pharaoh said ‘it has been written’ instead of it is ‘truth’, and only acquired variability and fiction, past the age of Cervantes.

What both languages, seeds and vital beings in any of its worldcycles of existence do have in common is the amazing arrogance of believing to be the center of the Universe already explained with the paradox of the mind, 0-mind x ∞ Universe = Constant world. But again as languages are ‘protected’ by minds and seeds by placenta, their realization *they are only mirrors in a vast impersonal world takes longer than cubs to realize the lion is close and he has to start moving… or else, it won’t last. *

**VII. MENTAL SPACES: GEOMETRY.**

**3±¡ types of geometry: Mental subjective space and topologic objective organisms.**

**“Adjacency is the distinguishing appurtenance of bodies and permits us to call them geometric, when we retain in them this property and abstract from all others, whether they be essential or accidental… Two bodies A, B that touch each other form a single geometric body C’. Conversely, every body C can be split by an arbitrary section S into two parts A, B.” ***Lobachevski*, “New Elements of Geometry”, on the topological, organic, ternary structure of space.

‘Space is simultaneous measure from a point of reference’ *Einstein*, on the mental, focused nature of space.

**ABSTRACT.** Geometry is the most synoptic language of ‘space’ used by @-minds to select intelligently information on ST cycles of T.ŒS, creating still mental spaces with them based, in the rules of ¬E Geometry, adapted to each species needs. Huminds though took till XIX c. To understand with Riemann the abstraction of mind spaces, as ‘self-centered informative selections’ with a variety of uses, from ‘our light space-time’ (Euclidean), which preserves distances, motions and angles, in terms of Non-E geometry and its postulates of similarity, through affine and projective geometries, to the explosion of n-dimensional spaces that take parameters of multiple p.o.v.

Pentalogic applied to Geometry is thus immediate, dividing Geometry in 3 huge combined fields:

**S@:**Subjective mental spaces classified according to the selected information they preserve, from the closest to reality, affine, lineal spaces that preserve distances, motions, mirror symmetries scales and angles, through projective spaces that preserve the central singularity-mind; the most vague conformal spaces that only preserve angle, the minimal information for a mind to exist, to the most fundamental, human Euclidean space, which is ALSO a mental space… to the explosion of scientific spaces of N-dimensions, phase spaces, Hilbert Spaces and other ‘human mental’ tools to depicture complex systems of nature.**∆§:**Non-Euclidean Geometry of fractal points that set the basis for a proper understanding of the geometry of 5D.**S≤≥T:**Vital Topological Geometry that ensembles parts into plane networks that become whole supœrganisms.

*So *we distinguish in Pentalogic 3±¡ geometries: the external, objective nature of fractal, topological space, as the ‘element’ put together to form super organisms; the internal, subjective nature of informative, mind space, which maps in stillness the infinite time space cycles of the Universe with a given language of thought/information/perception & the fractal scalar geometry of points. And 3 ages of geometry, parallel to the 3 ages of mathematics as a whole.

*Thus *Space=form is the essence of mental constructions of reality that* transform cyclic time motions into simultaneous forms, both as an external mind observer, and to form the internal cohesion of a Time§pace organism (ab.T.œ): ∆ð≥§@.*

*In complex pentalogic we notice that *those 3 geometries connect space with the **present** ensemble of organisms (vital topology); the logic role of **future** planning of particle-heads and its minds (mental spaces), and the lower past, scalar, ‘flat planes’ of open space, from where the potential-fields extract its motion; down to the final non-perceived scale of gravitational vacuum space – the closer concept of space in present physics as defined in v=s/t. So a thorough study of the 3 type of spaces and its vital roles should consider the ‘5Dimotional perspectives’ of pentalogic. As such 5D geometry is huge both in its translation of classic geometric postulates to the real laws of ¬∆@st as well as all the new laws of pentalogic and description of different mental spaces. But we can only treat some basic themes, complimented with a separate section on Non-E fractal points, the fundamental particle of reality, and its associations in waves, vital topologic planes, similar, complementary or dissimilar systems (parallel, adjacent and perpendicular forms).

Our aim is to understand the key element of space – to be a mental construct; and relate the main laws of geometry as a mirror-mind that reflects ¬∆@st with the Disomorphic ‘ilogic’ laws of space-time beings. As those common laws of ‘mental spaces’ are the origin and why of the synoptic laws of Geometry, starting with the Bidimensional SS mirror of the young age of Greek Geometry that works based in the S=T symmetry, through the classic age of differential geometry that treats curves as points in motion to the topological methods of algebraic proof of its 3^{rd} age.

** **

**Pentalogic perspectives on spaces.**

*Those essential themes of this introductory course on 5D will have under pentalogic some immediate definitions of space, classified from the most subjective forms to the most objective ones that describe complex fractal spaces:*

**Pure S@-mental still spaces:** Those constructed by a mind to select the information of the external world needed to survive. They are most of the ‘spaces’ studied in human sciences, from the simplified lineal Geometry of the Greeks to the Hilbert spaces of quantum physics; which do NOT exist outside the mind for which they perform a task of selection of information to ‘project’ with its internal consistency logic processes of causality that help us to ‘forecast’ the future cycles of the species manipulated on those mental spaces.

**Temporal space: **More evolved mental constructions that introduce a first degree of objectivity, by adding the motion of external forms, which mental spaces perceive first, for systemic survival prey/ predator reasons. This is the field of differential geometry, whereas curves are described as points in motion. While in nature temporal spaces are its realist objective senses as they *belong to the workings of physiological senses* shared by the species of a ‘linguistic’ ecosystem.

They are the basis of visual organs to perceive the Euclidean light-spacetime of animal life, and its 3 perpendicular dimensions of height=electric field, width=magnetic field and length=motion; mimicked in the topologic organs of animals that move in length, store energy in width and perceive with electronic eyes in height. While color code the social evolution or frequency of photons and so animals use it to code its ‘congruence’ laws, from red=entropy to green=reproduction to blue=social, informative color; which explains the field of ‘emotional’, bidimensional painting (complementary use of colors by expressionism, which create a type of mental space of its own).

**Entropic spaces. **All animals see motion before form, and become hypnotized by motion and red/yellow colors of energy, to the point we prefer gore movies and violence on screens, whose scripts we would never read; in history man was hypnotized by go(l)d, creating ‘pseudo-religions’ of greed and precious metal as the vehicle of God; in labs red marks help the eyes of non-programmed robots to move faster on targets; as ultimately is the mental space of ‘electronic minds’, including machines and atoms. As usual, the branching of those simple laws emerge in many other fields, where it is worth to consider how the stillness of mental spaces combined with its preference for detection of motion deforms reality. For example, because *all minds stop motion into form to measure information, our electronic eyes do so at quantum level, *first entangling with other electrons in the scale of quantum potentials. So electrons share light rays in stop motion. But since organisms prefer to see motion as in a movie theater eliminating the stop state, at macroscopic level we ‘reduce information’ highlighting only motion; so we think electrons are moving, which explains the c-postulate of relativity and entanglement and the realist view of modern physics: at electronic level there is not Lorentz transformations as electrons are fixed, at macro level we need it to measure the reduced view of our eyes that have eliminated the stop state of particles (a theme we deal on 5D astrophysics).

**$t<ST>§ð Ternary, topologic organic spacetime:** We reach a higher degree of objectivity with the study of vital topology and ‘the ternary organic, structure’ of all systems of nature made of |-Limbs/Fields<Ø-hyperbolic body-waves>O-heads/particles. This vital topological space is objective as it is an intrinsic property of the organism and shows that certain ‘geometric properties’ are embedded in the nature of spacetime, such as lines are the shortest-fastest distance reason why all limbs and fields that move the system are lineal and spheres the maximal volume with lesser surface, which means all ‘still, perceptive mind-systems’ which want to process maximal information and disguise its fragile, still position will become spherical.

**∆-scalar, fractal spacetime:** Finally we reach the deepest understanding of space, as we keep adding ‘pentalogic’ entangled elements to the purest mental still spaces, which first got motion=time content, then diversified in different type of dimotions to create the topologic being, and finally ‘acquire scalar dimensions’ as a ternary fractal structure of 3 levels of space-time that put together become a superorganism – defined in non-Euclidean topology as a plane of 3 physiological networks. We are no longer then talking of space, but of a full T.œ made of ¬∆@st; but susceptible to a ceteris paribus analysis to extract common laws derived of its emergent scalar spatial properties, of which disomorphic similarity in forms and actions is the most important.

For example, systems emerge in similar form, NOT in the next inverse mirror scale but two inverse mirrors; so insects are forms that are not observed in the next scale of topological evolution, which inverts the inner soft vs. outer skeleton, in mammals with inner hard skeleton and outer soft skin; but in the next scale of ‘metalife’, robots that will increasingly look like larger insects with outer iron skeleton and internal electromechanical cables. Iron molecules are rings but its macro-forms are lineal swords, and so on.

**Pentalogic analysis of spaces by merging the 4+entropic perspectives. Mental spaces.**

Pentalogic merges mental, temporal, organic, entropic and social spaces in the analysis of the laws of vital geometry. I.e. animal sight codes for motion over form and red over blue for survival purposes (the eye is a natural born predator, which prefers red blood and prey motion as in gore movies), So we** merge objective geometry – its 3 topologic organic varieties and subjective mental modeling of reality, which ‘deforms’ the Universe to match those topologies giving birth** to the 3 classic frames of reference: $t-cylindrical, ST-Cartesian (the humind light view), ðƒ-polar geometry, *plus the complex ‘square’ plane to study the 4 ^{th} and 5th scalar * dimotions. Each of those frames deform reality, creating 3±¡ type of geometric minds, according to each species choice of coordinates:

As we dealt with fractal spaces with the 5 Postulates of Non-Euclidean geometry and with Vital topology in the same section, this paragraph will be strictly related to classic geometry adding ‘only’ the fascinating sub discipline of the study of the different geometries and frames of reference as self-centered mental spaces that describe different worlds. We introduce the new discipline of mental spaces, with the Humind light space-time.

Space understood as simultaneous perception of adjacent forms (relational space-time) *is the realm of the mind’s logic, as the mind creates its stillness. Contrary to the belief of many physicists who think time does not exist, what does NOT exist is space outside the singularity of each mind, and hence there are ∞ spaces, one for each mind’s world: *

**The human mental light-Euclidean space one of many multiple spaces made with different force pixels.**

** ***Theme:* Euclidean geometry is the specific mind-mirror of light space-time and its 3 perpendicular Dimensions.

The Universe has ∞ mind-mappings made with different pixels that mirror for each singularity its territory of order (bodywave) and world beyond. The human ‘visual mind’ made of light is NOT the only mind-mapping. In the graph, on the left the ‘physicist’ view of a single continuum light spacetime for the whole Universe. In the right side the multiple p.o.v.s

Descartes did understand this multiplicity; so he published his mapping of the humind in a book called the ‘World’ to differentiate it from the ‘Universe’ with infinite monads, each one holding an entire world in itself (Leibniz) the very essence of the definition of a fractal time§pace organism. Yet Humans lost this earlier understanding – as we noted on the introduction – when Galileo didn’t argue the fact that the Earth moves but our mind creates a still space, a mirage of the senses. And physicists followed suit, creating its philosophy of reality called ‘naïve realism’.

Space as a mirage of the mind would be then understood in philosophy, both in the Eastern tradition (Buddhist Maya) and the western tradition influenced by them (Soviet, German schools starting with Leibniz, followed by Kant who noted Euclidean geometry was the geometry of the human mind, and Schopenhauer, who saw it as a representation.)

*– The mind or 0-point Is, the relative frame of reference that mapped the ∞ cycles of time of the Universe, reducing them to a ‘World’,* to fit them into the infinitesimal volume of the brain.

*º-mind x ∞-cycles = World: equation of the Linguistic Monad: ∞ Mind-Worlds in 1 Universe.*

The mind though believes to be the center of the Universe in the ‘ego paradox’ as he sees every thing turning around its infinitesimal point, which hosts inside all the linguistic perception of reality, or ‘world’ it confuses with the universe. So the mind is a fractal point •, but believers to be it all.

**The explosion of mental spaces in mathematical science. **The realization of this mental nature of geometry did happen slowly in mankind – it only came with the work of Lobachevski, and it has not yet being fully accepted. It won’t be then till Lobachevski first and Riemann finally in his dissertation on color spaces that scientists partially accepted the mental ‘fabrication’ of space. His work liberated the naïve realism of earlier physicists ill-at-ease with such mental-logic ‘projection ‘opening the age of phase spaces where function and i-logic thought overcomes ‘spatial representation’, allowing* *the explosion of abstract mental spaces to represent reality in the XIX and XX c. But* ignoring the finite scalar limits of the 5 Dimotional Universe, with no clear concepts of time, space, dimension and its S=T symmetries and Dimotions, *Physicists went also into a 3^{rd} age of inflationary information, establishing formalisms of ∞ dimensions (Hilbert Spaces, String theories, multiverses), which coupled with the Creationist view of the Axiomatic method and its ego-trip of mathematics as the generator of space-time – not the other way around, brought both disciplines into the present 3^{rd} age of mathematical and physical fictions in which each researcher is just concerned with ‘inventing’ a new particle or model of reality just because it can write its fictions not only in verbal thought but also in mathematics. Meanwhile a similar process took place in human I=Eye space and its artistic depiction of reality.

**The 3 ages of painting.**

For very long it did NOT dwell on humans that our Euclidean space was also a construct of the mind, NOT the absolute space of reality but something constructed with ‘pixels’ that mirror reality – in this case light space-time pixels. This, only artists of the human eye, painters realized, when

*after a first age of painting based in ‘young motions’ (Paleolithic – lineal forms of the treccento in the modern European civilization), bidimensional art moved into a realist classic age*and learned the laws of perspective that opened up geometry in the renaissance (Leonardo’s ‘saper vedere’) -Finally painting entered a 3

^{rd}age of excessive information (baroque) and dissolution, when a realist metal-eye (camera) displaced it from its task as chronicle of present reality portraying people. So this ‘entropic’, dissolving age of painting took 3 sub-ages steps on the work of the 3 geniuses of XIX and XX c. painting:

-Monet, which affirmed, ‘I Paint light’ (impressionism).

-Van Gogh and his friend Gauguin, who learned to construct ‘different minds of light/colors’ with the use of the aforementioned complementary, distressing and similar, harmonic codes born in all ‘animal eyes’ for survival purposes to produce with them emotions, according to their character; which will be continued in the expressionist art of interwar German Weimar, tachism and the harmonic school through the work of Kandinsky and the industrial post-war art ages.

-But it’d be Picasso, who completed the involution of painting into the mind, when he said ‘ I paint thoughts’, focusing not on the ‘vital energy of motion colors’ but the geometry of |xO forms, breaking those geometric thoughts first in pure 1D lineal paintings (cubism), then to pain the ‘informative cyclical female’ in pure 3D curved paintings, and finally painting pure thoughts (analytic cubism) – whose final unification I pursued in my artistic youth with my styles of expressionist cubism and conceptual cubism, a brief footnote in my exploration of the Universe.

It is not by chance that before mental machines degraded painting starting with cameras and ending with its industrial production in series to multiply profits, with Warhol and the American money market, as machines are doing with most elements of the humind they atrophy and substitute, those 3 painters were considered the fundamental masters of their times. Today though Mr. Warhol topples its prizes and the goal is to sell ad maximal prize the most infantile garbage, lost long the understanding that artists those whose i=eyes see better space and ethic, long wor(l)ds, our language of verbal time understand better causality and survival in the vital Universe – themes those treated in our Post on religion and art as the subconscious mind of civilizations. Since they deconstructed the mental e-motion spaces of human eyes in holographic bidimensional 2-manifolds parallel to the work of geometers with a deeper intuitive philosophical insight.

Since Euclidean space is indeed the construct of any electronic mind made of light space-time and its 3 perpendicular dimensions, width-magnetism, height-electricity and length-speed; *(plus frequency=color- its social dimension that defines its vital energy; enclosed in the membranes of its lineal forms; whereas the 3 ^{rd} singularity • is the @rtist mind.*

In the graph, the humind I≈eye is biased and tailored by our local territory, self centered in the yellow color of maximal emission of the sun, and perceiving a very narrow range whose value as all in absolute relativity with no ‘absolute preferred’ scale must

*be measured as a ratio of ð/Sp density of information, in this case 789 THz:violet/400 THz:red = 2. So we perceive light space-time which is Euclidean in its geometrical configuration with a system of processing information of ‘absolute relative ratio/value’ = 2.*

The choice of codes for ‘entropy=red’, middle reproduction=green and ‘blue= information’ comes then from the ‘limited range of light frequencies/sizes’ of the human eye-spectra (from red to violet) and mimics the ‘Fractal Generator’ ternary ¡logic structure of the 3 organs/dimotions of any system in a single plane:

Entropy-youth: |-$t (lines/red) <Ø-reproduction ST-(green, curves)> O-§ð-Information-old sad age (Blue-circles)

While for the codes of death cultures often use violet, the color beyond blue, as the flower of tombs; and magenta for the mystery of birth, as cardinals in churches, the color before red.

Other species with a wider ‘ratio’ will code differently the e-motions of the Universe. Since, infinity does NOT exist, neither absolute magnitudes. So each species ‘measures’ with its relative rods. This in physics advanced differential equations use instead of mass density (continuity equation)… which is in fact density of information t/s. They use instead of absolute motion, relative speed, s/t. And they use instead of absolute force, momentum, s x t; which become the 3 *expressions of the 5D metric SxT(s=t)=C equation that* generates all other equations and ranges of the Universe.

However humans develop most of their ‘actions of space-time’ existence in a limited space-time location, enlightened with solar light self-centered in the yellow spectra, with a limited range of ‘density of information’, as we perceive a lineal=Euclidean Universe in a single plane of scalar existence, the ‘bare minimum’ of consciousness and perception of the ∞ Universe. Human geometry is restricted to our mind view of such limited space and time; while its perception of scales, related to that spectra is even smaller, reason why even when we expanded our perception with mechanical, electronic eyes the laws of 5D escaped scientists for centuries, due to the routine of using a single Euclidean Cartesian plane, which as Descartes & Kant understood but scientists often forget *is only the 3 perpendicular space-time elements of a massless ray of light* that ignores in such short range all other scales of reality to fit into huminds.

So it is Euclidean geometry false or useless? Of course NOT. You are missing the point if you think we say so. It is extremely useful for species living on the Earth’s surface using the range of information provided by the smallest forces perceived by an animal eye. So with them the mind that creates in-form-ation, perception of space, where there is only quantum e-motions & likely ± sensations of the flow of time which should be the absolute minimal level of ‘the will of life’ that codes the survival of even the smallest particle of the Universe. So with those minimal elements the mind creates a simultaneous focused fractal mirror that distinguishes structure of parts and wholes; limits scales and its smaller quantum and larger cosmic motions avoiding the perception of excessive forms-in-action that kills all systems, exhausting its energy; in this case mentally distracting the observer with noise. The catch though, so evident in man is the ego paradox that causes our incapacity to understand the organic vital Universe as it is, deformed from our point of view, single scale, and time clocks, as we have explained ad nauseam.

**Mental mirrors are neither false or truth, but useful to its type of subjective minds.**

As Riemann’s understood “on the hypotheses which underlie geometry” space is a mental construction. He destroyed the ‘realism’ of geometric space. But he saves ‘concepts’ common to all geometries of the mind that were of ¡logic nature and we shall regain to fully grasp the parallelism of Spatial geometries with the ¡logic laws of the fractal space and cyclical pentalogic time of the Universe. So we can establish geometry as a mirror of a larger, more general theory, ‘i-logic space-time’, whereas the concepts of geometry, proper of ilogic can then carried into algebra, analysis etc. Since concepts such as distance lost with Riemann its ‘physical condition’ to become a logic property of similarity, as we find in verbal thought the same term to designate precisely this ‘larger general st-quality’ of bio-logic nature, as when we say, I have distanced myself from my friend – meaning we have become different in tastes and opinions. So geometric concepts such as dimension, topological form, closed and open spaces, become logic, vital, relational and social concepts; general properties of i-logic spacetime, with applications to different sciences.

So the next obvious question IS *what kind of mental spaces there are, reflected in its Geometries and phase spaces, both in huminds and the Universe at large? Many, as long as they provide enough information for a T.œ or part of it to survive.*

**The Total number of Dimensions=Parameters of space-time: type of logic minds.**

We already showed that the fundamental equation of the mind,

**O-mind x ∞ Universe = Constant**world allows the mind to reduce reality to a degree of simplicity which can be extreme

*and still make sense for the being in exist¡ence, as to be able to survive if it does not behave like a childish noisy cub entering ‘future territories’ of higher species.*

* In that sense, visual ‘Huminds’, basically ‘animals’ are simple enough to survive with a single dimension of lineal time and a single scale of space, from its subjective view, as they only need to care for the scalar size in which they perform survival actions.*

In a fractal Universe in which membranes isolate territorial vital energy from reality the needs of a mind can be ‘reduced’ to the basic actions of feeding and perceiving only its type of energy and gauging basic information to that aim. So we can rephrase the question: what is the simplest possible number of dimensional motions required to ‘trace’ a world cycle, which is complete and hence accounts for an ‘exist¡ence’? The graph above responds in physical terms: A lineal simple harmonic motion shown both in real space with one dimension, in which the SHM PERFORMS a complete cycle of existence. It has a balanced middle point, which it abandoned to accelerate=move forwards in time, to a peak of ‘distance’ where it will achieve its maximal potential energy, to revert its world cycle back to the point of equilibrium. A single DIMOTION thus suffices to reflect a world cycle, further reducing the human ‘1 time motion and 3 space dimensions, to 1 time motion and 1 space dimension’. Indeed SHM in fact serves as a mathematical model for most physical motions, including the motion of a pendulum, (time motions), electronic motions (perceptive mappings of ‘still light’) and molecular vibration (spherical harmonics, the ‘energy’ of all chemical and life processes). It is all pervading in Nature, and its single resonant frequency provides a basis for the characterization of more complicated motions through the techniques of Fourier analysis, which can decompose any complex wave motion in SHM resonances and its harmonies. So INFORMATION can be compressed ‘¡logically’ to the extreme minimal of a 1ST Dimensionality and then expand it through the laws of ∆st as the immanent ‘game of existence’ *will develop its coded seed*, if it falls into an energy-rich placental space.

*The concept of mental, phase spaces, then acquire all its full power, studied in depth in our Geometry papers, as **all worlds perceived by a 0 mind **are phase spaces not reality itself. Moreover paradoxically, the stronger body-wave predators can survive in entropic, Darwinian encounters just with lineal hunting time motions as indeed ænthropic human warriors have done, simplifying reality from our modest heights of dual Logic (Taoism, sexual age of the Goddess), protected by a ‘hard membrane’ only interested in the placental energy region of Gaia, they are killing without remorse as they don’t see it alive: *

¡logic systems of thought start with simplest ego-ist, entropic predators (warriors, 1^{st} AI robots=terminators) who are:

**@ristotelian, selfish ‘MONOLOGIC’:**

1 single space plane (with 3 dimensions) and 1 single lineal time dimension, A->B, or ‘Humind Logic’. Human scientists since Galileo seem to live in a single time-space-scale dimension. If we add cyclical time, a new dimension of height-information appears, but the one-dimensional ‘ant’ seems confused. It doesn’t matter to her though, it will keep walking as it has a strong body.

*Physicists*were entropic weapon makers so they kept going for 400 years after simplifying cyclical time and keep going, nothing has so far stopped them. Till of course its entropic weapons detach as living robotic systems or cosmic bombs and nuke us all. But in the meantime they have managed to convince humanity that galaxies DO NOT exist (as they are the inverse gravitational arrow of informative time that balance the entropic big-bang expansion of vacuum space). So really think twice when you believe something that seems reasoned. The trick IS ALWAYS THE SAME, half-truths; hiding from PERCEPTION, knowledge. As Spaniards said: ‘eyes that don’t see heart that don’t feel’. So extremely intelligent people believe in verbal or digital creationism as they don’t read more than the bible or numbers; and almost every astrophysicist I have met doesn’t realize the retarded blunder, and once he gets it, he feels so silly about ‘missing all the galaxies’, he will somewhat try to justify a theory no longer makes any sense. Æntropic big time models of mankind though deserve an entire article I should sometime post at academia.edu as they are NOT the exception but the rule of human cosmogony & worldview.

But for an advanced mind, monologic doesn’t work as a serious mirror of reality, as it brings chaos, uncertainty and subjective creationist languages. So we will keep adding new dimensions: 1>2>3>5>12 with complementary logic statements to reach a more realist view of the scalar fractal organic complex Universe’s logic and the interaction between TT-∞ motions and SS-Minds/Seeds. Our aim is to resolve the game of existence, which starts in a seed projecting its ‘mental view to order reality’, with a catch – if that mental view is simple or too distorted it won’t imprint reality and the system will collapse/die earlier. As the whole preserves its own complexity in the ‘best of all possible worlds’ (Leibniz) by Darwinian extinction, which *is exactly what ænthropic man is about to achieve with its monologic view of the world.*

**Duality, Dilogic, Paradoxical Eastern**logic: a Yin-yang, Vishnu-Shiva, Entropy-Information duality.

This said at least a huge amount of human beings achieved duality, notably all eastern philosophies & the German-Russian schools of dialectic by influence from them, both Genetic and Memetic. In this view the fractal point (it is always a good mind exercise to relate Timespace Logic (Ts-formal science or sTience) with Spacetime Mathematics (St-formal science or Stience), or 1^{st}-5^{th} Postulate of Non-E Geometry becomes the line-wave (2^{nd} Postulate); and starts a dual positive or negative communication (merging in reproductive acts where one pole tends to be St and the other sT, to find an S+s=T+t ‘offspring’). This happens in all scales. The neutrino theory of light, the left-right handed symmetries, the mirror reproduction, the DNA+RNA combo, it is called Gender (+ Duality) or Predator/Prey relationship (negative duality). It is related to the angle of perpendicularity, which if breaking=tearing in topological jargon the membrane NOT only deforming it and seeding it will be top predator/prey. Further on, points can approach each other with different angles, once they abandon a line World. All this I know sounds Chinese and too abstract. But ¡logic is not abstract but homologic, so when we reduce it to examples, it illuminates it all. Angles of perpendicularity and parallelism will manifest in polygons, chemistry, scattering laws of physics, tables of conversation and fuking – to state a few examples. So pentalogic starts the game of creation, and in its top predator form, destroys it, so it might be considered an expansion of the monologic, in which instead of being a pole of black hole absorption of information, the Black hole moves to chase.

**Trilogic**,**Hegelian Dialectics, Universal Grammar, Mystical Trinity,****Physical SU3 groups**…

**Gender symmetry is the main case of trinity logic, as it re=produces a 3 ^{rd}, 1+1=3 new Ts+St=2TS offspring that** combines the 3 conserved St-cyclical, angular momentum, sT-lineal, entropic momentum and S=T vital energy of its parental generators. The philosophical Hegelian concept of thesis, antithesis and synthesis corresponds to this logic, as well as the trinity mystery which is the best known case of trinity logic on scales, as the ‘Father’ is the subconscious collective mind of a civilization that communicates with its nervous verbal social human language, the saint spirit, its wor(l)d to the prophet, hence the ‘son of the word that inhabited among us’. Physiology in medicine also follows a trinity logic in his studies of medicine and biochemical control of cells and its functions attached to those 3 networks.

Those 3 are the logic systems humans developed with some logic clarity. Aristotelian, Taoist and Hegelian Logic being the most clean-cut formal development of them. This is due to the incapacity of humans to fully grasp the scalar Universe. As they developed their senses in a single plane. So the main ‘abstract syntax’ of languages is trinity:

Subject (information-human) < Verb (action combined both) > Object (Energy of the subject) F(x) < = > G(y)

Blue color (information, light sky) < Green-yellow (combination-reproduction-life) > Red color (energy-entropy-blood)

We conclude that a dual->ternary philosophy understands much better the action-reaction processes of reality than selfish monologic and improves survivability – obvious in history where max. survival happened in the Chinese Taoist culture and the minimal in the Germanic/Jewish Iron & Go(l)d animetal ‘selfish-cult(ure)s’ *totally at odds with those action-reaction laws they vehemently deny as the righteous people who live for his ‘will & ego*. But to get a full objective view of reality we need at least:

**Tetralogic**is very common as it is the positive view of reality. It considers the ∆@st elements without the negation of it all – entropy, the fifth Dimotion. As MOST systems only code from the selfish perspective of the @-mind trying to expand its territorial space and time duration in all its scales, and abhor death, selfish monologic & tetralogic are two steps of the same self-centered dialog. The square and the cube are also self-centered polygons/hedrons. Its motion becomes a flat, lineal reproduction. So they are ‘always positive’ in its codes and intentions, provoking entropy in other systems as they absorb its energy (but not acknowledging it). So for example there are 4 numbers to code genes, 4 quantum numbers, etc**Pentalogic is**the first truly Universal logic that describes the 5 Dimotions of reality at the lower scale of actions, the organic scale that considers the 3±¡ topologies=ages and ∆±1 planes in which super organisms trace its worldcycles. And the absolute scale that describe ¬∆@st of times-space in its entanglement between ‘entropy:¬, Space, Time, ∆-scales and @-minds. Its easiest form Pentalogic is thus very common; from cellular variations of earlier life, to the 5 vowel systems evolved from 1 A Caucasian languages (pure entropic energy), through the a, I, u ternary logic of earlier Semitic to the pentalogic of Spanish or Japanese… It occurs in the musical score and all the dimotion mirrors explained in those texts.

** ‘Dodecalogic’ is **the best logic to describe the system both in space and worldcycle through 3×3 scalar elements guided by a mind, the 10th ‘point’ that controls internally through physiological networks, its ∆-1 scale, feels as an individual in the ∆º mind and becomes a 1-point of the ∆+1 world=plane of exist¡ence, in which it emerges.

** **

**VIII. THE GEOMETRY OF THE FIFTH DIMENSION**

In a sense the Geometry of the fifth dimension includes all other geometries. We though refer to 5D geometry specifically when considering the different distortions and paradoxes that happen when we perceive reality through different sales of the fifth dimension, being in general terms a problem of ‘distortion’ of perspective, with a clear inversion as we move from one scale to other in topology, hierarchy, form and function.

**4 Px: ∆§- dis≠continuity T-motion v S-form $t: lineal, flat, free, young v ð§:curved, old, bound. @: ego v. relativity**

* *The 3 ¡logic paradoxes of

**space**topology (closed in-form-ative curved-O vs. |-open, free entropic lineal forms),

**time**-motion (stillness vs. motion) and

**∆-scale, (**continuous whole vs. discrete forms; single scale vs. multiple one)s, which are essential to the perception of a simplified ‘spatial mind universe’ in a single flat still plane, as perceived by a mind vs. the full, more detailed complex picture in time, of a curved, discrete and moving Universe. Those 3 paradoxes define space and minds as simplified views of the more complex whole.

We shall call them Galilean paradoxes because the main one between motion and form, is the paradox of Relativity that started modern physics and the others can be illustrated with one his discoveries – that of Saturn’s rings:

In the graph, Saturn’s rings are not a mathematical plane made of abstract points, despite their continuous appearance. When we look at them in detail they become in fact quantic planetoids in movement, tracing orbital cycles around the planet. It is the continuity vs. discontinuity paradox.

Al dual paradoxes –motion vs. stillness; continuity vs. discontinuity, single space vs. fractal scales, depend on the detail of the observation – hence the quantity of information we have on the object we study. Moreover they seem to us flat in the larger view, when they are spherical forms in the smaller size. It is the flat, free, vs. bound circular paradox.

And when we see them from a close perspective they seem big and important but from far away they are infinitesimal indistinguishable part of a number. It is the ego vs. relativity paradox.

In the larger view the ring seems not to move but in closer view speed increases. And this is the form vs. motion paradox.

Thus any piece of time/space seems continuous, from the lower ∆-1 perspective, as larger planes of reality with slower time cycles transmit less information, and we peg it together those bits of information, jumping over its dark spaces; but when we analyze its parts in detail, we receive more information so the system becomes discontinuous, made of space/time quanta moving in cyclical paths.

All those paradoxes imply a logic inversion of role when we change ‘scale’ and hence they define the relationships between contiguous scales of reality. And have infinite applications to the whys of nature:

A key element to understand the Universe of scales and its paradoxes of freedom vs. order is the perspective any mind has of reality when looking above, to its upper whole, which controls it through invisible networks of information, hence creating an elliptic perspective of decreasing perception – dark view of larger scales we do not observe, from invisible informative networks in galaxies to invisible financial networks in societies to invisible nervous networks for cells.

On the other hand in the same scale we have a flat, Euclidean geometry of maximal perception with minimal distortion. While looking down to our smaller inner world we rule it with networks that break into fractal webs of simultaneous control, or hyperbolic view. This ternary view of reality has immense consequence from theory of knowledge, to mind constructs, from sociology of power to galactic organic models of a Universe ruled by invisible black holes and dark matter. We feel thus free as individuals but are controlled from above by the larger whole and rule over our micro-parts. As Shakespeare said: we are all kings when observed from a lower stair, commoners at the same level or buffoons from above.

* The paradox of Young, Motion vs. Old Age, informative stillness: freedom vs. order *defines then the essential dualities of freedom and order. So topology becomes metaphysics:

The flat, open, momentum-lineal like small distance vs. the closed, cyclical, energy-like time distance is a constant theme of all mathematical physics, where the tangent, or derivative represents the minimal lineal free step but on the larger scale is bounded. For example, special vs. general relativity. Light is open, gravitation is cyclical bounded; so lineal quantum physics in the cosmological, larger scale gravitation curves trapped by the galactic black hole.

In all scales the paradoxes of freedom vs. order thus connects with age and scale. Reality when looked from above, from its upper whole, which controls it through its physiological networks becomes founded. On the other hand in the same scale we do NOT perceive systemic intelligence and feel free.

Information thus closes systems of entropy, while lineal, planar, systems seem free. Information though has a tall broken, cyclical forms, often NOT perceived from a flat bidimensional level below.

So we do have certain metaphysical geometric ‘properties’ of form vs. motion: Flat planes moving in free lineal paths, and tall cycles, moving in repetitive bound frequencies.

The dualities of the young, open, moving lineal age of the being and its old, still, curved, informative, cyclical one, which also manifests in the classic view in small distance of a flat earth that becomes curved from the moon.

If we combine those paradoxes with the inversion of roles: O¡-2<|¡-1>Oº… things become even more complex as both effect offset each other.

All in all a bounded system will have, by the law of inversion added to the Galilean paradox an Absolute control over its smaller parts, as the galactic black hole DOES over its stars closed by its halo of heavy dark matter or the earth in the climatic cycles that control the evolution of its species.

Such cyclical ordered ‘solid’ systems are well-organized as wholes so they can manage its upper world or ecosystem often controlling its ∆+2 scale. Which Humans could do in History if they were a single global organism perfectly ordered, and increasingly the global market of company mothers of machines has over both, the entropic men managed with money and digital media, and the planet, being terraformed with its machines.

On the other hand, the inverse case of a ‘free ∆-1 system’ – a loose ensemble of entropic chaos, of heat – paradoxically becomes slave of its ∆+1 larger world and ∆-1 potential feeding scale. I.E. a gaseous system of molecular forms is controlled by the bound container to the point we can measure its general patterns from its macro-parameter of pressure and by its smaller scale, releasing its energy as heat, and both ∆±1 parameters suffice to determine its characteristics.

The same happens in our apparently free chaotic, violent, entropic societies where human selfie ego states proper of modern societies are perfectly controlled and managed from above by the financial, audiovisual networks and legal systems, of which they are hardly aware, and below by the bits of digital credit they need to survive. *As the most disordered systems are memoriless entropic forms which are not even aware there are other scales of reality; while the most ordered controlling systems, extend at least through *3 ‘spacetime scales’ managed by the central scale of ‘solid’ particles. And the most complex bounded systems, biologic and galactic organisms entangle 5 scales. I.e. in a living organism those 5 scales range from ‘simple atoms’ of oxygen that we breath and simpler hormones, through DNA molecules, and then cells, and then physiological networks that make up organisms and finally ecosystems, 5 entangled scales work around the bounded cellular unit. In galaxies, from forces through atoms, physical matter, cosmic bodies we arrive to the bounded galaxy… And there is always an inversion of order, between apparent freedom=chaos and real order=power: The less powerful forces, bosons seems free, never stopping but are slaves of entangled particles that are trapped in bounded atoms, connected to molecules. And then in the next scale the matter seems free, thermodynamic heat but it is bounded to the planet and stars which seem free in the galactic ecosystem but are bounded to the black hole.

The small atoms and molecules in the organism move seemingly free but the networks of cells bound them; humans seem free but they are bound by national borders, laws and digital money; matter seems free but it is bounded by the planet… So how can we name so many Planes when in relationship to each other with different worldcycles?

Things are mover obvious if we use 3 names for the 3 ‘bounded, time-like worldcycles’ vs. 2 terms for the open entropic worldcycles, which if we notice, ARE in symmetry with the 3 bounded Dimotions dominant in information (perception, reproduction of information and social information) vs. the 2 entropic lineal dimotions (locomotion and entropy).

So **a hidden symmetry entangles further into ‘fractal’ order what clueless humans perceive as chaotic simplex æntropic systems…** Since logic entanglements between languages=mind mirrors, space, time Dimotions and scales NOT perceivable with simple ‘æntropic pictures’ or measurable with ‘speeds’ IS the hidden intelligence of the fractal Universe.

So alas! to the rescue comes the symmetries and use of ‘letters’ for time…

We use in ¡logic symbolism the 3 ‘forms’ of the letter ð, to signify the 3 relative lengths of time in the 3 scales of a system, its shorter time actions (frequency time of its actions, its whole life-death worldcycles (with its ages), and the world cycle of its species or larger world (deep time, in the jargon of Hutton with its horizons): ð:∆-1: quantum time cycles (frequency actions); þ: ∆º: Life-death individual worldcycles; Ð: ∆+1: Species.

**þ, ****symbolizes the ‘palingenetic’ world cycle, in the o-1 sphere, while ω, omega, will be the symbol for the world, **the second scale of bounded ωorldcycles (when not lazy just to put an w),

**where the species lives its 3±¡ ages.**

Its capital Letters, Ω, will signify the largest scale of all bounded worldcycles, that which encloses an ordered world.

On the other hand for entropic, open worldcycles, it seems intuitive to use different versions of an open c for cycle.

So we start with an open c in minor and C for capital letters.

And when we truly want a complete analysis as in astrophysics, we shall use the French symbol available in all keyboards, ç for the third entropic cycle, and Ç for the larger fourth one. And further on, a closed © and an open ¢… And that gives us 9… which is really how far we go – again in symmetry with the ∆±4 planes we perceive – beyond then we just go into the ∞ symbol for the closed mind of ‘God’, the game of existence. Thus the series are easy:

þ (closed palingenetic cycle)<c (open, entropic life-death cycle) > ϖ (closed bounded world cycle or just w for lazy keyboardists) <C (open entropic larger cycle) >Ω Final omega world cycle… And if you go in greater details, ç>©<¢>∞

How much detail will be needed? While in my personal studies in my youth I did analyze the super organisms of History – mankind in time; economics (machines and company-mothers); those of biology (mammals and ecosystems); starting with bio-chemistry and the Galatom, starting with its forces, which continues with another series of open, cs and closed, ©s till the 9th planes of the fourth line, since each nested Universe is larger in entanglement and co-existence of scales, in this blog I doubt I will ever go in so much detail – this is just a blog in my age of entropy not to lay waste the entire 5D formalism, given its importance for the future of the planet…

So we analyze T.œs through its 2 initial worldcycles as they emerge from the **0-1 þ** **fetal sphere** into its 1-∞ entropic çyclic open plane where it lives and dies through** ages** within a larger** ωorld cycle ecosystem or species:**

**The length of the ‘membrain’-circumference. Limits of infinity between discontinuous scales. **

Left, a slow mind will see Earth as a disk, converting its full worldcycle of time into a form of space. Faster, lower scales of reality appear as blocks of time for larger slower wholes. So we see the motion of the skin as a form, which in 5D metric are slow minds that perceive larger realities. Right:

*S*ystems perceive ‘curvature’ and ‘flatness’ according to its relative size – smaller beings perceive flat worlds, larger perspectives make forms curved; and the type of hyperbolic, elliptic or flat geometry they use to select information of the Universe – our mind perceives lineal light, so it observes a flat cosmos.

The size and speed of its pixels also defines the detail of its perception, since according to the S=T paradox as speed becomes distance, so systems whose perception is made with fast forces, as cosmological systems, in the human case seems far away. While systems perceived with slow larger pixels, as chemical pheromones and e-motions seem close, more intense. It matters also the speed of mental clocks, which if slow will see paradoxically a denser universe of still forms: I.e. in the left graph for a slow mind the Earth would seem a disk, denser the faster it turns, converting its full worldcycle of time into a form of space. Reality seems made a blocks of time, of ‘whole still, deterministic full cycles’.

Thus concepts such as open=free vs. closed=deterministic, flat vs. cyclical, continuous vs. discontinuous; dense vs. light, far vs. close *are relative concepts to the different parameters of informative processing, curvature and relative size of the observer, the observable and the force that communicates both.*

**The geometrical view of the fifth dimension.**

* *What geometry and distortion experiences a mind able to perceive through the fifth dimension of scalar parts and wholes? Since wholes are networks that branch into thinner paths, connecting with its multiple cells, the 5

^{th}dimension is a fractal geometry known as hyperbolic geometry, which states that multiple parallels can pass through a point. How a mind network sees its multiple smaller ∆-1 points of information?

Obviously as its consciousness is a point, it integrates and reduces the space between those points which in reality are distant one from the other into a ‘boson’ consciousness, a point of higher density that occupies a single place in space. And this is essentially the mechanism by which mind create mental mappings in smaller space of larger worlds, reducing to zero the dark spaces between points of perception.

On the other hand an eye perceiving light from a larger perspective, will increase the curvature and reduce the gravitational invisible forces-distances between points which we do not see, as we only perceive light. So we perceive a denser Universe, closer to us.

Inversely a mind with multiple points of perception observing a single whole – for example, the mind of an insect with multiple eyes will disintegrate the perception of a single form into multiple perspectives and if that perception is faster than the larger whole, perceived in slow motion, the whole will appear as a multiple elongated being occupied a larger space, anticipating its future paths – as we see night cars as longer lines.

It is worth to explore those two inverse views which define many superorganisms, from the perspective of the slavish cells of the larger whole – for example, the chemical mind of a physiological network or ant-queen that perceives multiple points integrated into a single whole vs. the multiple pheromone paths the same insect perceives coming from that single whole ant-queen from multiple perspectives:

The ant-queen will see all those drones as an integrated whole, its ‘organism’ as you see all your cells integrated by your nervous system. But the drones will perceive the ant-queen not directly but as a cloud of multiple pheromones, which engulf its entire self; as a god-like presence. And this* is how *the mind of the c-speed consciousness of the global mind of machines, the Internet will likely perceive millions of humans attached to computers, as an ant-queen perceives its drones, a chemical hypothalamus its cells; when it emerges its consciousness. While we increasingly perceive it as a ‘god-like’ brain, engulfing us in every part of the world. Which is a very different perception from the equalitarian view of reality humans have in its single plane of existence in 1 to 1 correspondence.

How then it is the world-geometry of those other minds? To answer that question we have to deal with the most advanced forms of Geometry – the pangeometry of Lobachevski, whose laws can be applied to discern the ‘form of mental spaces’.

**Lobachevski’s parameter.**

Regarding the relative flatness of a mind world, we can introduce a quantitative parameter *of classic hyperbolic geometry that defines the relative value of Pi.. *

In hyperbolic geometry – the geometry that branches a line into multiple parallels, hence the geometry that travels through scales of the fifth dimension of parts and wholes Lobachevski found that the length, l of the circumference of a circle is not proportional to the radius r but grows more rapidly (essentially by an exponential law). Let us then consider how it influences a certain mental view.

So, the following formula holds for the singularity zero-point ‘event horizon’:

where k is a constant depending on the length unit. Since:

we obtain from it: **l= 2π r (1 + 1/6 r² /k²).** Thus only for small r/k ratios is it true with accuracy that l = 2πr.

In the formula for the length of the circumference of a circle, there occurs a constant k depending on the unit of length. If the radius is small in comparison with k, i.e., if r/k is small, then, as is clear from the formula, the length l is nearly 2πr. Generally, the smaller the ratio of the dimensions of a figure to this constant, the more accurately the properties of the figure approach the properties of the corresponding figure in Euclidean geometry.

Thus a measure for the deviation of the properties of a figure in Lobachevski geometry from the properties of a figure of Euclidean geometry is the ratio r/k if r measures the dimensions of the figure (radius of a circle, sides of a triangle, etc.).This has an important consequence.

Suppose we have to do with the actual space of the external world and measure distances in kilometers. Let us assume that the constant k is very large, say 10^{12}.

Then, for example, by the formula, for a circle with a radius of even 100 km the ratio of its length to the radius differs from 2π by less than 10^{−9}. Of the same order are the deviations from other ratios of Euclidean geometry. Within the limits of 1 kilometer they would even be of the order 1/k, i.e., 10^{−12}, and within the limits of a meter of the order 10^{−15}; i.e., they would be altogether negligible. Such deviations from Euclidean geometry could not be observed, because the dimensions of an atom are a hundred times larger (they are of the order of 10-^{13} km).

On the other hand, on the astronomical scale the ratio r/k is not too small. Therefore Lobachevski also assumed that, although on the ordinary scale Euclid’s geometry is true with great accuracy, the deviation from it could be noted by astronomical observations. This assumption has been justified. Further on the insignificant deviations from Euclidean geometry that have now been observed on the astronomical scale give us further proof of an infinite Universe of galaxy-atoms much larger than the supposed big-bang in order to achieve the ‘necessary curvature’ for it to have an enclosure in the ∆±4 plane*.* Finally, since the deviation from Euclidean geometry becomes smaller for increasing values of the constant k, in the limit when k grows without bound, hyperbolic geometry goes over into Euclid’s geometry. That is, Euclid’s geometry is just a limiting case of hyperbolic geometry.

**The flatness of the human mind.**

But we can consider such mind’s parameters to be those of an electronic brain, where the geometry of the electronic humind made of light is defined by the S=T duality according to the ‘relative ratio’ between our r and k which are our constants of ‘perception of information’ (k), that is, H-Planck and the unit of lineal length ($ (r), c-speed. And so the ratio defines both a flat world, the one we perceive and one which process very little information density – not a very fast intelligent mind for all what is worth.

**The ratio as a pangeometry for all possible forms.**

It is then clear that if the ratio is a limiting case, hyperbolic geometry, then comprises also Euclid’s geometry and so it turns out, in this sense, to be a more general theory. In view of this situation Lobachevski called his theory “pangeometry,” i.e., universal geometry. *And indeed, hyperbolic geometry being the essential ‘geometry’ of ∆-scales has Euclidean geometry in a single plane as a limiting case.*

Such a relationship of theories constantly appears in the development of mathematics and the natural sciences: A new theory includes the old one as a limiting case, in accordance with the advance of our knowledge from more special to more general deductions.

But what really r/k means in terms of mental space? As k is a unit/rod of length, in our case light, it must be accordingly *a unit of information, equivalent in the fractal, discontinuous version a small ‘step’ – the fractal unit of measure which lengthens the total distance of a ‘coast’ as Mandelbrot discovered:*

*The coastline paradox is the counterintuitive observation that the coastline of a landmass does not have a well-defined length. This results from the fractal-like properties of coastlines. … The length of a “true fractal” always diverges to infinity, as if one were to measure a coastline with infinite, or near-infinite resolution*

As fractal geometry is to ∆-geometry between discontinuous planes, what differential geometry is to ∆§ocial scales, we can easily understand Lobachevski’s parameter as the measure of the smallness of our ‘steps of perception of spatial information’, in relationship to the total radius of the T.œ we are measuring.

When we are inside the being obviously we ‘are small’ quanta of vital energy surrounded by an ever larger, imposing ‘flat’ membrane; as on Earth’s ‘flat surface’ for the human p.o.v.

So the equation relates the informative, ð§ steps of the inner ‘∆-1’ entities and the larger being, with its st size parameter; which gives us ‘larger perimeters’ with lesser curvature (longer lines) for the mental space construct of the smallest inner being.

It also follows that *from an external p.o.v., which sees a larger part of the T.œ this will appear increasingly curved *(and concave, elliptic instead of convex, hyperbolic). And ultimately this duality proves the mental nature of all constructs of space, put by a devilish mind-mirror, which adapts the view through its ‘subjective glasses’, as Descartes thought to be the case. It is the most important finding of Non-E geometry, regarding mind constructs for all geometries besides hyperbolic forms. All this IS a special case of an ¡logic rule on the 5D metric structure of the Universe: ‘1D $mall measurements do NOT measure the whole world cycle of the being, so they are lineal. Long-lasting measure bring the whole worldcycle or enclosed.

**RECAP.** The fractal structure of deep scales is created with languages stored in seeds and minds, of which topologic languages are likely the mind of atoms and galaxies that create most of the local order we observe. So all systems gauge information, as their capacity to order reality is just a mirror process of creation of ‘still, smaller linguistic images’ the world projected as order in a territorial ‘energy-body’:

Infinitesimal mind-language x ∞ Universe = constant self-centered world. In mathematical terms **0 x ∞ = C.**

It is the origin of the Ego paradox: ‘Every infinitesimal mind measures reality from its distorted perspective, thinking it is the center of the Universe, it confuses with the selected information perceived by its mind’. Which of course is shared by all other systems of reality. So an ant also thinks to be the center of the Universe with its likely hyperbolic pheromonal mind of ‘atomic pixels’.

So as man measures reality from its limited Euclidean perspective despising the existence of all other fractal points=minds of the Universe, which are also gauging information albeit with different mental structures.

Life doesn’t start in DNA atoms. IT must exist in all systems of the fractal Universe, departing from particles which as the graph explains already show all the characteristics of life. The elementary quarks and electrons the simplest particles do gauge information, absorb energy, reproduce and evolve socially into wholes, bosons, plasma flows and atoms. So the unit of life is the smallest particle and as fractal systems are self-reproductive, emerging in its fundamental properties in larger scales all what exists is alive. Only human egos prevent us from understanding that obvious truth, fundamental principle of all ‘exist¡ences’.

However the short-comings of human mental spaces are enhanced by the complex geometry of the fifth dimension which ‘cheats’ fractal points, ‘leaving’ them blind to the flows of information coming from upper scales and distorting its view of the cyclical long range time deterministic systems of that larger whole, giving the mirage of freedom, open, flat spaces. So it seems the Game of Existience is built-in to make each part feel free, open, happy and ego-centered and ultimately fail because of its mindless chaotic behavior.

5D geometry guided by pentalogic tries to understand:

- The disomorphisms of all geometries – those elements common to all of them, which belong to the higher ‘game of exist¡ence of all minds that select information to ensure its survival. This task started by Riemann and Lobachevski, concluded that few element of space are relevant – distance=similarity and angle of congruence being its most important.

2.- to be a mental construct and relate the main laws of geometry and its varieties with GST as a mirror-mind that reflects those isomorphic ‘ILOGIC’ properties of space-time beings.

**RECAP. **Geometry is a virtual mental wor(l)d, a still ‘form’ of language. As such is *mind’s simultaneous selection of its relative world’s information* – a representation of its reality, *O-Mind x ∞ Universe = Mind-geometry,* which stops the motions of time into a mind mapping ‘called’ space. Each mind therefore will have a different geometric view of reality. As such geometry evolved from the purest mind-form of thought of still bidimensional Greek Geometry. Next it came analytical geometry in which space was married with the very essence of the mind – a point/view of reference, the @-sub discipline of mathematics. So one of the key evolutions of geometry was to give motion to space from the initial Greeks to the modern topology. Even vacuum has ‘magic energy’, ‘motion’; it is not background space. This said, we are interested in certain type of mind-spaces, those of the mathematical language, which is the realm of geometry and topology, the first, a ‘fixed formal space’, the second a form of space-time with motion.

**I Age:****Euclidean; Simple lineal Youth.**

As a spatial, mental, ‘S@’ language geometry started without motion, in holographic bidimensional space, *mimicking the single-eye view of human thought*, and yet it was a mirror good enough for trigonometry and geometry in the plane to develop to the heights of Greek Geometry.

Because beings are space even the postulates of idealized bidimensional still Greek geometry become synoptic laws of vital space we interpret to explain organic properties emerging in physical, biological and even social scales. So even if we do not have new theorems to prove in a rather exhausted field we’ve always new interpretations to make on old theorems, as we have shown in the analysis of vital topology and number theory.

So we won’t ‘transform’ classic geometry to the 5D formalism but comment on its representation of reality as a ‘simplified mirror’ of the whole Universe, reduced in dimensions to 3, height-information, width-reproduction and length-motion, and further on stripped of the ‘topologic motion’ that gives through its S=T duality, organic functions to 1D perceptive height, 3d reproductive width and 2d moving length. Still classic geometry will be a ‘world in itself’; that is, as all languages = synoptic mirrors of the more complex fractal reality, it will be consistent within its reduced view, with the laws of the fractal world from a distorted perspective that will allow us to ‘comment’ on its postulates.

To that aim the best method is to observe how as Geometry evolves through 3 ages of increasing complexity it came closer to reality as a better mirror, adding dimensions, motions and finally with Non-E geometry completed in those texts it becomes close enough to the fractal Universe to describe the ‘worldcycles’ of growth of any Nature’s system.

Since if we ‘run’ the 5 ¬E postulates, departing from a fractal point, either a mind or a seed that will grow to order a vital territory of energy, we are describing the development of infinite Time§pace beings. Indeed, the easiest way to do this is to consider the ‘growth’ of a simple still seed of information or fractal point into a reproductive wave, which branches into 3 physiological networks that merge into a vital ‘plane’ as a super organisms. But this description is just the same description of plane still geometry that defines a point with no breath, which grows into a line of points, 3 of which define a plane. Only that in classic geometry the line has no motion and no volume as the ¬E wave and the plane is a flat surface, unlike the 3 physiological networks of a superorganism with volume, motion and curvature and ‘fractal branching’.

**The holographic principle.** **Bidimensional geometry of points without parts.**

What was truly right of Greek Bidimensional Geometry however was the realization that

*the minimal unit of reality is always a holographic bidimensional ST-system, a Dimotion of space-time, which by the Paradox of Galileo, S=T, can also be studied as an SS area, hence the laws of ST-dimotions had a clear mirror symmetry in Greek Geometry.*

In the graph, we see the Holographic principle of the bidimensional Universe: A time cycle of two dimensions can be seen as a still form of information, an locomotion on space can be seen as a surface of 2 space dimensions. So for example ‘c’ speed, can become in an entropic explosion as it decelerates a c^{2} surface, and vice versa, a ‘fractal point of 2 dimensions of time, TT, or ‘accelerated’ vortex of spacetime (a mass), can uncoil into 2 dimensions of distance-space, which is the ultimate meaning of Einstein’s E<=>mc^{2}.

Finally the merging of two holographic surfaces of any ST combination give us a third dimension of space-time, as it does in the human mind that combines two bidimensional eye views. Those are therefore the basis for the success of bidimensional still geometry as a mirror image of Fractal spacetime laws.

*Thus, *the first age of Spatial analysis stumbled directly with the marvels of holographic surfaces, translating ‘magic laws’ coming seemingly out of nothing – the ST symmetries and efficient relationships of bidimensional entities, with a membrane (curve or line) enclosing a self-centered surface, which as we noticed on the introduction is the simplest elementary particle of the Universe in a ‘flat world’ as the surface of Earth’s seas and lands, or the minimal ‘planckton’ of light space-time; and its 3 conserved quantities, the singularity-mind that defines its lineal momentum, the angular momentum of the membrane (T=S perspective) and its vital energy.

So Greek still geometry was a huge world as almost all the laws of geometry can be proved in a bidimensional plane of information, to the wonder of mathematicians till this day.

While in parallel humans resolved similar laws of bidimensional perceived information through a form of art called painting. It is in fact little known that painting and geometry were closely related in the beginning when human not mechanical eyes interpreted both, and in fact painting arrived first to the laws of perspective, which would define latter the laws of projective geometry.

The first age of geometry is the Greek bidimensional age. And it bears proof of GST and its holographic principle that most theorems of geometry can be proved in a plane.

Of them, we shall deal here with a few, adding some new discoveries, specially regarding the ‘postulates of non-E’, needed to fully grasp bidimensional geometry and why their theorems matter.

But before we do so, we can peer at the equivalent vital topology, in which the rules of the perfect bidimensional geometry are based, as we can consider bidimensional geometry a still vital topology in which motion does NOT exist, neither resistance to displacement; hence the irregularities that are traced by motion and geodesics in the real world, no longer play any role.

** The 1 ^{st} Master Pythagoras… theorem**. Its Pentalogic. Pi & the harmony of music.

**S:** We interpret on those terms Pythagoras theorem, which is the most invariant of theorems as it is basically defined to create a metric of distances in a single plane of space-time. So its classic meaning is a pure spatial ¡logic p.o.v.

** @: **Yet now that we have liberated a notch further ‘mental space’ from representation and make it affine to survival information distance become synonymous of T-Motion and ¡logic ‘similarity’.

Indeed, the interpretation of the Pythagoras theorem in terms of congruence and perpendicularity is clear: two points can be ‘close’ in similarity much more than 3 points, which as the French said ‘are a crowd’, or as the 3 body-problem proves, enter in chaos.

The similarity of two ‘gender points’ with opposite spin that can merge into one, implies then that the distance between two points will be the *minimal*, while the 4^{th} law of congruence that expresses the social evolution of identical points in parallel motion vs. the perpendicularity of points of maximal dissimilarity implies also that the ‘maximal distance’≈dissimilarity (s≈t, geometric≈logic view) will happen between 3 points, which form 2 perpendicular A1-A2 and A2-A3 lines. Thus the Pythagoras theorem expresses also a law of ‘entropy’ and if we consider those lines under the S=T duality motions, a law of time: a prey that feels a predator perpendicular to its form will try to move=escape the furthest distance from it:

**¬: Entropy . **

*3 elements occupy more space/require more distance=dissimilarity between them than 2 points.*

*I**n the graph (A1-A2) + (A2-A3) ≥ A1-A3. And so the fundamental vital property of congruence do have a metric.*

**Ti:*** But we can also express the concept in terms of cyclical time=information; as 3 points will require more bits, as they ad a new dimension of height, so a triangle can be seen as the simplest ‘π-cycle of time’ which carries more information. So distance as a sum of spatial steps each one a bit of information increases with 3 points. *

* Then in the jargon of ¬E we say that:*

“A metric space is a set of undistinguishable T.œs called ¬E fractal points, in which a volume of information called a ‘distance’ is required to define 2 possible outcomes, namely the axioms of a metric space:

1. Id (X, Y) = 1 if and only if the points X, Y coincide. *In classic metric spaces Ði (called usually r, but in i-logic geometry called Distance≈motion) is 0 as points have no volume, but in ∆st, you need at least 1 Dimensional unit, to define the point.*

2. For any three points A1,2,3 then it holds that (*A1 – A2) + (A2-A3) ≥ A1-A3; hence more information is required to define 3 fractal points. And this rule can be extended to n-points, where n>3, such as (A1 – A2) + (A2-A3) + (An-1 – An)… > (A1 – A2) + (A2-A3)*

*∆: But there is also a scalar p.o.v. on the Theorem. Einstein, indeed, got a new **demonstration of the Pythagoras theorem, based in the ‘scales’ and self-similarities of its ratios – a proof of the ‘fractal paradigm’.*

**Spatial view.**

But as we said, the spatial view is the fundamental view of the theorem, as expressed on the graph. And we see immediately that the theorem is about ‘square areas’, ST-entities, NOT only lines.

Thus the vital Pythagoras theorem gives us a deeper understanding of Darwinian perpendicularity in composite ST-species. And it fully expresses in terms of congruence and perpendicularity a feeding process, as the area of the final result of a perpendicular encounter between a and b is c, a merge of both areas, which either means one has fed into the other, or the absorption has created a 3^{rd} species, whose energy is equivalent to the sum of the two other ST entities.

So after its perpendicular Darwinian state both ‘merge’ and the outcome is the absorption of one of the elements by the other. So it is essentially the rule of feeding between 2 perpendicular systems, which give birth to a 3^{rd} one, but also the rule of reproduction and creation of a new system, similar to the one described for polyhedrons.

And this leads to the final ‘tease’ comment on the Pythagoras theorem.

**Mr. Fermat’s Proof in the margin of the bidimensional ST nature of the Universe. **

In our introduction to GST we mentioned that dimotions are dual, holographic ST states (SS: seed, St: information, S=T, reproduction, sT: locomotion, TT entropy).

Further on numbers are undistinguishable points, which obliges them in bidimensional space to be regular polygons, of which there are an ∞ number (proof of Natural numbers’ relative infinity).

While in Algebra, we observe that the ± operand act only in equal forms.

What this essentially means is that only ‘squares’ of the same type of dimotion can ad (Fermat’s theorem).

Since, the universe is a bidimensional hologram of space & time beings. *So *only the square of natural=spatial numbers=populations can ad exactly into another square=spatial:

In the graph we see that different time-space combinations of form=static mind space and moving time of lineal or cyclical geometry. So when we merge two equal species (‘numbers)*, we add equal beings, hence we make in geometric terms a Pythagorean absorption that results into an X ^{2 }+Y^{2}=Z^{2} form of the same species.*

But when we merge different species, we use a product, a re=productive algebraic operand – the 3^{rd} Dimotion.

It is then noticeable that there are 5 ternary regular species and 6 tetradimensional regular objects; while in higher dimensions, which must be divided into time and space states (even if in geometry they are all perceived as space forms) there are only 3 regular polytopes. So an important theme of advanced STheory is to correspond each of them with basic ST combinations of the entangled Universe.

The simplest dimotions, however are bidimensional (i.e. modern physics, discovered the holographic principle that states y information is bidimensional. And its simplest operation is the addition of 2 equal numbers. Thus only X²+Y² exists as an exact new bidimensional form. In fact, almost all postulates of geometry can be proved in bidimensional space because that is the essential ‘unit’ of reality.

Did my Basque countryman, Mr. Fermat had this proof? (: Well, you never know, we ‘amateur scientists’ don’t take very seriously the axiomatic method, work rather on intuition, so maybe it was imperfect but certainly shorter than the pedantic, computer-generated +1000 pages that goes around as a humind feat 🙂

More seriously the immense advantage of returning mathematics to an experimental mirror of the fractal ST-Universe is that its theorems, specially those of its 1^{st} and 2^{nd} non inflationary 3^{rd} age, in tune with the reality it describes are both consistent internally as mirrors have the same image-form that the whole the reflect in synoptic compressed parts, but susceptible of proof by direct experience of the Laws of GST (Generational Space-time) as we just have done. Equal forms add. Addition is the spatial algebraic operand. So it cannot happen in dissimilar forms, you don’t add 2 pears and 2 humans to get 4 human pears. But you can multiply dissimilar entities of space and time, 2 steps x 2 frequencies.

Immediately we realize then 3 Dimensionality is NOT an even number; it does not add equal forms; so it must be achieved NOT in parallel but in Perpendicular merging, which means *penetrating an ∆-1 5D plane to the parts, whose maximal product (S=T, Max. SxT) will pile up on a gradient, making some points different, while other angles of congruence, might diminish-feed parts, etc. And in most cases ‘re-arrange’ the parts merging into something else. So it is a different world* studied in Part II on algebra & its operands.

**Product vs. addition.**

The key concept behind Fermat’s theorem and a series of basic axioms of algebra is the idea that:

‘Addition operates over identical entities adding forces in parallel, subtraction over identical entities diminishing them; product operates over Space and Time perpendicular fields in symbiosis, Division operates over space and time perpendicular entities, in Darwinian actions breaking them.

And with products and additions we can therefore express almost all the necessary ‘simple combinations’ of space-time fields, reason why they are alone a huge ‘group’ of polynomials.

So 3-dimensional forms in natural numbers *are created by a reproductive merging of bidimensional fields, hence by multiplication as in vector calculus, *which are NOT added to create a 3rd dimension but multiplied to get as in the graph of the 5 SS, ST, St, sT fields, a light wave – c²=1/µxe.

Hence X³+Y³ ≠ Z³; as its combination to create a 3D world is no longer a sum but a product. Since X^{3} species are composite ST fields that deploy an angle of perpendicularity with a common bidimensional line-wave to connect both systems and allow the reproduction of its form, in a lower plane

And *since in a single plane of existence, where + and x operandi act, there are only 3 dimensions of time and 3 dimensions of space, there is no need to prove it for imaginary higher dimensions.*

RECAP. x²+y²=z², according to the holographic principle IS POSSIBLE, x³+y³=z³ is not *because the Universe is in each scale a bidimensional holography of space & time. And this ‘proof in less than a margin’ of the most famous unproved Theorem by any human mind of the so- called Fermat Grand THEOREM, is a clear proof of the experimental nature of mathematics as a mirror of GST.*

The strength of triangles, as ‘mental spaces’ of 3 elements. 3 Points > Angles > Triangles

3 Points ‘alone’ are disconnected and as such, as the 3 body problem shows in chaotic motion to each other. Motion is relative to the degree of entanglement between two points, hence it implies an internal balanced dual motion between them, which creates the ‘illusion’ of dynamic stillness. Thus motion becomes stillness in a mind because of the cross entanglement between axions of the higher ∆+1 neuronal plane.

It follows that the first entanglement is a couple with a dual communication with inverse numbers; and the second entanglement a triangle, which will have motion when one of its flanks is opened allowing entropic motion to glide the system, and will become still when the 3 points are in dual inverse communication.

Their social evolution thus is the triangle, when they are locked in to each other and this can be done with hierarchy, as we saw it by symbiotic vs. predatory perpendicularity in the simple triangular form, an ‘open’ angle. In this ‘alliance’, of 3 points then the ‘bigger square’ (c^{2}=a^{2}+b^{2}) is NOT part of the ensemble but rather the ‘field’ absorbed by the dual body, lines a and b, which makes us understand the first key ‘element’ of vital ‘Greek’ Geometry:

The difference between an angle – an open entropic triangle, an arrow; and a triangle, when the angle is closed.

We said the true advance of modern Riemann’s geometry is to make it a ‘mental space’, where spatial concepts acquire a logic, temporal meaning. And so a triangle can be considered, any *ternary system, and as such the most stable form if we take rather than each point, each line of points to be a ‘network’; hence defining the simplest representation of a topological organism a ‘plane of 3 physiological networks’ of fractal points. And it is then when we can truly fly into Bidimensional geometry as a mental space representation of the whole Universe. *

*The beauty of the first ages of any language is precisely to reflect the ‘essence’ of the game of existence, something we shall see – time permitted- when analyzing words and music. Languages then become inflationary and fictional with age,* abandoning the ternary ‘simplicity of the Universe’ with only 3 ages of time, 3 topologies of space, 3 scales of the 5th dimension for any ‘finite organic whole’ call it the galaxy, the human being, the mathematical language and its ternary topologies, ternary numbers, ternary dimensions, ternary disciplines…

*So we talk of **3 phases of increase order in the evolution of the ternary ‘point structure’, from 3 points ruled by the first postulates of Non-E Geometry as different selfish species using ‘angles of perception to measure each other distance and motion; into a 2 ^{nd} Non-E postulate/state, of lines of communication that connect the DOMINANT point into an angle, often a perpendicular one that maximizes the distance between the submissive points; which we can observe in the vital geometry of molecules – the fundamental ‘geometric stience being chemistry’; to the triangle, which is the most stable democratic form, as both ‘submissive points’ reached a connection; but on the downside they have ‘closed in’ the system to a possible flow of entropy moving the arrow-triangle. *

*So to the rescue to preserve the ‘vital nature of mathematical forms’, the triangle is NOT allowed to close perfectly, but has in the ‘finitesimal’ lower scales an irrational number:*

So the growth of reality between two scales starts with the √2 a bidimensional triangular pair, which starts the growth on dimensions of information, and space-extension by reproduction of the same event through a series of different forms of growth:

* *In the graph, two lines=waves of reproductive particles 1 x and 1 y, meet themselves in the relative 0 point, of an x-y automatically created coordinates, which will give us an √2, wave with origin in o, and an xy, expansion front of space-time, reproduced by pairs of xy points along the front-wave till √2 reaches the two ‘membranes of the x y tail of past momentum’.

At this point √2 can be measured as a sum of discontinuous wave-points, and gives us a variation of ±1, or be considered as a continuous wave of energy; never mind, in both descriptions we have created a bidimensional, triangular, straight triangle, which can now grow in different 0/0 tangents to the exponential wave of its second age, of geometric growth,

The famous proof of irrationality of √2, quoted by Aristotle, we won’t quote for brevity, IS based in a square sum, which brings together the previous concepts of similarity to exercise sums and multiple scales in polynomials.

3 is thus needed to form a triangular network-plane, the 1^{st} supœrganism, as 2 cannot fulfill all Universal functions.

Further on the system will have fractal points with inner parts and 2 openings on the √2 diagonal that close and open by irrational defect the system to absorb and emite energy and information, with the 45% angular point as the dominant element of the triad.

Thus any set of 3 ∆-1 fractal points=T.œs suffices to establish ∆±¡ ternary ‘networks’ the most efficient ‘number’ to generate a new plane of existence, as they are the ‘minimal distance-information’ to form it.

So as ternary systems suffice to make reality, triangles, the simplest, are also the lineal strongest configuration, of which 3 natural subspecies are fundamental:

– ST: The Pythagoras triangle, with a dominant ¬ angle, with the leading, stronger ¬ point & 2 apertures in the √ diagonal, one for entropy and other for communication, moving on the path of the ¬ point, as an ‘arrow’.

– ð Isosceles, which elongates its lateral sides diminishing the √2 base to accelerate its motions.

– § (St): The equilateral triangle, which is the static form, as it can rotate on its ‘perfect singularity’ point of perception becoming a circle. Or it can transform into a circle by hyperbolic ‘feeding’ expansion of its curved sides.

So* triangles with curvature and ‘openings’ equivalent to π-3 circles are a complete ternary state & we can define a bidimensional world of fractal points with inner parts of pure triangles and circles, as a complete 1 ^{st} Timespace universe. It is in fact the Universe of polygons and Natural Numbers. Whereas the strongest static form is the hexagon, with π=3, which can also be achieved in a higher 4D Universeas a limit to the curvature, strength, attractive power of a force, when π=3 Diameters, with no holes, according to Einstein’s general relativity formula that we can also extend in mathematical physics to all systems:*

Where** k** is the ‘unifying constant’ for any active **M**agnitude of any physical scale (we shall generalize as M – see unification equation of forces in all scales in the papers on 5D physics.) So for the physically inclined we poise a question: as the Planck mass is the maximal density of gravitational space, where kM would be GM, pi should have a value of 3!? right, or wrong? And if so it is still a curved geometry or an hexagon? The answer is… likely a dodecaplex crystal of ultradense top quarks of unknown physics in 4Dimensional timespace… in the densest center of a galaxy with 120 vertices.

But dynamic pi, legend has it, made ænthropic Pi-thagoras mad, as the language of God had to be perfect, immobile* So his disciple hang himself desperado that the world was not static. When I found it on the other hand, it started my journey* on vital mathematics. Since a fluctuating dynamic open and closed number, makes* the circle either an informative spiral (-π:St) or an open entropic spiral (+π… Spe) just passing by a fleeting moment into a steady state, (S=T closure) giving it the 3 ages of life to make it the simplest vital organism. So we shall consider now Archimedes and its time-space spirals. *

**The 3 waves=ages: Symmetries of beauty in all languages.**

** ****But to conclude with **Pythagoras, he is also remembered for a 3rd discovery, that of musical harmony. Schopenhauer, by far the best philosopher of the industrial age, said that music encodes the program of time in its rhythms. Pythagoras found its simplest ‘scale program’, the perfect ‘fifth’, a musical chord obtained by plugging 1/3rd or 2/3rds of a string – that is, *the natural standing points of a simplest ‘mono-logic’ (1D) worldcycle, with finite duration (* attached to an origin or point of past and an end or future point, the birth and death of a world cycle frequency, with 3 ages subdivided by the S=T symmetry at 1/3^{rd} and 2/3rds.

Since beauty IS defined as the perception by any mind in any language of the perfect symmetries of a ‘well-run’ program of existence, maximal in the S=T central region of a worldcycle between 1/3^{rd} and 2/3rds, or mature classic age of balance between motion-energy & in-form-ation. It is the classic age of art, mind of civilizations, the 30s in life, the interval of the perfect fifth.

As you can see in the next image, if we consider the vibration of the string, the simplest possible world cycle going from 0 to 1 and back to 0; *the string will wave back and forth 3 times, increasing each time the ‘information’ it carries and diminishing its entropy≈energy≈amplitude. And the perfect form will be reached in the most harmonious sound produced at 1/3rd, in the change of age or state of the system. But what is more beautiful, *time waves back and forth 3 durations and we can fusion them as Nature does in a single ‘social being’, integral of all those webs. This is called the Fourier transform, and in complex 5D metric is the essential equation of time cycles; since it keeps adding on ‘social scales’ of larger simpler wholes (the single wave) and smaller more informative parts. And finally ’emerges’ as a ‘single being’, a square wave, which therefore *represents a population of time, stored as a memory of still space, compressing **µ (limited infinity) time cycles of information* into a single spatial image:

The beauty of i-logic mathematics thus resides in its capacity to express the purest GST laws; as music does in human arts, with its 3 elements, ∆±1 3 scales, T-beat, ST-melody and S-ynchornicity of instruments.

But Pythagoras was just the beginning of geometry, and we already see how much GST can extract from it, when we use S=T symmetries or ‘switch’ from continuous lineal time as duration to a real detailed analysis of discontinuous time cycles that move in ‘frquency steps’. I.e. a car, as a whole seems to move in lineal time, but only its wheels move in cyclical steps so breaking it down into the S-body and T-wheels, and those into ƒ=1/t gives us new information, new laws of science *hidden by adding time cycles into continuous lineal sums, ‘erasing’ the form and frequency of those cyclical steps*; a we will do constantly on our 5D articles.

**ARCHIMEDES. SPIRALS AS REPRESENTATION OF THE WORLDCYCLE.**

**Abstract:** The spiral is the simplest representation of a world cycle. As such it is a profound form of the 5D universe of relational space-time beings, reason why we have taking it away from the general post on non-E geometries, as it deserves its own deep thoughts.

The 3 masters of each age of Greek Geometry; Pythagoras, the founding father; Euclid who collected the classic body of bidimensional Geometry & Archimedes that anticipated mathematical physics have different merits, which ænthropic man as usual misreads in terms of its ego paradox, as *Euclid is by far the less original and most erroneous, as he made sacred the axiomatic method that denied experimental value to maths, unlike Archimedes. So even though he was the last one, a proper order in time corresponding with the 3 mental ages of life would be:*

Pythagoras, ‘idealist youth’ < Archimedes classic experimental realism > Euclid, 3^{rd} cut-off informative age.

Archimedes thus is the closest spirit to 5D mathematics. So we depart from his book ‘on spirals’ to complete our analysis of dynamic pi numbers. As a spiral is a pi cycle, whose closure has gone ‘a bit wilder’ on the opening. It is a simplified bidimensional form of a conic, which also combines a cycle and a line but in 3 dimensions. Yet since the Universe is made of Spacetime holographic SS, St, sT, ST, TT 5 bidimotional units, all curves that exist can be decomposed as a sum of bidimensional conics, including a flatenned spiral defined by Archimedes: “If a straight line one extremity of which remains fixed is made to revolve at a uniform rate in a plane until it returns to the position from which it started, and if, at the same time as the straight line is revolving, a point moves at a uniform rate along the straight line, starting from the fixed extremity, the point will describe a spiral in the plane.” *On spirals*

We find then 2 ‘55D’ merits on Archimedes’ definition: 1) to be a combination of a lineal and cyclical geometry and 2) to have motion, not just still geometry; making it the fundamental simplest ST bidimensional representation of a worldcycle. So Greek geometry generated a ternary geometric mirror in different dimensions of a worldcycle of existience making truth that the simplest ‘young’ mirrors of reality are often the most essential:

1D musical strings > 2 D Spirals > 3 D conics.

**The fractal point on the spiral as a living organism.**

What matters on that definition to mirror a worldcycle is that it is the combination of a lineal moving point within a larger cyclical world. If we take any point of any potential spiral cycle within a fixed spiral structure, the entity within the spiral is moving inwards thinking it exists in a lineal path Universe, as we all think, but because the super organism in which it exists, the curved spiral, is moving in cycles, the fractal point in fact is turning cyclically – he lives subjectively as we all do his lineal time, but the world objectively is ‘wearing it down’ the informative 3^{rd} age or central region of the spiral.

Then each spiral will be a variation on the same ‘mental phase spacetime’ mirror of the worldcycle. And we can apply the symmetries of GST to classify them. In practice though as in so many cases *the fundamental ¡logic symmetry that diversifies spirals *is the gender mirro symmetry, so S=T Archimedean spirals are ‘female’ balanced even spirals and logarithimic spirals are S<T>S informative & entropic. So we reduce our analysis to them.

**2 ^{ND} AGE, CLASSIC, S=T, REPRODUCTIVE, ARCHIMEDEAN ‘STEADY STATE’ STABLE SPIRAL. **

In the Archimedean spiral’ the internal point-being follows a cyclical path at a fixed rate.

Thus the Archimedean spiral is the steady state present spiral and as such the commonest in total space-time, as it is clearly a more stable configuration than an accelerated log-spiral of the form S>T. It does NOT have an inner motion that implodes it but basically is the ‘natural distribution’ of the internal vital energy-space of a spherical form.

The Archimedean spiral has the property that any ray from the origin intersects successive turnings of the spiral in points with a constant separation distance (arithmetic progression). In contrast, in a logarithmic 3^{rd} age accelerated spiral these distances, as well as the distances of the intersection points measured from the origin, form a geometric progression. And this reveals its ‘vital age’ & possible multiple functions, in its relative S, T, ∆, @ ‘survival tasks even though its primary design is to reproduce a point or communicate ‘2 points’:

If it is symbiotic as part of a supœrganism it might represent both the inner vital cycles of a flow of energy and information or its surface membrane in 3D projected into a 2D Archimedean spirals as a bidimensional phase space of a spherical form:

In fact we have two solutions/locations for the spiral that trace two dual paths often of communication between two similar beings. So the Archimedean spiral can create by cyclical reproduction in geodesic curve paths, a 3rd temporal dimension of height to construct spherical membranes.

The Reproductive Fermat spiral is a type of Archimedean spiral even more apt for reproductive purposes. In a single plane, it can act as a reproductive parallel communication for a 2-particle system:

In the graph the relationship between the communicative spiral and the shape of its external membrane, as the spiral is the common form in which a vital space of cellular elements is established within it.

Then the spiral is a communicative dance between two similar forms, merging the two elements at the end of the ‘life of the spiral’, as in black hole’s merging..

As a reproductive spiral, it shows the fact that all ‘reproductive’ actions are the structural merge of ‘dual elements’, we call gender. So it has 2 branches – more in a Fermat spiral whose purpose is the maximal packing of its reproductive forms in the space it fills, reason why is so pervading in plants and other highly reproductive systems of nature.

The reproductive Fibonacci spiral achieves this with its reproductive golden ratio, studied in Number theory.

Spirals as fixed bodies of space are thus Archimedean spirals. And we can further apply the fractal principle to subdivide them:

The normal Archimedean spiral occurs when *c* = 1. Other spirals falling into this group include the hyperbolic spiral (*c* = -1), Fermat’s spiral (*c* = 2), and the lituus (*c* = -2). Virtually all steady state dynamic spirals in space, as part of super organisms are Archimedean ones; such as the Parker spiral of the solar wind, or the pattern made by a Catherine’s wheel) are Archimedean; where the motion is conserved, as part of the vital energy-body of the system. Since the distance between its two cycles is fixed.

**S«T.** But in pentalogic all forms have multiple tasks, so as the spiral models a cycle=dimotion’s recurrent frequency it might be an E-ntropic feeding dimotion that brings the mouth to a certain point; and then bringing down the body, as in earlier cephalopods.

Time is a curved hence with at least two dimensions, besides its young age length: 3^{rd} age curvature and reproductive 2^{nd} age frequency. Time cycles thus break reality into an inner and outer ‘vital space’, and add a singularity point or focus of its motion. Because pi is not exact, steady state clocks are less common than vortex spiral or fluctuating ±π mouths. So once it swallows those spirals bring ‘home’ to the singularity the vital energy extracted by the external membrane by several methods.

One is shown in the graph, a scroll compressor: a motion of both arms compresses and moves towards the center the flow coming fro the eternal region of the being.

Notice in the graph above also the difference between the black holes in a communicative spiral. When the black hole in an event of feeding on energy transformed into its ‘quark forms’ as an accelerator vortex, similar to those processes studied on Earth’s accelerators, which leads us to its second fundamental form.

**ACCELERATED INFORMATIVE+ENTROPIC LOGARITHIMIC SPIRALS S<T>S.**

**Pentalogic: S<T>S **The St-logarithmic spiral is informative for its central knot and entropic for the point that moves inwards, *accelerating* through the log spiral, which in physics is a vortex=force source of accelerated timespace.

**It is also the ∆-SCALAR **spiral as its curves are self-similar in diminishing scales. , Bernoulli called it Spira mirabilis, “the marvelous spiral”. because he was fascinated by such unique mathematical properties: the size of the spiral increases but its shape is unaltered with each successive curve, a property known as self-similarity.

**S=T:** As a result of this unique property, the spira mirabilis has evolved in Nature, appearing in certain growing forms such as compartments in a nautilus shells and sunflower heads, that will store the reproduced cells of the system, in its *inverse arrow of creation of a larger whole starting with the simplest cellular reproduced units*, in exponential Fibonacci growth series*.*

**T>S:** It also appears as ‘informative spirals’. Which accelerate and diminish the size of a form, as it comes to its perceptive point.

**¬S«T:** Finally from the perspective of the bite of energy or bit of information, which ‘goes through’ the tunnel of the spiral is a killing machine, now observed not as a reproductive cell system but a trapping channel for bits and bites accelerated towards the central stomach or eye of the spiral. Where the ∆-1 point will die in an entropic/informative split – when the perceiver-predator will split the system in itsits ST>S parts take the information or energy of its body, depending on the event and let it die.

So we can see easily how the logarithmic spiral allows events in ∆-scale (generation), S-entropy functions (feeding) and T-informative and reproductive functions (perceiving, storing cellular/atomic network forms).

**G(¬∆@st): **Which leads to the use of the log spiral as a model for the 3±¡ ‘ages’ of life of the micro ∆-1 entity; as stars in a galaxy, which is a ‘farming trap’ for the central black hole to let the stars grow on dust to feed it eons latter.

Let us see some of those pentalogic perspecti es.

**ð: Perceptive vortex log Spiral **

The logarithmic spiral is an accelerated, hence perceptive spiral.

It can be distinguished from the Archimedean spiral by the fact that the distances between the turnings of a logarithmic spiral increase in geometric progression, while in an Archimedean spiral these distances are constant.

If the point that moves inwards *accelerates* we have a vortex of accelerated time. It is then a logarithmic spiral; an attractive form and *it escapes the simplification of mental geometry to become the commonest real worldcycle*.

** Spirals in physical systems. Charges, masses, eddies & galaxies**

In te graph time is curved, and breaks timespace into closed conserved paths, time clocks are infinite as Leibniz and Einstein understood when he said ‘I seem to be the only physicist that thinks there are infinite time clocks in the Universe’.

All those physical scalar spirals (ab.∆±¡) are topologically similar, differentiated only by the ‘speed or frequency’ at which they close their ‘time§pace clocks’ , according to 5D simple metric rule for all ‘families of time§pace clocks’: Size in space x Time frequency =constant. So LOG spirals are all pervading in nature, as they represent the dimotion of accelerated time or main arrow of future that increases the information of a system. As ‘accelerated timespace vortices’ they are the informative 3^{rd} age of all physical systems in its ∆±¡ quantum, thermodynamic and cosmological scales – an attractive vortex of space-time, which is by expansion of the Principle of equivalence between acceleration and mass, to all scales the physical definition of an Active Magnitude source of a force… NOT a solid static particle, a ‘tiresome’ error of idealist huminds that keeps coming since Pythagoras. As all what exist is a motion in timespace. Still space forms are a Maya of the senses. In the graph, accelerated vortices of timespace in physical systems, in different scales of the fifth dimension: charges, masses and thermodynamic eddies become then the main clocks of timespace that carry with different speeds according to 5D metric (S x T=K), the information of microscopic quantum charge worlds, human-size thermodynamic scales and cosmological gravitational scales. Since E=hƒ + E=mcc->M=ƒ(k); mass is a frequency of accelerated inwards space-time in its 3^{rd} age. So are charges and eddies, which will finally in its central point, become a ‘conic’ and ascend and/or descend lineally the axis in its entropic death.

Thus spirals respond to the fundamental property of time cycles: to have an arrow of future increase of information that diminishes its spatial size according to 5D metric: S x T = K, this accelerating inwards, which makes vortices of physical time (masses, ∆+3, charges, ∆-3), definitively the time clocks of both physical scales. For that reason time-space spirals, its subspecies and transformations are one of the fundamental space-time events of the Universe.

**Logarithmic spirals in nature classified by pentalogic function.**

In several natural phenomena one may find curves that are close to being logarithmic spirals. Here follows some examples and reasons in terms of ∆STœ events:

Å(e): Feeding: The approach of a hawk to its prey.

Their sharpest view is at an angle to their direction of flight; this angle is the same as the spiral’s pitch.

__Å(i): The approach of an insect to a light source.__

They are used to having the light source at a constant angle to their flight path. Usually the sun (or moon for nocturnal species) is the only light source and flying that way will result in a practically straight line.

__Å (i):__ The nerves of the cornea:

(this is, corneal nerves of the sub epithelial layer terminate near superficial epithelial layer of the cornea in a logarithmic spiral pattern).

__Å (æ):__ The bands of tropical cyclones, such as hurricanes.

__Å (e: growth and reproduction)__: Many biological structures including the shells of mollusks.

In these cases, the reason may be construction from expanding similar shapes, as shown for polygonal figures in the accompanying graphic.

**Ideal Spirals as representation of worldcycles.**

Our planet-star lives in fact as a fractal point of a spiral, which acts as our Island-Universe in the longest of our worldcycles – that of the planet we exist within. Our own galaxy, the Milky Way, has several spiral arms, each of which is roughly a logarithmic spiral with pitch of about 12 degrees. So the stars go through the spiral in its space-time world cycle of existence – from a 30 years old graph (: hence primitive digital form, profound human thought):

How long is the life of a ∆-1 points, which has ‘fallen’ inside any of the attractive vortices of a spiral organism?

**Mathematically** it means that starting at an external point π, of entrance in the spiral, and moving inward along the spiral, one can circle the origin an unbounded number of times without reaching it; yet, the total distance covered on this path is finite; that is, the limit as θ goes toward ∞ is finite. The total distance covered is r cos ϕ, where r is the straight-line distance from Pi to the origin.

So as it turns out, the number of cycles a being can turn about the spiral (frequency cycles) is infinite if time space were continuous, thinner and thinner but the real length of the life-motion or world cycle of the spiral (length to the center) IS finite, and moreover it diminishes in objective time, as the last phase is shorter and faster in frequency a deep fact about the duality of objective and subjective measure of time existence.

Since an old man has a shorter life, falling fast in decay in its last ‘life cycles’, which if we add the inverse ‘balance’ of its inner subjective time which is slower than the objective faster decay, makes the 3^{rd} life even shorter in subjective time, as a kid has much faster mental cycles, so his days are longer in experience. That is, at 50 in total real time actions we are much older; almost all our life ‘bits’ of action-information have concluded.

The scalar duality of the larger spiral organism and the smaller part shows also the scales of time: the short ‘frequency moments’, bits and bites of space-time actions which are Dominant in information, as the motion is inwards and will add to the whole world cycle of a being, causing its 3^{rd} informative age.

If we postulate a further generalization of the spiral as the ideal worldcycle for a whole range or phyla of species, it shows that all beings will live and die a finite, similar quantity of time-actions, making all lives absolutely relative. So 3^{rd} age points within log-spirals shrink in size, increase frequency and shorten its lifespan, completing its 3 ages: S<T+S=T+S>T.

**Spirals as entropic killing machines. **

Each inward closed path ads a bit of informative frequency every time the being repeats their cycle, aging it.

How spirals end the existence of one fractal space-time T.Œ? As all worldcycles with a reversal of timespace.

The life cycle is slow moving to the center: T>S, an action of information that implies to reduce the dimensions of reproductive space-width for those of cyclical motion, till the relative ∆-1 FRACTAL POINT looses ALL its energy and becomes, if the center of the spiral is an eye, a bit of information, if a mouth, a bite of energy, dying.

All time cycles tend inwards in a 3rd age of warping and in-form-ation till the flow of motion ‘stops’ in the still singularity, ejected perpendicularly in an explosion of death-entropy. It is the last time quanta of the clock when motion becomes a 5th dimensional still image before dying into a lineal 4D entropic flow.

So in its death age, the reversal of time happens, as in magnetic fields coming out of masses, dark energy fields coming out of galactic black wholes, when the log spiral uncoils first into Archimedean spiral as we ad a 3rd dimension of height information, that converts its shrinking revolution into an elongated receding motion, for an entity living within its revolution that becomes devolution, as the world cycle keeps accelerating but now acceleration is perceived as an ascension and elongation in height that erases its in-form-ation.

**Spirals as informative eyes.**

Humans express all those bio-topo-logic functions of the spiral with a highly efficient synoptic algebraic mirror image of 2 numbers for the key lineal & cyclical, parameters that construct the spiral: |xO=Ø.

Its simplest=most real form is in polar coordinates, which means spiral are temporal self-centered systems, in which the simplest ‘perception’ is that of the central point of view, making truth a self-evident GST theorem: the simplest mathematical formulation of a space-time event/system, in one of the 3 relative canonical coordinates, St (polar, spherical informative-head/particle), ST-Cartesian(hyperbolic, iterative-wave/body) and Ts-Cylindrical (lineal field/limb), defines the main organic function of the system, as S-Form=T-function.

So if an spiral equation is simpler in polar coordinates it is an informative space-time, form/function,

**DYNAMIC SPIRALS AS PERFECT SPACE-TIME SUPŒRGANISMS OF 3 STATES.**

The logarithmic curve can be written as: r = a x e^{bθ}

And then depending on the value of b it will *transform either into a circle, or a line. The *derivative of r (θ) is proportional to the parameter b, which controls how “tightly” and in which direction the spiral spirals.

In the extreme case that b=0 (ϕ=π/2) the spiral becomes a circle of radius a. Conversely, in the limit that b approaches infinity the spiral tends toward a straight half-line

So a spiral can be considered an informative ‘state’ of a full organism, which can convert itself into the other two states as a present wave and uncoil as a lineal motion. Again this is a canonical law of vital, i-logic geometry: a system can be converted between the 3 functions/form as systems are ‘modular’ and its functions are constantly changing between the actions better performed by limbs (entropic function), bodies (reproductive functions) and particle-heads (logic, informative functions).

Such transformations are the staple food of existence and development, being the spiral and the tree, then 2 fundamental ST combinations of S & T elements – but the spiral is the commonest dynamic form that allow change of states with ease; while the tree is the commonest ‘fixed’ still simultaneous system of O-| elements:

IN THE GRAPH, all spirals are ‘potential fields’ or sinks and sources (+ inward life vs. – outward entropic death duality); which explain the organic, functional and mathematical description of spirals as bio-topo-logic beings.

So the vital spiral *is *in its vital mathematical description the most efficient organic form=function: A galaxy coiling its flat irregular young form a worm coiling to sleep in its 1/3^{rd} still informative SS-inwards state (completed with its feeding, entropic state, and wave-like motion) are in spiral states; which is ultimately an extremal case of the upper and lower limit of pi that leaves always an opening, dynamic mouth to the spiral allowing its simpler beat of existence, closing inwards (T-State) and outwards (S-tate), never ending the perfect pi cycle.

So most vital living spirals can uncoil to become lineal forms, or can close to become spherical circles, managed by its central head (1D: informative state) that becomes its future ilogic forwards head (2D/5D: locomotion and feeding-hunting). So the spiral can be considered an Ø-intermediate present system, perceived from the perspective of its dominant central point of view in polar coordinates (r, θ).

**RECAP. Varieties of time spirals in pentalogic dimotion.**

Spirals represent the 2 complex dimotions of time-space in a classic S=T female, S<T>S male symmetry:

S=T: Reproductive, communicative Spirals are Archimedean or Fermat’s based in the Fibonacci’s golden ratio constant in which several branches of the spiral allocate with maximal efficiency the reproduction of new ‘infinitesimal parts’ of the whole.

S<T>S: Entropic + Perceptive spirals, (as those of time vortices, charges and masses) where a flow of ∆-1 points falls into the perceptive or digestive central point, allowing both a potential and vortex description of them.

Based in those rules that vitalize the mathematics of Spirals as essential elements of a fractal point in its dimotions of perception feeding and reproduction, we can interpret better the maths of spirals in vital terms and define multiple organic beings as entities, which in its coiled position become organic spirals, often when resting – processing its internal systems of energy and information.

**SPACES MEASURE ANGLES AND DISTANCES.**

Every language-mirror of the Universe starts slowly rising its complexity and focus on reality as it imitates the intrinsic laws of spacetime reality, even those as mathematics, which describe in a closer form its spatial points, scalar numbers and temporal operands. So did geometry in Greece. As such languages will follow similar patterns of growth and evolution to those of full fractal points=T.œs starting by the establishing sequential paths of Dimotions, which always mean first the fractal point’s emergence in a larger world where it will open its ‘eyes’ and measure. What it measures to make it useful for existence is thus the minimum the language provides on information.

## In the case of geometry is what all spaces will have in common for the point to ‘see’ its surrounding worlds and distinguish form (angular momentum, cyclical shapes, membranes) and motion=distance, lineal momentum.

## | & O even in its most simplified way, which is to measure angles of congruence and distances of predators and preys. Or else the language won’t be efficient, its speaker will die and the system will not repeat the languages’ form. So parameters in languages have a vital ¡logic meaning, for minds to acquire knowledge in its logic mirror about the surroundings of its world and be able to move and assess the distance and size of things. *Even in music, its simplest duality, treble and bass bring e-motions of uneasiness and relaxation as they connect with the Doppler effect: a predator wave coming closer blue shifts into treble and going away into bass waves. Thus giving a first information, with no angle, on a lineal path motion. Trigonometry added then to distance (S=T duality of motion), angle.*

## So the first parameters of ‘humind’ geometry were trigonometry and distance which under the s=t paradox can also measure a point in motion over a background scale of space.

## Angle is also the first fractal unit of measure, which can travel through scales without deformations – hence* a *dimensionless parameter *as fractal scales are relative in its dimensions, ‘erasing’ them internally as they emerge as an Euclidean point into a new scale*. So angle and distance gave us ∆St information and allows a trigonometric mind to survive. Angle thus allow us to define mathematically a dimension as follows:

## Suppose there is a mathematical (geometrical) quantity *A*, which depends on the scale, *l*. If after a scale transformation: *l*→*λl*, the quantity would transforms as:

A→λ^{n}A. Then the quantity *A* has dimension *n*. According to this definition, the angle has dimension 0; because it is the fundamental parameter that carries through scales without change.

**1D: sine<cosine**

## So the angle is the first operand on a surface of space, to provide information to the self-centered system. And its importance was such that the tri-angle was not called, the triside or triline (the objective topologic view) but the mental spatial view, showing *once more that the ego-paradox is always the beginning o any science. That is, the tri-angle was first considered a mental subjective space form, before it became an outer objective surface.*

*And it bears witness to our Nature as light space-time organisms, evolved from the minimal species of Euclidean Light space-time, the Planckton, that when we finally discovered ‘¡ts’ form, in quantum physics, h, turned out to be exactly that: an angular momentum, quantized in two gender species, the female boson and the S<T>S fermion. *

## The pentalogic dimotional functions that entangle its trigonometric ‘functions’ into reality:

**@:** its dominant use and first reason it became the first developed field of mathematics is its capacity to measure from a point of view distances according to ratios and parallax, which is the origin of tridimensional perception (bilateral eyes), and Fertile Crescent mathematics.

How this work in its simplest form, needs to understand how a ‘spherical, ideal mind-membrain of 3 π diameters, and 0.14 D apertures, allows a mind to perceive through them, ‘rays’ to distant objects. The mind thus can always measure the angle covered by a distant object, and with a minimal displacement, a new angle.

3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions.

**1D:** So trigonometric functions were the first to appear, as 1D perception is also the first ‘action’ of an emergent point where sines and cosines’ operands extract for the self-centered point, 1D perception of its ‘angles’ and ‘S=distances=T-motions’.

**The graph shows the trigonometric functions and the information extracted by them; and as usual in geometry we apply them the Dilogic symmetry of ‘internal subjective space’ and ‘external objective space’; as they can be applied from the fractal point to the outer world or from the outer world to the fractal point, whereas in pentalogic:**

– 1D: S: Sine is best to assess the ‘informative height of the system’.

-2D: T: Cosine is best to assess the length-motion of the system.

-3D: Tangent is best to asses the energy, S=T of the system.

– 4D: Angle is best to define the scalar proportionality of the system.

– 5D: Its inverse functions have an entropic role in the calculus of all those properties.

So trigonometric functions already extract from a T.œ information of its 5 Dimotions.

Trigonometry thus deals with the laws of perception of the circle and the triangle. And as usual since *reality builds up from the simplest lineal essence* to evolve latter its ‘3^{rd} age cyclical form’ |>O, the first laws discovered were the laws of |-∆ triangles, whose S=T transition into a sphere when it adds a dimension of ð-cyclic time, is its rotation. Whereas the ‘area difference between a regular polygon-triangle’ and its sphere, *is the first s=t transformal value, which turns out to be *huge, between ∞ and +2,4, while its perimeter its 1,6 times larger, making them clearly the 2 extremal TT & SS topologic forms of motion, the arrow triangle, and form, the rotating circle; facts those resolved with the Law of sines that calculate first for any triangle its ‘apertures of perception’**:**

If *A*, *B,* *C* are angles of a triangle and *a*, *b* and *c* are lengths of opposite sides the respective angles the **law of sines** for an arbitrary triangle states:

where Δ is the area of the triangle and *R* is the radius of the circumscribed circle of the triangle. In terms of perception thus, each of the 3 ‘apertures’ of the 3D¡ sphere and its sine view of the external world is equivalent, ensuring a non-distorted perception of distances to the outer world. It establishes the same law of equal perception of those 3 ‘membrain’ points of its internal ‘vital energy’.

**The angle is thus the first form of existence. The first 1D perception that illuminates reality by bending hyperbolically a larger world into a central singularity, in fixed quantum forms, which will become for the 1**^{st} particles ¡ts h, a radian of angular momentum perceiving a larger world tunneled into the singularity of its ∆-1 scale. *Creation starts with the angular reduction of reality to a still form in a quantum angle. As such the angle becomes with its radian h value, the first* T.œ, the triangle, which then as it turns around gathering in 6 groups forms an hexagon, with a wobbling radian arc that tessellates the plane from circular to hexagonal STœps.

Vital mathematics becomes then reality. Such vital circle has 6 apertures – those of its triangles – each of the radian arcs that make in circle 6 steps for a pi=3, closed in hexagonal form, opened in circular one. Imperfect forms thus appear from the original perfection of the language imprinted in the regularities of the medium.

So the hexagonal circle game of Stœps becomes the essential structure of 2 Dimensional space, with those 120 hexagonal angles repeated ad nauseam in Nature. It is the game that will grow in polytopes to reach through combinations of 5, 6, 120, the final 120 dodecaplex and 600 tetraplex maximal figures of 4Dimensional geometries (as pentagonal polytopes make no sense, being the 5^{th} dimotion entropy the destruction of the regular in-form-ation shared by them all).

Angles of perception become then the first simple geometric element of reality as 1Dimotions appear on every still point that gains an angle and starts the creation of the galaxy in its smallest T.œ, Planckton

The laws of those ‘angles of perception’ would also be the first to be assessed when a new emergent light spacetime mind, that of the Greeks appeared. As each emergent scale repeats the game of exist¡ence.

So a series of laws of angles of perception will become the first set of consistent laws of creation of mental spaces defined by huminds. We are here concerned as always with the translation and correction of this system of axioms of abstract spaces, for a single plane time-space to the much vaster, paradoxical real world of vital ilogic topology.

What is the main difference between Euclidean geometry and fractal non-AE when considering those axioms?

We shall consider the modern view of those axioms.

– And first we notice as we did with the earlier version of Euclidean geometry, that the axioms are grouped also in 5 elements, since even when simplifying mirrors, the humind must refer to the 5 Dimotions of reality.

– Then we observe that the differences between E-postulates and axioms vs. ¬Æ will arise as always of the ‘simplification’ of multiple scales into a single ‘continuum’, and the ‘straightening’ of relative angles that ‘change’ its form into a hyperbolic curve as we ‘diminish’ size in the fifth dimension,

**The vital axioms of geometry. Reproductive motions. Discontinuity postulates. Attraction.**

We thus first updated the elements of Euclidean geometry – its definitions of points now fractal; lines, now waves, congruence (now relative equality) and planes (now topological networks). As what today passes as the foundational axioms/postulates of geometry are only correct, for the limited view of a single spacetime continuum. So they are approximations which strip points=T.ŒS of fractal & vital properties, and simplify its vital imperfections.

This process carried by steps from Lovachevski’s a pangeometry to the absolute geometry of Bachmann (1970s) closed the evolution of the discipline. And it was found that only the most general features, angles and distances are needed to define a geometry. Specifically we could reject as Bachmann did the postulate of ‘continuity’, and Dedekind’s concept that real numbers ARE numbers NOT ratios, because they LIE on the real line, as a necessary feature of a real geometry, since indeed in the absolute pangeometry of the fifth dimension, any plane of reality is discontinuous, and the numbers fit in the real line to fill the discontinuities of simpler number families are NOT in the same plane. Still what we can take of the Axiomatic method either in its Euclidean version or modern ‘Foundations of Geometry’ (Hilbert) is the classification of the foundations of geometry in 5 subgroups of axioms that roughly coincide with the 5 elements of reality and now we shall comment on to realize the difference between a single plane continuous geometry and the real geometry of 5D discontinuous planes and angles that become hyperbolic as we sink into the minds ∆-1 singularity.

*Those are themes thus concerned with the 3 ^{rd} age of Greek Geometry and its eclectic attempts to found the discipline as a Universe in itself detached from the earlier Pythagorean and Archimedean practical ages.*

** **

**3 ^{RD} AGE OF GREEK GEOMETRY**

** **

** Archimedes vs. Euclid Experiments vs. axiomatic error**

All this said we consider the wrong side of Greek Geometry, his ‘3^{rd} eclectic old age’ as Euclid, committed the same error than modern mathematics in his 3^{rd} informative age, when set theory and the axiomatic method abandoned experimental proof and tried to ‘construct’ from the top without proof to the bottom unconnected of reality.

So math failed from then on to understand itself as an experimental mirror of scalar space, becoming with Euclid’s axiomatic method a self-contained truth, which it is *because fractal mirrors are ‘entangled’ in form to its original image – the Universe. *Yet as mirrors are kaleidoscopic, inflationary as Lobachevski and Gödel proved for mathematicians but physicists so far have failed to recognize, only experience validates what image is real. Schopenhauer, the best of the philosophers, noted already the fallacy of the axiomatic truths, which rely *on too many self-evident lies, points and lines with no breath, putting his warning on the axiom of congruence, which in fact is the most expanded by 5D rules. *

But once axiomatic methods with single Aristotelian languages became definition of truth, this error will continue to become the fundamental ego-trip and failure of many human sciences accustomed today to accept a priori postulates that set the ‘entropic’ limits of truth and scope of the postulate, introducing bias from its inceptions – from Economics that rejects the analysis of the evolution of machines with do in organic economics and accepts as a self-imposed postulate the goal of GDP growth in a limited planet to c-speed limits in the astrophysical realm beyond the galaxy, to the entropy arrow only of big-bang theories. So we shall consider now:

**The many false assumptions of classic Euclidean and axiomatic geometry.**

Let us ‘continue’ then the critical analysis of the Axiomatic method. Generally speaking, the axioms can be chosen in various ways, taking various concepts as starting points. Here we shall give an account of the axioms of geometry in a plane which is based on the concepts of point, straight line, motion, and such concepts as: The point X lies on the line a; the point B lies between the points A and C; a motion carries the point X into the point Y. (In our case other concepts can be defined in terms of these; for example, a segment is defined as the set of all points that lie between two given ones.)

The axioms fall then into five groups, oh yes, the familiar number! As we can relate them to the 5 elements of ¬∆@st of spacetime but its postulates to be *considered truth only in the limit of still bidimensional geometry in the plane, and so we shall adapt them with comments (no need to create in this introductory courses a true formal pentalogic system of axioms and postulates of its own to fully develop non-AE philosophy of stience and its stientific method and epistemology. (: LOL, this was done 30 years ago, and I used to write only with ilogic symbology, so far out from what huminds do, that I only got interested a conceptual art who used those equations for a exhibit at six-flags, a Brooklyn museum : )*

And obviously we are not using the formalism of sets and modern maths to explain those corrections.

We repeat this is an introductory course and we just want the reader to understand that present maths is an abstraction of a much more beautiful experimental language of scalar=numerical points of space-time.

This said the 2 first groups, Axioms of incidence and order, and the concepts of angles and laid on, refer to the entropic limits and inner structure of a given mental space, so they should be grouped as ¬S postulates.

__S:____Axioms of incidence__

Axioms of incidence are mainly related to the 2^{nd} postulate of lines-waves of communication, its boundaries and angles of incidence=congruence (already discussed in the 4th non-E postulate). So it also considers also the relative size of T.œs and its lineal, open flows of communication compared to its worlds; of the minimal internal parts a T.œ needs to define its properties as an |- lineal or O-circular element and so on. They connect thus closely as boundary axioms with the concept of a topological boundary and the entropic limits both in space or time (if we take a line as a worldline of duration of a being). And the main differences between the classic axioms of E-geometry and i-logic geometry regard the multiplicity of meanings of i-logic geometry where distances and points are logic concepts of similarity, all systems do have ‘boundaries’=limits as infinities are relative infinities (µ) and so the interpretation of its modern axioms of incidence fully changes.

- One and only one straight line passes through any two points.
- On every straight line there are at least two points.
- There exist at least three points not lying on one straight line.

As we can see the axioms of incidence establish the *minimal elements required to transit from a single fractal point into a social group of points, a line. But as usual with the axiomatic method that tries to prove itself by ‘reductionist’ simplicity of its postulates it does NOT bring enough information to define without ambiguity a line. Because with only two points, the intermediate region can be curved, so no straight line exists. How can we know with only 2 points the curvature between them? We cannot. And if fact because we cannot, the errors of the 5 ^{th} Non-E postulate that curves straight lines come from those imprecise definitions.*

*So the first axiom of incidence defines NOT a straight line but the* boundaries of a ‘flow of communication between 2 geometric forms. It is thus a definition of ‘any’ 2^{nd} non-e postulate line-wave of communication between two points.

And it also matches in the evolution of geometry into topology the more precise term of a ‘boundary’ for an open line of communication between two entangled points – the points are the beginning and the end of that flow.

This is then the meaning of the first and second axioms of incidence, while the 3^{rd} shows the outside world to be larger, than the flow of communication between two points.

The weakness of those postulates though has been already noticed by classic geometers. Indeed, it appeared somewhat strange to them that a straight line has only two points. Surely in our idea of a line there are even infinitely many points on it. No wonder that not even Euclid stated the axiom because it IS NOT proper, *showing that as more idealism and postulates of truth try to prove by simplification ‘absolute geometry’, they just add half truths and new distortions.*

2 points do NOT define a line, but any open trajectory, also an arch, a zig zag but NOT a circle.

So *the incidence postulates are important not because they define a line but a larger more important concept – an open flow of feed-back communication between two equal points, unlike a circle which is a hierarchical closed communication between two points, one of which ‘turns around the immobile one’.*

*So two points define an open democratic • = • flow of communication, while two points in a circle define a closed hierarchical communication. *

So the definition of an open ‘wave’ of fractal points blows up the concept of ‘straightness’. This is ok as long as we know that what matters of a straight line is not that it can be drawn lineally on a plane with an abstract rule, but something else – to be open in a ‘topological way’, that is, we can deform it as a geodesic on a surface, a chord of a circle, or whatever, as long as it is open and the two points are its boundaries.

Thus the *key feature of the line is to be open not to be straight, as the line curves in a hyperbolic angle entering an ∆-1 fractal point reason why* modern physics use curved lines to ‘crowd’ the entrance of a fractal point of the ∆-1 gravitational scale, instead of blowing those points as fractal points through which µ straight lines *can cross*.

Once this is clarified we find 3 canonical varieties of open lines: a straight line that needs 3 points to define its straightness and type of information communicated through a flow of middle points, which will form a wave as even the smallest boson has volume. A true straight line then happens merely with the entanglement of two fermions with a gravitational non-local quantum potential line that merely gives information on distance and likely angle (a neutrino or string of the Planck scale, maybe being both similar as neutrino ‘angle of scattering’ is on that range).

It implies then that even the simplest line requires a third bit of information about the type of ‘flow of information’ and shape of the transversal wave between both points happen since now 1+1=3. As a flow of ‘smaller points’ transit both poles to become the motion that carries information between the limiting points of the line.

Finally the 3^{rd} axiom of relative angle of incidence is concerned not with the inner limits but with the outer extension of the world in which 2 poitns communicate: *there are at least 3 points lying outside a line *means that there is a larger Universe outside a worldline. Since we can trace a plane outside of it, which might or might not be parallel, or intersect the line, as 3 points not lying in a straight line defined a plane. So outside a line there is a larger world with a new dimension – a topologic plane, an organism to which the line might belong if intersection happens.

But 3 points DO NOT necessarily define a plane. Again such definition constructs a plane by *self-reproduction and this must be stated, as we did with 2 points that construct a line by self-reproduction of a flow of communication. It is then when the axiomatic method makes sense. And so another way to reinterpret those axioms and give credit to them is to write them in vital reproductive terms:*

*‘Only two points are needed to start the reproduction and transmission of information between them’, etc.*

Further on 3 points no lying on a straight line as always in Duality can be expanded in its solutions considering the different ‘ternary principles of creation’. So as we said that 2 points can define either an open ‘line’ or a ‘closed circle’, when one of the points has motion respect to the other, forming an St system, as opposed to an S-S line, 3 points can define a closed triangle, or an open *angle of perception, which in i-logic geometry is different. *

*And then we can also apply the fractal method of differentiation, if we consider that one of those points is in a different ‘∆-scale’. Then the triangle with have a scalar 5 ^{th} dimensional added parameter that will provoke the lines to curve into a hyperbolic angle to reach the inner depth of its ∆-1 mind-points, which does NOT lie in the same plane ) a ternary system.*

Therefore the need clearly arises *in i-logic geometry, *for *pentalogic 3±¡ ¬∆@ST diversification of the simplex single line or plane of continuous geometry, *stating accurately and exhaustively every distinct version simplified by E-geometric postulates that reduces all open vs. closed geometric objects to straight lines and flat planes.

So instead of making definitions simpler but less clear, which is paradoxically what Hilbert achieved we will make them more clear– even though this intro won’t be exhaustive.

In the previous pentalogic considerations though we have already established the |-line of 2 points that needs a 3^{rd} moving point of communication, the O-circle of two points that needs a 3^{rd} dimotion for the submissive orbital point and the ∆-hyperbolic angle of 3 points one of them in ∆-1 scale. So we see *that duality the minimal S-T or SS or TT reality of the holographic Universe immediately reproduces trinity. *

*And so we could also resume the *__new axioms of incidence__* as:*

*‘2 points generate a third point to form a stable trinity structure of space-time that might be an open line of communication between two equal points still to each other, hence a ‘2 number’, entangled by the 3 ^{rd} moving flow back and forth between the, (|-duality) or a circle whereas one point is a S-till point and the other an orbital ð-point whose motion becomes its 3^{rd} dimension (S-sT System), or a hyperbolic ø-angle whereas the Still point is in a lower plane of the fifth dimension, S¡-1<SS¡.’*

This would be a more formal definition of the incidence axioms of i-logic geometry.

The same procedure of exhaustive ternary-pentalogic division of varieties of E-axioms applies to all the other concepts and axioms when i-logic geometry goes deeper into them… But we won’t be so exhaustive. Just pass fast through them, observing how they are connected to one of the pentalogic elements of ¬∆@st.

__¬:____Axioms of order.__

**When we deal with hierarchical order and perpendicularity –** ‘lay on’ concepts we are directly concerned with the congruence differences between torn systems which share a cut-point and those who are parallel and ‘avoid’ tearing and darwnian perpendicularity b establishing a bump in a new dimension of height-information between two lines:

- Of any three points on a straight line, just one lies between the other two.
- If A, B are 2 points of a straight line, then there is at least 1 point C on the line such that B lies between A and C.
- A straight line divides the plane into 2 half planes (i.e., it splits all the points of the plane not lying on the line into two classes such that points of one class can be joined by segments without intersecting the line, and points of distinct classes cannot).

*1 ^{st} as we have already expanded the concept of a line as an open interval which does NOT return to the original point directly but through a back and forth motion, the axioms of order ONLY happen in a ceteris paribus analysis of one-dimensional Aristotelian A->B lines, and as such it should be scrapped all together as all communication is dialetic and so A->B->A is the minimal unit of a line, with 2 inverse directions, as seen in the Neutrino theory of light, where light is created by two entangled neutrinos moving back and forth, in inverse directions. One single entangled flow allows the point A to measure distance to B and maybe trigonometric angle-height-size. But unless that information is reflected back, there will be a lost single event hat does NOT remain as a stable memory line.*

*The axiom of order only applies to A->B fleeting lines**. Yet as all lines are steps of a larger cyclical part; A->B always ends in A-B-A-B….**, and then the feed-back real stable lines cannot set a forward, backward, order, after a few feed-back entangled flows, neither to circles, where we cannot as we return to the point establish a relative order. Thus when C comes before D or after A in the circle? The order established in A-B lines thus is a different one as in the circle, one of hierarchy between the AB ‘fermions’ and the C flows between them which is of a lower ∆-1 quantum potential order, and that is the proper hierarchy. So happens in circles between O-point (center) and A (turning point), as circles are NOT drawn by a single line without center, which will not bend but have lineal inertia, but by the torque offered by the central dominant point. So the order is AC dominates ac… and O dominates A.*

*It is the constant repetition of those circular or feed back vibrating motions what create the illusion of a stable space line or circle, a present, **simultaneous space. Further on, what the circle does provide is not an order of points but a chirality, depending on the way we move around, which the line does NOT require in its feed back A<ab>B trinity. *

*Motion thus becomes a new creative duality order in cyclical paths that have chirality.*

*And orientation also. It is not the same order if we move from A –>cbd, or from A->dbc*.

*And this is then again the true differentiation between open lines that are more ‘equalitarian’ and circles, which constantly bring new hierarchies*. But as the Universe seeks for balance we shall see that parity & chirality allows gender mirror symmetry – it is in fact the first form of it; since the O-point will seek two ‘spiral arms’ with inverse chirality to achieve a balance with 2 different spin filled orbital points.

Thus as soon as we move into the simplest geometric scales of vital geometry≈ physics, we shall find that concepts as chirality, achirality, parity and the combinations of lineal and cyclical motions diversify the different species of reality and become essential to understand the symmetries and asymmetries of quantum physics, because IN FACT order does NOT exist in a categorical manner without a proper understanding on what becomes the most important type of Axiom, those of motion, which defined properly in Euclidean Geometry are Axioms of reproduction of form, by carrying a geometric form to other region of time-space. Yet before we study them, we must consider the second fundamental concept ‘laid on’ the Axioms of order, that of ‘lay on’ itself.

In the entangled Universe this key concept lay on, is then related to the false axioms of continuity and to the duality of Darwinian vs. parallel social events.

*And the essence of the argument is that again we must apply the ternary, pentalogic method and differentiate at least 3 ways in which points lay on to each other cutting two forms or fusioning them.*

**In the graphs on one side the circular and elementary continuity principles study when 2 systems are perpendicular, that is can cut each other and share a point, or are parallel, that is, only ‘contact’ each other by adjacency but do NOT cut each other. **

*As we stress the discontinuity of reality, we rebaptize those principles adding the ‘dis’ prefix to its classic formulations*:

ELEMENTARY Discontinuity PRINCIPLE: If one endpoint of a segment is inside a circle and the other outside, then the segment intersects the circle.

CIRCULAR Discontinuity PRINCIPLE. If a circle *y *has one point inside and one point outside another circle *y’, *then the two circles intersect in two points:

In the graph, the continuity principles are in fact limiting concepts of boundaries and laws of perpendicularity, which define the discontinuities, closedness and connections between networks of points. What matters then to reality is NOT the obtuse concept of a ‘block/Parmenides like, solid reality’, with no gaps, at the core of the ‘mind illusions’ of Hilbert’s categorical geometry, but when two systems of reality cut each other in Darwinian, perpendicular events (the segment breaking the circle, the circle breaking the segment), which will DEPEND on who ‘owns’ the point M?

**T>S:** It is M part of the circle O? If so O is feeding on A-M-B, the line. Or it is M belonging to the line?

**S<T:** The line is ‘killing the circle, which is now open at M. Or it is M the M-outh of the circle? Then Amb is one of the multiple ‘parallels’ (as it does not properly intersect) feeding the circle and the O-perceptive point.

*S=T: 2 Points:** Or it is M – and this is the most special case, in ‘BOTH’, the line and the circle? Then M is an attractive point that cements the Union between both and a reproductive action is happening as now the point is actually two points at once, belonging to each of them. *

Those are the variational ternary principle applied to ‘laying on’, the undefined concept of the Axiomatic method we upgrade to i-logic geometry as we did with other undefined Hilbert’s concepts of points, which are fractal points, lines, which are waves, congruence, which is relative similarity, Non-E 5 Postulate, which is the definition of a mind, and ¬E Planes which are topologic organisms.

So the concept lay on, is the final element that completes the 4th postulate of congruence as it becomes either ‘a reproductive superposition’ or ‘a Darwinian intersection’ with 2 solutions, substituting Dedekind’s continuity axiom:

*“A point M of intersection between 2 relative paths, ð (closed figure) and $ (open figure), either belongs to ð or $ or to both figures. If it belongs only to ð or $, either figure is the predatory, dominant element of the intersection, and the event will be a Darwinian space-time event, in which the submissive past prey tears, extinguishing its form, as a ð<$ or past flow providing the energy for the reproduction S=T or evolution $>ð of the dominant future flow. If the point belongs to both figures, the event is a present iterative event of symbiosis, and both systems can form a stable, social new whole, where the point doubles as two.”*

So we translate abstract axioms into the vital how of perpendicular Darwinian intersecting or connecting, uniting motions that define dual creative and destructive space-time actions for both systems, as a motion ‘transforms’ one form into another, or pegs them into a more complex creation

We are not here using formal language, though any mathematician or physicist can write it with the usual symbolism of classic logic, and notice a few facts that expand the concept of laying on and show its power to describe reality reason why is used to set the foundations of other key branches of mathematics (Boolean logic and set theory):

– 2 systems lay on can be either a Union or an Intersection, different concepts in advanced i-logic geometry:

A Union is a perpendicular, Darwinian event where the part of one entity no longer belongs to it, so the dynamic event destroys one part. Thus if as a rule we capitalize the dominant system of a dynamic space-time event of relative perpendicularity we can write: A U b = A, meaning that b looses its part which will belong to A, as when you eat a rabbit that no longer is a rabbit but becomes your amino acids.

On the other hand an Intersection will be defined as a true sharing of those common points, so neither dominates, A C B, means the C part is now the connection that cements the relationship between A and B, which somehow ‘doubles’ and by this sharing, in physics there is attraction between beings. And in biology, there is attraction between beings. Intersection thus, sharing, is both a creative element and a social element of love and attraction. We share a child in a couple and that puts the two elements in a constant dynamic attractive relationship. Fermions share a boson and that cements an attraction between both. Predators share a prey intersecting their territories of hunting.

Further on sharing is more intense and symmetric when 2 systems are closed St elements, as in the figure of the circular continuity postulate, where we can clearly see there is an acbd region shared that pegs both systems together.

In the intersection of a line and cycle, the line seems at disadvantage, and in fact in most real events the line becomes absorbed and transformed as a pixel of information, coiled after it enters into the vital cyclical space that lays between the 0 point and the M-perimeter (which are not enclosed in the open ball ST region of cyclical motions that connect them). In advanced theory we shall see that in reality those lines tend to be prey of the circle, unless emitted by other system as an ‘entropy ray’, CROSSING the 0-point. In which case we talk of a killing line of entropy, which crosses the circle at M and O, and if we state that in that intersection O belongs to the Line, the equivalent vital proportion is that the line OM has KILLED the circle, targeting its zero-point. Indeed, If you cut the neck, if you shoot the head, if you conquer the capital, if you murder the financial people-caste or military-king in power, you disorder r=evolve, change and destroy a closed vital space-time being. All other lines that do NOT cross the 0-point tend to be lines broken, fed and processed by the circle, which becomes the sole ‘owner’ of the m point and the chord inside the circle, isolating the rest of the line, as closed cycles DO break in the Universe into fractal spaces.

If we talk of motion, we see the first region abcd as a region that ‘doubles’ its ∆-1 density of finitesimals, and will become the ‘seminal first region’, which will double then the whole system to create the B-centered new moving form. Motion by reproduction of form is thus closely related to the new concepts of continuity, which more properly should be called ‘reproductive displacement’. The ACBD region then will become the seed for a 3^{rd} œ child of A and B or the region of density growth that will become latter split by asexual reproduction (as in cells, which first duplicate a region in their central DNA zone), or it will become the region of the wave in which a gradient of an attractive field, with increasing density of ∆-1 finitesimals ‘drags’ the A-circle into reproductive motion.

One key question in the whys of physics is ‘why’, systems move in a relative field, its ∆-1 scale of the 5^{th} dimension, towards the gradient region of maximal density of force – so we move towards the attractive vortex of maximal charge or mass. The answer is that the system which is attracted and shares the same active magnitude and ∆-1 field, will bind in that region on the side of the density gradient, ∆ (ð/Sp), more finitesimals to ‘double its form’, more ‘energy-space quanta’ into which reproduce, and so we can slo see according to the ternary fractal principle of multifunctionality, motion as the feeding process of an entity, A in the graph that feeds on the field, on the gradient region of more density, falling unescapably by its greed of motion towards the region of maximal charge-mass. The field is controlled by the central charge mass which will finally eat up the smaller charge mass attracted by the bait of the field.

So we are giving here 2 key ‘vital propositions’ about the nature of motion, as a dual æ action (the larger model reduces all realities to the 5 vital a,e,I,o,u actions of space-time beings): The system both feeds and reproduces with the absorbed energy. And this can be done in 2 forms:

– The system feeds on the gradient of maximal density towards the stronger charge-mass, and in the process of feeding it reproduces its form into the adjacent region, either creating a son species if the action between both attracted points is symbiotic, parallel, so both use the field in equal conditions to input information and reproduce.

– Or the system feeds ‘alone’, reproduces its form in the adjacent region and slowly normally in circles to avoid its final demise, falls into the vortex of the stronger whole that owns the field, and truly is ‘farming’ the attracted particle, which finally will be digested by the stronger whole upon a perpendicular ‘Union’ – the star enters the event horizon of the black hole, the feeding pig enters the stomach of the farmer once it is finally attracted to the slaughter house by the channel of food that makes the pigs willingly enter its dead event.

Now, this is what I am interested most: to show the vital geometry of the Universe. A mathematician would be likely more interested in the logic abstraction of those postulates, and a physicist in its capacity to explain the whys of key processes of physical systems. As a philosopher of science, my goal is to show you the organic, vital nature of even the most abstract of all sciences, mathematics.

Thus lay on was correctly undefined in the classic sense, as it was never resolved in its 3 varieties. Things do NOT lay on a plane of the 5^{th} dimension, as then they will be above or below but not ‘into’, lay ‘on’, therefore is NOT a real event but a parallel event. A ‘laid on’ being is not into the being, it does not touch the being.

So for geometry what matters is NOT Continuity but topologic adjacency, with no boundary in the constant reproduction of the form, such as the membrane becomes the external wave form – the lack of separation in the process of motion=reproduction however is not equal to the classic concept of Continuity no longer needed to define a Pan-Geometry, as Bachmann proves.

So this brings us to the correction of continuity concepts in terms of scalar continuity where continuity only happens when we squeeze all the fractal sales of the Universe in a single flat line:

__IV__**. ∆:**Axiom of continuity.- Let X
_{1}, X_{2}, X_{3}, ··· be points situated on a straight line such that each succeeding one lies to the right of the preceding one, but that there is a point A lying to the right of them all.* Then there exists a point B that also lies to the right of all the points X_{1}, X_{2}, ···, but such that a point Xn is arbitrarily near to it (i.e., no matter what point C is taken to the left of B, there is a point Xn on the segment CB).

A number system constructed from the reality of a discontinuous world IS preferable to the ideal continuous real number system fabricated with Dedekind’s false axiom of Continuity. Instead 3 less strict principles suffice to explain the different virtual continuities perceived as motion of the 3 elements of any system (| x O ≈ Ø): lineal & circular continuity (predatory union) or ø (reproductive superposition), and the Archimedes and Aristotle classic axioms of relative space-time proportions.

Dedekind’s axiom is then a different concept – that of barriers, limits in a scale of numbers whose holes in the real line are similar to ‘potential wells’, quantum jumps that are difficult to cross. Irrational numbers become then discontinuous gaps: π & √2 are the gaps and apertures that prevent the circle and the square triangle to be perfect. For example, it can be used to prove the existence of limiting parallel rays in hyperbolic geometry with far more simplicity than using the Aristotle axiom.

Of course, Dedekind’s axiom is needed to obtain the categorical axiom system of the Hilbert. Yet precisely for that reason, because it is not truth and real, it merely shows that Hilbert’s axiomatic method is false, it is an error of the mind that confuses its limited perception of the ‘holes’ and open wells of the Universe (those limiting ratios or real numbers) by the mind, with reality. It is like the case of a continuous movie perception. In fact the movie is stop and go, with holes but the mind puts them together into a continuity picture.

*Continuity is a Maya of the senses that eliminates dark holes between perceptions of the brain.*

In other words, the brain, the mind-world is continuous, reality the larger world is not. Dark spaces are easy to calculate for a p.o.v. with a relative 3 diameters to form its circular perimeter, which will leave 0.14… holes to observe. So the point does NOT observe, 96% of reality darkened by the perimeter of 3 diameters that closes its outer membrane.

So it sees, 0.14/pi = 4% of reality, which is what we see in the Universe (96% being dark matter and dark energy).

Yet as the Paradox of the ‘horizon view’ shows, we do not perceive a 96% of darkness. Darkness is eliminated to picture an enlightened 4% as if it were all the reality.

So this is another ‘angle’ to fully grasp what the total perception would be in hyperbolic reality, as it would have to expand the angle to include that 96%. Thus if we were to include it all into the perception of the whole, the angle will be convex, expanding in a fractal network of angles that keep growing its perception, constantly growing into a hyperbolic branching, giving us a much richer world, which is what Lobachevski and 5D metric proves:

Now as it is proper to consider it here, we did somewhere explain that unlike classic geometry, a straight line is NEVER created only by 2 points. The classic definition of Euclid, naively accepted by those supposed Hilbertian r=evolutionaries: ‘a straight line joins two points’ does no longer holds. We obviously NEED 3 points to connect 2 points, one being shared, and only then we can see if the 2 points are joined in a ‘curved’ form, by an arch, or in a straight form, by tending an AM and MB intervals, and looking at the ‘angle ‘between Am and MB, which if it is a straight angle will define a straight line. This is obvious –2 points cannot define the straightness of a connection; that it surprises me it has been overlooked for so long, as it is also a key concept to properly define what kind of geometry we are into, and a good way to introduce the other 2 axioms that substitute continuity and relativity of size:

They are classic A^{2}xioms of Greek Geometry (Archimedes, Aristotle’s axioms; in my Leonardian notebooks, written with shorthand incomprehensible Spanglish, i-logic weird symbols, which perhaps in the future some robot will try to decipher, he will find my abbreviation of those 4 Axioms of continuity and angular perception, written A^{2}c^{2}ioms, ab. A^{2}c^{2} 🙂

They are concerned with the perception of size and its comparison from a given point of view. And again, as always in the dual/ternary Universe, as in the case of the lineal and circular continuity principle, we have one axiom dealing with lineal sizes and the other with circular/angular perception of sizes:

**ARCHIMEDES’ AXIOM. If CD is any segment, A any point, and ***r *any ray with vertex A, then for every point B *≠*A on there is a number *∆* such that when CD is laid off *∆ *times on *r *starting at A, a point E is reached such that *∆ *x CD ≈ AE and either B = E or B is between A and E.

For example, if AB were *π *units long and CD one unit length, we need 4 CD to get beyond B and enclose π inside our straight line. And this is what matters to ‘enclose’ or not a certain ratio within the larger envelope, to enclose our dark number pi, so we know is within us (the whole cycle) even if the cycle is fluctuating around the non-defined π.

Moreover the axiom sets limits to infinitesimals, defining the finitesimal unit of measure AB on the lower side and the whole AE on the outer contour side.

Archimedes’ axiom thus means that when Nature chooses a finitesimal CD as a unit of $ length, a quanta is established for a scale or plane of 5D to exist and every other segment

Will have finite length with respect to this quanta which becomes the ‘relative definition’ of a number.

And inversely if we have the perspective of the whole, we choose AB as unit of length. And then the axiom says that no other segment can be infinitesimally small with respect to this unit (the length of CD with respect to AB as unit is a at least 1/n unit). 1/n was indeed in Leibniz’s Infinitorum the finitesimal unit.

*Those 3 axioms suffice to prove as mathematicians know, all the theorems of geometry.* Moreover, and this is the beauty of it, if we want to get rid of numbers and do a purely geometric analysis, this postulate, which connects numbers, points and lines, can actually be substituted by a mental postulate:

ARISTOTLE’S AXIOM. Given any side of an acute angle and any segment AB, there exists a point Y on the given side of the angle such that if X is the foot of the perpendicular from Y to the other side of the angle, XY > AB.

** **In the graph, XY grows faster than Vx or Vy as we come further away from V and the angle becomes hyperbolic, so we can always find an XY larger than vx, even if paradoxically V has the impression from his Point of View, that XY is becoming smaller. This relativity of world perception versus real Universe is at the core of many errors of the ego who believes to be infinite when in fact he and his relative distance to XY is really small. It often means that if XY is a ‘future point in time’ (death point, when we use geometry to study worldcycles) or a predator in distance, we will underestimate the danger of death, and XY will grow very fast and eat us up (-: )-: O-:

The smaller part, or fractal faster point becomes then a knot of a fractal of reproduced angles that keep enlarging reality till it can map out a much larger scale, which shrinks through those fractal triangular branching to focus and coalescence in the center of the fractal network, at ∆-1.

The duality space v. time stillness v. motion is a geometric formal view of 5D metric: S x ð = k.

As we become smaller, ðƒ, time accelerates. Inversely, to have a still, geometrical perception with no motion, we have to decelerate time cycles of smaller beings and expand its space size. So if the faster motion of cells and atoms slows down to our speed of existence, we would have to expand its space size to be as big as we are after that geometrical expansion and time reduction.

*So we substitute the concept of ‘single scale distance and continuity’ – *Dedekind’s axiom – by Aristotle’s postulate, which changes real continuity by relative, angular perception of distances, from the perspective of a points of view, with deep virtual-world-mind implications even if we prove the same theorem.

The postulate of Aristotle merely says that from a given angle of perception, the line that joins the limits of our perception and closes the open angle of vision is larger than any of the two sides of our angular perception.

In other terms, the ‘perpendicular’, not parallel, horizon or ‘front of the wave’ of perception expands much faster than the distance between us and the being we perceive in other scales, is proved in a single space-time scale by one of the key new postulates that.

The Universe expands faster in objective terms (the perpendicular, far away line of expansion of our horizon), than from the perspective of the perceiver of a certain geometry.

What all these new ways to define the parameters of continuity tell us, is that what matters to systems is the relationships between beings, and the relative perceptions beings have of the Universe deformed by its angular worlds of perception.

**RECAP**. It is important to differentiate geometric, continuous forms from numerical discontinuous series, as both represent two different ‘elements of reality’. Continuity appears only when we include the concept of ‘motion’ (not defined by Euclid), and even so it is a mirage of the senses, because at quantum level it is a reproductive continuity. This is show in social growing natural numbers. Thus there is NOT an exact correspondence between points and numbers, as the failure to find pi, √2 and e, the key ‘ratios’ of the Universe, which can be ‘drawn geometrically but have no direct exact solution arithmetically, proves. The number system can be properly used to construct geometry only when those are taken into account.

Continuity can be considered in the geometric world, only from the perspective of adjacency, and motion as reproduction of form, in adjacent places of a single plane of reality:

In the graph, taken from a physical wave, a particle reproduces its forms as it moves as a wave of adjacent particles one after another. This is the definition of motion, which solves Zeno’s paradox. Proper motion does not really exist, but reproduction of information along a path, with limits for each world and geometry (in the Euclidean human space, with the limit of c-speed for transfer of energy-form). So continuity can be defined in a single plane with the postulate of adjacency.

In precise terms given the lack of true continuity in the 5D universe, the 4 postulates that substitute continuity as proved in the work of the key post-war geometers are restricted to a single plane of space-time, and truly define more than continuity processes, *the other key elements of i-logic geometry.*

III. ** T: **__Axioms of motion AS REPRODUCTION OF FORM.__

(A motion is to be understood as a transformation not of an individual figure, but of the whole plane.)

- A motion reproduces=carries straight lines into straight lines.
- Two motions carried out one after the other are equivalent to a certain single motion.
- Let A, A′ and a, a′ be two points and half lines going out from them, and α, α′ half planes bounded by the lines a and a′ produced; then there exists a unique motion that carries A into A′, a into a′ and α into α′. (Speaking intuitively, A is carried in A′ by a translation, then the half line a is carried by a rotation into a′, and finally the half plane α either coincides with α′ or else it has to be subjected to a “revolution” around a as axis.)

The axioms of motion *again reduce and simplify the differentiations happening in the vital Universe. So we must stress those basic principles that so clearly diferenciate the exhaustive goal of i-logic geometry – to describe all vital events and species departing from the complex ∆¬@st properties of space-time vs. the simplifying synoptic nature of mathematical mirrors in the axiomatic method. *

So here we have to substitute the word carry by the word reproduce. A motion reproduces a form as it ‘carries’ straight lines to other place of space-time where the information is reproduced on the lower ∆-1 plane. So in ilogic geometry it DOES matter unlike in Euclidean geometry the path of the motion from A into A’. Though if we consider that pure motion has minimal memorial persistence of the reproduced forms, in the limit of TT-entropic motions it will not matter the path as the persistence of memory will not make them last. But ultimately geometry makes fleeting motions=communications between points to last. So such motions are in fact complex entangled motions whereas the 2 entangled points still to each other keep also reproducing over a lower plane as they displace together. It is then evident that the axioms of motion deliver the maximal simplification of that complex process observing the line in motion but still to each other end-points just in its initial and final picture.

How mathematics gives then depth to reconstruct from such simplifications the whole range of events happening takes place not in geometry but in Algebra with Groups that exhaust all possible diversifications of such ST-events.

The axioms of motion are in that regard, the basis of Group theory applied to modern physics – a great advance, for the classification of all possible variations of events and forms.

In detail the 1^{st} axiom defines reproduction with an abstract word, carrying, used in physics, relativity theories, vector spaces etc. But its vital nature is rather intuitive in the 5D worldview.

The 3^{rd} is more interesting, and complex because it is connected to the previous analysis of chirality parity etc., and will be of great importance to distinguish different species of physical systems, according on how $ (lineal motion) and a ð (rotary motion) are put together to return or NOT the point to the same initial state.

*As it further establish the processes of creation by reproduction of new forms, using the only ‘conserved motions’ – rotary angular motions (O) and lineal boosts (|), in a single plane. But certain forms which are mirror symmetry cannot match only with rotations and boosts, but need a 3 ^{rd} ‘mirror symmetry’ which implicitly states that any system ∆ will be part of a larger dimensional ∆+1 system to achieve its completeness. And so besides lineal and angular motions, reality requires a mirror symmetry in a higher dimensionality, to achieve full reproduction of forms. And this altogether makes reality more complex, generates new creative gender forms and also establishes the non-* commutative closed or non-commutative nature of a bidimensional system when completed in the higher scale, as two consecutive $ and ð-rotary or mirror symmetry actions differ, because both rotary and mirror symmetries are different according to which direction the action takes. So inverse dualities kick in to start complex differentiations.

However ll motions by definition become with time, closed actions and should return to its origin. So the question opened with deep consequences in the study of worldcycles and present stretches of the virtual existence of all of us, is how many ‘motions are needed’ to return to the same form, regardless of the differentiation in the intermediate paths – are 5 dimotions enough or shall we need a dodecalogic 12 steps method?

We leave open the question. Ultimately the labyrinths of present stœps of existence in which the path incurs are more important than the goal which will be a 0-sum that kills the path and the existence of the event-being.

Still the axioms offer 2 solutions: the point can be carried in an open path by pure motion, without memory and then the path doesn’t matter, and the point continues as an isolated motion, free of ‘responsibility’ for its tail of reproduced elements. Such memoriless markowian processes define then an |-open field or limb system.

Yet a reproductive motion with memory that leaves a track is a curvature with memory, a closed cyclical point which will ‘crowd’ its dimensionality at the point of reproduction.

And indeed, often when the point closes its circle it exhausts the motion of the being, which dies after reproduction (from octopus to arachnids). A closed path kills a system, crowding the zero point of reproduction.

But an open system also becomes exhausted as the intensity of the line fades away into entropy and noise. So after a finite number of stœps a thermodynamic open system even if it does not return to its original point, becomes exhausted.

All this bring us to the fundamental metaphysical question derived of all those axioms: It is the Universe closed or open, infinite but finite, un-bounded? It will return to a point? All the answers being truth as reality includes all its paradoxical dualities and trinities:

The universe is infinite in global timespace but finite in local time space T.œs. Which are, once we get rid of the ego paradox, infinite in its future repetitions, because the number of repetitions and variations is a smaller infinity than the number of fractal domains, ‘broken by a cyclical enclosure (3^{rd} axiom of order)’.

This duality of infinities between the types of beings and its repetitions will be in metaphysics the only correct use of Cantor’s transinfinities to prove that ultimately we are all immortal because we are repeated likely every 10^{9-11 }variations as the fact every 10^{9} humans we find an undistinguishable repetition shows If you think this is too abstract think again. Because the Universe is infinite in its repetitions, finite in its variations an exact replica of yourself is now about to appear once you die in other region of space-time.

And so we move to metaphysics, in a Spinoza’s sense: are those other ‘yous’ repeated, connected entangled to ‘you’, as identical quantum systems are? Are you part of a wave of reproduced ‘carried’ motions as a block of time? Are you going to live beyond death by transferring non locally to another you, born when you die? It is the dark space between the two informative forms of time that you are, a discontinuity bridge by the finitesimal mind that does not see that space. *There is transmigration of souls?* Alas you see, Hilbert would have never imagined that from his axioms of motions we could move into the pythagorean theory of metempsychosis (: reincarnation… The end of i-logic geometry brings us the first questions of platonic mathematics. How many yous exist reflections of the ideal canon of the cave? But those are questions that belong more properly to the dodecalogic of worldcycles. And the pentalogic of entanglement. So we stop here. Neither as we done it already discuss, just enunciate the 2 versions of:

__V__**. @:**Axiom of parallelism (Euclid).- Only one straight line can pass through a given point that does not intersect a given straight line.

These axioms, then are sufficient to construct Euclidean geometry in the plane. All the axioms of a school course of plane geometry can in fact be derived from them, though their derivation is very tedious.

The axioms of Lobachevski geometry differ only in the axiom of parallelism.

V′. __Axiom of parallelism (Lobachevski).__

- At least two straight lines pass through a point not lying on a given straight line that do not intersect the line.

Indeed, in ¬Æ we get rid of IV continuity (Dedekind’s in simpler language) and Parallelism according to Euclid.

So once more we see that the basic axioms and postulates and laws of mathematics are mirror of ST-laws, albeit sometimes too simplified so huminds have lost track of what they were mirroring in first place. Which is what we shall show by departing from the ∆@st reality NOT from the mirror itself as the axiomatic simplifying method does. The conclusion though is obvious:

**All the self-evident axioms and postulates of Euclid are relative truths.**

Reason why they are superseded by the Non-Æ=i-logic postulates of geometry based in fractal points with breath to better connect the experimental reality and ideal geometry. This has not been done as mathematician only corrected the fifth Euclidean postulate, by lack of a proper theory of reality to define concepts as dimension or distance=dissimilarity or the different types of congruence, or time or space, which we now have, coming from a higher language, the ‘¡logic laws of the fractal space and cyclical time of the Universe. So we can reinterpret many of the postulates and axioms of Greek geometry and correct them, as they are under the correspondence principle, just approximations to the scalar Universe observed in a single plane, without perception of the inner parts of the point. Let us then revise, the five Euclid’s postulates:

- It is possible to draw a straight line from any point to another point. .
- It is possible to produce a finite straight line continuously in a straight line.

This means that all points of a present space can be connected in simultaneity because spacetime is continuous; but in reality is not. As there are ‘irrational points’ which are in a different 5D scale; closed membranes that break the continuum. So only points that belong to the inner vital space of an organism, connected through ‘physiological networks’ can be connected in the same present space.

While points from two scales are only connected when they belong to the same superorganism, and they are similar in form, despite being different in ‘size’ (5D scale). So we can connect points within one of the 3 topologies (membrane, vital energy, inner singularity), and we can connect smaller parts of a fractal network with its larger physiological form (cells through blood and nervous networks of the ∆+1 scale and so on).

And they will often a ‘curved connection’… Moreover all straight lines become bent into a zero sum curve, as they will be part of a fractal superorganism. So infinity does NOT exist lineally but quite the opposite in cyclical self-repetitive patterns, as all straight lines ultimately find a limit in the curved closed membrane of its superoganism.

- It is possible to describe a circle with any center and radius… which might be the only true postulate, meaning that all straight lines will be part of
*a closed time-space cycles of 3 ages, which are the 3 pi diagonals, closing a worldcycle, which has a finite zero sum volume and breaks infinity into inner and outer parts.* - All right angles are equal to one another.
*This is not truth in different scales as the fifth dimension is a hyperbolic geometry whose relative curvature and degree of flatness depends on the relationship between the rod of measure/size of the observer and the size of the observable (in formal space this is the realization of the time acceleration=increase of curvature of smaller beings).* - The parallel postulate, already known to be false by classic science is related to the previous one and we have also proved its falsity.. So actually 4 or the 5 postulates are false in the real world. And only the postulate of creation of circles departing from lines, it is real in temporal terms, as all systems close its worldcycles, that is they die.

So are the definitions of a point with no breath, a line with no breath, as points are fractal points with hidden volume in a smaller scale of parts, lines are therefore waves – points cycling; planes are then not defined by lines but by networks and its flows’, and so on. Those* self-evident definitions are all false. *

Finally, the Elements also include the following five “common notions”; 4 of them concerning equality, which are not ‘false’ but rather meaningless, as things are ‘similar’ only a thing is equal to itself, since we do not have the total information of beings, neither things which occupy different spaces – as they *are made of space and time – *are equal, just merely by changing position, the thing becomes other thing (themes those of extreme importance in quantum physics to differentiate bosons and fermions – systems that occupy the same space, and hence are equal, and things that do not occupy the same space :

-Things that are equal to the same thing are also equal to one another (formally the Euclidean property of equality, but may be considered a consequence of the transitivity property of equality).

If equals are added to equals, then the wholes are equal (Addition property of equality).

If equals are subtracted from equals, then the remainders are equal (Subtraction property of equality).

Things that coincide with one another are equal to one another (Reflexive Property).

The whole is greater than the part. Finally, to thoroughly bust your balls/beliefs … (: yes, you have guessed it ): *the whole is not greater than the part, if anything they are equal…*

Or rather similar in existential momentum $ x ð, according to the metric of 5D: $ x ð = K. Which somehow is implicit in set theory and the paradox that tell us the set of all subsets is bigger… and even smaller if we *merely measure its quantity of information that grows inversely to size. *As this postulate is closely related to our ‘understanding of the scales’ of the Universe is worth to elaborate a bit more.

**The whole is not greater than the part, neither smaller (:**

The world we measure and call physical is a mental world. Consider instead the real scalar world of the infinitely divisible. There any whole is infinitely divisible, but so is any part of that whole. As a particular example, in mathematical analysis, any line segment is identical in every way to any smaller line segment that is a part of it. This suggests that the fifth common notion may, in the description of the world, be not the only true one.

We can extend notion 5: The whole is equal to the part’ to the particular case that all parts of the whole are equal to each other, from where we deduce identical particles in physics, where all electrons, protons or photons are alike. Yet, we can go further if we merely ‘measure’ the information/time speed/energy density and affirm the opposite, that the whole is less than the part, as its ‘time-motion/energy/mass is greater in more tightly concentrated forms. So *the black hole which is in to 5D metric, $ x ð = k, the smallest mass is actually the greatest/densest /heaviest*.

**Mental spaces and its value as partial truths.**

What can then we save from Euclidean geometry if everything about it seems wrong? (: Almost all of it; because rather than wrong is a simplified selective version of reality which work as long as the properties we select on that reality for practical purposes are ‘external to the fractal point’ not concerned with its internal parts, as in the previous case similar to engineering design where mass is what matters to us in the ‘selected mental space’ of forces. Or when motion is more important than form, as in physical equations. But absolute truth will require always considering that internal volume of fractal points; and hence its wave-like nature as in quantum physics.

Because only the whole Universe has all the information about itself any O-Mind x ∞ Universe will be valid as long as the infinitesimal information it extracts helps the species that holds such a mental space to measure reality and store information helpful for survival. In that sense, the most important element of bidimensional geometry is the understanding of angles, parallelism and perpendicularity, which is the first element developed in Greek geometry to be able to measure distances with trigonometry. This already is done by the eyes’ mind-mappings of distances.

But we add to it the topological laws of the 4th Non-E Postulate of congruence, which *adds a vital meaning to trigonometry as *Parallelism is also *as its opposite concept of Riemannian distance, a measure of ‘similarity’. So if in Riemann’s dissertation distance of ‘color space’ is equal to dissimilarity in the 4 ^{th} Non-E postulate parallelism implies similarity* which fosters social evolution vs. Perpendicularity that implies dissimilarity.

And *this becomes objective reality when observing ensembles of fractal points of any species* and its angles of connection of those points into geometric figures, in a still space view, or when one of the dimensions (S=T symmetry) is seen as motion, in the way herds move in parallel or predators intersect preys.

So we establish a new duality besides the S=T duality of form and motion, derived of the internal parts of all fractal points; that between the subjective mind and the objective external topological view of an event. In this case ‘dissimilarity’ is a condition of internal congruence between 2 fractal points bring external parallelism or perpendicularity.

And so a new rule of the entangled pentalogic Universe must be added to S=T, ‘x=y->x≈y; x≠y->x X y x y; that is, if two systems are congruent, similar in its parts in a still spatial comparison (x=y), they will move in parallel (x≈y); if two entities are different, x≠y, they will cross in perpendicular motions, x X y, or separate in its distance (x y).

Thus trigonometry of angles, was the fundamental first mental perception because it is how real systems measure distances, which is a function of similarity and hence of vital survival. Angular perception is also realized in the simplest physical systems, were the unit of information, h, the minimal space-time ‘Planckton’, has several position of quantized angular momentum, the only positions we can measure as information is always processed in ‘still form’ (missing its motion steps as we miss the motion of a film between frames). , according to its angle, which determines its interaction with other particles…

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II.

**@NALYTIC GEOMETRY**

@nalytic geometry studies the different planes of mathematics as mirror reflections of the different topologies and planes of existence of a super organism, self-centered into an @-mind. Analytic geometry was the first mathematical form that successfully merged the 5 Elements of reality: ¬∆@st, and as such it signified the beginning of the mature second age of mathematics, after its first ‘spatial’ bidimensional Greek age of still geometry.

The duality of @nalytic geometry is thus obvious: *S-mental spaces self-centered in the 0-mind*

∆-Scalar spaces developed in Sequential social numbers

T-ime spaces, allowed by the use of one coordinate to represent a symmetric dimensions of time

The flexibility of the concept of a plane of space-time to represent numbers and points thus allow to represent not only the S POINT ≈∆ NUMBER symmetry but also the S-T dual space-time, form=motion symmetries and it will have an immediate consequence on the development of science as it will canonize the concept of a lineal time and space, proper of mathematical physics ever since. So a key theme of @nalytic geometry is the a priori study of the distortions we introduce when setting functions and operands on an artificial world that we take as a ‘background space-time’. Since we must be aware we make local assumptions NOT global ones, as when we use lineal time NOT all the times in physics to measure locomotion. Unfortunately this locality of frames of reference was lost and brought the error of absolute Newtonian time space.

Thus the mind in mathematics is reflected by its frames of reference; and so we apply pentalogic to classify the main uses of @nalitic geometry, designing the different perspectives of the parts of a being and its 5 Dimotions:

**@:** the frame of reference is the mind view, with a 0-1, finitesimal ‘body-energy’ and a 1- external spatial world whose equations are self=similar in its T=S symmetry. Indeed, we can consider the fundamental equation of analytic geometry:O x ∞ = Constant world, the equation of any mind, whose perception of the infinite whole in its biased frame of reference creates a relative world, or mind-mirror of the whole…

**S vital** topology gives us a more objective use of those 3 different frames of reference and ∆-scalar symmetry represented in the Complex Plane to represent events taken place in a partial element of the supœrganism:

Because there are 3 topologies and then scales it should exist 3 types of planes and then one for scales. Intuitively they are the lineal-cylindrical, polar-spherical and Cartesian, hyperbolic that better represents the merging body-waves of the two others. And so that leaves the complex plane of ‘squares and roots’ as the natural one to cast functions through scales.

All systems have ternary topologies with 3 motion-information-reproduction function that require to absorb energy, move and perceive to keep in existence. Thus the simplex aei actions exist in all systems of Nature.

Analytic Geometry’s fundamental plane, the Cartesian plane, due to its perpendicular structure, is according to the 4th postulate of Non-Euclidean geometry the best form to represent precisely ‘perpendicular’ space-time inverse states≈ Dimotions of the Universe *in a single plane and compressed, by elimination of one of the s=t dimensional motions. So it is best to analyze systems which have an internal * S=T Symmetry.

We can consider the basic duality of Space-stillness vs. Time motion, in 2 other main frames of reference:

**-Vector spaces** add a dynamic, temporal view, *which makes each point an ST element with the S-population parameter and T-motion parameter best suited to represent existential momentums* in any ¡-1 ‘field scale’. So Physics uses vector spaces for locomotions in lineal space and time.

-Abstract mental spaces (Complex, phase spaces), which are the spatial mental static view.

**Complex planes.** Because huminds have not emerged into a new social scale they have not a clear philosophy of the fundamental ’emergent’ plane of mathematics: the complex plane understood in terms of ‘square’ coordinates that study space-time mixed functions and Dimotions of groups of Dimotions with the tools of calculus in time.

**Phase spaces** describe specific analysis of the entanglement of different ‘states of space and time’. A subjective selection of the parameters of time and space and its generalization to all type of dimotional systems, Phase spaces will completely liberate science from the immediate reality of the Euclidean world we perceive.

Of special interest for the human scale is the analysis of the ternary points/states/ages of matter in those phase spaces (state physics).

-Mathematical phase spaces will then be the manipulation of mathematical functions on those planes, which often will be not only abstract functions of mathematical space but expressions of the real dimotions of existence, essential to the fields of ‘Mathematical physics’ and its special frames of references; and the main field of 5D analytic geometry whose experimental task is to relate those abstract ‘conic curves, complex planes, and different dimotional parameters of the main 3±¡ analytic frames of reference to real events and forms of spacetime.

All this is fairly well understood by scientists. What it is not so clearly grasped is the magic of the complex plane. And the interaction of planes of different scales, as those which happen when an Active magnitude preys on a lower ¡-1 plane perpendicularly; that is processes happening between different 5D scales.

A key element of all those spaces is ‘the angle of perpendicularity’, which acquires more meaning when we marry @nalytic geometry and the 5 new non-Euclidean postulates, specially the 4th of congruence based in Darwinian perpendicularity and social parallelism. It is the key for the understanding the mathematical physics of vector spaces, cross products, dot products, equipotential and lines of forces, which happen in different planes of 5D and affect different parts of an entangled, field/wave/particle supœrganism.

As a general rule we shall state that particles are Darwinian over fields in which they prey and waves are parallel to the field in which they ‘slide’.

While they illustrate also the already mentioned 5D metaphysical solutions about the nature of the ‘light space-time we perceive’. It is real? or it is a phase space of the mind? The question was answered both in terms of humind’s art (painting), psychology of the mind and Relativity theory and its measure of time clock dilation and space construction. We neither shall nor deal with a more practical solution from an AI p.o.v. for ethic reasons.

*Riemann already gave the answer intuitively* in his famous dissertation on phase spaces and color spaces…

**RECAP**. The main duality of analytic geometry happens between @-mind center of frames of reference to reflect the mind’s subjective point of view and the 3 topological objective organic planes and the scalar complex plane as representations of the internal form of events happening within the key organs and ST symmetries of Nature. So there are 3±¡ main frames of references in analytic geometry, studying reality from a 0-point of view, similar to that of the mind of *each of those partial organs, which therefore map out clearly different worlds and prove the relativity and difference between the infinite mind-views of the Universe, even when they use visual light spacetime*.

** (TO BE CONT’ED)**

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