**FOREWORD**

**The i-1 mathematical perspective: points**

In terms of i-scales, mathematics is the i-1, ‘point-like’, ‘atomic-cellular’ analysis of beings.

What complexity does, is to highlight those limits of mathematics and widen its ‘perspective’ by considering that the ‘points’ which are units of mathematical languages, are only ‘apparently’ points-with no breath (Euclidean points).

Generally speaking, physical laws are given in local form Physical laws are given for mathematical points but applied to extended objects. while their application to the real world requires a departure from locality.

For instance, Coulomb’s law in electrostatics and the universal law of gravity are both given in terms of point particles. These are mathematical points and the laws assume that. In real physical situations, however, we never deal with a mathematical point. Usually, we *approximate *the objects under consideration as points, as in the case of the gravitational force between the Earth and the Sun.

Whether such an approximation is good depends on the properties of the objects and the parameters of the law. In the example of gravity, on the sizes of the Earth and the Sun as compared to the distance between them.

On the other hand, the precise motion of a satellite circling the earth requires more than approximating the Earth as a point; all the bumps and grooves of the Earth’s surface will affect the satellite’s motion. All points thus have parts and details when we come closer to them, and this fact can only be explained accepting the complex metrics of 5 dimensions.

When in fact they do have the closer we come to them, ‘widening’ our ‘scalar perspective’, points with volume with inner parts, fractal points. And so a new/old form of mathematics, the study of the Universe as made of fractal, non-Euclidean points with breath, means a r=evolution in mathematical thought similar to the one complex sciences will cause in all other disciplines of human knowledge.

Consider for example a human being, in a match stadium. From above and far away it is a ‘point’, and it does matter little. As O.Welles said in the 3rd man, you can look at them, they don’t matter, if each of those points were money and you could grab them, how many would you take? Distance and lack of information make us indifferent to far away entities. yet if we come closer to the stadium we will observe them as i-ndividuals – this is the biological perspective – and as parts of a social organism, which move together with similar actions. This is the i+1 social scale and its laws of common motion, and social interaction are again of a different quality to those of a mathematical point.

So despite the common belief, since Galileo that the world is ‘written in the language of mathematics’, we must consider a Upanishad’s version ‘the languages of god are infinite’, or rather, the scales of the universe are infinite and when we look from above, we make a mathematical description and when we look from below we make a social one.

**INTRODUCTION**

**I-LOGIC MATHEMATICS: NON-ARISTOTELIAN, NON-EUCLIDEAN FRACTAL WORLDS OF I-POINTS.**

Mathematics is thus the perception of the universe with a distant perspective. Its units being points, which represent any form of the Universe, reduced to its i-1 minimal, ‘energetic’ perception.

Then the second level of structure of mathematics, is a collection of points, which mathematicians tend to refer to with two ‘structures’ of algebra, the set and the group, which is basically a set (a spatial collection in ‘present’), subject to temporal motions, caused by a ‘symmetry’.

Thus a set is ∑i, a sum of points…

And a group is STi, a set in ‘motion’ due to a symmetry between its dimensional scales.

Thus we can write an equation with ‘existential algebra’ to describe those mathematical levels:

∑∑i-points=∑Sets-> ∑Set x Symmetric Motion = i+1 Groups…

which define the 2 main scalar structures of mathematical systems.

Indeed, it has been said that mathematics is all about geometry, its initial configuration, whose fundamental unit are points; and in the modern age about sets and groups, the fundamental tools of algebra. And today they are used to define mathematics. Now, we will understand them in harmony with all other sciences, as a ‘manifestation’ of the 10 Dimensional Universe. To notice though that mathematical analysis has only ‘2 fundamental scales’ – that of points and its elaboration by geometry, and that of groups of points, sets and groups with operations. Why there are not an intermediate i-ndividual scale? The reason is the aforementioned lack of ‘inner structure’ of classic mathematics, which we partially will remedy with our r=evolution of the concept of a point. So far, this means that the i-ndividual scale matters not in mathematics.

Mathematics is about ‘sociological’ properties, as ‘numbers’ are social collections of points where the individuality disappears. And so i+1 social scales are indeed an overlooked property of at the Universe, denied by the ego-centric human being, which the mere existence of mathematical numbers prove.

Mathematics thus is the essential, ‘primary’ language of the simplified i-1 vision of the Universe, but also fundamental to understand the sociological, i+1 qualities of ‘numbers’. Whereas, points are the ‘spatial’, geometric perception and ‘numbers’ the ‘ordered’, ‘causal’, hence temporal analysis of it.

We thus have defined for mathematics and its sub disciplines two fundamental symmetries of the Universe:

– The symmetry of its scalar dimensions

– The correspondence or ‘isomorphism’ between its spatial and temporal vision (points and numbers).

And both prove or rather ‘reflect’ the laws of scalar space-time (as mathematics as all other languages are secondary to the Universe and its fundamental language, the ‘existential algebra’ of the 10 Dimensional super organism).

Mathematics also reflect the properties of at the 3 dimensions of space, considered ‘alone’. In fact in combination with physics, (which adds motion, the dimensions of present-time duration to the static geometry of space), this field is the most exhaustively studied of all the aspects of the Universe by the ingenuity of man. If there is a subject where all the details are known is on translations, rotations and symmetries in space and at the motions taken place within it

But what about the symmetries and ages of time? How mathematics has dealt with it? It has done so in steps.

First mathematics dealt with the dimensions of present time, or ‘duration’, v=s/t, and its inverse frequency, 1/t and again it has done so exhaustively through the development of calculus and the corrections for ‘simultaneous’ measures discovered by Einstein’s relativity. But by definition calculus studies ‘continuous’ functions, which therefore do ‘not’ change phase, state or age, that is do not ‘abandon’ the relative duration of a present, in which the inner forms of the system do not change.

In other words mathematics has not properly understood the causal processes from past to future, which in any case are *much better understood by Logic. And so if we were to consider the ‘limits’ of mathematics, they would be found in the proper understanding of ‘evolutionary’ and causal processes and the past and future dimensions, which are far better analyzed by logic, causality and biological theories of evolution and organisms, so we shall refer those dimensions and processes in those other sections. *

However in the XX century the last branches of mathematics to be developed – topology and fractal mathematics, have made inroads on different temporal motions than ‘duration’, which are essential to the understanding of the morphological processes of change – essentially directed by at the 2 manifold, membranous structure of biological organisms. While fractal mathematics is essential to understand the repetitive scalar structure of the Universe. So we shall also consider those themes in depth, in this web ‘always a work in progress’, as i keep pouring the thousands of pages of analysis of all sciences done with the formalism of 10Dimensional Super-organisms.

Let us then start as the History of Mathematics did, with the initial postulate of geometry, the definition of a point, upgraded to includes its ‘inner, informative and energetic parts’. Since now the 5 postulates of ‘fractal, non-Euclidean geometry’ will further enlighten our compression of the world in one of the main ‘languages of God’ but not the unique, mathematics.

** 1****ST**** POSTULATE: POINTS WITH PARTS: 3 FRACTAL, ORGANIC TOPOLOGIES**

**Topological Spaces. The why of geometrical forms.**

Reality is made of fractal points, which are knots of Time Arrows, able to perform energetic, informative and reproductive functions. As complex as one of those points-entities might be when observed in detail, *any fractal point is made of 3 regions whose geometry responds to the topological forms of a 4-DimensionalUniverse, the convex plane, the torus and the sphere.*

The Universe comes down to two bidimensional elements, energy and information, and its 4-dimensional combinations. Thus all entities can be described as wholes made of 3 internal parts whose geometrical properties maximize their energetic, informative and reproductive functions:

– Max Ti: 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.

– <=>: 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. E: An outer membrane of energy, described by Riemann’s spherical geometry.

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, exi.

It follows that the first part we observe in a point is the external membrane, which without detail seems to have a continuous, energetic appearance. But on close view, we observe most external membranes store and/or absorb information (Holographic principle) due to its fractal geometry. This can be generalized to any membrane which shows a ‘bidimensional surface’ that acquires more form, more ‘fractal steps’ when we come closer to it. Thus, the Holographic principle, which physicists know in the restricted field of black hole theory but can’t explain why exists, is both explained and extended to any bidimensional membrane of information. We find bidimensional, warped, fractal membranes that store information not only in black holes, but also in the development of organic senses, departing from the exoderm (external membrane of the fetus), in the seminal cells that reproduce life, formed as an outgrowth on the body surface (genital systems) or in the complex forms pegged to the surface/skin of the Mandelbrot and Julia sets, the best known mathematical fractals.

This external, energetic membrane has the topology of continuous surfaces called *the Riemann sphere*, which is the external surface of any point – the skin and limbs and any of its multiple self-similar entities, some of which are drawn at the bottom of the graphic. In any system of the Universe, the membrane acts as the energetic, external surface of the informative, point through which the point absorbs energy from the external Universe, to process it into information.

– The hyperbolic, which is the body or reproductive region (seen as a cyclical path of space-time), which fills up the space-time between the nuclei and the external membrane. It is the zone where the reproductive organs of the system exist, and where the information of the system is born.

– Energy and information systems have inverse properties: energy is expansive, external, more extended, and information is implosive and smaller. Thus we find the informative head or system either on top of the reproductive body, with a spherical and smaller size or in the center, and its topologies will correspond to those of maximal form; the so-called *toroid topology.* In the upper graph, it is the double ring, or convex, toroid surface, with maximal form, or informative center of any entity of the Universe, the point in which the fractal reproduction of information reaches its zenith. It is the same form than the hyperbolic but reproduced into a higher content of information, by doubling the initial form.

– Finally between both topologies of energy and information, the point will have a middle region of exchange of energy and form, made of cycles that go back and forth between those regions, which correspond to *the 3**rd**canonical topology*. It is the torus, plane or Klein disk – a curved region of energy & information quanta, with cyclical motions, confined by 2 limits, an external, spherical membrane of max. energy and an internal informative nucleus. It must be noticed that according to Klein, the topologist that studied better this type of surface, the hyperbolic is NOT really a fixed form of space, but we must consider those cycles’ motions and add the parameter of speed. Thus Klein introduces the Paradox of Galileo to describe the Non-Euclidean geometries of the Universe as we have done in this book. Cycles are mere static perceptions of motions and we must always consider distances as space-time distances. So we say: London is at 4 minutes distance because we consider distance and speed together. This is ultimately the meaning of time in physics, a measure of the speed of motion as a way to gauge space-distances: v=s/t.

In the graph, we observe several non-Euclidean points created by those 3 canonical topologies that can adopt multiple forms by deformation, but suffice to construct all the shapes of our Universe. Indeed, a topology is deformable. So an external membrane, which corresponds to the topology of a sphere, can become any shape, as long as it is not torn up, to enclose a reproductive and informative zone. So your skin is in topology a sphere, which encloses the complex forms of your reproductive organs. Those organs are Klein cycles of great complexity that exchange energy and information. Since those 3 topologies suffice to describe any 4-Dimensional form, it follows that the Universe is merely a puzzle of energetic, informative and reproductive parts, associated in ‘numbers’, groups and all kind of entities as those shown in the previous graph, from different sciences, which are created with those 3 topologies.

The 5 Non-Euclidean postulates and 3 topologies common to all species prove that all fractal points are complementary, made of regions that process spatial energy and regions that gauge temporal information, both of which exchange motion and form in cycles mediated by its intermediate, dynamic, reproductive ‘body’ region. This is what Lobachevski, Riemann and Klein discovered when they invented Non-Euclidean Geometries in the XIX c.: Euclidean space is a simplification of a moving space, made of points with volume that constantly trace cycles.

In a 4-D Universe there are only 3 topologies of space, which actually display the properties of energy, information or a combination of both, the 3 canonical topologies structure the inner geometry and functions of the organs of any species, proving the homology of all space-time fields in the Universe. In the graph, an animal, an embryo, an electromagnetic flow, a galaxy, a proton, a seed, a planet, a cell and a boson display the 3 topological zones of a fractal point, each of them performing 1 of the 3 functions/arrows of space-time.

Topology *describes the internal parts of Non-Euclidean points when we come closer to the ‘fractal point’ and see its formal parts in detail, as we do in the last graph, with several* systems of Nature made of those 3 topological regions that correspond to the 3 main arrows of time of each of those systems:

*Information x Energy = Reproduction*

Time arrows are performed within each space/time point by one of the 3 topological regions, which explain causal processes of transformation of energy unto information as topological transformations. While the 4th arrow of social evolution, which requires more than a point, is described by the 2nd and 4th postulates of lines and planes.* *Thus Topology confirms a fundamental tenant of Multiple Spaces-Times, the 4-dimensionality of the ‘Holographic Universe’ made of bidimensional energy and information, which combine its properties to create the ‘visible’, bulky regions between the informative center and energetic membrane of those points.

Each of the 3 elements of fractal points displays the properties of one of the 3 Time Arrows: the center has height, it accumulates information and it is small. The external membrane is larger, continuous and protects the system. The intermediate zone reproduces the information given by the center with the energy absorbed by the membrane creating new energetic and informative bites and bits, which latter migrate towards the other 2 zones.

– Informative particles accumulate in the inner region as units of the central brain. Or, if they are seeds, they migrate to the surface of energy *where they become fractal systems of information, senses in organisms, informative human beings in the Earth-crust, ovules in mothers*. And once detached, they start a fractal, reproductive, palingenetic process.

– Energetic particles migrate to the membrane, becoming parts of a discontinuous protective shield. So in a cell we find mitochondria that produce energetic proteins and RNA for the surface membrane.

The internal, reproductive region happens also in all systems. In the human body the organs reproduce the cells needed for the blood network and the hormones and products used by the brain. In a galaxy, the intermediate region produces energetic stars and informative black holes, which migrate to the central region; in an ecosystem, the territory of the informative center, the predator, produces preys in which the predator feeds, bringing them to its central den.

*Recap*: The fundamental Particle of the Universe is neither physical nor spiritual but logic-mathematical: the fractal point described by the Laws of Non-Euclidean, fractal Geometry and topology, as an entity which becomes more complex when we come closer to it, till we can differentiate its 3 regions, corresponding to the 3 topologies of a 4-dimensional Universe: an energetic membrane, an informative center, and an exchange zone of bites of energy and bits of information between both.

**Formalism of i-logic geometry. Causal Algebra.**

The ternary structure of all points formed by the 3 topologies of a 4D Universe can be expressed with the symbols of the ‘Generator Equation of energy and form’:

– Max. Σ S: An external membrane or of max. extension, described by Riemann’s spherical geometry.

– <=>: A middle, reproductive plane or *hyperbolic*, whose cyclical paths happen between the skin/boundary of a sphere and a central hole, described by Klein as a disk, made of quanta in cyclical movements that communicate energy and information between the inner and outer zones.

– Max. Ti. The inner, dual center – a convex toroid, informative, central couple of disks, which display maximal form in minimal space, with a growing dimension of height that touches the poles of the sphere and channels the flows of energy and information of the system. 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.

Since function is form, those 3 topologies are perfectly suited to perform the 3 temporal functions of any super-organism: energy feeding, reproduction and information.

So we can write with causal arrows the topology of each point and its 3 regions: ΣS < EXI> Ti.

The 3 non-Euclidean geometries structure the geometry and functions of the organs of any species, proving the homology of all space-time fields in the Universe. In the graph, an animal, an embryo, an electromagnetic flow, a galaxy, a proton, a seed, a planet, a cell and a boson display the 3 topological zones of a fractal point, each of them performing 1 of the 3 functions/arrows of space-time.

There is in that sense only a small correction to classic topology needed to fully understand the structure of organic systems with the 3 topologies of reality. In topology we distinguish two types of spaces called closed balls and opened balls. Open balls are spheres with a center ‘a’ and a maximal distance/radius, r (from a to its surface), defined by all the points x, such as x < r (therefore an open ball does not include the surface or perimeter r); and closed balls are spheres which include all points x≤ r.

In i-logic geometry, all organisms are both, closed and opened balls, depending on our perspective.

As closed balls they include the 3 regions, previously described which are, a, the toroid center; r, the perimeter; and x, the points of the intermediate space.

The intermediate space is on the other hand, an open ball, which does not include the membrane, r, as it constantly exchanges energy and information with the external universe; *but it does not include either a, the center, so it is also opened inwards, to the information system.*

This is for example, the case of a black hole, which is wrongly understood in classic physics, *since the black hole has 3 regions, a, the singularity, ∑ x, the quark-gluon soup of extreme density, whose cyclical mass vortex are the black hole in itself, and r, the event horizon, which is not the black hole per se, but the membrane of exchange of energy with our electromagnetic space the black hole warps and feeds on.*

*Recap*: In a discontinuous Universe there are infinite fractal points, but all of them are composed of the same 3 ΣS, EXI, Ti zones; the external membrane, the intermediate region and the inner center.

**Mathematic description of the 3 regions of a st-point**

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. 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 toroid 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 toroid geometry till becoming the informative center or ‘brain’ of the system, attached to the informative network.

*– Max. Space: 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.ΣSe=Min.Ti*.

Thus *all Fractal points are ‘inner worlds’ whose membrane creates a discontinuity that defines an External Universe or* outer world from where the point obtains its energy and information. However the membrane is also the zone through which the point emits its reproduced micro-forms of information, and 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. +ΣSe: A* ‘*mouth*’ or opening that absorbs energy.

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

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

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

*– Max.Ti*: 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.

*– ΣS*<=>*Ti: 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.

**Dimensions of the 3 regions: Holographic Universe.**

In terms of dimensions the spherical membrane and inner informative singularity are bidimensional fields: The central singularity is a bidimensional surface of convex in/form/ation that curves the external spatial energy coming through the bidimensional membrane, creating together the 4-dimensional quanta of the intermediate region. For example, physical models of the inner nuclei of atoms made of informative quarks, define them as bidimensional, convex singularities. Black holes are said to store bidimensional information in its external membrane or black hole horizon. The computer screen or sheet of a book stores bidimensional information. The vacuum energy of the galaxy has a planar form, etc.

A 2nd consequence of the inner volume of points is a rational explanation of bidimensionality. Since now points have a minimal volume a bidimensional sheet of information has in fact a minimal height, but since reality is fractal and size is relative, from the ‘giant’ p.o.v. of our perception that depth of a sheet of paper or computer screen is a relative ‘zero’, yet makes bidimensionality ‘real’.

*Recap*: informative regions have more dimensions of form than energetic topologies, more extended in space.

**Morphological change and informative dominance.**

Natural organisms start as spherical seeds of information and then through morphogenesis differentiate in 3 functional regions that ensure the capacity of the system to process energy and information and reproduce itself, surviving in the Darwinian Universe.

As the species changes and evolves into more complex shapes those functional zones are kept. For example, the informative egg evolves into the energetic, lineal larva or young phase of most insects, by *translating *the inner, informative center to the dominant, forward head region and the external membrane to the ‘tail’, but both functions are preserved. Thus the inner informative regions migrate through the informative dimension of height to the dominant zone of the system. Yet the dominance of the informative center is not compromised:

* *In all systems we find a core/brane that acts as the dominant region of the organism and display paradoxically less spatial extension than the other regions they rule. It is the smaller nucleus of cells, humans and galaxies generates its information (DNA, human brain/eye system, galactic black hole, CPU (central processing unit) in computers, etc.)

In a galaxy the halo of dark matter and the central black hole dominate and seem to feed and form the radiant matter of which we are made. In man, the informative brain, extended through the central spine and senses, dominate the reproductive body and guide it.

The 2nd region in importance is the ‘body’ or reproductive region, which absorbs energy from the limbs that become imprinted by the system’s information.

And finally the 3rd region of the system, which is easier to renew, due to its formal simplicity is the energetic region, the skin, limbs and membranes, which brings energy to the intermediate region to allow the reproduction of the information stored in the center or head of the system.

Given the ∞possible deformations of those 3 unique topologies of 4-D space-time, reality creates an enormous variety of species, from an original seed of spherical information that develops those 3 regions in morphologies that soon resemble energy lines, reproductive cycles and information centers.

Our energetic region extends in lineal limbs; reproductive cells group into organs, becoming the body; while the informative centers move to the height dimensional and multiply its cellular forms in the sensorial boundaries of the head.

All those processes studied by morphogenesis, can now be explained not only in its how but its why.

*Recap*: The 3 relative forms of energy (external membrane), information (dual ring) and reproduction (the cyclical paths that exchange energy and information between the external membrane and the inner convex form), can be found in any system of reality. In all those systems, the informative region dominates the bigger, simpler reproductive body.

**The ternary structure of all Universal systems.**

The 3 functional topologies of a Non-Euclidean point become the 3 regions of all Natural organisms:

Atoms have a central, informative mass of quarks, spatial, electronic membranes and fields of gravitational and electromagnetic forces exchanged between them. Those 3 topologies also describe the galactic structure: the central black hole is the toroid, informative topology, the Halo of the galaxy is a Riemannian, spherical form, and the stars in the intermediate region, which feed the dark matter of black holes and reproduce the atoms of life, turn in cyclical, hyperbolic paths around the central black hole.

Physical space-time is the simplest world where the most basic morphologies play that same process of transformation of external energy that converges and reproduces cycles, attracted by a Non-Euclidean point, charge or mass: E=Mc2.

Cells have lineal, external membranes of proteins, which are a deformation of a Riemann sphere, an informative nucleus and in between they are invaginated by all kind of e<=>i cycles that transfer energy and information from the outer world to the cell.

Finally, a human being has a reproductive body, lineal, energetic limbs and a cyclical head, with an informative, smaller brain, composed of two hemispheres, which are toroid, convex, warped forms, corresponding to the informative dual ring of a Belgrami cone. The toroid, highly warped brain is a double hyperbolic, self-similar to the toroid topology of informative cycles. And so the brain hosts more information in lesser space than the body, as a mirror of its functions. Man though, while responding to the same canonical topologies in his organs, is by far the most complex being of information known in the Universe. And so his topologies are immensely more complex than the simpler physical particles and its transformations just described.

*Recap*: All systems of the Universe are made with the 3 canonical regions of a non-e point, which perform the 3 arrows/functions of all existential beings. Galaxies show the topology of non-Euclidean point, with a complex informative black hole of maximal mass, an energetic membrane of dark matter, and an intermediate region of reproductive stars that create the atoms of the cosmos. Human beings have also 3 ternary regions, the cyclical, toroid informative brain, the reproductive body and its organs that produce the energetic bites and informative bits of the organism and the lineal limbs that cause our motions.

**The fundamental unit of the Universe. The Non-E Point as a time cycle, its derived actions and integrated worldcycles. **

Now Non-Euclidean fractal points are time cycles. It is therefore necessary to have a dual conversation through this blog and reality- between the idealized logic-mathematical world of i-points of view creating the Universe in any plane of existence, and the multiple varieties of them we find in each plane.

Indeed, what in common have a human and an atom? Now if we compare only the head/particle point of view, we suddenly realize we do have much in common. We are two spherical systems with an aperture to absorb energy from the Universe in the equator region (our mouth, the atom’s main S-orbital plane), and two spherical points of maximal informative activity, contently absorbing and emitting waves, which act as our eyes/receptors of information, whose information regulate the behavior of all our systems.

To which extend we want to ‘make up’ this reality with niceties is up to us, the ‘deep 5D coverings’ we add to the bare bone atom to become molecule, cell, human, is of course of interest, but it is ultimately a repetitive game.

So the point, number, Œ, starts a game of actions.

First it observes. It is the observer, the primary being and as such it needs one better 2 eyes, to perceive with an excellent angle, in the tropic North, of the sphere. Here the point achieves rotation, which means without disturbing truly the outer world, except for what the pi sphere can implode as bits of information in its rotational motion, the being perceives.

Rotation thus is the first form.

It is not lineal motion which occupies other’s space and enters in conflict

It is rotation.

And so the point said, let me be a point of view, let me rotate, let me have angular momentum.

And existence was born

We shall call this first existence which we know spin and give it value h/2, the first known angular momentum of the cosmos.

That the point perceives is clear. It is an oxymoron, as we perceive light, our eyes are made of light and light is made of h/2 angular momentums, 2 of which make an h, dual point of view, which scans a 3-Dimensional world.

For a while perhaps the point vibrated, but certainly it was only too soon that the point made an entire π circle, and so as H turned two pis around it became a Planck constant.

Those minimal forms, the radian of perception, the Planckton with a full rotation, the first time cycle with a complex M x V x R 3 variables – the point, the angle of perception and the moving object with p-momentum, which the point observes, are already a game of gauging information and perhaps future energy, a first act of communication and so it enters the second postulate.

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