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# Non-E Fractal Points

Fractal points gauge information with infinite parallels.

The fact that all points interact with the external Universe is explained by the 5th postulate of Fractal Logic, which completes the 1st postulate.  The 5th postulate defines a point as a knot of communication of multiple flows of energy and form, which the point absorbs to obtain energy and information for its organic cycles of iteration.

Thus we introduce a new perspective in our study of a Non-Euclidean, fractal point of view: its capacity to gauge information and absorb energy, acting-reacting to the Universe. Since beyond those 3 internal regions the point feeds on an external world of relative forces, which provide the point with energy and information to map out reality and feed its body with motion. Thus, while energy and form will be different for each point according to the scale and form, all entities of the Universe will be able to absorb a relative number of ‘infinite parallels’ to create mental fractal mappings and keep on moving: light feeds particles, intergalactic gas feeds stars; plants feed animals that feed man; stars feed the black hole, and so on.

In science, mathematicians and physicists seek to describe reality as ‘made of mathematical entities’; while logicians try to explain reality as a series of causal events. But only when we use i-logic geometry we can mix both approaches, describing reality as a series of ‘mathematical points of view’, or entities that ‘gauge’ the geometry of the Universe, constantly mapping out the universe, to achieve a ‘selfish’ will, the desire of all entities to feed on energy, gauge information, reproduce and evolve socially. Those points of view are thus ‘mathematical minds’ because they gauge and somehow ‘perceive’ the geometry of the Universe, acting according to that geometry as they gauge and move with an ‘automatic’ behavior, trying to absorb energy for their body, information for their gauging mind, and combining both, to reproduce ‘self-similar points’.

So we shall consider that the Universe is made of i-logic, fractal ‘points of view’, akin to the concept of souls and Atmans in western and eastern religions’, of ‘Non-Euclidean points of view’, ‘monads’ or ‘Spinoza’s geometrical entities’, in western philosophy of science and mathematics; of ‘gauging particles’ in physics, or ‘informative DNA nuclei’ in biology and perceptive heads in biology. The size and form of all those points varies but all of them are spherical knots of information with sensorial apertures to absorb and emit energy and information, from boson particles to sound waves or light signals.

THE ZERO-POINT OR MIND, CENTER OF THE WILL OF ‘ACTIONS’ OF THE SYSTEM

Foreword. The equation of the mind and the 5th postulate of Non-Euclidean geometry.

Now a hyperbola is perceived as a cycle from the 0point in the centre of reference. that is how we are. We are a zero point but perceive a hyperbola of the Universe. We are nothing and we perceive it all – the entire curve of the hyperbola has 2 focus, but the point that matters is the zero point.

In the beginning there was an observer, a focus point that let itself cross by infinite parallels and perceived a mapping of the Universe.

This original point might be considered infinite or infinitesimal, God or microbe, the unmoved one, of the eternally moving. But there was no other inertial frame of reference but the one he perceived.

And it was of course, a polar one, the perfect form, the 2-sphere.

Did he started the Universe by the mere fact of co-existing with it? Was he looking at its inside forms? We do not know this metaphysical questions but we can describe bio-topo-logically the first point in existence.

The infinitesimal point, O, saw the infinite Universe, ∞, and created a constant mapping of it, O x ∞ = Constant, in the stillness of the mind.

The point, the Observer the measure is the first action that starts any activity in reality. Prior to motion towards a field of energy or information that interest us or away from it, the point must perceive it. And for that to be possible, the point must have parts, an informative mind –like capacity to gauge information, and rotational motion to change the angle of view.

The point thus have to have angular momentum, the minimal unit observable in the Universe, the h action, a point of view a momentum moving and a radius of distance.

The fundamental Unit of the fractal universe thus is also a point that makes up an absolute space-time, but such points unlike Euclidean classic points of mathematics are fractal non-Euclidean points, which have breath, hence are cycles, the minimal unit of the Universe a time cycle, seen in statics as a cyclical point, dynamically as an attractive point-particle in a static position or a particle with momentum, neutral to a background universe.

Infinite of such points exchanges energy and information in infinite scales is the departing point of any scale of realty to understand how it builds above and below itself O±∆ superorganisms.

But the infinitesimal point to be noticed and exist must at least ‘think’, observe, perceive with a certain angle a part of reality converted into mind mapping.

And this can be measured with the simplest, smallest quanta, the angular momentum of a system, the h-derivative increase, the ∆minimal action of perception:

∆ï: angular momentum = h-derivate.

Objective vs. subjective perception

If the fundamental differentiating, ‘external’ property of species in the Universe are the Metric of the 5th dimension (i=SP x Tƒ), which makes them belong to different inverse scales of size in space and energy volume (SP) vs. speed of temporal information), the essential equalitarian feature is the co-invariance of that product, which gives them the same value when considered, internally’ from the inner point of view of the system,

Since from the inner ‘measure’ of time, the smaller, faster clocks of a diminutive being, process the same amount of information in its life span than the larger being. So an insect which lives a maximal of 7 years (queen ants, elephant cockroaches), since it processes information 10 times faster than we do – reason why we don’t catch a fly – it actually lives as much as we do, 7 x10=70 years.

This is the wisdom of the 5th dimension: all selves feel the center of the Universe they measure from their Non-Euclidean fractal point of view, which has from ‘its’ perspective as much information as the entire Universe do.

The mind self-equation:

Tƒ(infinitesimal point of view) x ∞ (infinite, perceived Universe) = ∆ (constant mapping of the whole)

nicely puts it in perspective.

Topology explains it: the sphere is the only topology that can shrink without limit of tearing, allowing the reduction of the infinite into the infinitesimal (Poincare conjecture).

The 5th postulate of non-Sp geometry also defines it.

Let us then start with the usual methodology of ‘building dimensional blocks’ of ST theory, by considering the Spatial location of Minds, as the center of maximal communication of a system.

The Mind’s point of view

In the graph, a mystical representation of the mind, as the central triangle of a tetraktys. The mind is the ‘fractal point of view’ (mathematical interpretation) or zero point, in which maximal flows of information converge, both from the external and the internal regions of the being, where the will of the ‘4 åctions’ resides.

The Mind is the ‘point of view’, site of the will, that perceives each element, each part.

It is the ∆-centered dimension, in the sense that it is the most ‘connected’ point, in the center of the system, which ‘reflects’ upon the existence of the whole.

In mystique terms it is in a way the soul, the atman. It is the central number of maximal communication in the tetraktys that Pythagoras rightly considered the perfect number, illustrated in the Kabala graph above.

In Physic-mathematical terms it is the final black hole, non-Euclidean point of view of the system. It is the 10th or 11th dimension of Time on the string formalism, etc.

In biological systems it is the nervous, ’10th system’ that integrates the other 3×3, energy/reproductive/informative physiological systems of the human being.

Depending on which ‘human jargon’ with more or less errors – lacking the ∆ST formalism and ∆ST philosophy – we talk of, it will be ‘named’ with different forms.

I often use the term the 5th Dimension, of all ∆-scales of the being, to represent the ‘whole’. And so we have termed it also as the 5th ‘invariance’ or ‘isomorphism’ that all beings share, since the mind-self, the centered dimension of all scales, the 5 isomorphism of nature, the point of view, the soul is the most important dimension of all beings.

It wraps up all the other dimensions, isomorphisms and scales.

And it is therefore the center of its 3×3 ∆±4 scales, its 3 topological space organs, and its 3 time ages (the being is at its maximal force, in the SxT (s=t) steady state mature, adult age of balance, ST, age of maximal duration.

In man obviously is the brain, expression of the ‘Whole’ or ‘Soul’, which perceives internally the system through its ∑3×3 networks (our sensations of the energy, digestive system, the reproductive, endocrine system, and the mind thoughts and emotions) and externally through the senses. But in many other systems is a very different site.

So perhaps the best definition is that of Non-E mathematics: the ‘point of view’, the part of the fractal point that gauges maximal information.

I.FRACTAL POINTS.

THE FUNDAMENTAL PARTICLE: POINTS->NETWORKS->SUPERORGANISMS OF SPACE-TIMES

Time motions break absolute space into bidimensional space-time membranes.  Œ: fractal, i-logic, non-Euclidean, cyclical points of view: the fundamental particle

We exist in a Universe, whose fundamental particle is not a material form, as simple science thinks, but points of time space, which mirror in their inner, i-logic, fractal topologies the information of their world, acting-reacting to the Universe. Thus only by mixing perception and motion, mathematics and organicism, we can define those points of existence, as the fundamental particles of reality and:

‘The Universe has a body and a soul called logos.’ Plato

So we describe a Universe with a geometrical how, the laws of Non-Euclidean points of views, the cycles they trace, the mathematical laws that describe those cycles; and a bio-logical, temporal why: the will to move, trying to fulfill the 6 arrows of time. It is the game that all P.O.Vs of all planes of existences, from atoms to human beings perform.

Those logic and mathematical laws of points of view are the metaphysical laws of the Universe – the only essential events/forms that repeat unchanged in all forms. And so, a proper description of the Universe should start with the laws of i-logic geometry; then apply them to the 2 simplest motions of reality, lineal forces and cyclical masses/charges; and then, once those ‘immortal laws’ are defined for any system of energy and information, science should use them to define the specific events of each scale of reality and its species, as we all play the same Game of Exi=stence.

Mathematics, as a language that represents reality with simplified symbols, has a limited capacity to carry information. Its symbols, geometric points and numbers simplify and integrate the fractal, discontinuous reality into a single space-time continuum, the Cartesian Space/Time graph, made of points without breath.

However the points of a Cartesian plane or the numbers of an equation are only a linguistic representation of a complex Universe made of discontinuous points with an ‘internal content of space-time’. In the real world, we are all pieces made of fractal cellular points that occupy spaces, move and last a certain time. When we translate those space-time systems into Euclidean, abstract, mathematical ‘numbers’, we make them mere points of geometry void of all content. But when we look in detail at the real beings of the Universe, all points/number have inner energetic and informative volume, as the fractal geometry of the Universe suddenly increases the detail of the cell, atom or far away star into a complex complementary entity.

What mathematicians did was rather a runaway path. Because once geometers realized that multiple lines can cross a point, they could not use any longer the Euclidean definitions.

Now points, lines, planes and equality could no longer hold under the definitions of Euclid (points, lines and planes with no breadth), they remained undefined, when Hilbert, the father of modern mathematics affirmed ‘I imagine lines, points, planes…’.

Instead the proper solution was to redefine them in terms of their ‘inner parts’ as fractal points that grew in size when we observed them with a more detailed ‘hyperbolic geometrical view’, as that unveiled by non-E mathematics. So we ‘create new postulates of geometry’ by defining its essential elements, points, lines, planes and the laws of congruence (similarity) between them, as well as re-interpreting the concept of a Non-Euclidean point crossed by infinite parallels (5th postulate) in terms of a mind that absorbs flows of energy and information: O-mind x ∞ time cycles = Spacetime mapping.

So T.œ redefines points, planes and line-waves and their relative laws of equality (also called congruence) and continuity (which no longer holds), to refound mathematics on the basis of a far more realist view of its laws as ‘experimental laws’ that reflect the geometry of the 5th dimension and the complex casualty of a world with multiple time motions. And consider that those points with volume form social ‘numbers’, whose laws are therefore related to the laws of the 5th dimension that brings parts into social wholes.

So we propose a new Geometrical Unit – the fractal, Non-Euclidean point with space-time parts, which Einstein partially used to describe gravitational space-time. Yet Einstein missed the ‘fractal interpretation’ of Non-Euclidean geometry we shall bring here, as Fractal structures extending in several planes of space-time were unknown till the 1970s.

So Einstein did not interpret those points, which had volume, because infinite parallels of ‘forces of ‘—motions’ and information’ could cross them, as points, which when enlarged could fit those parallels, but as points in which parallels ‘curved’ converging into the point.

This however is not meaningful, because if such is the case parallels which are by definition ‘straight lines’, stop being parallels. So we must consider that what Einstein proved using Non-Euclidean points to explain the structure of space-time is its fractal nature: points seem not to have breath and fit only a parallel, but when we enlarge the point, we see it is in fact self-similar to much bigger points, as when we enlarge a fractal we see in fact self-similar structures to the macro-structures we see with the naked eye.

That is in essence the meaning of Fractal Non-Euclidean geometry: a geometry of multiple ‘membranes of space-time’ that grow in size, detail and content when we come closer to them, becoming ‘Non-Euclidean, fractal points’ with breath and a content of ‘—motions’ and information that defines them.

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

Euclid 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:

Abstract, continuous, one-dimensional point:

. ____________

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.

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. The maths are the same, the interpretation of reality changes, adapting it to what experimentally we see: a cell-like point enlarges and fits multiple flows of ‘—motions’ and information, and yet it has a point-like nucleus, which enlarges and has DNA information, which seems a lineal strain that enlarge as has many point-like atoms, which enlarge and fit flows of forces, and so on.

So each point is in fact a 3-dimensional point, and if we go to the next scale, a 3×3=9 dimensional point and so on. Yet those dimensions are the so-called fractal dimensions, which are not ‘extended to infinity’ but only within the size of the point. In Euclidean geometry, a point has no volume, no dimension, but string theorists say that even the smallest points of the Universe, cyclical strings, have inner dimensions that we observe when we come closer to them. That is the essence of a fractal point: Tƒ be a fractal world, a space-time in itself.

‘Any Non-Euclidean point is a fractal space-time with a minimal of 3 internal, topological, spatial dimensions and an external time motion in the ∆+1 ecosystem in which it exists’

This simple law is the most important law of the 4th paradigm, foreseen by Leibniz in his Monadology, the foundation of the mathematical model of Multiple spaces-times that completes the 5 Postulates of non-Euclidean geometry and gives us the tools necessary to create a complex new logic and new mathematical model of the Universe, easy to connect through topology with the isomorphisms of the previous paradigm of a single metric space-time continuum.

Further on those points must be described always in 4 dimensions, with motion. This should have been obvious, but abstract mathematics simplifies entities into numbers and static forms, and organic motion properties disappear. Yet we still say ‘San Francisco is at 8 hours from LA’, because we mean that journey is a combination of the motion of a car and the spatial distance. Thus we measure reality in Time-space, not only in space as Euclidean maths do.

Thus, in the same way Saturn’s rings stop being planes without volume when we come closer and observe them as fractal points, called planetoids; Non-Euclidean points acquire both motion and volume when we approach to them. In words of Klein, a sphere is not a continuous static form, but a group of points in cyclical movement. So in the same way the Saturn’s rings are a group of planetoids, a Klein space – the space-time that fills a point has motion – it is the sum of a series of cycles5.

Einstein didn’t go further, adapting the other 4 Euclidean definitions to the new Geometrical unit: a fractal point with volume. Only then we will be able to define the 2 planes of physical forces, the plane of gravitation and electromagnetism, or any system in which several planes of space-time co-exist together (as in a human being extended from atomic to social planes of cyclical existence).

In all those systems planes are made with cellular points, Riemannian spheres with volume that form lines, which are waves between points that exchange ‘—motions’ and information and planes, which are organs of self-similar points that process ‘—motions’ or information in parallel networks.

Thus the 5 Postulates of Non-E Geometry vitalize the Universe as a series of networks of ‘—motions’ and information of self-similar cellular points. Since the line and the plane acquire volume and become self-similar to the commonest forms of the Universe, the wave and the network of points with a 3-D volume.

This simple fact explains one of the most important discoveries of modern physics, the Holographic principle, according to which information might be bidimensional, as in the screen of a computer or the page of a book. Now bidimensionality no longer becomes ‘magic’ since the 3rd dimension is the relative size of the ‘fractal point-particle’. Thus bidimensional sheets of information do have a minimal 3rd Dimension; the inner content of the point, which in a relative universe of infinite sizes seems to us a particle-point without volume, as we don’t see either the volume of a sheet of paper or a pixel.

All in the Universe are thus complementary systems made of networks. Now this might sound absurd to the anthropomorphic reader that thinks humans are different from the rest of points of the Universe, but it is a fact that those points obey in their åctions and communicative flows within a network the same isomorphisms: humans and electrons behave the same when they move through slits or in herds, the geometries of social groups are also the same and the ultimate purpose of those points, to feed on ‘—motions’ and information, whatever kind, is also the same in all networks of the Universe. And so that group of isomorphisms of networks becomes a primary why for all beings of the Universe.

A fact the leads us to the final element needed to understand the why of the Universe: ‘non-Euclidean points’ organize networks that become points of a higher scale, which reproduce and organize new networks; and so the Universe keep growing in fractal scales, from particles that organize networks and become atoms that organize networks and become molecules that organize networks and become cells, that become organisms, that become planetary societies and planets and stars form gravitational networks that become galaxies, organized by dark matter into Universal networks.

Because all entities have motion reproduction is merely the repetition of a motion with form. Because each entity has 4 time arrows, all of them trace multiple trajectories in search of those arrows, hence they realize multiple time cycles.

For example, a human feeds on ‘—motions’ and information with body and head, reproduces through multiple social cycles and evolves into societies. Our åctions are more complex but essentially the same of those of any particle.

So the unit of reality is a space-time cycle, and many of them create a knot of time cycles or entity of reality, which will be reproduced by repeating those formal cycles with motion in other region of space-time; and many of those knots of time cycles, which are self-similar, since they are born from the reproduction of a first form, come together with self-similar beings into networks. Some of those networks are spatially extended with a lot of motion (fields in physics, bodies in biology) and some are very tight, formal, with a lot of in/form/ation, (particles and heads or nuclei in physical and biological jargons). Both types of networks together then create a complementary organism, which is fitter to handle both ‘—motions’ and form; hence it survives better, it ‘exists’.

Today scientists of measure scorn philosophical and logical analysis of causality in time because it cannot be easily put in numbers. But numbers are only one of the languages of information in the Universe, and many of its properties of bio-logic nature are better described with logic words. In that regard, we can now fusion philosophy and science answering the fundamental question, ‘why we exist’; since once we realize that we are ‘made of time cycles’, knots of time cycles and networks of knots of time cycles, an intelligent, informative, eternal universe of motions and wills of existence makes the dogmas of deism and mechanism, childish myths.

‘What is existence’ cannot be revealed from the simplex point of view of a mechanical world, which cannot explain the fact that we are made of motions with form that leave a trace on space but are essentially åctions in time that have a social finality – to create more complex networks, chaining knots of åctions into systems. This social will of every point and entity of the Universe is completely at odds with a mechanist, fixed, solid, senseless, dumb Universe. Motion in Time and social evolution are concepts that require the capacity to gauge information and interact with other self-similar points to create those organic networks. Further on, dual networks tend to evolve and reproduce new points through exchanges of ‘—motions’ and information.

The result is the creation of 3rd network/system: the reproductive network. And so most systems of the Universe are organic, ternary systems made of points (which can be anything from atoms to cells to human heads) organized in 3 networks. We exist as organic networks, to sense flows of ‘—motions’ and gauge information. Existence justifies itself.

Those Non-Euclidean points crossed by infinite parallels are able to gauge information, which implies a perceptive, intelligent Universe in all its fractal, self-similar scales of reality – a world in which even the smallest atoms can act-react to the environment, ‘apperceiving’ light and gravitational forces. Aristotle and Leibniz, the 2 foremost predecessors of the 4th paradigm of biological whys distinguished conscious perception from vegetative and mechanical perception.

It means perception has degrees of complexity. So the simplex particles of the Universe act-react in a mechanical way; yet they still gauge information, reason why quantum physicists called their theories gauge theories, and they still have 2 complementary networks of ‘—motions’ and information, reason why quantum physics is based in such complementary principle.

Yet this intelligent, active, temporal, informative Universe can be described with the formalism of logic and mathematics because its fundamental unit – a spacetime cycle – can be explained with ‘feedback’ equations, used in system sciences to explain the back and forth interaction between two poles or elements of an equation. Se<=>Tƒ where Se is a component of spatial ‘—motions’, a motion element, body or field and Tƒ is a cycle of time that carries information, particle or head, becomes the syntactic, logic, minimal unit of reality.

These simple first elements of reality – points with volume that exchange ‘—motions’ and information, creating waves of ‘—motions’ with form (no longer lines – 2nd postulate), according to a set of isomorphisms based in their self-similarity, (no longer equality – 3rd postulate), that makes them evolve into different topological networks (no longer planes – 4th postulate) – make mathematics an organic language able to describe the logic of creation of all systems of the Universe made of infinite fractal, organic networks intersecting and creating when we put them together under those isomorphisms and topological restrictions, the puzzle of reality that the simpler, 3rd paradigm called the space-time continuum and becomes now a General System of Multiple, Fractal Spaces with vital ‘—motions’ and Time cycles with information, the two substances of which all beings are made.

And so with those 3 scales of ‘existence’: time cycles, knots of time cycles and networks of knots of time cycles (Non-E Points) we can explain all the ‘åctions’ and systems of reality made of those cycles, knots and networks; and describe a complex Universe that exists ‘in time’ more than in fixed space, since it has always motion; it is also dynamic, made of cyclical, feed-back equations whose causal relationships, forms and trajectories are the essence and purpose of existence.

We thus consider a more complex analysis of time arrows, beyond the duality of ‘—motions’ and information, which combine creating a reproductive arrow, Sp X Tƒ, and further on socialize, ∑Sp X Tƒ, creating networks. And so the universe has also an organic will: to create networks of self-reproductive points of ‘—motions’ and information.

Yet the most astounding property of those points is to be points of view, points with will, which perform åctions with the purpose mechanical or not, but probably felt in all scales as the inner freedom of the point, of obtaining ‘—motions’, information reproduction and social evolution. The 4 wills or whys of the Universe are indeed embedded in the postulates of i-logic geometry. The point to exist has to be complementary, to feed and gauge ‘—motions’ and information and to last beyond its wearing it has to form part of bigger social networks or reproduce itself to last beyond death.

This simple program self-selects those species that reproduce and evolve socially even if that contradicts the primary individual arrows of the point. Thus the engine of the contradictions of behavior of points is that tug of war between the Galilean paradox of all points which gauge bigger his nose than Andromeda but need to hunt in herds and control the forms of the Universe with self-similar minds, joined in networks, this eternal duality of freedom vs. order, individual ego vs. collective spirit.

Tƒ exist is to act with motion and form, trying to achieve the ‘arrows of time’ or will of the Universe – feeding your ‘—motions’ network, absorbing information for your informative network, reproduce your system and in doing so, starting an external process of social evolution with self-similar entities to yourself.

Those processes can be described with mathematics but we have to accept an intelligent, perceptive, fractal, self-similar Universe of infinite points of view gauging reality in a mechanical, vegetative or conscious way to explain why it happens. Their mathematical description stems from the duality between geometric form and logical function (hilomorphism).

Thus, the postulates of fractal i-logic geometry define also the basic arrows= cycles/dimensions of the Universe: the 1st and 5th postulate define a point as a system whose inner parts are able to transform and emit ‘—motions’ and information, Sp>Tƒ<Sp; the 2nd postulate defines an Sp X Tƒ wave of communication that reproduces ‘—motions’ and form between 2 fractal points; the 4th postulate defines the social evolution of a herd that creates a fractal plane – a network with dark spaces; and the 5th postulate explains a point mapping reality, as it absorbs ‘—motions’ and transforms it into information through its small apertures to the Universe. Since even a minimal quark, as Einstein affirms, should be crossed by a relative ∞ number of strong forces.

It is the organic will of all systems that search for those 4 arrows what makes the quark to exist as a knot of such flows of time arrows: a physical particle traces energetic cycles described by the principal quantum number; shapes the form of its trajectory, a fact explained by the secondary quantum number; iterates along the 3 coordinates in similar shapes, an act described by the magnetic number, and gauges information to evolve in social groups, a fact described by its spin number. And those 4 numbers define it as a Non-Euclidean quantum knot of complementary ‘—motions’ and information with a 4D will of time. And as it happens they express the 4 arrows of time: Sp->Principal number; I-> secondary number; Re-> spin number; 4-> social evolution: magnetic number.

Further on, we can reduce all those topologies of social numbers and networks to the canonical 3 topologies of a 4-dimensional Universe, proving that those 3 topologies have the properties of ‘—motions’, information and reproductive events.

And so we talk of 4 ‘arrows of time’ or dimensions of change that create the future: energetic and informative systems and events, which reproduce a wealth of self-similar beings that organize themselves into social networks, creating bigger wholes – new scales of reality. And this simple game of complementary beings that in favorable conditions reproduce self-similar beings, self-organized into bigger social networks becomes the why of all realities. Even the simplest particles, quarks of maximal information and electrons of maximal spatial extension and motion ‘decouple’, reproduce, when absorbing ‘—motions’ into self-similar forms, and associate in complementary networks called atoms, made with a central informative mass of quarks and an energetic, electromagnetic, wider body of electrons.

The 3rd paradigm of metric measure is not at ease with such ‘dynamic, spiritual concepts’, even if they can be described with the same mathematical formalisms as the previous example of the quantum numbers show. Those apprehensions however are dogmas, which stem from anthropomorphic beliefs.

Fact is that even the simplest complementary systems (quarks and electrons) interact together and if they can absorb more ‘—motions’=motion they are able to repeat=reproduce the cycles of its system. And so we talk of a 3rd reproductive system: from quarks and electrons, the fundamental particles of the Universe that decouple in new particles when they absorb new ‘—motions’ to living organisms, the fact that all is motion with form makes easy to reproduce those formal motions in an organic way. Thus the new concept of a world made of formal motions brings about also a more complex philosophy of reality – organicism. Organicism and its mathematical units, fractal points, that gather into social networks called topological spaces substitute the restricted concepts of Euclidean points, continuous spacetimes and mechanism, explaining why all those time cycles exist, guided by 4 time arrows: ‘—motions’ feeding, information gauging, reproduction and social evolution.

Those 4 categories are the so-called drives of living beings, the quantum numbers of particles, the 4 dimensions of our light space (electric-informative height, reproductive magnetic width, energetic length and social colors). Thus, there is a ‘Universal Plan’ with an existential finality: to create organic systems, departing from ‘—motions’ bites and information bits evolved 1st into social networks, then into complementary systems and finally into organic systems, news points of a bigger fractal whole: particles become atoms that become molecules that become cells, organisms, planets, galaxies and Universes. It is the 4th organic why that completes the adventure of science and this work explores in all its consequences.

Reality can be resumed in 2 words: networks, whose flows of exchange of ‘—motions’ and form creates the patterns and events of reality and organicism, the philosophy of reality based on them. Organicism means reality and all its fractal parts are made of vital spaces (bodies and forces) and time cycles (informations). We do not exist in an abstract background of time and space but we are made of time cycles and lineal spaces, cyclical and lineal strings if we were to use the restricted jargon of physics, a specific case of the wider jargon of general systems, which evolve socially to create the complex systems of each science.

Those wider, more complex definitions of time and space will substitute and absorb according to the Principle of Correspondence that makes each paradigm a particular case of the new, wider view, the limited concepts of a single space-time continuum and a mechanist description of the Universe, proper of the age of metric measure, which the pioneers of systems sciences and complexity have wrestled with throughout the XX century.

Recap. A Universe made of motions in time, extending through multiple scales of relative space, requires evolving the human languages of space (Euclidean geometry) and time (Aristotelian logic), used to describe the simpler space-time continuum of classic science. This task started by the pioneers of modern science (Riemann, Darwin, Planck and Einstein) must be completed before we can study the properties of physical and biological entities with the new tools of Non-Euclidean mathematics, which describe a world of multiple, relative space-time networks made of points as Complementary entities with energetic and informative system or ‘Points of view’ and the new tools of Non-Aristotelian logic, which describe the creation of the future by multiple causes – all the points of view or agents that create a certain network space.

The universe is a game of creation and destruction of networks of fractal knots of time cycles=st-points.

The minimal unit of the universe is a Non-Euclidean point/number, which classic mathematics defines as void of inner form and organic properties, to simplify the networks of numbers and point-like entities of the Universe for its geometric study. In reality though, points have breath; that is, they are real entities with ‘—motions’ and information parts, and so we have to upgrade Euclidean definitions of points, planes, lines and equality with the new tools of Fractal and Non-Euclidean mathematics to make the language of geometry closer to reality.

What mathematicians did was rather a runaway path. Because once geometers realized that multiple lines can cross a point, they could not use any longer the Euclidean definitions.

Now points, lines, planes and equality could no longer hold under the definitions of Euclid (points, lines and planes with no breadth), they remained undefined, when Hilbert, the father of modern mathematics affirmed ‘I imagine lines, points, planes…’.

Instead the proper solution was to redefine them in terms of their ‘inner parts’ as fractal points that grew in size when we observed them with a more detailed ‘hyperbolic geometrical view’, as that unveiled by non-E mathematics. So we ‘create new postulates of geometry’ by defining its essential elements, points, lines, planes and the laws of congruence (similarity) between them, as well as re-interpreting the concept of a Non-Euclidean point crossed by infinite parallels (5th postulate) in terms of a mind that absorbs flows of energy and information: O-mind x ∞ time cycles = Spacetime mapping.

So T.œ redefines points, planes and line-waves and their relative laws of equality (also called congruence) and continuity (which no longer holds), to refound mathematics on the basis of a far more realist view of its laws as ‘experimental laws’ that reflect the geometry of the 5th dimension and the complex casualty of a world with multiple time motions. And consider that those points with volume form social ‘numbers’, whose laws are therefore related to the laws of the 5th dimension that brings parts into social wholes.