II. THE FORMALISM OF FRACTAL NETWORKS: NON-AE ORGANISMS

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The minimal Universal event communicates 2 Non-E points,: a Non-E point with more energy, Ei, that each science defines with different slangs, (‘a white hole’, ‘energyzer’, 'past form', ‘male’, ‘body’, ‘yang’, ‘moving field', etc.); and an informative point, Ot, the smaller form (‘a black hole’, ‘codifier’, ‘future form’, ‘female’, ‘Head’, ‘yin’, particle or 'center of perception', etc.) Both become united by a dual wave that transfers energy from Ei to the informative point of relative future, Ot, and information from the future point, Ot, to the relative past point, Ei, creating together a cycle of temporal energy, the minimal ExT, stable structure of the Universe. When those cycles organize themselves in more complex feed-back loops, the result is a system, network or super-organism.

Non-Euclidean points. The fundamental particle of the Universe.
The formalism of super-organisms describes them as networks of non-Euclidean points, a concept developed in my earlier work on quantic space-time (2) that describes the smallest scale of the Universe as a network of spins of temporal gravitation joined by strings of spatial energy. Those microcosmic networks have evolved into the bigger, more complex networks of the macrocosmic world through a process of social evolution, triggered by the higher efficiency of networks in all processes of survival. Yet, despite growing in complexity, non-E, informative points and energy flows have maintained the original properties of any quantic space-time network.
So we can define in mathematical terms any super-organism as a quantic space-time network extended in i-scales of information and composed of  n, non Euclidean points or individual cells: ΣEn=ΣTi. Thus any super-organism is a particular case of the Generatrix Equation of quantic space-time fields, in which networks of spatial energy or bodies and networks of temporal information or Heads exchange lines of energy and information. Yet to understand that formalism in detail we have to digress for a moment and consider the structure of any quantic space-time network by defining a Non-Euclidean point, the fundamental element of a quantic network:

‘A non-Euclidean point is a knot of spatial energy and temporal information, joined by a non-Euclidean line or geodesic flow of energy and information’.

Non-Euclidean points were philosophically described by Leibniz in his Monadology. They appeared first in science in the incomplete work of Mr. Einstein, regarding the Non-Euclidean structure of space-time. Euclid said 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 Universe followed a Non-Euclidean 5th Postulate:

"A point external to a line is crossed by infinite parallel forces".

There are 2 interpretations of this fact. We could consider that lines are curves, thus Non-E points are focus of multiple forces that bend to flow into the point. This is Mr. Einstein’s preferred solution that defined a curved space-time. Yet Mr. Einstein’s solution brings 2 logic contradictions: Lines stop to be lines and become curves, and still those curves have to become fusion into one as they approach the point. A second, more logical conclusion determines that the 5th postulate can only be truth if a real point has an inner space-time volume through which many parallels cross:

Real, discontinuous, n-dimensional point:            o ===========

Hence, as reality proves, any point can be enlarged when we observe it closely. Indeed, since reality follows that Non-Euclidean 5th postulate, it follows that 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, points with breadth, with space-time parts... Abstract points are fantasies. Anything real has dimensional size. Thus the 5th postulate defines a fractal Universe of super-organisms that grow in scale when we use thinner forces to observe them. And since there is not limit of size for electromagnetic and gravitational waves/forces, the Universe become a fractal structure of unlimited microscopic and macroscopic sizes.
Thus, the first important change that quantic space-time introduces in previous Non-Euclidean Geometry is a more realistic definition of its 5 postulates that define now a point with parts, a line-force of communication, a topographic plane or network of energy and information, and the relative equality between 2 of such non-E structures:

1st Postulate: ‘A point is a discontinuous form with an inner content of temporal energy’.
2nd Postulate: ‘A line is a wave of points’
3rd Postulate: ‘2 points are equal when their inner temporal energy is equal’
4th Postulate: ‘A plane joins n points through flows of temporal energy’.
5th Postulate: ‘All points have apertures to the world through which multiple waves can cross’

Thus now a point will be a one-dimensional ‘form with parts’, as it has ‘inner energy and information’. A line becomes then a bidimensional wave of points exchanging flows of energy and information, as points with parts become cycles. While a plane becomes a 4-dimensional ‘holographic topography’ with temporal height and movement, a quantic network of social points put together by n-flows of temporal energy… with ‘dark spaces’ in the holes of the network, and energy and information ‘dimensions’ to absorb energy and information. Since a dimension is a flow (dynamic version) or network (static version) of energy and information around which quantic, cellular points gather into bidimensional waves and 4-dimensional space-time fields. Even the smallest points gather in waves which are lines of points that scientists call strings with 3x3 inner dimensions. Those lines then gather in herds called planes. And planes gather in topographic, relative, ecosystems, in which those organic points communicate energy forces and informative bits, showing in this manner their systemic nature.
For example, each human is made of many quantic points of space-time, that in the specific case of a human being we call cells, gathered around informative, nervous networks and energetic, digestive and blood networks that give 4-dimensional volume to the human space-time field.  But there are many types of points with parts gathering about networks of temporal energy: atoms in a molecule, stars in a galaxy, strings in a microcosmic piece of vacuum, etc.
In fact, once we have clarified that the Universe is made of points filled with temporal energy, it becomes clear that the fundamental mathematical property of those points-beings is its social nature. Since a number - the social unit of algebra - is a social set. While a plane - the social unit of geometry - is a social herd. Thus, the fact that all beings have mathematical properties means they associate in herds called numbers and planes, which are quantic networks made of Non-E points with a certain content of spatial energy and information.
And the sum of all those herds of points that aggregate in a series of systems and structures with discontinuous parts (humans made of cells, made of atoms, etc.) is the fractal Non-E space-times we call reality. Even the Universal vacuum, which seen from a macroscopic point of view seems a continuous, static emptiness, when observed in its microscopic scale becomes a 'foam' of space/time quantic energy, moving and fluctuating constantly.
In terms of size, the biggest of those space-times is the Einsteinian gravitational space-time, in which the luminous quantic space/time network seems to ‘float’ as a net in the water, occupying a mere 4% of it. Then we find ‘floating’ within that light-space a huge number of smaller, discontinuous masses and charges, or physical space-times points, some of which organize in complex, social networks which form the organic space-time of living beings like us. However all those partial quantic space-time points and networks are composed of 3 clear regions, which correspond to the 3 topologies of any 4-dimensional Non-Euclidean space-time:

2. The 3 classic Non-Euclidean geometries are the 3 zones of a Non-E point.

Non-E points were discovered in the XIX C. as ideal geometries of space. Einstein used them to explain gravitational space/time; but its quantic , empirical understanding is poor due to the error of continuity that considers only 1 Non-E space, the Universe, when there are infinite discreet Non-E space-times. In the graph, the 3 zones of a Non-E point:
- An informative center described by Belgrami as a cone of ‘height’ with a negative curvature.
- A central reproductive zone, described by Klein as a disk, made of quanta in cyclical movement that communicate energy and information between the 2 discontinuous inner and outer zones.
- An outer membrane of energy, described by Riemann’s spherical geometries.

Lobachevsky was the first geometrician to define a Non-E geometry that Riemann generalized to all relative space/times. Latter Einstein used Riemann’s geometry to describe the gravitational space/time, assumed, due to the error of continuity, to be the only real space/time of the Universe. Yet in a discontinuous reality, there are infinite Non-E space-times; and all of them can be analysed with the mathematical tools of Riemann, Klein and Lobachevsky, that describe the inner geometry of quantic space/time fields (physical jargon) that we shall call in the mathematical jargon Non-E points and its 3 quantic zones. If we put together the work of those 3 authors we obtain a complete description of any Quantic Spaces-Times field and its fundamental properties:
- According to Klein, Non-E space-times have movement. So their ‘speeds’ measure also their distances; as physicists do in Cosmology with the expanding speeds of spatial vacuum, which are proportional by a ‘Hubble constant’ to the distance of stars; or as people do in life with real space-time. So we say that Brooklyn is at 5 minutes by train from Wembley not at 5 miles.
- According to Riemann, space is quantic, made of herds of points with similar ‘properties’; and so it has a quantic density determined by its number of quanta, like the space defined by light does. So when a volume of spatial energy is very dense, it is very difficult to go through it, as it happens in the gravitational space close to black holes. Also Riemann’s space-times might have beyond its discontinuous borders an inner space-time with several dimensions, one for each of its ‘energetic or informative properties’ as it happens with the points of physical reality, from strings to super-symmetric fermions. Hence Riemann through the concept of ‘properties’, defines a ‘dimension’ as a certain energy or informative network (static point of view), which ‘puts together’ points that share an energetic or informative ‘property=flow’ (dynamic p.o.v.).
- Finally Lobachevsky and Belgrami proved that space is curved and indeed temporal information curves the spatial energy of any real space-time. So points move in curves, which are ‘cycles’, gathering energy and information through their ‘dimensional networks’.
In the graph, those 3 abstract analyses of Non-Euclidean space-time define 3 different regions that structure all space-time fields (Physical jargon), all Non-E points (Mathematical jargon), all organisms (Biological jargon) and all entities (Philosophical jargon) of the Universe:
- Max. Information: An informative, quantized center or temporal brain, the so-called hemisphere of Belgrami, a Non-E space-time, with a dimension of height that transforms energy into information, absorbed or emitted by the central singularity. It is a quantized, informative region similar to a black hole structure. Since it follows the ‘black hole paradox’: Max. T x min. E. So according to the inverse properties of space and time, it has max. temporal information and minimal spatial energy.
- Max. Energy: An external, continuous membrane or Riemann’s sphere of maximum energy that acts as a relative infinite, unreachable distance, creating the discontinuity between the inner parts of the point. Since the internal quanta 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 quantic, discreet elements: Max. E=Min.T.
- Energy=Information: A middle volume or intermediate territory, discovered by Klein, which combines the energy coming out of the membrane and the information provided by the center, in quantic herds of ‘points with parts’ in perpetual movement, that draw cycles of "parallel lines". Those inner quanta are confined within the Klein’s disk by the 2 other regions, which have more energy and information and might destroy them by overdrive or absorb their energy and information at will. Thus, Klein introduces the concept of an infinite, relative distance measured in terms of time and movement. Since in any discontinuous world the infinite are those 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 0K temperature and a man beyond the Earth’s atmosphere. Further on, since those 2 relative infinite limits make any temporal movement of an intermediate point towards the inner informative singularity or the external, energetic border, eternal. As in the myth of Achilles and the turtle, Achilles never arrives because every time it moves it crosses a smaller spatial distance. The same happens in a Non-E space-time, when a point moves temporally towards its inner or outer spatial limits 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. So the intermediate, quantic cells of the point circulate in parallel cycles always inside the interior of the sphere with contact zones of the type "A" (central figure). For that reason Klein interprets the intermediate region as a circle with an invisible, unreachable membrane, equalling the ‘space-time distance’ between the intervals B1-B2 and B2-B3, despite being increasingly shorter in space. Since the quanta take longer in each step and don’t reach the membrane.
When they do it at point C they become destroyed or deviated. Thus, the energetic membrane and informative center are the discontinuities that isolate their cellular quanta of temporal energy and create a discontinuous ‘World’ within the point. Those discontinuities are called in Geometry a relative infinite, in Biology a membrane, in Sociology or Topology a border. They explain the existence of many different species, which occupy a limited region of spatial energy and last a limited time. So we define a Non-E point as:

‘A curved region of energy & information quanta, with cyclical movement, confined by 2 limits, an external, spherical membrane of max. energy and an internal informative nucleus.’

3. All known 4-dimensional space-times can be reduced to those 3 Non-E regions.
The 3 topologies of a 3-dimensional space are the 3 zones of any quantic space-time of the Universe.

The capacity to describe all space-times of our Universe that those 3 regions have is ‘absolute’ as Topology, a branch of mathematics that classifies all possible space-times, has discovered.Since all possible forms of a 3-dimensional space can be reduced to those 3 zones that build the quantic geometry of any form of reality, whose inner parts can be transformed into the 3 Non A-E forms:
The graph shows how any form becomes a Non-E zone through topographic symmetries:
- A sphere or volume "without holes" becomes a Riemann’s external sphere or membrane.
- A toroid or any volume with a hole can be turned into an intermediate plane equivalent to the central Klein disk (if we cut it we have a cylinder that we develop into a plane).
-  While a volume with multiple holes can be turned into a topography with negative curvature and a convex surface, homologous to the central hemisphere of Belgrami. And so our 4-dimensional Universe is indeed a sum of those 3 Non-E zones with clear geometrical forms and properties.

4. The equivalent Non-E regions and their organic ones.

The variety of real organic spaces that Non-E geometry explains is enormous. In the graph, an animal, an embryo, an electromagnetic flow, a black hole, a proton, a seed, a planet, a cell and a bosonic condensate: all of them can be described with the ternary structures of Non-E geometry.

In the graph, Universal species are space-time fields, structured in 3 Non-E basic formal regions:
- A central, very quantized, evolved, ‘future’ zone, dominant on temporal information. Which invaginates into ‘informative networks’ that spread throughout the other zones.
- The external membrane that isolates the being: A relative, simpler, energetic, ‘past’ zone of lineal structures with little information and a lot of energy.
- An inner, reproductive region of mixed S-T quanta, which perform the 2 arrows of the universe of the system, mixing external energy coming from the membrane and internal information from the center to create new informative and energetic quanta, specific to the space-time field.
As points evolve and develop specific structures, they become ‘invaginated’ by energy and information networks departing from the outer and inner regions of the point that spread through the intermediate zone to control the S-T quanta, creating in this manner the morphological variety of quantic space-times we observe in the Universe, which however can be reduced to the previous functional elements. Those 3 zones are homologous in form and function to the 3 internal parts of most real, quantic space-time fields, from the cell to the human organism, from an atom to a star to a galaxy, which have an informative center, an energetic membrane and an intermediate zone.
For example, the human being is a non EA space-time with 3 inner networks-dimensions: the physiological networks of nervous information, the digestive system of energy feeding and the blood network that combines energy and information, to which are attached a number of ‘space-time’ cellular quanta that reproduce all kind of glandular substances, creating together a human being.
While a nation is also a sum of human quanta, (its citizens), joined by networks of energy and information (legal, religious and cultural networks and economic systems).
 In the graph we show the Non-E structure of Universal species according to those 3 quantic zones:
- The singularity of max. quantic information: the knot of quantic information ‘or brain’ of the system is located in the center or z-axis: it is the spin axis of atomic particles, the seed of fruits, the crystalline nucleus of a planet, the DNA spiral of cells, the height dimension where bosons condensate in matter, the spinal chord of the nervous system in living beings, the genetic nucleus of the embryo, the yolk in the egg, the axis of any electrical current, the vortex of the black hole in the galaxy, the nucleus of the atom...
- The continuous membrane of max. energy and strength that limits and isolates the system. The Non-E point reaches its max. extension, structural hardness and linearity (that is, the minimal curvature of its cycles) in the external surface of the sphere, the skin that exchanges energy in contact with the external organic territory of the point. Some examples of those membranes are the electron orbital in the atom, the skin of the tomato, the crust of the planet, the membrane in the cell, the placenta of the embryo or the halo of dark energy and matter in the galaxy.
- And between both limits, the intermediate space-time filled with reproductive, cellular quanta that combine energy bites and information, absorbed from the external world by the other 2 regions. They are the cells of the body, the stars of the galaxy, the carbohydrates of the cell, the planetary magma, the pulp of the seed or the electromagnetic forces that circulate within the atom.
Thus all Non-E points are ‘inner worlds’ whose discontinuity defines an External Universe or outer world from where the point obtains its energy and information. Since most points, despite being discontinuous, have in their external membrane several generic openings or ‘senses’ joined to the informative networks or ‘brains’ and energetic, ‘digestive networks’ of the superorganism:
         - Max. +E: A ‘mouth’ or opening that absorbs energy.
         -Max. –E: A ‘cloaca’, through which the cyclical body expels its temporal energy.
         -Max.+i: An ‘eye’ through which the informative center receives external information.
         -Max. –i: And an ‘antenna’, through which it emits information.
Those apertures vary in their number, location and size, depending on the form of the point. In the simplest, initial spherical ‘seeds’ of most species, they are mostly situated in 3 regions:
       - Max. E: The Equator of the system, through which the membrane absorbs energy.
       - E=T: The Tropics, in systems that use the same opening to emit and absorb temporal energy.
       - Max.T: 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.
Some examples of those apertures are the senses, the sunspots, the magnetic poles of a planet or black hole, the spin or radius of a particle, etc.

Non-E points evolve into morphological organisms that keep its 3 functions.

In the graph, the informative egg, and energetic, larva phase of most animals translate the inner, informative center to the Head region and the external membrane to the ‘tail’, inverting their positions as the informative function of the egg is substituted by the energetic function of the larva

Yet morphology is flexible, adapting itself to the 3 possible, E, et, T environments:
- Max. T: The spiral or spherical ‘still’, cyclical form that packs and focuses the maximum information in minimal space is preferred in a 3-dimensional, homogeneous, ‘safe’ space or when the form is born as a ‘seed’ of genetic information. Those specialized, informative, still points have an external, spherical, rigid, still, symmetric membrane that can ‘focus’ information into an accurate, reduced image of the Universe. On the other hand, since energy-information flows occur within the borders of the world, the informative singularity that orchestrates those flows, occupies the central axis with a dimension of height, moving linearly as it orders the inner world. Some examples are: a cell with DNA, or a spiral galaxy with a tall black hole and a Halo.
- Max. E: Lineal, spatial, energetic, moving forms are common in moving worlds with directionality. Then the energetic system occupies the ‘tail’; the intermediate organic elements, the center; and the informative senses occupy the top ‘Head’ in the direction of movement to absorb and control the ‘future’, energetic and informative flows that come from the forthcoming region, which the organism will occupy in the future. Some examples are: a train wave with a singleton ‘Head’, the glycineor a shark. All of them have their Heads upfront, with their energy cells on its tail.
- ExT: In complex, space/times existing in the limiting surface between 2 discontinuous regions, the form mutates between energy and information morphologies in 2 different ages as in the graph, or depending on the environment and function it performs, as spirals do; or it becomes discontinuous with quantized triangular legs to process energy, hands to emit information, tails to control directionality, eyes and antennas to absorb information, etc. It is the case of life beings that exist in discontinuous surfaces between the atmosphere and the ground or the water. Thus the morphology and relative position of the energy and information elements of a Non-E point adapt to the ecosystem in which it exists. Yet its 3 Non-E regions and its parallel energy, informative and reproductive functions are maintained.

A galaxy, a DNA molecule, a tornado, a shell and a sunflower have all a spiral form, the most efficient and hence commonest form of any space-time field, since it can easily mutate between a lineal and cyclical form, opening or closing its arms. The homology and similar structure in time and space of all Universal entities shows the existence of an impersonal design
in all forms of reality that obey the quantic laws of space and time.