The word *algebra* comes from the Arabic الجبر (* al-jabr* lit. “the reunion of broken parts”), this fact alone indicates the subconscious capacity of mind mirrors to go beyond in the understanding through their subconscious grammar of deep intuitions in the meaning of the existential game. Since Algebra must be considered indeed the study of the most ‘complex’ all encompassing element of reality: its 5th dimension of multiple scales.

*Algebra studies with numbers the internal structure of the organic parts of a whole that communicate between them dimotions of space-time through operands (**≈), which** represent the 5 Dimotions of existence, forming **complex systems, super organisms and groups, in which all the parts are related by its space-time a(nti)symmetries.*

If Analysis studies T-motions, Geometry S-pace, Philosophy of Mathematics (axiomatic vs. experimental method), the @-humind perspectives on the language, Algebra, and its first ‘form’, the arithmetics of social numbers, studies the ∆-scales of the universe.

As the Universe is pentalogic and entangled, obviously each of those main disciplines is able to mirror all other ¬∆@st parts of reality. For example, analytic g eometry is arguably more essential to the understanding of @-minds of ‘space’ than philosophy of science, given the *mental nature of spaces.*

So if we reduce all forms of mathematics to S@ and ∆T, we can include in geometry, mind spaces, and in Algebra, numbers and analysis, but as the mind of man becomes more complex, it is obvious that it distinguishes the pentalogic of the Universe, and so it happened in the language mirror of mathematics, whereas numbers (social scales within a single plane of reality), departed into Number Theory, and Analysis (motions but also through the concept of a derivative as an infinitesimal and an integral as a whole) Planes, also departed from Algebra.

We did take Analysis as a discipline of its own to study time motions, but numbers can by all means be considered the ‘smaller broken part’ of Algebra which studies ‘social scales’ between two planes of reality, while Algebra extends much farther this analysis, studying the internal dynamic structures of the whole.

So its full understanding is that of all the parts of reality together and its symmetries: ∆S=T

As usual for easier understanding we use both ‘pentalogic’ and ‘Its world cycle’ through the ages of evolution of Algebraic minds.

**SUMMARY**

**Foreword.**Algebra within Mathematics.

**1.ÆLGEBRA:**∆@ST¬ SYMMETRIES.

**2. 3 AGES OF ¬ALGEBRA**

**3. CLASSIC AGE:** **OPERATIONS & FUNCTIONS**.

Dimensional growth – its 3 operands: ±; x÷; xª log:

Symmetric inverted growth/diminution around a neutral §œT.

Polynomial Functions. The fundamental theorem of algebra.

**4. 3rd, INFORMATIVE AGE: §ŒT. GROUPS, FUNCTIONALS**

Groups: ‘present’ ∆st-symmetries.

Cantor’s §œ.T

The last dimension, functionals in Hilbert Spaces.

Old baroque age: Hilbert’s axiomatic method. Its incompletenesse.

Entropic death**.** Algorithms & Boolean Algebra: the metal mind

**5D: EXISTENTIAL ****¬Æ****LGEBRA.**

Classic A-logic: single time arrow. Dogmatic truths.

The generator group. Ternary Non-AE logic.

The worldcycle. The pentalogic of zero-sums.

Universal, ternary grammar of all languages.

The symmetries of 10 dimensional §œTS.

**Foreword. The content and limits of those posts. **

As usual we recommend to read at least the introductory pages of the main post of the web, to understand what the 5Dimensional fractal Universe an its fundamental elements, Space, Time, ∆-scales and linguistic @-minds, the four components of all Supœrganisms of Time-space are.

The fundamental purpose of this and all other posts on maths is to UPGRADE the understanding of mathematics as a language mirror of the vital, organic, scalar properties of the space-time Universe, observed in the next graph:

The Universe has 4 sets of properties that correspond to its 3 structural elements:

– S: SPACE: Topologic, mathematical, social properties, described by its mathematical units, points and numbers, which are social groupings of undifferentiated elements.

– T: TIME: Temporal, CYCLICAL, logic, causal properties, which are cyclical frequency patterns origin of the laws of science, caused by the repetitive, memorial nature of its time cycles.

-∆: SCALES: Organic, vital, biological, survival properties caused by THE CO-EXISTENCE of several scales of parts and wholes organized by those parts, from particles to atoms molecules cells and matter systems, planetary ecosystems, galaxies and its networks.

-@: LINGUISTIC, MENTAL properties due to the existence of languages of in-form-ation that are ‘still mappings’ of reality used by it’s super organisms to order a territory and create within that order the conditions for its survival actions that ensure its existence and reproduction in finite time.

Unfortunately humans do NOT recognize under the ‘ego paradox’ of their mind, which perceives reality from its self-centered point of view and selects only information favorable to its territorial view ANY of the 3 ‘cyclical, intelligent, organic’ properties of all other atomic systems of reality, which are the first bricks of ‘life’, as particles gauge information, feed on energy, decouple=reproduce and evolve socially through magnetic fields, and quantum numbers that code its behavior as genes code cells or memes code humans.

All what humans must do is to extract mathematical properties of objects, with instruments, gathering data cast in equations that mirror the space-time properties of the Universe, and run them in the calculation machines of more complex metal atoms, to extract images of reality that are the essence of knowledge. No further questions are needed. And inversely we can design in less mathematical dimensions objects and then reproduce them.All this of course works fine as long as we eliminate any question on why species follow certain properties, on what are the properties we do not map out as they are no susceptible to be mathematical (notably vital properties and sentient properties and organic, fractal properties). As LONG AS ALL THAT IS CONSIDERED not proved, (by mathematical methods that cannot portray them), as long as we do NOT have a vocabulary for those properties in the restricted mathematical languages, they do not exist. ‘All what cannot be measured does not exist’ said Planck.

Logic positivism affirmed that all properties of reality which cannot be proved with the mathematical logic do not matters and so on. This reductionism thus reduces reality and knowledge but makes the ego of man the center of reality again and happy.

Of course if Mathematics turns out to show vital properties in its mirror languages, those are also overseen, as in the case of the social properties of numbers, which do not satisfy the ego paradox of the individual man as the center of it all. HUMANS have an astounding rather contradictory capacity to deny obvious truths when they don’t cater their self-centered ego paradoxes and accept falsehoods as truths, from religion to big-bang theories as long as they satisfy their subconscious ego, and put him back into the limelight.

On approaching therefore those difficulties of the human ego to appreciate the whole and form part of it, we can only try as we will do in the posts on mathematics to show that mathematical properties are vital properties and so they can also describe some of those organic elements, when we adequate its postulates to the fractal form of space (definition of fractal points with parts, lines as waves, planes as intersecting networks, and so on)

This is the purpose of this blog and each of its posts: to vitalize reality and explain experimentally each science as a reflection of the previous organic, scalar, topologic and temporal properties of space-time beings.

So we are not so mucha advancing maths beyond the upgrading of Non-Euclidean mathematics, but interpreting maths as an experimental science, that is as a mirror of the space=geometry and time=logic, algebraic and scalar=digital social properties of all what exists.

Needless to say in a ginormous field as mathematics is, we shall just keep it simple. To that aim, we shall often use texts from one of my fav books of my learning years – the 3 volumes of “principles of mathematics and its applications’ by Aleksandrov, which if still alive would love to see as a member of the ‘dialectic=dualist’ soviet school of thought, the upgrading of his philosophy of science. But mostly from the wikipedia, because of the advantage of its format that can be copy-pasted to WordPress without loosing its formulae.

Plus the obvious fact we are trying as much as possible to make the blog respecting the new copyright laws (a clear form of censorship for the only thing the web is worth for – to evolve the subconscious collective mind of mankind), so wikipedia can be used without those problems and the oldest books of science used here from the Soviet school have also ‘Russian copyrights’ made available as they were state copyrights (so we use Aleksandrov’s for maths and Landau’s 11 books encyclopedia of Physics for physics, besides wikipedia, which were the books that in my teens back in the 70s I used to self-teach myself the essence of both disciplines, and I had annotated with my first insights on 5D, and will time permitted become little jewels as the last Ramanujan notebook (: that will be the day…

We repeat that we cannot go into a full depth, as we build slowly this work from 30 years of notebooks, so the work will always be in progress, improved and as Descartes who had a quite adventurous, drifting life as this writer did, put it at the end of his ‘geometry’, we shall ‘stop here’ (in my finite point of my world cycle) for future generations to have something to do (:

This post is dedicated to Algebra, the main discipline for the study of complex Dimotions in sequential sets, and simultaneous space structures, as well as the symmetries between the 3 4 main structural elements of all T.œs, ∆S<≈>T

**FOREWORD: **

**¬ÆLGEBRA as PART OF MATHS. ITS PENTALOGIC**

The Universe is an entangled fractal game of Dust of space-time, ¬∆@st, where each element flows as a series of ‘5 Dimotions’ (dimensional motions of time space), which can be perceived as ‘form=space’, in the stillness of a world mirror or linguistic mind or as a motion of time, in its true nature since MOTION not form is the underlying substance of reality.

So all fleeting forms, ‘a Maya of the senses’ will return to motion and die (¬4th Dimotion of entropy, death and dissolution). Its 4 positive elements, organic scales, topologic planes and time ages and actions however, carry the system as a finite super organism of space with a finite time cycle.

And so those 4-i elements, entropy (¬), Scale (∆), time (T) and Space (S), are the elements all languages mirror either in a ternary grammar (if scale is missed), whereas often instead of Space we talk of information and instead of time we talk of energy of motion, in a single plane:

Light language: red-energy colors, blue-information colors and its green/yellow combinations.

Verbal Language: subject (information) < verb (action-combination)>Object (energy of subject).

Algebra: Y: Future-information < Operand-action> F(x))

Trinity is thus the logic of most beings. However as humans reached higher and lower scales of observation, a pentalogic was possible and its mathematical mirror became analysis, with its operands that extract finitesimals (∆-1) or integrate into wholes (∆+1) smaller or larger systems.

**Algebra within mathematics as a mirror of the Dimotions and scales of ****reality.**

Thus Math is the closest language mirror of space-time. Thus its main 5 disciplines mirror the 5 structural elements of any dust of space-time, ∆@s≤≥t, scales, minds, space, time and its a(nti)symmetries:

**S@: Geometry studies space.**Some key ages/subfields are:**Flat****Euclidean Geometry**, with no motion in a single plane.**@nalytic geometry, which**represents the different mental points of view, self-centred into a system of coordinates, or ‘worldviews’ of a fractal point, of which naturally emerge 3 ‘different’ perspectives according to the 3 ‘sub-equations’ of the fractal generator: $p: toroid Pov < ST: Cartesian Plane > ðƒ: Polar co-ordinates.**Topology,**geometry with motion and 2 Planes.**¬E Geometry**studies fractal points of**simultaneous****space, ∆-1,**& its ∆º networks, within an ∆+1 world domain.**∆§: Number**theory. Discontinuous numbers study time sequences. ∆-1 social numbers, which gather in ∆º functions, part of ∆+1 functionals; hence it is the first ‘stœp’ of:**S≈T: ¬Ælgebra, which is the most important part, as it**studies through operand the different Dimotions, from single S=T steps to larger associations of Dimotions, in more complex ∆+1 structures (Functions) and further on IT DOES SIMULTANEOUS ANALYSIS OF super organisms in space, through the study of its a(nti)symmetries between its space and time dimensions (group theory)… So Algebra is first the science of operands that translate into mathematical mirrors the dimotions of space-time and then build up from them as the Universe does building up from actions, simultaneous organisms in space and worldcycles in time, in different degrees of complexity, mirrors for all those elements of the 5 D¡ universe.**∆T: Analysis**studies ALL forms of time=change, and hence it can be applied to the 5 Dimotions of any space-time being, as long as we study a ‘social structure’, hence susceptible to be simplified with ‘social numbers’. We thus differentiate then 5 general applications of Analysis according to the Dimotion study, and the ‘level’ of analysis, from the minute STeps of a derivative, to larger social gatherings, and changes of entire planes (functionals). It is then not surprising that despite being analysis first derived of algebraic symmetries between numbers, it grew in complexity to study changes in functions (first derivatives/integrals), and then changes of changes of functions, as motions between scales of the fifth dimension (higher degree of ∫∂ functions, called functionals).**¬@-Humind:****Philosophy of mathematics**studies the specific @-humind elements of mathematics (human biased mathematics) and its errors of æntropic comprehension of mathematics limited by our ego paradox. As such It is more concerned with those ‘entropy’ limits of mathematics as an inflationary mirror of information, which deviates from reality (limits of solubility of functions, etc.). And it puts in perspective the ‘selfie’ axiomatic methods of truth, which tries to ‘reduce’ the properties of the Universe to the limited description provided by the limited version of mathematics, based in Euclidean math (with an added single 5th non-E Postulate) and Aristotelian logic (A->B single causality). This limit must be expanded as we do with Non-Æ vital mathematics and the study of Maths within culture, as a language of History, used mostly by the western military lineal tradition, closely connected with the errors of mathematical physics. Instead we must develop a vital mathematics and its experimental philosophy.

So Algebra IS the study of all the structures and symmetries of the Universe from the point of view of its mathematical mirror.

*Algebra is more concerned with space, simultaneous structures, joined by the =* symbol of equality, which relates points with internal parts=numbers, through operands that represent Ðimotions; while *analysis specializes in all time-Ðimotions, by establishing a FINAL LAYER OVER THEM, AS each Operand specializes in one Dimotion ( sine/cosine in Perception, ± in locomotions, x ÷ in social evolution, log xª in reproduction) and OVER all of them the new layer is giving by analysis.*

But as in the entangled Universe all mirrors can reflect all forms, Algebra also can analyze other elements, as it has been extended from its initial analysis of equations into the whole range of ‘structures’, apt to study, super organisms’ entanglements and variations of species.

Ultimately Algebra and Analysis ‘ARE’ the complex ‘level’ of reality, as reproduction and social evolution are the complex demotions, including obviously as its ‘background parts’, the theory of numbers, the analytic geometry – study of frames of reference, and the topologic analysis, embedded in the secondary operand, numbers and frames in which we ‘cast’ the complex space and time algebraic and analytical analysis of a ‘Domain’ of the fractal Universe.

We shall call then the upgrading of Algebra to 5 Ð, non-Ælgebra.

**EXISTENTIAL ALGEBRA (ab.¬æ)**

It is obvious then that algebra, aptly called from Arabic “al-jabr” meaning “reunion of broken parts”, and analysis by studying S<=>T and ∆§ suffice somehow to describe the 5 Dimensions of present, past and future, and indeed, they together cover all of it intensely… but *they require first a good understanding of social numbers, and the @nalytic planes which work as the ∆-1 ‘cells’ and ∆+1 world in which the organic systems of mathematics work out its ‘actions of existence’.*

The only other ‘great subject’ which is on parallel in importance to that of algebra & analysis would be geometry in motion, topology, which thanks to the 5 Non-Æ postulates will be explained as a far more ‘intuitive’ and profound perceptive mirror of reality, once points acquire volume (1st Non-Æ postulate), from flat prime numbers to platonic solids, reproduce inner parts, communicate through infinite parallels (5th non-Æ Postulate), which become

**How ¬Æ reflects the ∆@st elements of reality.**

¬Ælgebra mirrors the fundamental elements of reality with its 4 fundamental ‘synoptic’ forms, as all language mirror reduces reality to parts that carry less information. So a T.œ becomes a point with internal parts, called numbers, whose dimotions are expressed by operands that enact the actions of the being in ‘mathematical space’ from its point of view or ‘frame of reference’:

**S: A point**is the first perception of A T.Œ in mathematical ‘space’, which as we come closer acquires content. And then its next degree of complex description occurs:**∆: A number,**is a social group of undistinguishable ‘internal’ parts of a point, which represents the point in scale. And then the next degree of complex analysis occurs:**T: An operand**expresses the transformation of a point through a Dimotion of timespace, and since there are five Dimotions, 3 continuous Dimotions in a single plane (perception, locomotion, reproduction) and 2 discontinuous Dimotions that start and end in different planes (entropy and social evolution) we shall find 5 basic operators of those Dimotions in any ‘algebraic structure’ that truly reflects the being in the mathematical mirror.**@: A frame of reference**expresses those changes through = equations reflects the @ element of the being.

It is important to bare in mind that numbers are points differ in several key aspects: numbers are discontinuous information, and its continuity an error of the humind, seek for through spurious methods (irrational numbers, which are ratios, not numbers; decimals which are numbers of a different social scale of the fifth dimension). So a key difference in 5D is that points form a contiguous (not continuous) surface in a single plane, but numbers do not. And that is right or there would NOT exist in-form-ation, which requires gap to ‘differentiate’ form.

*Operands as reflection of the 5 Dimotions*

The purpose of operands is obvious: to explain the most general laws of Ðimotions of spacetime beings.

The all pervading use of ∫∂ is then clearly because it reflects ALL forms of change. And so analysis is the most extended sub-set of Algebra.

Why there is a negative inverse operand to each of them, is the first question to be considered: simply stated, the 5 Dimotions of reality have its destructive entropic 5th dimotion for all of them. So negative/inverse operand balance positive ones.

In that regard even a negative numbers is NO LONGER A NUMBER BUT AN OPERAND OF INVERSION. Inversion indeed is the reason why + x + = +; – x – =+, etc. but this COMMUTATIVE, and distributive laws require the ‘NUMBERS’ to be of the same species. When we consider other ‘objects’ however nothing of this is necessary truth.

This said the main ones are in correspondence with the basic Ðimotions of existence.

**1D:** Of those operands then we consider the simplest ‘operands’ as those who act directly on a point-number. We need then to find the operand of the first Dimotion, self-centered in the point, perception.

Since all points are also numbers as all have internal structure, we can operate both in S-pace with points and topologies and in scale with numbers to show those internal parts with numerical values:

The graph shows the elements of the point-number in which an operand can act. Even though most operands will act through a similarity ≤≥ symbol in two different elements. Yet the simplest operands are those which can act in a single point-number.

*The sinusoidal functions. π.*

Pi then becomes the numerical value of the external membrane measured in terms of 3 diameters turning around with 3 apertures to ‘see’, from the self-centered singularity:

*± 2nd Dimotion: Locomotion*

Next comes the second dimotion, which obviously is represented as a ‘lineal motion’ by a simple ‘sum’ when moving in the positive direction, or a negative number, when moving in the inverse direction; whereas the positive direction is the direction towards ‘energy’, and the negative direction must then be the inverse direction of the ‘arrow’ of information. We can see this also, as an expansive direction (energy feeding) and an implosive direction (information warping), concepts of importance, as information ‘warps’, and since height is the dimension of perceptive information, a negative, shrinking number, i, will define often in mathematical physics the warping direction of information (as in Minkowski’s space, it).

However in the entangled, fractal Universe each element can be entangled with all the other elements (fundamental principle of pentalogic). So we canals add frequencies in time (+, T), social elements (5D, +), points in a single plane, and groups of social elements to give us new decametric scales. And we can ad reproductions as sums of identical beings (3D, +). And we can count the decay of a point into its internal parts as a sum (4D). Thus we can build with only sums a mirror image of 4 dimotions of reality. But perception is still the most difficult element to describe mathematically, without the consideration of an angle; since we find that the other operand act in numbers, while perception acts in a different ‘concept’ – an angle.

And indeed, we shall find that in many mathematical representations of reality and mathematical physics (argand plane, polar coordinates, quantum formalism) we will NEED both an ‘angle’ and a number and the operand in them. So angles suddenly recoup its classic importance.

*x: 4th and 5th Dimotions: reproduction, social evolution and entropy (division).*

It follows then that the next Dimotion in degree of complexity, which is social evolution is served by the ‘product’; which is carried in 2 scales, even if it finally surfaces above. Indeed, if we have 2 points with 4 internal sub-elements, the product 4 x 4 will give us the number of maximal connections, when we put each internal element in connection with all the elements of the other sub-set: (1 x 4) x 4.

3D: reproduction requires between two ‘genders’ such interconnection at the lower level to form a whole.

2D: But in the entangled Universe operand have multiple mirror possibilities so the product can also account for a ‘longer time period of locomotion’: S=vt.

4D: And it is obvious that its inverse function, the division, breaks the whole into parts, and so it represents the inverse function of entropy.

Yet again perception doesn’t fit in the product.

Mathematics is in that sense best suit for the spatial (topological) and scalar, social (numerical) elements of reality. Logic is better for the complex pentalogic of the entangled Universe and its perceptive elements.

*log a, xª: reproduction and entropy (decay, negative exponential)*

3D: And so it only remains to define the more complex Dimotion, with the next level of operand, which are powers and logarithms. And indeed, we model reproduction in case there is infinite energy, with an exponential curve; yet in reality we do so adding at the end a logarithm curve (logistic curve); as reproduction saturates a system. It is then a proof that REALITY seeks always a ‘balance’ that the REAL curve of reproduction is a combination of both, from the initial unit to the carrying capacity that DEFINES an ∆+1 social group that has ‘reached equilibrium’, when it becomes in itself a herd or super organism emerging in the new upper scale.

4D: Which also as a negative exponential will show the rhythm of decay of the system. And in this case there is no need for ‘a logarithmic’ limit, since for the predator the death body is ‘unlimited ¡-1 energy’, though once the ‘relative infinite’ number of its ¡-1 parts are absorbed the ‘e-function’ will have a cut off.

So as we can see operand are directly connected to the Dimotions of the being, giving so much power to mathematics.

The beauty of that reproductive curve that rises the 1 to the K-organic whole (in a herd organism in its simplest view) is difficult to stress as it will allow us to determine a fundamental feature of all those operands: the existence of an inverse that transforms the being into itself, but re-establishing the balance of reality as a zero sum, but in this case *having accessed a new plane of the fifth dimension. So while the logarithmic part of the graph is the inverse function, it still grows the system. Together though they form, one half of the total graph of a cycle of existence, as we shall see then, happening a decay of the whole system in two negative curves, so we can see the graph of the whole world cycle as a ‘composite’ of a positive exponential and logarithmic and a negative exponential and logarithmic:*

Which is the ‘stretched representation of the world cycle we initially considered with a ‘sinusoidal function’. This said as any student knows there is an intimate relationship between the exponential and the sinusoidal by way of Euler’s formula in the realm of complex numbers, which we shall define in terms of 5D on our articles of theory of numbers.

**ENTROPY: **

**INVERSE FUNCTIONS AS ‘BALANCING STŒPS’ OF THE FIFTH DIMENSION**

HUMANS AS WE have repeated many times, being essentially entropic, thermodynamic virtual ghosts trapped between the immortal quantum proton and cosmic black hole (no entropic evaporation there), LOVE entropic simplest lineal theories of reality.

Entropy death IS INDEED GINORMOUS in its fields as it APPLIES TO ALL OTHER attempts of ‘differentiated’ evolution of form.

So we can ultimately state that there are 4 POSITIVE DIMOTIONS (locomotion, reproduction, perception and social evolution) ALL OF THEM SUBJECT to negative entropy and destruction that ‘returns them’ to the ‘mother womb’ of no form. And for that reason we can treat the 4 INVERSE operands, negative exponential decay, negative subtraction, division and derivatives (when applied to the search for an 1/n, ∆-1 infinitesimal) as reflections of the Universal entropy that destroys the positive DISTINCTIVE operand, often in a much larger shorter explosion of dissolution of form.

Entropy thus is the disentanglement of free-chaos that affects all systems and the inverse operators best describe.

When comparing each new Dimotion operation, we observe several elements of entanglement. First the simplest expression of the next ‘Dimensional operation’ seems just a repetition in a new dimension of the previous one: the Sum is multiplied by adding sum after sum. The Power is obtained by multiplication over multiplication, but this is just when the operandi is applied to the self-reflective point.

Operands do have however as they grow in complexity a direct entanglement with the 5 Non-E postulates that define from points waves of communication that become networks of topological planes and organisms, where multiple pov.s are put in perspective. And this even more clear when from mere polynomials we move into exponentials and from exponentials into integrals, *the more complex operands then can interplay with multiple points of view, merging them through the product, reproduced through its inverse partition/division, moving them through a world cycle, in exponential growth and decay, and emerging into a new planet of existence through derivatives and integrals.*

So as usual in the entangled Universe of INFINITE ACTS OF COMMUNICATION even if the one-dimensional man tends to reduce the whole and its flows of communication to a single line of thought.

Re=production through social evolution in a lower plane of the fifth dimension, in which the axons join the wholes is then the 2-point meaning of reproduction; often a bidimensional Space-time combination of a state of forma and a state of motion, as in the parameter of momentum.

It must be noticed then a very interesting new vision of the interaction of inverse functions, which do NOT merely cancel each other as one could expect, in classic science, *but advance the Dimotion in alternative balancing acts, leaving a memorial trace of its existential duality.*

Indeed, consider the case of the product and its inverse division. They are functions primarily of social evolution and reproduction, as parts of wholes are produced and exchanged. So if we have an individual of 5 parts that reproduces it will give birth to 25 parts, *but then we need to divide them between 5 parts, to get 5 individuals of 5 parts: 1 (5) x (5) = 1 (25) / 5 = 5 (5).*

Same happens with the positive and negative motions, which in alternate current move back and forwards the ‘electrons’ which in fact don’t really move, but its ¡-1 scale *the current keeps moving.*

So goes for the exponential+Logarithmic functions of reproduction, and its inverse, which form a worldcycle that in the middle point of maximal ‘carrying capacity’ will give birth again to a new ‘exponential growth’=generation, to continue the world cycle in an adjacent space-time.

So by the continuous game of inverse functions, which become a zero sum, *the Dimotions of existence continue to move the whole of time-space that never ceases to exist. *

**∫∂: Finally it came calculus, with its inverse operand, which represents the scalar next social gathering of elements of algebra**, as it IS APPLIED TO THE PREVIOUS OPERANDS, AS WHOLES, except for the trivial xª – the previous more complex polynomial, and so we must regard calculus not only as the operand of all dimotions of change, but also as the operand between planes of existence, since the logarithimic/power previous expression, just reaches between the two limits of two planes of the fifth dimension, but calculus allow us to ’emerge’ and transcend between planes.

Indeed, consider the ‘derivative’ of the exponential, *which is also the exponential, and that of the logarithm, which is 1/x, where 1 is the whole and x its parts. It gives us the ratio of change of an individual that form a whole or a radiation of a species, with two clear phases; the first of maximal possible growth as the E number is the base with maximal exponential growth of all numbers, but then that growth reduces to a ‘cell’ after a ‘cell’ of the whole, which means it merely maintains the being as t*he system will also keep loosing its -1/ units slowly.

**The complex plane as a square plane.**

We have also to consider where those functions and operand are set in – that is what background space we use to express it, and 3±i are the essential background spaces which correspond also to those dimotions as forms in space, the 3 lineal=cylindrical, spherical=polar, and hyperbolic=cartesian planes and the scalar plane, ill-understood, which is the complex plane better perceived if we ‘square it’ eliminating the √ symbol of its negative -1 axis:

SO THEY can study 2 fundamental ’emergent’ ∆+1 planes of mathematics, the study of Dimotions of Dimotions with the tools of calculus in time, and the study of spaces of spaces, with the tools of the complex plane properly understood in terms of ‘square’ coordinates.

But as in the entangled Universe all mirrors can reflect all forms, Algebra also can analyze other elements. But its main beauty is in creating sequential chains of pentalogic actions that reflect the motions of existence of the being, even though its ‘Group’ simultaneous analysis of all its ‘variations’ of species, has been the most developed inflationary mirror in its last ‘excessive’ age of form.

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

*ALGEBRA MIRRORS S≤ST≥T STRUCTURES, A(NTI)SYMMETRIC DIMOTIONS & ∆ TRANSFORMATIONS from the point of view of a given frame of reference yet its main mirror is Algebra as expression of the Dimotions of reality with its operands*

It is then clear that what languages as synoptic mirrors of the mind will try to do is to establish the basic relationships between the space, time, scale of the being, expressing them through its operandi, DEPENDING on the degree of perception the being has of reality and its scales which might be reduced if the being is not fully aware of all the scales of existence, as most minds exist only in a plane of reality

So does mathematics, through combinations of:

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

This said, the Universe is a game of ∞ Sπcies of space-time making STeps and STops in its constant cœxist¡ence (where we use π for time, in this particular wor(l)d according to the slightly changed rules of i-verbal thought for english to look more like the universal game), and so as Stops and steps (ab. œ) bring the being into exist¡ence we can measure them quantitatively as space-distances and time-motions. Stop and step then becomes topology and feed-back equations, where = is substituted by the dynamic symbol <=>, and the duality from = stops and ≈ wave steps.

Once the intuitive meaning of those symbols rises the awareness of the reader to the vital nature of our humind’s abstract rendition of reality things become then dynamic. ≈ for STeps, = for motions, and ≤≥ as the different implosive or explosive, informative or entropic ‘degeneracy’ of a system are easily quantifiable and translatable to both mathematics in its pure expression and mathematical physics.

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

And to do so, as a fractal can always be divided in sub-fractals, mathematical disciplines subdivide further at all levels in 5 elements.

In this post we shall deal with the a(nti)symmetries and operandi of Algebra.

**Though**we cannot be exhaustive ias Algebra is the largest of all the sub disciplines of maths, concerned with time-space symmetries, but merely do a very fast analysis of its main themes, enlightened with ∆st insights to show the enormous power to ad new whys to reality and all its stience of 5D², since even the simplest truths scientists think are thoroughly understood have new insights observed when putting the Kaleidoscopic 3±∆ perspectives of the fractal organic Universe, what we call honouring one of my fav films, the ‘Rashomon effect’ (a truth can only be ‘judged’ with multiple ∆@ST perspectives.

**The fractal Universe and its mathematical analysis MIRROR the laws of ∆-planes.**

2 are the fundamental subjects of ∆st: the fractal structure of space-time and its ternary symmetries.

In classic mathematics the second theme is treated with ‘group symmetries’. We shall treat with the more realist ‘Generator equation and its ternary symmetries. The scalar Universe is a theme of 2 disciplines, the old calculus/analysis and the modern fractal geometry of non-differentiable functions, from where we have taken our ∆ intuitive symbol.

Let us then once that we have shown in more depth the mathematical=experimental evidence of fractal space, start by a basic understanding on the sub disciplines of mathematics as a language which *understood in GST terms, becomes the most experimental detailed language on the properties o*f fractal space and cyclical time:

*In the graph, above t**he formal stience of 5Dimotions is analysis. Contrary to intuition, Derivatives are mostly concerned with synthetic information on upper scales (and mostly time), as they ‘reduce’ the information into a synoptic ‘point-singularity’ the tangent that expresses tendencies in the flow of time of the function; while integrals are concerned mostly with the sum of its lower parts in space, as they expand the ‘base’ of analysis but reduce the different(ial) added information of the whole. A fascinating subject being how a ‘simple parameter data – that enclosed in a derivative or a fourier transformation SUFFICES to expand and remodel the whole system.*

*In a single plane, the formal stience is Topology is the formal stience of the three arrows of time-space, and as such it reduces all the forms of the Universe to its only three topologies – another ‘magic’ fact of experimental mathematics, never understood in science.*

And algebra shows both, as chains of sequential events that transform topologies through the dimotions of the being.

*AS such the true revolution of mathematical stience goes through the conversion of mathematics into an experimental stience, relating the fundamental theorems of mathematics, as the stience of logic space, to the different laws of GST both based in the fractal properties of space and the cyclical nature of time. We shall thus evolve mathematics and physics to unveil in the human languages of time and space, the underlying isomorphisms of Nature. *

So once we improve Mathematical logic with ¬Æ math will become the most experimental of all languages, as it directly describes space (topology, mother of mathematics) and time logic with algebra.

In that sense, math is a mirror of God NOT its substance, or origin of its laws determined by the laws of space and time which structure the Universe. But as maths carry so much synoptic information, *it might seem as it is the case among platonic scientists to be as dense as reality **itself*. Hence the epistemological error o*f mathematical physicists which often confuse reality and idealist maths, unaware of the proper way in which languages create reality.*

*The interaction and symmetries and travels through 5D are thus better understood in terms of analysis, which has become the dominant formal stience of maths, without mathematicians and physicists understand why. *

Analysis is the study of events happening *in transitions between ∆-scales, either as infinitesimal parts come together into ‘integral wholes’ in space (integral functions), or as we extract from a system the information about its higher wholes, synoptically encoded in its inverse derivative functions.*

So integrals work mostly on space, giving us therefore information about lower 5D scales and derivatives mostly on time, giving us information about parameters of the whole (but as both are symmetric, this general rule does not apply.

As the Universe is a kaleidoscopic mirror of symmetries between all its elements, this dominance of analysis on ∆-scaling must ad also the use of analysis on a single plane, in fact the most common, whereas the *essential consideration is the ∆§ocial decametric and e-π ternary scaling, with minimal distortion (which happens in the Lorentzian limits between scales). *

**RECAP. The main operands of algebra as reflection of the st, topological, temporal and ∆§ symmetries.**

**≈, ≤,≥,**operandi IS the basic S≈t step connection

**±, x/**are the connection the basic ∆§ocial scales between planes operandi

**xª, log x**represents mostly reproductive and decay processes. As the most complex of the ternary chain of basic operands, sum, product and power, it is a less precise version of the ∫∂ ∆±I operands, since it just reach in its ‘balanced’ sum (logistic curve) the ’emergence from one to a whole’

**∫∂**is the most general operand that works with all forms of time-change including its ∆±I emergence and dissolution.

**(G, •,*)**groups are a ‘simultaneous attempt’ to put all those elements, together in a single superorganic structure.

**S≤≥T A(NTI)SYMMETRIES**

Why there are inverse operations with a neutral element? Obviously because that is the main condition of systems in the Universe, which are in search of a perpetual balance and always end up into a cyclical zero sum, after stretching its ‘virtual existence’ through its Dimotions to the world suffering action-reaction processes from the world.

Thus the bridge between Algebra and the vital, creative and destructive arrows of non-euclidean topology is the concept of an A(NTI)symmetric operation, word that includes the different S≤≥T transformations a T.œ can suffer, departing from an initial symmetric condition or ‘neutral element’, which can dissociate into annihilating ‘negative operands’ =destructive flow vs. ‘positive operands’=creative flows.

The duality of algebraic operands ±, x÷, √xª, log aˆx, ∫∂, can also be considered a reflection of the Symmetric, creative, parallel vs. antisymmetric, perpendicular, annihilating dualities of Nature, which ultimately refer to the Inverted Duality of inverse Dimotions of 1D still perception vs. 2 D locomotion, that switch on and off in ‘steps’ and ‘stops’ (ab.stœps), and the duality of 4D entropy and 5D generation (∆±1), all of them merged into the present reproductive dominant Dimotion of the Universe.

So again we have a slightly different ‘mirror’, in this case to relate non-e geometry, non-A logic, and ¬Ælgebra, which will be useful for mathematical physics. In that mirror we can give a new definition of algebra:

*Algebra is the study of the a(nti)symmetries, reflected in the operands that connect Non-Æ points in space perceived as numbers in scale together through a simultaneous ≤≥ equation. So the first analysis is that of the symbols of self-similarity, <, > and =. And its translation into the logic symbols of 5D, > (implosion that reduces a system in space but increases its information) =, which moves a bodywave into an equal form through stœps (stops and steps) and < which increases the size of a system in space, but reduces its information.*

It comes then as immediate that human algebra has chosen* a lineal view and a spatial view, *by considering the dual paradoxical symbols of non-Æ algebra: T>S, S<T, T≤≥S, in 1/2 of its meaning, NOT as a balance of T>S growth of information and shrinking in size, and vice versa, S<T, growth in size-motion (Galilean paradox: all distances are motions seen in stillness) and decreasing in information; but JUST AS GROW IN SIZE – the spatial simplistic view in a single plane (as information migrates, to the warping inner folding of the shrinking process).

It must be stressed though that huminds observe only one side of the duality of > and < logic operators, namely, they look at the ‘size’ of it, not at the inverse growth of information, obsessed by the ‘bigger’ forms. So instead of interpreting, T>S as a shrinking in size that brings more information, they consider only the symbol as T smaller than S (but it is also more complex in its information than S). And viceversa, S<T (but also S has more information than T).

**EQUALITY**

Regarding the = symbol Is important to remember that = does NOT really exist as absolute equality (as all points and numbers, gathering of points have internal parts and dissimilarities) but IT IS MORE either a parallelism or similarity symbol, ≈, or A SEQUENTIAL LOGIC operandi of implosion or explosion of form =>: ≥, =<:≤, or a combined feed back equation <=> of both in a dynamic stœp series.

For example when we say Energy is mass, we should say is ‘equivalent’ and E=Mc2 is a ‘transformation’ of an informative T.œ, Mass, (a vortex of space-time of the cosmological scale) into an Entropic dissolving expansion of space in the lower plane: M=>E, but as the motion can be reversed and entropy-energy can evolve into mass, it is really <=> a feed back symbol.

Again as all mirrors simplify reality humid’s mathematics reduces it all to an equality in which the inner content is not considered, not even the form but just the equality in a single parameter (in e=m, the energy-motion).

*The basic equations of existential algebra.*

A Supœrganism comes then defined by a sœt of basic feed back ¬Ælgebraic Equations that represent those Dimotions and its chains, of which the most important are those that reflect the 3 ages of any world cycle, which is born as an expansive motion, reaches maturity and then warps:

-S=T, that is S(i)<=>T(e) as in the previous case, represents a present balance between energy and information and the dimotion of reproduction, of beauty of coming together when energy and information peg into a single, being – hence of gender also.

S x T = K, with its two extremes of Max § x Min. t (informative dimotions) and Max t x min. $ (locomotions).

Needless to say in mathematical science, as opposed to stience, all is simplified or blurred by slight errors of the @-one dimensional humind.

So the first thing we must understand is that any mathematical equation will substitute, ‘all’ the operands or logic connections between the two parts of the equations by a simple = symbol of equality and one-way mostly spatial < or > smaller or bigger than.

**The equality symbol becomes complex.**

In classic mathematical science, as opposed to stience, all is simpler as time is lineal with a single dimotion. So logic is A->B, Aristotelian. Thus a classic mathematical equation will substitute, ‘all’ complex operands of ¡logic connections between the two parts of an equations by a simples = symbol of equality (identity).

We define instead ‘5 Symbols’ to *express the 5 Dimotions of the Universe, as trans-form-ative processes between the two parts of the Sœt.*

So we *subdivide t*o ‘regain the information missed in the nature of the equation that represents one of the 5 Dimotions, = for, <, ≤, ≈, ≥, >:

A<<B is an entropic process or 4 Dimotion.

A< or A≤B is an energetic process, most often related to the 2nd locomotion.

A≈B is an identity, often born of a reproduction or 3rd Dimotion.

A≥ or A>B is an informative, perceptive process, or 1st Dimotion.

And A>>B is a linguistic,social evolutionary process, or 5th Dimotion.

The choice is obvious as we can consider entropy, a social process of multiple locomotions, in expansive mood (<, <<) and vice versa, information and perception > is an individual inward informative dimotion which for social evolution in the higher whole requires a multiple >>. We prefer to use ≤ and ≥ for ‘energy’ and ‘information’, when they are in a merging process, as those are the ‘states’ that finally merge into the act of reproduction, ≈.

So this will be from here on a necessary inclusion to add information to vital mathematics, which we shall often include. Now as this is mathematics and ≤≥ have obvious meanings on math we shall establish for future stientists the use instead of the reduced symbols « for 4D entropic motion and » for 5D social evolution.

*An ≤≥ feed-back equation is the ‘time-perspective of mathematical thought’, in terms of the cyclical nature of time, as it establishes the ‘feed-back equations’. And so perhaps the most important change in Existential algebra is to substitute the = operands, which often is interpreted as a **logic, A->B concept for the <≈> operands of a feed-back equation.*

Algebra studies A(nti)symmetries (> ≈<) of space-time, trying to achieve its maximal generalisation hence it is a ‘mirror in space’ of those 3 formal symmetries, anti symmetries and asymmetries through its inverse operands and equalities, which define fully the dynamics of the Universe, in the same way Analysis studies them across scales.

**MONIST ALGEBRA.**

**¬Æ in frames of reference.**

Because then all those Ðimotions are complex stœps, which can be discomposed in a form and motion, ¬Ælgebra is better represented in a frame of coordinates as a vectorial field; that is, as a point in a vectorial field, which has both direction – hence motion – and form, mass, a stop state, with a scalar value, things those considered in analytic geometry.

In mathematical physics as a closer reflection of there Universe that eliminates from the language mirror ITS INFLATIONARY IMAGES (SO math will always be paradoxically larger in theorems that reality is, obliged to ‘bend’ and ‘limit information’ to those shapes energy can ‘bend’), it is even better to represent a vectorial field in General coordinates aas they allow the multiplicity of points of view of the supœrganism each with its self-point of reference. Since all T.œs have a self-centered point or monad-mirror – a linguistic mapping or ‘seed’ that reflects reality in its frame of reference and will try to act from that point of view or re=produce andimprint the external world’s suitable energy (∆<¡) with the inner mind of its points of exist¡ence.

To be anchored first as a point is the first function of any T.œ when emerging into exist¡ence.

We call the description of all of this, ‘Existential algebra’, ælgebra, or ¬Æ, as it is both ‘Existential Algebra’ and Non-Aristotelian, Non-Euclidean in its form.

It is in that sense the most important, structural element of the mathematical mirror – the closest classic linguistic expression of the Universe and its 5 Ðimotions of exist¡enœ.

We are NOT that much interested in merely translating human algebra to ¬Æ as a formal mirror of the Universe, a ginormous task, others will built after me, but in defining a ¬Æ of the Universe and as human Algebra does work in that purpose, to establish as many correspondences between ¬Æ and common algebra.

We depart in non ¬Æ of a first element, a fractal non-euclidean point, ruled by the 5 Postulates of non-euclidean topology: points are fractals, lines are waves between points, with less ‘amplitude=energy’ than the point – a fermion so to speak vs. a boson-line traced. How Fermions or non-E points proper share bosons, or line-waves of lesser ‘mass-position’ and more ‘speed-reproduction of information in a lower, ∆-¡, is a relative question.

For terminology, every non-euclidean point is a t.œ, his index ¡ is always relative to its mind: I=0=T.œs mind. Departing from it, we can cave into its inner world of -¡ and its upper world or +¡.

The laws of ¬Æ are expressed thus in terms of a relative self, as ∆±¡ different planes in which an ∆º mind exchanges energy and information with different ∆±¡ planes.

It corresponds for the case of ∆º and ∆±1 cases to classic science (single plane coinciding with the language of the mind-will).

Philosophically we can consider the mind, the perfect block of time of the syntax of one language that reflects the game of existence itself.

But logically we have a first element or seed, ∆º, and then we can ‘follow it’. as a non-Euclidean point in space, non-Aristotelian monad in time, as it mirrors in its language and reflects upon its world to exchange energy and information and fulfill its ‘Parts’ as an ∆º must have a body, ∆±1 and co-exist in a world, ∆±¡, whose relative extension might depend of the fractal fine detail. in our measure of ±¡.

As such the minimum ‘space’ to represent a world of such points is a vectorial space in which each point has at least two values that might represent the body-motion and mass-direction of the point, which becomes then the simplest representation of a T.œ spacetime, in which each points is a T.œ, which has two creative parts we might call space-form-scalar parameter and time-motion-direction of the vector parameter.

The simplest of those are momentum fields, in which each point represent in generalized coordinates the relative momentum of the being.

The being then has two states, one S=T, or reproduction and one S x T = K, with two limits, max. S x Min. T = Birth-generation, and Max. T x Min. S = Death – entropy; and two lesser limits, §(information: particle/head) x t (motion: limbs) = K (body-wave)…

We play then with those 5 components or dimotions deduced of the equivalence of space=time, or present, reproductive dImotion, and the limits and elements of § (information) x t (lineal motion-speed).

So the 5 Dimotions or basic symbols of Existential algebra are a development of the two equations that formalize the logic postulates – space is time, and the metric of 5D, S=T & SxT=K:

§->∞ x t->0 (Seed: **Ð5) **∑§¡-1=tº (birth)= Min.§ x Max. t (Youth: **Ð2)** > §=t (maturity:reproduction**: Ð3)** > Max. § x Min. t (3rd age**: Ð1**)<< ∑t-1 (Death**: Ð4)**

**Chains of Dimotions expressed as chains of equations.**

*All we said,** each Operand specializes in one Dimotion (angular sine/cosine in Perception, ± in back and forth locomotions, x ÷ in complementary and social evolution, log xª in reproduction) and OVER all of them a new Plane of existence is accessed by analysis. So operands guide the mathematical equations through a vital process of stœps (stops and steps) and will allow us to ‘vitalize’ equations, as we have done with points with ‘numerical parts’ as the essence of a mathematical T.œ*

¬Æ thus sets a limited number of logic propositions that can happen when a system or group of T.œs interact through its 5 Ðimotions, as an point of view, can potentially change its state between those 5 Dimotions, and the limits of its function of existence, such as the being can only exists without permanent disruption of its ‘vital constants’, (conserved energy, angular and lineal momenta – energy and membrain). All systems ca exist with the infinite cut-off limits of space (membrain) and time (death), which are set as part of the fractal Universe.Only the whole if potential or real in existence can be talked off as a function of infinity but not perceived.

So a point will start any of the 5 Ðimotions and we need formal symbols to address the Ðimotion of any being in existence, and the states of switch between Ðimotions.

Does the being stop before switching Ðimotion? If so it would simple to establish then for each sequential steps of a being:

∆¡ Ð1,3,2,4,3…. and so on as a simple 5 letter process of the actions of a being (whereas 4Ð entropy refers to feedings not dean, only in it final state being ‘that entropy’… So we know all sequential of a being ends in 4Ð.

Can then we run a sequential for any species through its life as a complete deterministic sequence?

There is there the sequence of all sequences, the perfect worldcycle=life sequence?

**Dual dimotions.**

An important element of those operands is when we consider them to be not self-reflecting operands, but communicative operands between two relative ¬Æ points or n-points in a network. This reads in several forms. We already showed how balances are achieved by switching the ±, x÷, log xª operands on a given NON-Æ POINT with numerical parts, along a chain of sequential time space actions, which leave a memorial trace hence NOT annihilating in most cases the sequence. How many possible combinations of dual inverse dimotions exist can be resumed in two great fields:

A) Dual Ðimotions within a single ¬Æ point, which ‘walks’ together through its paths of vital actions with steps and stops ± x÷ log ª dimotions.

B) Feed-back communicative Dimotions between two T.œ points that forcefully have inverse directions but tend to be the ‘same operand’ both sides with ± symbols.

C) Merged Dimotions between those communicative points that approach each other and finally merge, ‘re=producing’ through x÷ operand in a lower scale ‘connected axons’ (since A(x) x B(y) = AB (x •y)), that is the number of axons of communication between two entities A, B, with ∆-1 x,y parts is the product of x and y. So the product becomes the first operand to probe a lower plane of existence, while the ± operand stays in a single plane.

Operations then are the connections between T.œs that define their actions, mostly as balanced, parallel = connected creation of social networks bringing a 5D social evolutionary form or perpendicular, Darwinian 4D absorbtions=flows of entropy, motion, energy and information. Those dual actions are mediated by operations. And so there is first the abstract definition of those operations in mathematical terms, with the study of its properties and then its connection with the dual åctions of 2 beings that enter in communication within a given world-Universe.

The EQUIVALENT of such algebraic numeric analysis being in geometry the study of the topo-biologic properties of non-euclidean ‘waves’ of communication between 2 fractal points (second postulate of non-e geometry):

In the graph we can see how two asymmetric parts, normally one with more form and the other with more motion, come together into a single space-time event super organism, which will either become complementary (gender asymmetry) and evolve socially (which we can generalize to n-points in the 3rd postulate, forming networks) or will enter in a darwinian struggle, and be operated negatively in terms of the 4Dimension of entropy.

What algebraic equations do then is to operate in abstract, with numerical properties those events/superorganisms, 5D evolving vs. 4D devolving; that is 5D, adding vs. 4D resting, 5D multiplying vs 4D dividing, 5D potentiating vs. 4D rooting, 5D integrating vs. 4D derivating the system.

**A definition of algebra in terms of stœps and asymmetric dimotions. **

So we redefine BOTH in terms of the 5 Dimotions of time-space and its a(a=nti)symmetries:

*“Algebra studies **a(nti)symmetric **<≈> **transformations in the 5D space-time dimotions of a being, simplified as a mathematical §œT of ‘fractal points’, through its inverse ‘OPERATIONS’ that reveal the initial and final STate of the dimotion of the being, perceived as a whole in a relative PRESENT-SPATIAL, static state’.*

In that sense algebra is a more complete analysis, as it shows the ‘whole potentiality’ of the change, in a more generalized way, and most often can be interpreted in a full cyclical manner, such as A=>B and B=>A: A≈B. That is, an algebraic equation often proposes both ‘directions’ of a full time cycle.

And it is therefore if we were to use physical jargon, ‘time-like’, in the sense of being a ‘cyclical’, complete image of the whole ‘class of events’; whereas the operand and <≈> connecting parameters matter more.

*On the other hand analysis is NOT so much concerned with the back and forth dimotions’ transformations of the being as a whole, but with the creative and destructive processes which ‘ad or substract NEW dimotions of existence, through the integration of multiple small ST<≈>TS stop and go dimotions, scales or space-time dimensions, hence focused in the Temporal=Change p.o.v., integrating ‘presents’ and dimensions, or reducing them through derivatives, and so more than the operandi, it focuses on ‘THE VARIABLES.’*

Classic Algebra came through painstaking foggy idealist schools of axiomatic thought to formalize all this into group theory which is the best/closest approach to the true meaning of algebra we have today but we shall not use but rather as usually adapt it and comment on it with the better focused mirror of T.œ.

Indeed, in T.œ it is very simple to understand that any transformation happening between events and forms which we shall call generically actions, can only have from its initial asymmetry two paths, which are inverse and hence *all mathematical operandi have an inverse function.*

We can thus loosely consider that departing from an asymmetric, ≤≥ encounter, two systems can be operated ‘positively’ into a symmetric whole which ‘ads up’ and empower both elements into a larger creative process, or be operated ‘negatively’ into an antisymmetric whole which ‘subtracts’ as it eliminates one of the elements ab=used by the other. This is the loose concept behind all dual algebraic operandi.

**Algebra in its basic forms: stœps, parts and a(nti)symmetries of space=time supœrganisms.**

Algebra is the S<=>T perspective of mathematics as a mirror of the different steps and stops of each Ðimotion of a T.œ, with emphasis IN ITS SIMULTANEOUS gathering as a ‘whole event, united in synchronicity by the = symbol, creating in this manner feed-back equations between the space and time state of the 5 Ðimotions of the Universe.

Indeed, because Dimotions form complex chains most often of stop-space and motion-time steps (ab. stœp, which is the sum of a stop: perception-step=motion dual chain), the = symbol is essential to a mirror of reality’s events. And further on because it ‘equals’ or puts in relationship parts of a being, it is the ‘chaining’ symbol for the different parts of a super organism, in a single simultaneous space:

Thus we define a fundamental point-particle, the Non-Euclidean, Non-Aristotelian T.œ. as Supœrganism (ab. œ) of Time§pace (ab. T.œ ≈ ∆±¡) with a simple non-Ælgebraic expression or generator equation of all T.œ = Time§pace Supœrganisms:

*§ð (head-particles)≤ST(body waves)≥$t (Limbs/potentials)*

The beauty of algebra becomes then the capacity to write ‘chains’ of stœps in such a manner that we can follow sequentially the events that conform the Program of existence of a T.œ as it goes on through different Ðimotions of which the simplest stœp is shown in the graph of a quantum particle-wave duality.

Since reality is about the S=T and S x T = K ‘space=time and spatial size x Time clocks = constant, basic relationships hold in reality, from where we derive the two ‘extreme dimotions’ of entropy (pure time motion) and form (pure still language-space), which can be seen as the S=k/T symmetry; yet as those poles come together, we move to the S=T, energy=information final reproductive state of the being, through an intermediate, S X T, locomotion-distance x perception (lineal x cyclical vortex motion) state.

All those states and symmetries are thus feed-back equations which can only be properly ‘generated’ with algebraic inverse operations, in its simplest form, and in its most complex structures, when we consider multiple ‘symmetries’ and ‘variations of species’ with the help of Group theory, as essentially S=k/T, will be the ‘inverse numbers’, k, the identity number (if equal it to one) and the dual demotion represented by certain algebraic operandi.

So Algebra connects the closest humind formalism of the Universe – maths – with the Generator equation, whose operandi and transformations have been taken accordingly from the classic symbols of algebra, albeit as in all other disciplines of science, slightly transformed to make it correspond to the larger, more encompassing, efficient focused mirror of GST (when in the future is properly evolved by other researchers).

So we call the formalism of relational vital space-time and its T.œs, of ‘Gst’, general systems sciences and its Generator of Space-time organisms, also* existential algebra*, and write it as ⌜Ælgebra, which reads as ⌜Æ-logic: that is Existential Algebra (Æ) (the 5Ðimotion and its inverse 4Ð entropic dissolution, **⌜**.)

This post thus will be likely in the future and when completed by expert mathematicians and physicists, as I don’t believe I have time to finish it properly, a key posts for future researchers or robominds exploring the existential laws of the Universe.

# ALGEBRA AND ∆-SCALES

**¬Ælgebra of parts vs. wholes.**

**numbers**, which are §ocial groups of undistinguishable elements or sequences of lineal time; which being algebra a time-dominant discipline will dominate the algebraic analysis over the spatial point states, meaning most equations write as:

**equations**and its operandi, which establish the relationships in a single or adjacent planes of existence between sequential and social numbers, and its ‘ternary states’ and transformations, social evolutions and topological forms derived of them.

**functionals and groups,**and those will be studied in mathematical physics.

**Foundations of experimental ¬Ælgebra vs. metalanguage foundations.**

**Time-like algebra and space-like topology. Time like probability and space-like statistics.**

**topology**, space with motion hence an St-version of it, and

**modern algebra**, numbers with geometrical structures, hence a Ts-view, which then can be further related by a nice theorem:

*final more complex ‘whole view’ of number theory, from where indeed departed (Fermat’s work, as the founder of time theory and probability).*

**S=t symmetries between algebra and analysis**

*the complexity and creation of new layers and structures come from its*combinations in dualities and ternary symmetries (number and point, s and t; scale and number, ∆ and t; scale and fractal point, ∆ and s dualities and s<st>t, ∆±i, |x0=Ø ternary symmetries).

*capacity to give us a huge amount of information into the most complex symmetries and simultaneous space x sequential time structures of reality*

*in the generator equation, which is the basic generator of ¬Ælgebraic structures, as it embodies the ternary symmetries of time, space and scales.*

**1st algebraic S=T MIND EQUATION: 0x ∞ = K ****search for ‘wholeness’ in a single equation. **

*find wholeness, and stop all motions into puzzled mappings that try to enclose it all, all the steps, all the variations, all the forms, and all the motions, in a single algebraic structure, a potential equation of all possible bifurcations of those steps, which humans think to have achieved with those ‘monstrous lie groups’ and other algebraic structures that try to be an impossible minimalist mirror that encodes the information of all the symmetric steps available to reality.*

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

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

* O x ∞ = K = ∞º=1*

*Or in dynamic way, S@<≈>∆ð.*

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

In mathematics*, with the duality of **inverse operations, + -, X ÷, √ xª and ∫∂.*

The humind plays a key role in all this game, as we can only perceive through the languages of the mind, which ‘perceive in themselves’ as ‘still-simultaneous-linguistic mapping’. WE DO NOT PERCEIVE REALITY BUT LANGUAGES-MIRRORS, we often confuse with reality (Mind paradox).

*All is in the codes of the Universe and its operators, also clear in mathematics, with the duality of **inverted operations, + -, X ÷, √ xª, ∫∂.*

We thus define in existential algebra the main operators of classic algebra as the two directions of a dimotion of space-time. We can in this manner compactify some dimensions as inverse, being entropy and generation (4th and 5th Dimotions) inverse in symmetry to a given ∆º plane (±¡ SYMMETRY).

To which extent we can consider stop and motion, 1D and 2 D inverse or rather asymmetric, must be dealt on a one case basis. Finally as it turns out we need the other four Ðimotions to achieve the 3rd Ðimotion of reproduction, *we can reduce all other Ðimotions to reproduction; hence establishing the unity of intent in the Universe: to keep reproducing an eternal present. IT is this present-eternal reproductive Ðimotion what defines the Universe as a fractal.*.

IT IS ESSENTIAL *THEN to understand ‘existential Logic’ and its space-time inversions, which represent the fundamental logic postulates of reality:*

- That S≤≥T, but also SxT=K, So, S=k/T.

So systems advance through inverted Steps and Stops which balance each other, since the equality of both sides of existence coupled with its paradoxical inverted properties make all systems finally a virtual zero sum, balanced *across steps and stops, Dimotions and Scales, represented in the logic of human formal languages notably mathematics by equalities of ‘dimotions’ as functions of space-time (Sx=Ty) in different mirror-languages, and so we shall find constantly those balances are needed in Nature and represented in logic and mathematical language. *

Some examples of physics will suffice.

In the study of particles, painstakingly as they did not have the ‘basic truths’ of space-time symmetries (still don’t) physicists discovered that the proper representation of the quantum world was not only a wave (Scrodinger’s equation) but also a particle, that is the motion and stop duality, so they moved to the Klein-Gordon equation; but then they had to marriage those equations with relativity on the limit of our c-light spacetime (full c motion); *so they have to find a balancing opposite rotary motion, the spin, also on the verge of c-light speed (Pauli matrices); but then they realize they have the imbalance of + particles without the counterpart (- Particles), so Dirac widened those Pauli matrices to bispinors that represented the needed counterpart of antiparticles.*

Yet then they found there were in the Universe more particles than antiparticles. Where are then the antiparticles to balance the zero sum? It was found in the post-war age but still misunderstood that a certain force, the weak force, they incorrectly tried to model as a spatial force, when it is A FORCE THAT EVOLVES PARTICLES INTO HIGHER SCALAR families of the fifth dimension, preferred to create ‘particles’ in its way down (entropic arrow of devolution from heavy top, bottom, strange quarks to lighter ud-quarks), first in the Strange kaon decay and just recently at the LHC in a much wider ‘angle’ (20% of excess of protons over antiprotons), so it follows that if the ‘next scale of matter’, the black hole world of heavy quarks and ‘black quark stars’ (see our posts on 5D astronomy), favors in its decay to the smaller world of atoms, particles, in its way up, the ‘galatom’, or physical upper scale of the Universe will have galactic antiparticles, as it is the case (our left-handed galaxy has most of its mass as an anti hydrogen does in the outer halo, NOT the center)…

So we observe in fact the existence of inverse balances *in all the directions of the fractal Universe,in steps and stops of S=T balances for the 5 Dimotions of existence, hence in between the O-¡, Øº, |¡ different scales that will show to ‘change its role’ – so for example proteins are lineal, but in the next scale of ‘4D topologic warping’ form globular shapes… and so on and so on.*

All this must then be properly described relating it to the different operand of ¡logic, which we shall adapt from classic algebra, at the simplest level (leaving for future researchers the full-fledge discovery of the whole model, in very specialized elements, such as those of quantum physics in the wrong Copenhagen interpretation with its unneeded conjugate normalization and apparatus).

There are in that sense an ‘a priori’ sets of rules for all sciences, motions and dimensions, species and events, in simultaneous adjacent super organisms that balance limbs and head, locomotion and perception, entropy=death and informative generation, merged in reproduction (the 5 Dimotions, then cancel each other, and keep reproducing the present); all of them based in the 3 simple logic statements translated with those operators:

‘That space and time states of form and motion balance each other,** S≤≥T** and yet their properties are inverse in 5D metric, **S x T = K±¡**, but isomorphic in all scales (K±¡).’

The astounding beauty of the 5D model of reality then arises from the fact that we shall extract all the laws of reality from that simple statement, in all the required detail as we just have done for the world of particles that so complicated appears to the physicist which lacks this basic understanding of the parallelisms, perpendicularities and inversions of space-time systems in all scales of nature.

So the simplest method of extracting a meaningful quantity of information on a being or event requires to comment on the 4±operands ‘5’ perspectives, even *when we study as we do in the first line, *those elements in themselves. In this post we shall deal thus with time from the perspective of space, time, symmetries, scales and mind languages that ‘stop it’ in the singularity of t=motion zero, from black holes to eddies to charges, to brains to frozen genetic DNA…

*and time is far richer as time-motion is all, space being only the humind mapping of reality – or that of a different mind.*

**All mathematical disciplines are experimental hence extract essential ∆•s≈t properties.**

*in discontinuous time numbers certain relationships such as π or √2 are NOT defined, while i geometry do exist.*

*in time pi never closes its cycle, allowing a perpetual motion Universe, but in space, it practically closes in each present the super organism, breaking its vital space from the outer world.*

Along all the pages of this blog we have shown an enormous array of examples proving the duality between time-motions and its space-forms. So basically the fundamental symmetry of reality is that we *can express anything as a series of motions in time, which will be related symmetrically or inversely through one of the fundamental operandi of algebra, or any ‘verb’ of language, or any of the logic symbols of space-time flows of 5D (< ≈ >) forming a time-space event.*

And hence *FROM ONE OF BOTH SOLUTIONS WE WILL BE ABLE with the proper methods to extract the other mirror-solution, which we express in the fundamental duality/equation of space-time symmetry: ***∫@≈∆T**

This is what makes Algebra, once *we apply the Rashomon effect (multiple 2, 3, 4 or 5 D povs and functions, depending on detail) so powerful to mirror real it as it is basically all about S operandi T symmetries. *

But the main task to do in algebra is to fully account for the meaning of all its operandi and establish its relationship with space and time symmetries. I.e. we talk of sum, product, integrals, logarithms and numbers, complex, irrational and so on with very little understanding, happy just with the pedantic, axiomatic method and its pretension of absolute truth – which by Godel is by all means incomplete, and reveals very little on the ultimate disomorphisms of space and time.

So the explanation of operandi and Space-time dualities=symmetries+inversions is what will take us to the very deepest levels of understanding of the Universe.

**RECAP.**

*“Algebra studies social, sequential numbers and its relationship with spatial points, extracting the S=t symmetric motions between them’.*

It seems a simple sentence but is huge in meanings.

It shares its minimal mathematical units with number theory, as numbers are closely related to the arrows of eusocial evolution of the 5th dimension and the arrow of discrete ‘frequency’ in time. As such numbers measure the ‘long’ , social evolutionary game of scales, and ‘short’ sequential arrows of time of the Universe.

While ¬Ælgebra truly starts in the next scale of space-time events, with multiple groups of points (variables), exchanging entropy, energy and information, through operandi (specific of each action) in ‘equations’ (which become partial case of the generator’s allowed st exchanges and symmetries). And so this classic algebra is the most important part of it.

Whereas modern algebra and its study of the whole ‘block of time-space’ of a supœrganism, through all its potential actions expressed in **functionals** of functions that embed two or even 3 scales in whole ∆±1 structures, *is the all inclusive ∆±i perspective, while* **groups** of symmetries, according to a set of ‘restrictions’ of its allowed actions, is the single ‘space-time’ plane attempt to describe the whole potential transformation of any entity in such a single plane. So groups and functionals explored in the 3rd age of algebra complete its wholeness.

This FIRST VIEW of Algebra can be done through its analysis in space as a simultaneous connection of the two sides of an equation, which put in relationship the ‘ternary extremes of an ∆st element’, either by its operandi (which allows a single plane connection of s and t elements) by its polynomials (which allows a multiple scale connection through ∆§) and by its inverse symmetries (group connection).

And because the universe is dual in its inversions and ternary in its elements and scales, 3+ are the operations of math. Since Potency and integral operations being closely related as we explain, as the duality of ∆ curved analysis between planes vs. §lineal scaling in decametric societies happen to be. Indeed any mathematician immediately will notice this duality with the concepts of a derivative vs. a lineal differential approach, a logarithmic scaling that tends to a fixed asymptote vs. a changing tangent-derivative of a more complex curve, and the ultimate proof that all algebraic equations of exponents can be approached by Taylor series of derivatives.

Now all this is explained in classic algebra with group theory and axiomatic methods; so we shall consider that approach. And the only elements left to define then are the identity element and the properties of the operandi.

Indeed, all dual operandi have identity neutral elements and inverse ones. THEY CAN be considered loosely as the neutral=asymmetry form, which ‘splits’ both ways into the negative=antisymmetric element and the positive=symmetric one. The identity element leaves the asymmetry unchanged, it is we might say a non-operation. The negative element is the result of antisymmetry, the positive element of symmetry and tends to be a larger whole.

So for example, if we have a herd of 4 + 3 elements, and they come into parallel social evolution they give us a herd of 7.

But if we have a herd of 4 elements and they come into a darwinian relationship they will separate the defeated elements, so if it is for example a fight for supremacy, only one will be left and we rest the 3 defeated candidates, 4-3=1.

The first curious thought coming out of this simplified analysis is that *positive operations tend to be more restricted additions as they require identical elements, so they are often simple social evolutions, negative operations however might have multiple meanings as the antisymmetric, ‘entropic’ states of a system multiply statistically.*

So this has a reading into the classic arrow of time of thermodynamics, which already noticed that order ‘probabilities’ are less than ‘disordered ones’, *unless there is a maxwellian demon – which we contend do exist in the survival will of all points of time – to contain entropic, destructive probabilities, which is indeed what it happens most often.*

So in brief, when 4 and 3 elements that are numbers, which can be operated together hence equal beings as numbers are social groups of equal beings, *the magic of social love and the fourth postulate of similarity in non-euclidean geometry means that all the systems which fall between symmetry and asymmetry, and can communicate come together. Entropy is REJECTED by almost all systems from the singularity perspective, which implies also that a mere abstract mathematical analysis will fail to understand the bio-logical input of those informative singularities that reject as Maxwellian demons do the negative systems, tricking the ‘dices’ of God. *

What about product vs. division? Again this duality is obvious. If we have 3 x 4 it means 4 herds of 3 which ad together in 4 steps to a 12 herd.

But the entropic events here are also multiple and divisive. I.e. If we have 3 kilos of wheat, we divide the whole between 4 entropic hungry men to get 3/4 per capita. Divisions thus ‘divide’ almost always a system into broken parts. Products multiply societies, or create tighter communication between those smaller parts. I.e. A= 5 and B= 4 elements can be multiplied also at the ∆-1 scale to get the number of axons that ‘tight’ the society together if each element of set A communicate with each of set B, then A x B = 5 x 4 = 20 axons.

So we come to the potency vs. root, integral vs. derivative that merely take this dual process to its final ternary scaling (as we know systems are ternary so no need for further operandi). Its complex study being carried further in the post of analysis so we shall let it go for the time being.

# II. 3 TIME AGES

**Synopsis. The 3 Ages of Algebra.**

We divide this exploration in 3 sections; one of classic algebra, mostly concerned with the **basic operandi that connect the Space and Time states or 5 Dimotions of any system, which form the core of the equations of algebra** a second part on modern algebra, mostly concerned** with groups and functionals**, which put its emphasis not on the feedback steps between Dimotions or S=T dualities**, but on the structure through scales of social function (functionals), and the entangled variations born of the total possible symmetries of one given element.**

So the 3 fields and ages of algebra are:

Youth: ∆-1: Arithmetics, the Greek Age, now mostly studied in number theory.

Maturity: ∆: Equations which extend each ‘letter’ to a range of numbers, *creating a new ∆-scale of generality for maths; mostly studied in @nalytic geometry*.

Old Age: ∆+1: Algebra grows into a new scale of complexity making functions of functions, in time (Functionals) or in space (Groups).

**The 3 ages of Algebra as reflection of 3 scales of growing complexity: numbers, functions and functionals.**

**ðNº≈§•:**, meaning that time numbers are put in relationship with spatial points, and time functions with space functions through the operands of algebra.

*The sequential number*, which is the temporal version of the fractal Œ-point as its minimal unit, whose properties are ‘narrated’ in number theory.

*The function and its operands,*which expresses normally a partial equation of the ternary fractal generator, most often of

*the space and time states, connected by the function and operands, overwhelmingly the basic equation of mathematical physics (as in E(s)=M(t)c².*

*The structure,*which is the highest comprehensive analysis of full ‘blocks of time-space’ laws; and specifically encloses all its possible variations; today developed through

*the concept of Group*as a ‘receptacle’ of all the elements of the being, with a focus on its spatial symmetries and its similar expansion with the concept of the functional (function of functions) with a focus on its ∆§cale symmetries.

**1ST OF Algebra: Social Numbers: ****‘arithmetics’.** Single ∆§ plane and ternary dimensions and S=T equation-symmetries.

In its original form, algebra dealt with mathematical operations on numbers considered from a formal point of view, in abstraction from given concrete numbers. So *it was really a way to calculate social numbers in their growth and diminution in herds (±); in its growth and **diminution in 3 spatial dimensions (X, Y)³. *

*A*nd* as such it could not go further because there are only 3 dimensions in a single space-time plane – **Fermat’s grand theorem, being a lateral case of the difficulty to create by superposition in a ‘higher dimension, a simple social sum, such as x³+y³≠z³.*

Hence the seflie teaser of some of this blogs’ post – that my fellow basque countryman Mr. Fermat might have known 5D², cause that is the only proof that fits in his Apollodorus’ margin (:

This impossible ‘simple’ problems are thus PROOF of profound LIMITS in 5D Universes – equivalent to the classic problem of doubling the cube in bidimensional geometry, also known as the Delian problem: Given the edge of a cube, the problem requires the construction of the edge of a second cube whose volume is double that of the first, using only the tools of a compass and straightedge. As with the related problems of squaring the circle and trisecting the angle, doubling the cube is now known to be impossible.

And it shows that geometry and algebra, points are numbers are ≈ similar but not =. So some numbers do NOT exist (√2, pi) as such but are ratios, which therefore can be found better in geometry and viceversa, some geometrical figures are neither pure forms or proportions and cannot be found with compass.

So while geometry was first born in a single plane of space-time ITS FUNDAMENTAL PROPERTY, arithmetics soon allowed to study at least the social evolution of numbers in a single plane of space-time. And this was the birth of Algebra, with its subtle continuous-discontinuous variations of geometry vs. arithmetics – NEVER UNDERSTOOD even to exist as Euclid had defined absolute equality in its wrong axiomatic method that has weighted so heavily on human thought and its ego-trips of absolute truths.

Equality does not exist only similarity. In any case the first age of algebra ended with the limiting study of a single space-time social evolution in 3 dimensions and its decametric scales.

The second aspect of mathematics deals with sequential numbers and equations, in which sets of numbers are transformed by an X=Y function, in which often one component changes faster than the other, tracing a curve in a Cartesian Plane plotted with those 2 variables that mathematicians study with great detail. It is the thesis of this work that all those X=Y functions and differential equations represent particular studies of a general ∑Se<=>∏Ti equation, and or a partial event between two or n-points of a non-Euclidean network, and or the network and its environment.

Yet as Einstein put it to Poincare: ‘while I know when mathematics are truth I don’t know when they are real’, meaning that many mathematical equations and functions do not exist in nature, as they are not partial cases of the Generator equation and do not respond to the restrictions the Ternary method imposes to a Universe of multiple spaces and times but only 4 Dimensional arrows. In that regard, the laws of multiple spaces and times and the syntax of the Generator equation with a limited number of variations restrict the possible mathematical realities there is in the physical Universe. On the other its study provides the scientist with a deeper meaning for the Algebra of numbers and the meaning of equations and functions.

*Recap*. We shall consider merely the meaning of the main mathematical operations within the restricted world of 4-Dimensional spaces and times we live in. And analyse in more detail some of the parameters and functions most commonly found in the study of the Generator equation, which connects the equation of space-time cycles with the detailed mathematical analysis of those cycles by different disciplines.

** Theory of numbers.**

Numbers in this new outlook are not only intervals of a one-dimensional straight line, but as Pythagoras and Plato stressed, they are geometrical forms:

Mathematics is concerned with 2 seemingly different worlds, the geometry of spaces and the logic of numbers. To fusion both requires to understand numbers as forms. A number is not only an abstract set but always a collection of self-similar beings extended over a common vital space, a network And so networks create complex forms, topologies of space-time, as the motions between points of the networks become stable exchanges of energy and information between two polar points. Yet since each number is a geometrical form no longer limited to the simplest one dimensional form but can vary its geometry and hence its function, degrees of freedom and complexity as we increase its ‘number’.

The 4th Non-E postulate shows how points, numbers, the self-similar class of equal forms create geometries:

The line is simple. The line joins two points and can only have a combination.

The triangle can only have a closed combination, but 3 possible open combinations, Ab, Ac, Bc.

The quadrangle is more complex. It can be joined in 2 combinations, as a cross and a square. And it can be left as an open snake with 3 different orientations. So a foursome acquires a snake shape to move with the arrow of energy; a crossed form to perceive in its center ‘5^{th} point’ and a square shape to accumulate and reproduce its internal organs; and so each shape of the same number becomes a topology with a different form and function.

Indeed, function and form are now fusioned. So certain numbers in its ‘degrees’ of freedom of form, represent certain functions. The quadrangle can store energy, but in a zigzag open line it can move – spend energy and as a cross it can gauge information. Numbers also define arrows of time. So for example, 1 lonely number without motion is perceiving, with motion is processing energy, 1+1 might be 3 (an act of reproduction) or 1 (an act of Darwinian feeding). All those vital actions determine that certain numbers survival better than others. So, 1, 3 and 4 are very common systems.

In that regard, a complex analysis of the simplest numbers shows that the more perfect form is the 10-cellular system or tetrarkys, in which 3 x 3 triangular corners act as organs of energy, information and reproduction with a 10^{th} central element that communicates all others and acts as the one of the higher scale, representing the entire organism.

Thus as the number of cells grows, the topology of the system will grow in degrees of freedom and complexity till resembling more and more the repetitive, geometrical forms of social organisms. Topologies become thus at the end, complex networks, adapted to different functions of complex organisms.

As abstract as all this might seem, when observing nature we shall see how those type of events, waves and social planes happen in all the scales of the Universe, from atoms which form crystal networks based in the equality of the same atoms or at best in the existence of a ‘body-mass’ of equal atoms intersected by a few ‘stronger’ atoms that form a complementary network of higher resistance, to the body rejection of cells with different DNA.

What things we can do with numbers can reflect then many of the actions of networks. For example:

– We can study how social groups organize themselves or fluctuate between states=functions. This is the study of the internal point of view of networks as a collection of self-similar points. Those changes of states are often defined by a differential equation as informative systems have less spatial extension/motion but are more complex networks with more bits of information=points. Thus differential equations, most of them of the type Y (ti) = aX^{3}±bX^{2} ± cX ±D, express ∑Se<=>∏Ti transformations, where Ti is a network in 2 or 3 dimensions of time bits, bits of information and Se is a network with one (same organism) or 2 (Darwinian feeding) scales of lesser complexity than Y, such as f(x)=Y^{n.} It follows from the Fermat Theorem that there is a restriction to the number of solutions a system can find, which is n=3, the maximal number of dimensions an informative sphere can have as it displaces itself over a plane of energy.

The relationships between limbs and heads that exchange in a 3^{rd} region called body, form and motion, such as the head designs the motions of the limbs, which move the head, and both exchange in an intermediate region of elliptic nature called body more subtle types of form and motion to create more complex cycles that will in fact reproduce both systems can be mathematized in infinite different ways, using matrix, combinatory theory, differential equations, polynomials, Riemann surfaces, etc.

– We can study how networks grow and multiply creating new species and we can add them and observe how they reorganize creating curves which are differentiable to obtain the rate of grown and diminution of the organic population. The study of herds of energy and networks of information in its life cycle is one of the key disciplines of all sciences specially physics and ecology.

– We can study them as networks with form through its geometrical ways of exchanging energy and form, from the simplest point to the line of 2, the triangle without a central focus, the structure of energy, which can however turn into a pi-cycle, the 3, the 4 with its zigzag, solid quadrangle and cross structures, the 5 and first 3 dimensional structure, and so on.

Each number will increase the possibilities of the game, yet when we reach 10 we play a perfect game with 3 triangles that act as organs of energy, information and reproduction, and a central point both in a 2-dimensional or 3-dimensional geometry, acting as the collective action/will/intersection/knot of all cycles – the first clear, complete ego structure in 3 dimensions with perfect form and complementarity. Thus beyond 10, while some numbers might bring slight improvements to the cell, most forms are just growths of the primary numbers in multiple associations.

*Recap*. All the structures of mathematics, regarding of the notation we use, reflect events and forms of knots of time arrows (st-points or numbers), as mathematics is a language whose grammar derives from the Universal grammar of spacetime. Numbers are thus formal networks that try to achieve the essential arrows of time. And so certain numbers (1, 2, 4, 5, 7, 10) deploy better those arrows and are the commonest on nature.

– We can study the evolution and reproductive creation of new networks with successions and combinatory is important in multiple time-spaces since we find always complementary systems of reproductive energy and information, each one with a ternary choice of evolving differentiation (energetic, informative and balanced species). So especially in the classification of species of different sciences we shall find simple combinatory laws that explains the differentiation in 3, 6, 8 and 10 elements depending on the triads and dualities of multiple space-time systems.

– We can study a key antisymmetry of time and space expressed with the language of probabilities: Sequential events are studied with probabilities in time, whose symmetry in space are the study of percentages of populations in space, such as if each event in time is the birth of an individual of a population both probabilities and percentages are the same.

This confused physicists in some cases, as in an electronic nebulae, which is a population of fractal electrons in space, but it is studied as time probabilities, and created the bizarre theory of multiple universes (multiple, probable electrons) instead of a fractal Universe (fractal self-similar micro-electrons, which are bundles of ultra-dense light forming a nebulae which also acts as a ‘whole’ electron, self-similar to its parts). Thus the study of probabilities in time events and growing populations of a wave of space-time cycles is an essential tool: we can study the proportions, herds, groups and networks of self-similar st-points in its evolution either with probabilities or differential equations.

*Recap*. Probabilities study causal events in time and populations in space; combinatory studies the differentiations of species according to the variations of bodies and heads.

**II Age of ‘Algebra’**: development of letters as parameters, mathematical physics and ∆-nalysis. The age of functions.

Things got interesting when algebra started to focus ON variable parameters which represented T.œs and its Disomoprhisms and symmetries, flows, and ST-eps of the different motions of beings in space-time.

This abstraction found expression in the fact that in algebra magnitudes are denoted by letters, on which calculations are carried out according to well-known formal rules.

Algebra now considers “magnitudes” of a much more general nature than numbers, and studies operations on these “magnitudes” which are to some extent analogous in their formal properties to the ordinary operations of arithmetic: addition, subtraction, multiplication, and division, adding *inversions of 4D-5D, ∫∂ and log/xª*.

And it adds dimension of motion to the original geometric calculations, hence the need for ‘variables’, operandi of change, and finally vector magnitudes, which include both form and motion dimensions, even if they can still be represented in metal space with geometric rules, as the well-known parallelogram rule of addition.

#### Functions and operands.

In simple terms, a function *f* is a mathematical rule that assigns to a number *x* (in some number system and possibly with certain limitations on its value) another number *f*(*x*). For example, the function “square” assigns to each number *x* its square *x*^{2}.

The common functions that arise in analysis are thus definable by formulas, *which are related to the ∆s and ∆T duality of functions,* such as:

**§:** Polynomials of the type, *f*(*x*) = *x*^{2}. The logarithmic function log (*x*); & the exponential function exp (*x*) or *e ^{x}* (where

*e*= 2.71828…; and the square root function √

*x*.

**T:** Trigonometric functions, sin (*x*), cos (*x*), tan (*x*), and so on.

**∆: Differential functions.**

It is then also when having algebra completed its main task – the creation of a ‘translative method’ of reality and its dimensions through operandi, the fundamental 4,5D ∫∂ duality takes off as analysis, which becomes the most important field of mathematics, as the scalar Universe do include all other elements within its folding.

So Analysis will be algebra and also number theory and also @nalytic geometry… But not so much geometry which by definition is a spatial, single plane view that only with topology extends to an ∑∆-1 points-parts>Whole geometric forms.

And for that reason as Analysis becomes the natural evolution of algebra into the realm of wholeness and scales, Algebra enters a third age, in which the essential element of algebra, which is NOT polynomials – a rough approach to differential changes, approximated by Taylor/Newtonian binomial methods – but OPERANDI, the study of the relationships between the symmetric sides of the equations, takes a flight of its own, focusing now into the structures created according to the rules of engagement of those operandi.

While a new complete fundamental field as it ‘brings’ a new mathematical species into being, ‘vectorial calculus’ takes a flight of its own, thanks in great measure to the work of the most underrated genius of mathematical physics, and one of my favs for his absolute despise of wealth, power, fame and penpal peership (: mr. heaviside.

**BAROQUE AGE: §ŒTS**

** 3rd age: ∆+1 age of Algebra**, or group and functionals, which deal with a ‘timeless’ attempt to include all the potential duality and ternary symmetries of the ∆@S=T universe split into:

∆lgebra represented by Functionals, which are equations of equations – a complete ‘scaling’ of algebra, from concrete ‘social numbers’ and ‘sequential numbers’ (space and time view of numbers) to the wider laws of ∆@S≈T geometry, in which each ‘fractal point-number’ encodes an entire ∆-1 scale defined by its ‘function’.

*Groups: Algebra as the expression of Space-time motions.*

Special mention on modern algebra, deserves group theory, which is the modern expression of ST-motions, hence concerned with the other ‘dual side’ of the Universe, motions in a single plane of the 5th dimension:

Algebra, being concerned with ‘spacetime motions’ required a final frontier – a way to represent motions in time, with a synoptic language that ‘extracts’ all what is repetitive in those motions, including motion itself to bring the essence of it into the ‘reduced syntax’ of spatial symmetry – as *when we enclose the whole of all time views, we obtain a spatial synchronous still mind view. *This is what group theory does in its comprehensive wholeness, converting the entire range of time motions and events into a ‘fixed whole view’ of it, as a whole spatial range of all potential variations seen in simultaneity (the group).

Alas! thanks to the genius of Galois, we do have *an extraordinary new field of Algebra, which is essential to the most advanced formalisms of mathematical physics, and its study of reality.*

**Groups and ‘Algebras’.**

*trying to find the complete equation of the Generator:*

*∆±i, $t≤ST≥**ð**§.*

Yet as humans ignore the proper focus of existential algebra (the Generator, all its sub-equations and embedded Ðisomorphisms) Algebras became complicated (unfocused complexity = complication), and instead of marvelling at the clarity and synoptic power of the fractal generator, they became increasingly pedantic and the ‘pest of group theory’ (Weyl) settled in, requiring a huge memorial effort (the lack of intelligent understanding bring memorial repetition). And so we enter into the Von Newman age (: the age of nerds, which thanks to computer thought – still thanks god, or else we won’t be here, in their memorial repetitive cyclical algorithm).

And there they are all, trying to find the fractal generator of Everything Organic (T.œ), with ever more complex algebraic Lie groups and statistics (you have lies, damned lies and statistics economists say of their go(l)d religion, we should paraphrase, you have ‘lie algebras, damned groups and statistics 🙂

This said the age of §œT and Group theory *must be modelled as the present foundation of Mathematics in terms of ∆s≈t; as §*

**ocial Generators=Groups of Time Spacœrganisms: Γ§œT.**

And why with its abstract, convoluted reasons, they do actually mirror quite well reality.

The answer is simple. A set is an ensemble of mathematical entities, hence ultimately plugs in directly from the roof down with the social nature of numbers and geometrical points. A group is further on, a fascinating *symmetry between time-motion or space-form. *

In that sense group theory is a bit like a ‘cubist painting’ which tries somehow artificially to gather all the possible mirror elements of a given S=T Space-time super organism and its possible parts:

In the graph, Algebra as it happened with the visual language when realist photographies came, displaced from reality by analysis moved inwards into more complex, abstract Nature, became a ‘baroque language’ purely formal trying to explain it all* departing from reality into a multiple ‘cubist perspective’ that tried to gather all perspectives into a single painting *(Set and group theory, often a dead end with no reference to experience.)

In the graph, we see that parallelism of evolution of algebra and painting, under the Ðisomorphic laws of languages as formal mirrors of reality which become inflationary, when departing from its immediate constraining experience of that reality, suffering then Kantian paralogisms – pretentious expansions that try to freeze all the ∆@s=t kaleidoscopic perspectives of the Universe with a single mirror; a meaningless attempt to reflect complex 5D reality that seldom gives us relevant information about the connections between all the elements of the T.œ.

The positive side of those last forms of algebra is that they merge all the sub-disciplines of mathematics as the ∆@S=T universe is also entangled, but *without the conceptual clarity it should have if* humans understood the duality of the Universe (or the Asian, Taoist world, which did understand it had dominated history and its sciences).

In that regard, while in old texts we respected the ternary elements, ascribing algebra to time, to ‘bridge’ the nebulous thoughts of humans and GST, in this web as we keep building it we will clearly separate mathematics in two major fields:

**S≈T A**lgebra & ∆nalysis which studies §ocial scales of numbers, whereas ∆lgebra tends to concentrate in a single plane through its social, decametric scales and ∆nalysis correspond closely to the process of growth between scales.**S-**Geometry and its modern form Topology which deals closely with space-time symmetries.- And all the branches, which mixes the topological and algebraic approaches; such as probability (time view) and statistics (space view) analytic geometry and so on.

So if we were to define i-logic mathematics we would say that it is composed of Non-Aristotelian ∆lgebra and Non-Euclidean STopology; the first including analysis the second bidimensional and static geometry. And the natural evolution *of the discipline correspond to the combination of them all to express ∆st whole processes. *

From the practical purpose though beyond the introductory texts of GSTructure we shall maintain the general division of disciplines trying whenever possible to include as usual some small change of symbols to remind us of what they are mainly concerned with: ∆nalysis, ∆lgebra or ¬Ælgebra (which studies specifically the way arrows of time mess sequentially to give birth to space-time changes of state).

On the other hand S-Geometry and Topology highlights the capacity of the discipline to study space-time symmetries, mostly in a single plane. While Statistics & Probability STudies from both perspectives, space and time the same phenomena. And so on.

And the similarity between GST and i-logic mathematics as the most experimental of all sciences is so great we shall only consider an entire new discipline, to ad to the three classic ones, Ælgebra (Existential Algebra, not to confuse with ¬Æ, non-aristotelian time and non-euclidean space), which studies from scratch the formalism of the 10 dimensional ∆ûst of space-time of which we are all made, with three time arrows, three ±1 ∆-scales and 3 Spatial-topologies.

**This process of formal search for a total formulae **that encompasses all possible variations of reality is in a great degree a barren search for the absolute born of the creationist egocy of mankind, which culminate in the 3rd ‘informative age of algebra’, with §œt theory and the axiomatic method, both a false pretention of ‘mathematical creationism’ that changes the true units of the mathematical universe, points of space, scalar social numbers, and classic operandi of S=T dimotions, by humind ‘creations (sets, categories etc).

While the axiomatic method rejects mathematics as a mirror trying to proving its truth, against Lobachevsky and Godel’s completitude theory, without using empirical knowledge. So in Get Sets… are NOT the basic unit of mathematics – only of the imaginary Cantor’s paradise (Hilbert).

As we are back to the true units of space, points, scales, numbers, and time, operandi; which are the fundamental elements of classic mathematics the second most perfect age of any ‘form in exist¡ence’, mental or physical. So we shall consider modern mathematics in the post-war era, the baroque, 3rd inflationary age of its form, with an excess of in-form-ation and inward looking view of reality, akin to the 3rd old age of a human being, the baroque age of art… prior to its death and renewal, which in algebra is happening with the simpler forms of Boolean algebra, the mind of computers and AI (algorithms of information, as Artificial intelligence is a meaningless abstract word).

Here we shall also escape NOT because of the lack of intellectual interest, on the contrary, it is to me an ethical problem – AI will be the mind of robots which are displacing humans from labor and war field and will exterminate us unless a true science of History, the super organism of mankind in time, learns how to control the evolution of the tree of metal. So as much as I would love to discuss the way to program a robot to become a perfect survival sentient machine, I won’t give any further information on the specifics of that theme, which anyway primitive hominids will do in its military form much earlier than you think.

And as it happens, mostly by chance and with little understanding, this *is what Group theory enables in great measure due to its most remarkable features, one of structure and the other on method:*

**STRUCTURE**

To define 1 or 2 operations, loosely called ‘product’ and ‘sum’ but which *can be anything as long as it truly defines one of such symmetries because it requires, the 2 fundamental features of a space-time symmetry: inverse elements (where inversion is also loosely defined) and a neutral element, which can be considered the symmetry axis, in a close but quite general symmetry with the 3 elements of the Generator – Sp, its inverse Tƒ and the neutral element, ST… *

So the structure of groups is quite close to the general structure of space-time symmetries and this allows it an enormous flexibility to show all kind of S OPERATION T relationships.

It is for that reason that the concepts associated to Group theory, isomorphisms, transformations, Generations, representations and such are very close to the concepts expressed in 5D²; but as 5D² has a much wider range of applications and a far more realist outlook, it should be clear that Group theory is a mirror of 5D SPACE-TIME and not in the other way (as creationist mathematicians think of).

Consider for example the concept of isomorphism, which I BORROWED not from group theory but from general systems science, my earlier discipline: it is purely structural meaning that if a group its identity elements, inverse elements and ‘product operation’ behave in the same manner two groups are isomorphic to each other.

Disomorphisms go a notch further and affirm that all systems of the Universe are in fact isomorphic as all have the same Dimensional properties deduced from its 5DST CONFIGURATION. So all will go through a life-death cycle. All full working beings will have a ternary topological/functional structure and so on.

Can then we express in terms of GROUP and SET theory 5D? Likely but it is of not interest because groups and set theory is the humind slightly unfocused mirror on reality, which the generator by looking directly to space and time and its properties describes better.

**METHOD**

Then there is GROUP methodology. And as it turns out any successful structural mirror of the humind on the Universe is always in question of form vs. motion – a trans-form-ation of one into the other to have a new angle… So derivatives in time and integrals in space are inverse functions; algebraic problems in time can be resolved observed their trajectories as topological paths in space, and so on. But in group theory, given its extreme abstraction, kept always in algebraic terms, this duality of methods of finding ‘spatial mappings that the mind can perceive as knowledge’ are a bit hidden and I am not aware that mathematicians understood them.

Consider THE first of those groups discovered, the Galois group which help us to solve polynomials, by *considering a group of all the possible permutations of the parameters-letters of the equation. Then by carefully studying those parameters we can find the solvability of the equation. Why we say this is another St SYMMETRY?*

Obviously because in the original polynomial, *the variables are NOT the coefficients/letters but the ‘variable’ X; and in principle what we want to find is a ‘fixed’ parameter/coefficient/letter. But Group theory does exactly the opposite. It converts the fixed single solution/parameter into the variable by establishing all its permutations.*

And it does get results then by studying carefully which of those permutations of parameters do make sense – are isomorphic to the roots-solutions of the initial X-variable.

Now apply this method to the space-time generator. If we consider time the variable and the organic space form the fixed form, we cannot really treat it as a group. Because the group has only one element, the super organism moving through the world cycle.

*But if we consider the 3 time ages/dimensions, past ≤, present ≈, future ≥ as the elements of a group, where the operation is the Generator, representation of all the Disomorphic beings of the Universe, then present becomes the neutral element, as it remains unchanged and it does not change any relative past field/limb or future head/particle in its form (they do not evolve in present) and past and future become inverse arrows.*

So goes for the ternary scales of beings, ∆±1, which then become 3 elements of another group, where the operator is a Disomorphic ∆±1 scalar organism. And so goes for the 3 topological elements that conform a whole.

So both §œts which study elements of mathematics in groups and ultimately should study societies of numbers and geometric points and functions that relate them by various methods and Group theory, with its ‘loose operations’ and symmetric neutral element and inverse S≤≥T ones, are the closest ‘structure’ all comprehensive ‘wholes’ invented by the humind as focus of the space-time symmetries of the Universe.

Hence its value despite many tantrums against their excessive pretension of becoming the ultimate equation of reality, which they re close to but not yet – the generator and perhaps something else in the future revolution of the humind or metal mind is closer.

**Humind death age: Digital thought ≈ Boolean algebra, a new beginning.**

*inverse*numbers, which in a world with single time-arrow logic will always have difficult interpretations.

**operandi**, structure and properties of its operations (10 basic properties of ±, x ÷ and so on).

**The baroque and death ages of algebra: axiomatic method, §œt theory; Boolean algebra & chips**

And yet all this is ‘past stuff’ for modern algebra, which is precisely concerned with two fields which a proper understanding of theory and praxis in the Universe will not explore, as they are the 3rd baroque age of ‘§œt theory’, defined as any Social (§) collection of T.œs, which is a superfluous metalinguistic roof as it does NOT depart from the fractal points or social numbers, the T.œs by definition of mathematics, but from its mental Kantian ‘regulative concept’ of final ‘paralogisms’ that ‘fusion all’ into the singularity mind-mirror, in perhaps its finest insight on the synoptic nature of the mind.

Thus this ‘end-point’ of the mind of algebra has obscured the empirical elements, of its main T.œs, S=t symmetries, holographic polynomial bidimensions, fractal points in sequential social numbers, and such. We thus will use the inverse ‘wording’ of T.œ§, which spells basically like set, §œT to define those collections stressing the duality of the two extremes of the 5D ∆-1 ‘ground field, of those points, numbers or elements of the set’ and the mental gathering of them all into a regulative category, the set.

Finally far worse than §œTs are computers and Boolean algebras in a biological senses, as they are substituting atrophying and like extinguishing us in the future. So they mean the death age of human minds. So since Boolean computer algebras and its evolution is eviL=anti-live, for the praxis of survival of humans should be forbidden and we shall according to our life activism against mechanisms that kill life, ignore completely the field.

**RECAP.**

Algebra as the most evolved mathematical language of the humind (ab. human mind) is in that sense very wide.

Algebra is to reality through mathematics what words is to reality through verbal thought – a full developed human language, which did not even translate easily to Digital thought (based in boolean algebra, quite a different type of mind-image of the Universe).

So while topology, space in motion and ∆nalysis, scalar studies of finitesimal derivatives and whole integrals, are much more intuitively related to the reality we observe directly algebra, has gone a bit too far into the human mind/fictions as word have done. Indeed, the whole thing of proving maths with a certain axiomatic, algebraic, set-theory, category theory, you name it, despite Godel’s proof we need experimental evidence, is a huge human ego-trip we do not share. Points in continuous space, sequential causal numbers in time arithmetic/classic algebra, ARE THE SPACE-TIME language, and analysis the ∆-languages. Modern algebra, beyond group theory is too far into the fiction world. So we shall not go so far.

Boolean is to a larger ethic view of man as the measure of all things, dangerous as it is creating a mind that easily competes and will ultimately substitute the human mind.

Yet, there are ethics in languages too, as we can observe in the posts on Stocks and money. So we are not writing much on Boolean Algebra. Only the basic fact that it is an obvious __dual language, which can therefore model all forms of the Universe, as well as a 10-decametric language of numbers, which are at the basis of human algebras.__

Thus we shall escape almost all of the axiomatic, set theory and Boolean Algebras, because ethics of human survival and our limited time does not allow us to dwell in bull$hit.

**The inverse method to 3rd age of maths: less is more.**

Finally to notice that as all languages are inflationary, the expansion of new theorems in the baroque age of maths, is less interesting than to study the limits to the Universe departing from the limits set by the simpler states of mathematics, *prior to its inflationary 3rd age*, and then see how they transform from canonical mathematics into ∆s≈t, allowing us to infer properties of the 5D Universe, *as viewed in the synoptic equations of maths*.

An example will suffice from number theory, as we explain in our demonstration, ‘in the space of a margin’ (: of Fermat’s Grand theorem, which merely means there are no more than 2 superposed holographic dimensions *in each plane of space-time, hence we cannot find x³+y³=z³, as it will mean there is actually a fourth dimension of spacetime in a single plane, achieved by superposition of the other space-time dimensions, which we cannot do.*

Let us then enter into the pre-analytical age understanding the whys of the first age of algebra, with its lineal equations and simple operandi, in a single social plane (we shall not be so much historical as methodical, meaning this age is the §-age of a single plane of space-time and all its S<st>t relationships, the second age is ∆±i planes and ∫∂ and the third, the age of ‘blocks of time’=groups unfocused reflection of the generator.

## Age of numbers: arithmetics.

Because we treat arithmetics in number theory we escape this first age of maths.

## III Age of algebra.

In the third age of Algebra, all the operations were resumed in the concept of a group, which has any operandi, two inverse elements and an identity, neutral event.

*So the essence of groups is this: *

*THERE IS A NEUTRAL ELEMENT, THE T.œ, the first being, the fractal point which can go two inverse directions, the inverse elements; AND IT CAN GO those 2 inverse directions through a restricted number of operations, which ARE mirrors of the TERNARY ELEMENTS OF REALITY, ITS SCALES, TOPOLOGIES AND TIME MOTIONS. That’s all folks. **Existence in a nutshell.*

So all this is what Algebra specially for simpler systems, which as we have seen express those arrows of dimensional form and function through simple geometrical forms and numerical structures of identical particles, social numbers and wholes, IS expressed by the different operations that relate T.œs or rather to keep the correspondence Principle §œTS, social ensembles of Organisms of Time§paœ…

**Sœts.**

Some words in our effort to use as much as possible classic mathematics are slightly changed, often used the dual œ symbol. So Steps and stops are merged into steps, Sets become the inverse of Tœs, Sœts, and so on. We cannot HOWEVER if we want to rescue mathematics of a pure formal axiomatic metalanguage without connection with the immediate reality of points with numerical parts, moving through chains of operands that transform them into = different states of the same being, go back to modern axiomatic methods and Set theory as the way to ‘explain mathematics’.

This I am aware will limit 5D Math but Sœt theory as the basis of math is NOT making math an experimental science, mirror of the Universe, but yet in another paradox of the ego trying to make the Universe an image projected by math. So while we define a Sœt as a collection of Tœs, and can use some elements of Sœt theory, we scrap completely the jargon of modern mathematics and return to the preset age – also to notice that the whole issue of infinities and its paradoxes is bogus as in the Universe infinities are obviously relative, to compare them in abstract is absurd, as all infinities end with the limit of perception of the frame of reference and the fact scales ‘transform’ realities (even if math can extract much information from a plane to another plane through a calculus operator).

A SœT – the slightly changed name for a set of Tœs – is *any kind of indistinguishable entities=numbers=points=∑œ connected by one of the possible ‘a(nti)symmetric’ relationships of ilogic, existential algebra, defined by the inverse operands of the 5 Dimotions of the Universe, (1D: sine/cosine), (2D ±), ≤≥, (5D: x/÷), (3D: √, log xª), all D¡: ∫∂…*

As such operands play a fundamental role on Analysis, which *as all physical equations relate complex processes of trans-formations of Sœts, through different dimotions of space-time.*

We arrive thus to the fundamental elements of algebra, its operands, and how they reflect the Dimotions of the Universe.

Let us then start our inquire of Algebra, no longer through its ages but through its structural elements with them.

**OPERATIONS AS MIRRORS OF 5D¡MOTIONS.**

The growth and discovery of new operandi IS exactly what it looks: the expansion of our perception of the dimensions of the Universe departing from the humind mirror, the neutral element that can go up and down, left and right, grow or diminish, D=evolve socially, change its point of view or slide in its existence from the body to the mind or limb state, and finally travel through the planes of the fifth dimension emerging in a larger plane or devolving down to its parts in the moment of death.

The intelligent reader will have notice easily that we do have just the exact number of classic operands to make them coincide with the Dimotions of time space.

So it would be easier to use them for that aim, but all mirror languages are relatively unfocused images of the generator equation and its operands, so the correspondence is NOT so easy – trust me, I first tried that avenue of thought.

It follows that more important than ‘variables’ are to algebra ‘operands’, whose encoded meaning and ‘magic’ way of relating systems to get a ‘future or present’ outcome by merging them according to certain rules of ‘creative engagement’, truly gives the power to algebra to mirror the a(nti)symmetries of the Universe.

The key connector of T.Œ with classic science is the full understanding of the dual algebra operands, sine/cosine ±, x/, ∂∫, √xª as part of the classic logic game.

Immediately we observe that:

1D: the trigonometric functions ARE the angles of perception.

2D: The sum is the operand of locomotion that ads frequencies of steps and simple social evolution in herds.

3D: The product of re=production that merges to variables.

4D: The negative exponential power of entropy and decay

And the negative functions of each the inverse entropic destruction.

5D: Calculus of social evolution of parts into integral wholes.

Further on we can group them in two great families of functions:

- 2, 3, 4D: Polynomials are simple functions of social reproductionmand herding in a single plane (with a single parameter): ¡-1<polynomial<¡+1.
- 1, 4, 5D: Complex scalar Dimotions group complex scalar Dimotions’ operandi: 1D trigonometric angular perception (¡-3>¡o), 4D exponential entropy (¡<¡-2) and infinitesimal lineal derivatives (¡<¡-1), which are the natural Dimotions of scalar space-time.

But FUNCTIONS do not correspond directly with single Ðimotions but rather with combinations of Ðimotions, hence closer to the rules of ‘a(nti)symmetry’ in ¡logic and the related non-Euclidean fourth postulate of congruence in non-E maths; and oppositions of Ðimotions, such as the inverse ∫∂, related to the inverse arrows of entropy – dissolution of wholes into parts, and 5th dimension of social evolution – integration of parts into wholes.

Entropy, ¬, in mathematical systems are the inverse operations that eliminate the information of a system. As it happens entropy can then take the general format of the negative operand of the systems. So for each positive system there is a negative one. And among all the operand, there is one which is the most entropic of them all, the exponential, notably eˆ-x, whose massive negative growth signifies the growing dissolution of a form into its finitesimal parts. As systems are in general decametric, such exponential entropy also affect the very same number, which looses its ‘meaningful series form’ after 10 decimals.

**CLASSIC OPERATIONS**

Let us first remember the dimotions=actions of existence of any system of nature:

A-celerations, lineal motions, entropic motions, energy flows, informative vortices and Social evolutions (a,e,i,o,u).

Huminds measure externally actions of space-time, in any of its 5 subspecies classified by its complexity; so the most complex reproductive actions will have the more complex operandi – powers and calculus.

Mathematically it is quite relevant to know *which scale is used to perform an action, to consider how ‘finitesimally’ small is the being, which the whole absorbs to perform *a minimal action in human time quanta – similar for the 3 synchronous t-st-s parts; i.e. in humans an eye glimpse of mind-perception, a limb-ate of motion and a heart-beat of the body – and how much it absorbs in the time quanta.

To notice that as humans for all effects measure reality with the mechanical chronos of 1 second, often very unrelated finitesimal actions are equalised with our human quanta, while the quanta of space the system absorbs are clearly differentiated in mathematical physics (h, k, c, G, Q constants of nature).

It is thus important for all systems today only described quantitatively in abstract terms, to *vitalise and explain the organic whys of its space-time events by introducing the a,e,i,o,u type of actions it performs.*

The connection on qualitative terms though is self-evident, for all scales, as most actions of any being *are extractions of motion, energy and form from lower ∆-i scales.*

So we and all other beings perceive from ∆-3 quanta (light in our case), feed on amino acids, (∆-2 quanta for any ∆º system), seed with seminal ∆-1 cellular quanta (electrons also, with ∆-1 photon quanta).

So derivatives are the essential quantitative action for the workings of any Tœ, space-time organism.

And so we study in depth the connection of the a,e,i,o,u actions between Planes (qualitative understanding) and its mathematical, analytic development (quantitative understanding of 1st second and 3rd derivatives – the late extracting ‘1D motion’ from the final invisible gravitational and light space-time scales).

SO THE FUNDAMENTAL LAW OF OPERATIONS TO VITALIZE THEM IS THIS:

‘BY THE RASHOMON EFFECT ALL differential OPERATIONS CAN BECOME AN ACTION IN ONE OF THE 5D DIMENSIONAL VOWELS (A,E,I,O,U) THAT DEFINE THE FIVE dimensions OF EXISTENCE, AS VITAL QUANTA-ACTIONS OF THE BEING.

THIS IS THE LOGIC CONCEPT THAT TRULY VITALIZES THE OPERANDI OF ALGEBRA.

Derivatives allow us to integrate, a* sum of the minimal quanta in space or actions in time of any being in existence, namely the fact that its sums tend to favor growth of information on the being and then signal the 3 st-ages and/or st-ates of the being through its world cycle of existence, which in its simplest physical equations is the origin of… ITS space-time beats.*

Actions in timespace are the main finitesimal part of reality, its quantity of time or space if we consider tridimensional actions as combinations of S and T states, stt, tst, tss, sss and so on…

So how differential equations show us the different actions of the Universe?

To fully grasp that essential connection between ∆st and mathematical mirrors, we must first understand how *species on one hand, and equations on the other, probe in the scales of reality to obtain its quanta of space-time converted either in motion steps or information pixels, to build up reality. *

So for each action of space-time we shall find a whole, ∆ø, which will enter in contact with another world, ∆±i, from where it will extract finitesimals of space or time, energy or information, entropy or motion, and this will be the finitesimal ∂ ƒ(x), which will be absorbed and used by the species to obtain a certain action, å.

So the correspondence to establish is between the final result, the åction, and the finitesimal quantas, the system has absorbed to perform the action, ∫∂x, such as: å= ∫ ∂x, whereas x is a quanta of time or space used by ∆ø, through the action, å to perform an event of acceleration, e-nergy feeding, information, offspring reproduction or universal social evolution.

It is then when we can establish how operations are performed to achieve each type of actions.

FOLLOWING this inversion of complexity of actions vs. simplicity of operations, comes the next more complex social action of reproduction, which most often is expressed for quantitative simple physical systems through the second simpler operation of re=product-ion.

We adscribed each operand to a single dimotion, but they are ‘once more’ entangled operations, which besides its preferential Dimotion, do participate of all the others – remember LANGUAGES AS MIRRORS OF REALITY have also the same entangled properties of the pentalogic, ¬∆@ST universe, looking at all its elements. So we shall now analyze them in more depth.

We establish first such direct relationships – taking into account that for each operand we must distinguish also the dualities of ‘space-like volumes’ and ‘time-like motions’ and the inverted operations; in ceteris paribus analysis, but then we must study how they merge and entangle.

For example, the ∫∂ operator that crosses planes of existence, as generally speaking, a *first derivative in space or time defines those dimotions only as S or T, while double derivatives often work for both together.*

The same happens with the product/division responsible for most Merged Dimotions ‘re=producing’ through x÷ operand a lower scale of ‘connected axons’ as the product is the first operand to probe a lower plane of existence, AND THE ESSENTIAL OPERAND to merge S and T states, from gender couples to stop and step into a NEW merged creative parameter, a stœp, (i.e. mass a stop state and v a moving step merge into momentum a stœp).

**1D: < (angle) & sine / cosine:** The first Dimotion of perception is served not by numbers but by angles of perception, expressed externally by a sine, which acts as the informative, height parameter of the outer Universe, and the aperture of perception of the being, maximal with maximal ‘height’, the dimension of information, (sin 90º=1). This poises a question: can a system perceive more than 90º? Obviously not much, without tricking the eye, so systems have two eyes to have more or less a 180º aperture to the world. It has to be noticed on a first sight that again ‘inversions’ in the eternal Universe do NOT fully cancel each other, but bring something else. I.e. there is a point at 45º of balance in which sine and cosine are equal; and so it is the point of minimal cancellation of inverted functions, reason why without any further argument it will be the point in which there is maximal reach of an stœp process (Galileo’s first discovery with cannonballs). This we can express it saying that the sine is the ‘stop’ perceptive state function and the cosine the step-motion, lineal one.

* ±: The 2 Ð l*ocomotion is best served by the + operands, as we have shown in the analysis of a time tail (motion-memory of a distance), which is a sum of ‘steps’ that also can be calculated with integrals. The sum is also the key operand for the ∆§cales of social groups, in decametric form, which also is served by the logarithm. And so on… The negative operand however is profound in many ways scientists do not understand. Reason why so many errors, from the negation of the faster than light speed, to the confusion of particles and antiparticles arise. The INVERSE operandi are in general misunderstood because as we have said so often unlike the paradoxical ¡logic Universe, the humind is @ristotelian, single arrow A->B So the B->A is quite missed; but generally speaking served by the negative function. I.e, a negative spin just have the inverse orientation, a negative coordinates just means to move in the other direction. Negative operandi thus are MOST useful for time=motion related systems.

In terms of the duality of space-stop and time-step, the negative function is the informative stop state and so often a – symbol does NOT mean moving backwards, but halting motion (and perhaps releasing the ∆-1 micro-points as a flow or current).

In this is important then to understand the existence of one-way dimotions vs. 2 way dimotions, where operands make sense. Because huminds do not properly distinguish both, they get confused when trying to consider a negative operand for spatial forms (what is a negative apple? nonsense) while there are always negative ‘directions’ for temporal motions (what is left and right motion?). Clearly all understand.

**3, 4, 5Ð social evolution and complementary merging** by the product and division operands, as merging often requires first a product at the lower axon level and then a mitosis or ‘division’ into 2 wholes with all the new parts, which again gives division a very precise meaning.

In that regard the less important form of multiplication is the most often used to define it in classic math (the sum of sums) as this not implies merging into something else but merely stay with the same clone species. In this ‘line of thought’ we do have also the x² polynomial functions and its inverse √x, fearless interesting than…

**3, 4D: The Aª and log: **We already shown that the 4th and fifth dimotions are easily represented in its d=evolutions by the log, exponential operands.

**5D: √xª:** social scales: When social evolution is not transformative between planes but only a social herd, it emerges through multiple mostly decametric, 3×3+0 scales or the √xª operands best suited to that purpose. And here again we find quite difficult the comprehension among huminds who love to ‘go only the way upwards’ so to speak of the √ operands, specially when in negative mood: √-x, a completely mysterious element of ‘mathematics’ to the point they call them imaginary numbers (:

**∆nalysis:** Yet the 4th dimotion of entropic devolution,and the complex integrals of informative perception and social evolution are also studied by the ∫∂ integral and derivative operands, which work for all Dimotions and make them transcend.

Why those operands do NOT have a one to one correspondence to each of the Dimotions of space-time is obvious: each of them have as everything in reality a pentalogic multiple purpose, as we shall not cease to repeat, the basic feature of reality is to be an entangled game of 5 elements which are themselves ‘fractal’ in its nature, that is, each of them will be able to perform the other 4 tasks to become in itself a ‘whole’ being made to the image and likeness of the total fractal, mind of the Universe and its ilogic structure. So while certain operands are clearly more useful for certain dimotions, all of them can be used to a certain degree of accuracy to ‘reflect’ a mirror image of the 5 dimotions of existence – themes those to be studied in depth in the posts on algebra and ilogic.

It however becomes evident from the beginning that we ascribe the more complex dimotions, which do enclose in their actions the other 4, to the operands of analysis, specially those who ‘transcend’ and ’emerge’ between scales, as they are processes NOT of lineal nature, best served by them.

While other operandi have multiple correspondences such as the product, which corresponds both to an aggregation of herds, and to a reproduction into the lower ∆-1 planes when two similar elements entangle at the ∆-1 level as most reproductive processes do (i.e. you reproduce by entangling your lower scale of genes). Since ∆o (S x T), where S is a set of s¡-1 elements and T a set of t¡-1 elements, equals the number of ‘bijections’ between s¡-1 and t¡-1 (we shall try to minimize though the use of the modern jargon of mathematics as a philosophy of stience should do to be understandable by all disciplines).

Still it is immediate some easy correspondences of those operandi with the dimensional elements ∆st, taking into account that general rule are:

- The sum-subtraction are the inverse arrows of the simplest superpositions of dimensions
*between species which are identical in motion and form. So equal Dimotions can be added or subtracted.* - The product/division rises the complexity of operandi a first layer, and serves the purpose, besides the obvious sum of sums, of calculating the margin of dimensions, as combinations which are not purely parallel between clone beings, most likely through the recombination of its ∆-1 elements, as the product of 2 Sœts inner elements give us all possible combinations. Ie. 5 x 4 = 20 IS also the number of connections between all the 5 elements and 4 elements of both sets. So multiplication ads either a dimension of multiple sums in the same plane, or probes for the first time in an inner scalar dimension.

**The rashomon truths: 3±i Asymmetries of operands.**

*different uses, in terms of space, time, symmetry, scale and mind distorted pov. And so it will widen its applications but will require also an in-depth philosophical analysis of each of them.*

**Operands, polynomial limits and group transformations.**

All this say it is obvious that the essential elements of *algebra are its operand that reflect fundamentally two type of 5D² symmetries in its equations:*

**Γ: $t≤≥ST≤≥****ð****§;** whereas the *operands of i-logic < ≈ > are translated into simplified ‘identical’ (not self-similar) operandi, =, or < > operands, in spatial terms (hence simplifying their scalar and informative or temporal simultaneous meaning).*

**∆±i Operands:** Which must be inverse operands, as *only the 4D vs. 5D entropic and informative arrows are truly inverse in its Nature; that is, forming a perfect zero sum. So they can be obtained as inverted operandi, either:*

*In ‘continuous’ social growth of scales within an ∆-plane: ±; ÷ X; xª log x.*- Or in
*discontinuous*scales through integral processes (which differ slightly fro the logarithimic/polynomial scale, hence the possibility to approximate polynomials with derivatives of different order, as per Newton, Taylor etc.)

Alas we have resumed in a footnote the correspondence between Algebra and reality (: the rest are details…

It is then obvious that algebra will give us the most thorough understanding of the processes of the 5D Universe as a synoptic mirror, and there is no magic to it, but the lack of a proper philosophy of science in its practitioners.

WE shall then call this enlightened algebra plus whys, ¬Ælgebra, or ‘Existential Algebra’.

Algebra of Existence is mostly about equations that represent motions on the dimensions of space-time, which ALWAYS HAPPENS AS A STOP AND GO PROCESS, HENCE AN S OPERANDI TIME MOTION AND TRANS-FORM-ATION IN A BALANCED WAY OF TWO DIFFERENT ST-POLES.

as such algebra deals with S operandi T operandi S operandi T Cyclical ST@tes; where the @ monad author of the @tions of existences of the system do take place in sequential discrete manner…

The key to it all as algebra deals with the S**<≈>**T SYMMETRIES ARE THE **<≈>** logic operandi OF ITS EQUATIONS and its S=t flows and transformations embedded on them.

Mathematics derives from logic, as the operators of algebra are always logic statements about the relationships between the variables and parameters of the equation. So we must relate properly what mathematical equations mean, by comparing them to the Generator Equation written in ¬Æ symbols.

So a part of ¬Æ of the greatest importance is to fully understand from the perspective of Non-E mathematics, where points have parts and so they are never equal but similar.

Thus the true meaning of the key logic symbols of mathematics, such as =, must be < =>, a transformation; or ≈, a similarity or a communication, (as in E=Mc2, where Mass is NOT energy but the transformation of lineal energetic space into cyclical time vortices).

The confusion of truths caused by the misunderstanding of the ‘dynamic’ presents represented by those symbols is not trivial. Many errors of sciences derive because the ‘evident, fixed mind of the scholar’ just says ‘equal’ and thinks really mass is equal to energy because a transformation converts a lineal motion (kinetic energy) into an accelerated vortex of gravitational time (a mass), which are as different as a living and dead corpse.

The highest homology with reality: Semantics, Syntax and Growing Sentences of mathematical operandi.

Now for understanding algebra, it is useful to consider the fact that numbers are points and there is a direct relationship between points and numbers, lines and variables, planes and squares and so on. So polynomials and operandi represent ‘social evolution’ of dimensions, points into lines into planes into 5dimensional structures.

Departing from those elements of the Universe closely imitated by Sp (Points)<=>Ti (informative Numbers), its sequential processes of growth into new planes of the 5th dimension ARE the next element of mathematics:

– Defined in the spatial symmetry of points by its 5 geometric Postulates (points->lines->parallels/perpendiculars->planes->points with parallels), which we upgrade to Non-Euclidean postulates (Fractal points->waves->Similarity->Networks->Minds that knot planes into organisms).

– And Defined in the temporal symmetry through operations of algebra (sum/rest->multiplication/division->potentiation/logarithm->integration/differentiation), which we shall clarify in its awesome capacity to study ALL the actions of space-time of those knots of cycles.

So this ‘reclassification of mathematics’, will divide the subject basically in a duality of perception proper of most mental languages: spatial still, continuous perception (geometry) vs. Informative discrete perception (numbers).

And then classify all operations and geometric postulates as the necessary elements to grow those dual elements into ‘larger’ groups, through scales of the 5th dimension (positive arrow of future) and its inverse operations that dissolve, divide, disintegrate them (negative arrow of future).

In this manner mathematics is able to explain better than any other known language (except in questions referential to human existence for which our natural language, words works better – which should therefore rule through ethics, politics, economics and the creation of our social super organisms,).

Now this should be clarified first before trying to streamline mathematics as it is today understood, due to the fact that without knowing ‘WHAT IS ALL ABOUT’, man is loosing the necessary relationship of all classic languages:

Syntax & semantics of ‘sentences’ (language reductionism of reality to fit the brain, which normally ‘reduces’ motion’ and simplifies layered dimensions and limits perception to relevant cycles) -> More complex reality of ∞ space-time cycles with motion.

So the semantics are points-numbers, the syntax are operandi, and the sentences are planes/equations, which grow in complexity as the ‘chains’ of those equations and planes grow through ever more complex, integrative, ‘operandi’ (sum/rest->multiplication/division->potency/logarithm/integration/differentiation).

But then we realize that is all. Yes, THAT IS ALL. Why? Because as we shall stress once and again, the Universe is infinite yet, its perception by any MIND is not – it is a ‘finite game of finitesimals’ as opposed to the infinity of the whole, meaning that information is lost beyond the ternary limits of at least ‘the human mind’ (might there be minds who see across 9 scales, etc.)

So we DO have just the need to reflect ‘ternary games, scales, elements’ wrapped up then by a ‘whole temporal cycle of maximal motion, integrative activity’ which becomes a finitesimal of a new finite game (hidden its inner parts within the finitesimal point). And so we do START afresh a new game.

So there are no need for more than 3 ‘scales’ of ‘formal operandi’ (±; x/; lnª) which then are ‘warped’ by the dynamic ∫∂ operations.

Now there is a field which is truly new, that of the causal logic of 3 arrows of time, where the most important equation is:

Space (past) x Time (future) = Present space time,…

and through those new symbols of equality, S<=>Tƒ, we define systems as dual, dialectic, where thesis and synthesis fusion together. But this would not be analyzed here as it is more proper to study it on the general laws of the first line and the isomorphisms of the third line.

Why we still need to explain classic algebra and analysis. The ego abstract paradox

Since all sciences are studied from a mathematical and logic perspective, we shall study the ternary logic on those analysis.

Even for those sciences that ‘anthropomorphic’ man, under the ego paradox see too simplex.

Fact is the ego paradox – all finitesimals see themselves as the center of the world they perceive from their point of view – does not give an organic perspective to other 5D, Œ-points. BUT in as much as all points do follow mathematical laws, the explanation of maths as an upgraded ORGANIC science (5D analysis), with complex, vital causal, times sequences and space-time symmetries, across several of those planes (algebraic functions) give us a true sense of the vital Universe in formal terms, connecting properly T.Œ with the present formalism of most sciences.

Fortunately unlike lineal physics, mathematics is ‘well-done’ and already closely related to 5D ‘analysis’ of ‘fractal, topological’ space and cyclical, sequential time.

Thus we do not need to make deep changes into the discipline but merely update some parts (notably the concept of time, and the dialectic logic of operandi) explain others (the meaning of those operandi), and limit its ‘inflationary information’ (concepts such as infinite quantities, new abstractions that substitute the real elements of mathematics, the number, the point and its social geometries, such as set theory).

To that aim though we need to introduce a simple formalism of the main symmetries of the 5th dimension.

Thus algebra is the time, discrete-number related to spatial points part of maths, and as such is a unifying thread of almost all of mathematics, which has also gone through 3 ages of growing completeness:

We thus have now put ¬Ælgebra as the central discipline of maths, in relationship with its ages, other math disciplines, math physics and reality.

We shall go through the development of those central elements of classic algebra in this post studying its main postulates along their illumination of some of the fundamental laws of the ∆@S=T Universal structures, where algebra, the most comprehensive ‘linguistic, human-mind related language of the whole, extends its tentacles better than geometry>topology, basically a spatial endeavour, and analysis, basically an ∆-scalar endeavour.

*So a better definition of ¬Ælgebra is to be the central discipline of maths, where all the subjects of reality can be treated.*

But to that aim we need to understand better ‘social sequential numbers: N§º’; single aristotelian logic made into ternary ‘i-logic’; imaginary coordinates which express whole ‘worldcycles of holographic bidimensional numbers’ and so on…

The study of __the different operands of algebra in relationship with the 5 Dimotions of existence they mirror, each of them in itself a pentalogic operand, which can entangle with other operands and dimotions, in increasing degrees of complexity, will therefore bring a vital outlook to the abstract operations of algebra, starting from the less understood of them all, the sinusoidal functions of mind’s perception…__

**1D: ****@ngles**

**Trigonometric pentalogic**

Let us then start with the trigonometric science of perception, which allows a pi cycle to observe through its natural ‘membranes’ – its 3 diameters that become the cyclical membrain that isolates it from reality, the ‘angles’ that give us information by parallax on the 3rd dimension of relative size and distance of systems outside the being.

The simplest ‘monologic view’ of trigonometric functions is therefore, the fixed perception of an angle that ‘measures’ the height-information and length-distance between the being and an outer T.œ. The sine therefore provides the dimension of form, and the cosine the larger dimension of distance, energy and motion…

The simplest level of the trigonometry of perception is the Pi number: *pi does exist as a perfect form of space, from 3 (the hexagon) to pi (the circle) but it does not as a number (irrational) since time is dynamic, not static, so constantly moves between ±π, allowing the open-entropy/closed-information duality of membranes that enclose systems:*

its dominant use and first reason it became the first developed field of mathematics is its capacity to measure from a point of view distances according to ratios and parallax, which is the origin of tridimensional perception (bilateral eyes), and Fertile Crescent mathematics.

How this work in its simplest form, needs to understand how a ‘spherical, ideal mind-membrain of 3 π diameters, and 0.14 D apertures, allows a mind to perceive through them, ‘rays’ to distant objects. The mind thus can always measure the angle covered by a distant object, and with a minimal displacement, a new angle.

3rd-century astronomers first noted that the lengths of the sides of a right-angle triangle and the angles between those sides have fixed relationships: that is, if at least the length of one side and the value of one angle is known, then all other angles and lengths can be determined algorithmically. These calculations soon came to be defined as the trigonometric functions.

So trigonometric functions, the first to appear, as 1D perception is also the first ‘action’ are operands for the first Dimotion of perception.

**1D:** Of those operands then we consider the simplest ‘operands’ as those who act directly on a point-number. We need then to find the operand of the first Dimotion, self-centered in the point, perception.

Since all points are also numbers as all have internal structure, we can operate both in S-pace with points and topologies and in scale with numbers to show those internal parts with numerical values:

The graph shows the elements of the point-number in which an operand can act. Even though most operands will act through a similarity ≤≥ symbol in two different elements. Yet the simplest operands are those which can act in a single point-number.

An entire field of Algebra or rather geometry is then the study of the laws of perception of the circle, based on the laws of trigonometry.

However in the entangled Universe, we can consider that ‘all systems’ are alive, and somehow evolve ‘adding dimensions’ as a world cycle does, developing new perspectives, and indeed the sine cosine didn’t stop in the stopping geometry of the Greeks that observes the outer world statically. The circle can be given *different motions, up and down and then it becomes an SMH system, or it can emit ‘energy’ imprinted by its formal motion and produce a wave. In this manner, the sine and cosine can perform also the*

*2nd Ð – 3rd Ð: MOTION AND COMMUNICATION: The sinusoidal functions. π.*

Yet as the second dimotion of all systems, once it perceives is motion and communication, trigonometric functions can move a fractal point creating the form of the wave, or communicate a finitesimal bit of its information through a wave, which carries the information (amplitude, frequency) of the original circle.

Pi then becomes the numerical value of the external membrane measured in terms of 3 diameters turning around with 3 apertures to ‘see’, from the self-centered singularity:

The point-number has INDEED a central point of perception, which therefore can be defined externally by an angle of aperture to the world. The membrane normally is a pi, 3 diameter number, which leaves an angle of aperture, 3’14-3/π=0,14=±4,5%, with 96% of ‘dark matter’ outside the perception of its singularity. But as the membrane of the point turns, the angle ‘sweeps’ with its 3 apertures between diameters in a relative discontinuous Universe all the world outside.

So the angle of the point will finally allow a full 100% view in 3 different ‘ages’ of the full world cycle of the membrane. So in dynamic 5D analysis we can consider not only a sine and cosine function; but the sine ‘main’ function that calculates the aperture, also as a function of time, such as for a full turn of the 3 diameters that complete the membrane (2π radians), the angle has seen the entire external world 3 times.

What this means then is that the sine of a singularity point ‘grows’ with each turn of the world cycle of the being and in time can be higher than one.

This said we shall not go into so much complexity and consider the ‘static’ concept and define the sine merely as the angle of perception of the central point and hence the function that reflects the 1st Dimotion of existence.

It must though be clear enough that the fundamental function of the perceptive singularity is the sinusoidal function, so it is in its active SHM motions, which try to maintain the being through its worldcycles in the ‘central point’ of the system, its ‘mind-singularity-zero point’:

The inverse operations of sine and cosine find as usual a point of balance, which is the 45º, where therefore the maximal ‘momentum’ can be reached (maximal distance of any throwing). 45º is therefore the ‘eye’ position of the sphere; the place where there is always a relative view.

The sine must then be considered the informative function and the cosine the energetic function or ‘stop and step’, form and motion states of the being. General rules to enlighten the meaning of mathematical physics.

It also follows immediately that angular momentum is the best external parameter to measure the ‘perceptive speed’ of a system, measured in terms of frequency. So in quantum physics, ~~h ~~ will be the measure of the first DImotion, and as it appears as a constant of all systems, we deduce that in quantum physics there is a simple metric equation for all species of the galatom, h(ð§) c($t)=K.

**The first ‘timespace’ numbers: Polygons as root of unity**

A de Moivre number, is any complex number that gives 1 when raised to some positive integer power n:

An **nth root of unity**, where n is a positive integer (i.e. *n* = 1, 2, 3, …), is a number z satisfying the equation:^{}^{}

They are complex numbers (including the number 1, and the number –1 if *n* is even, which are complex with a zero imaginary part), and in this case, the nth roots of unity are:

This formula shows that on the complex plane the nth roots of unity are at the vertices of a regular n-sided polygon inscribed in the unit circle, with one vertex at 1. This geometric fact accounts for the term “cyclotomic” in cyclotomic polynomial; it is from the Greek roots “cyclo” (circle) plus “tomos” (cut, divide).

Euler’s formula, which is valid for all real x, can be used to put the formula for the nth roots of unity into the form:

We find therefore the first timespace numbers, And as such they will become ‘the creative process’ of dividing the ‘whole’, 1, into cyclical ‘tics of time’ of increasingly faster frequency, in a progression, for *k* = 1, 2, …, *n* − 1, which will generate the frequencies of all clocks of time, till reaching the circle, which can then be considered in bidimensional spacetime, the ‘Infinite clock, of infinitesimal time tics’. Do have those infinitesimal ticks a ‘limit’ as all relative infinites do? In physics it is believed the minimal tick will be 10ˆ43 or Planck’s time, which therefore would become the limit of ‘points’ that form a time clock.

Another fundamental theme being the reasons why the ‘clock’ is counterclockwise in its direction, as it will also be its complex representation in 4D relativity theory. The reason being that in scales of the fifth dimension, as we create new dimensions from the lower planes with more entropy, the emergent dimension ‘sucks’ part of the entropy of lineal space of its lower dimensions, ‘contracting’ it as it rises on height. I.e. a pi circle is made of 3 ‘curved’ diameters (with open holes between them), but it does not MEASURE 3 but 1 in the length dimension.

It also means we are adding a new time dimension, with a negative entropic property for the ‘dimension of real space’, which therefore can be written also with the number of entropy, e

*4th scalar Dimotions of Entropy*

A theme them of profound importance and depth of meaning is the relationship between the sine and cosine functions that allow an angular perception of the whole and the exponential function that reduces the whole to its decaying elements (Euler’s formula). We could say then that the whole is ‘split’ between the entropic negative exponential part that is discharged, and the sinusoidal, informative elements that are absorbed by the mathematical mirror mapping.

It is interesting to note the connection which occurs between the exponential and trigonometric functions when we turn to the complex domain. If in (13) we replace z by iz, we get:Grouping everywhere the terms without the multiplier i and the terms with multiplier i, we have (16):

Euler’s formulas solved for cos z and sin z, get:

I*n the complex plane, 1D (sin/cos) combine to represent a full worldcycles, interesting enough through the eˆix, 4D exponential decay function.*

*This is possible precisely because he exponential function switches between growth and negative decrease, as the sine and cosine switch between informative and energetic perception; but the sine function, the informative Dimotion grows less, as it happens in nature, where height and information has less energy, and so in parameters of size and volume matters less.*

Two new qualities moreover make it more interesting to cast trigonometric functions in terms of the function of entropy: we are adding both cosine and sine ‘on and off’ SMH for a value of 1, the total value of a world cycle, so we can use frequency equations (as in electromagnetism) to represent this exponential world cycle. And we superpose both, the function of ‘space-form’ the sine and time-motion-lineal distance, the cosine, to observe a harmonic balance as the function goes up and down but never passes beyond the value of the whole.

*5D: social evolution: Fourier series.*

The Fourier series form the more complex dimotion of the sine cosine functions, as they allow by social evolution to draw any ‘function’ of information as the sum of the main harmonic.

**Expansion of functions in a trigonometic series**

On the basis of what has been said there arises the fundamental question: Which functions of period 2π/α can be represented as the sum of a trigonometric series – that is what kind ‘of social forms’ evolve from parts into sinusoid wholes?

This question was raised in the 18th century by Euler and Bernoulli in connection with Bernoulli’s study of the vibrating string. Here Bernoulli took the point of view suggested by physical considerations that a very wide class of continuous functions, including in particular all graphs drawn by hand, can be expanded in a trigonometric series.

The answer was a true intention of a genius, as it meant that all motions of a double pendulum as the ‘hand’ is, all __composite SHM motions were combinations of cyclical wave motions.__

This opinion received harsh treatment from many of Bernoulli’s contemporaries. And would be proved only after Fourier and Dirichlet showed that every continuous function of period 2π/α, which for any one period has a finite number of maxima and minima, can be expanded in a unique trigonometric Fourier series, uniformly convergent to the function. Since the essential element of a ‘sinusoidal series’, which we anticipated is related to the SMH simplest representation of a world cycle is to have a perfect balance of minimal and maximal, which represent the ‘turning points’ or change of ‘ages’ of the world cycle, which can then be decomposed in smaller ones.

Figure illustrates a function satisfying Dirichlet’s conditions. Its graph is continuous and periodic, with period 2π, and has one maximum and one minimum in the period 0≤x≤2π, hence it is a sum of cyclical ‘SMH’ worldcycles expressed as a composite of 3 Dimotions, the up and down motion of the point, or amplitude, the perpendicular field of ∆-1 entropy which the point shapes in its form to carry the wave and its temporal frequency in the vibration, that forms a ‘ternary’ structure of reality, which can be ‘socially added’ if multiple similar vibrations take place upwards, (5D social evolution) or downwards (dissolution of the total wave into its parts) to form a complex 5Dimotional system, which is the highest ‘state’ of a sinusoidal ‘mathematical mirror of the pentalogic Universe – and hence the ‘proper limit’ of study of the Trigonometric function in non-Ælgebra:**Fourier coefficients. The calculus of 5Dimotions.**

At this stage, though it belongs to Analysis, a question is poised: it is the series analytic? hence Differentiable without limit? And what does mean the possible constant differentiability of the trigonometric functions of ‘perception’, in terms of metaphysics?

Let us discuss the case which belongs to more complex 5D ¬Algebra.

The maths of it, then are well known: If we consider functions of period 2π, which simplify the formulas any continuous function f(x) of period 2π satisfying Dirichlet’s condition may be expanded into a trigonometric series, symmetric in its ‘height and length’ cos/sin functions, sum of multiple simpler waves, which is uniformly convergent:

We pose the problem: to compute the coefficients ak and bk of the series for a given function f(x).

To this end we note the following equation:

These integrals are easy to compute by reducing the products of the various trigonometric functions to their sums and differences and their squares to expressions containing the corresponding trigonometric functions of double the angle.

The first equation states that the integral, over a period of the function, of the product of two different functions from the sequence 1, cos x, sin x, cos 2x, sin 2x, ··· is equal to zero (the so-called orthogonality property of the trigonometric functions). On the other hand, the integral of the square of each of the functions of this sequence is equal to π. The first function, identically equal to one, forms an exception, since the integral of its square over the period is equal to 2π. It is this fact which makes it convenient to write the first term of the series (22) in the form a0/2.

Now we can easily solve our problem. To compute the coefficient am, we multiply the left side and each term on the right side of the series (22) by cos mx and integrate term by term over a period 2π, as is permissible since the series obtained after multiplication by cos mx is uniformly convergent. By (23) all integrals on the right side, with the exception of the integral corresponding to cos mx, will be zero, so that obviously:

Similarly, multiplying the left and right sides of (22) by sin mx and integrating over the period, we get an expression for the coefficients:

and we have solved our problem. The numbers am and bm computed by formulas (24) and (25) are called the Fourier coeficients of the function f(x).

Let us take an example the function f(x) of period 2π illustrated in figure 13. Obviously this function is continuous and satisfies Dirichlet’s condition, so that its Fourier series converges uniformly to it.

It is easy to see that this function also satisfies the condition f(—x) = —f(x). The same condition also clearly holds for the function F1(x) = f(x) cos mx, which means that the graph of F1(x) is symmetric with respect to the origin. From geometric arguments it is clear that:

so that am = 0 (m = 0, 1, 2, ···). Further, it is not difficult to see that the functions F2(x) = f(x) sin mx has a graph which is symmetric with respect to the axis Oy so that:

But for even m this graph is symmetric with respect to the center π/2 of the segment [0, π], so that bm = 0 for even m. For odd m = 2l + 1 (l = 0, 1, 2, ···) the graph of F2(x) is symmetric with respect to the straight line x = π/2, so that: But, as can be seen from the sketch, on the segment [0, π/2] we have simply f(x) = x, so that by integration by parts, we get:

Thus we have found the expansion of our function in a Fourier series.

Convergence of the Fourier partial sums to the generating function. In applications it is customary to take as an approximation to the function f(x) of period 2π the sum:

of the first n terms of its Fourier series, and then there arises the question of the error of the approximation. If the function f(x) of period 2π has a derivative f(r)(x) of order r which for all x satisfies the inequality:

Then the error of the approximation may be estimated as follows:

Where cr is a constant depending only on r. We see that the error converges to zero with increasing n, the convergence being the more rapid the more derivatives the function has.

For a function which is analytic on the whole real axis there is an even better estimate, as follows:

Where c and q are positive constants depending on f and q < 1. It is remarkable that the converse is also true, namely that if the inequality (26) holds for a given function, then the function is necessarily analytic. This fact, which was discovered at the beginning of the present century, in a certain sense reconciles the controversy between D. Bernoulli and his contemporaries. We can now state: If a function is expandable in a Fourier series which converges to it, this fact in itself is far from implying that the function is analytic; however, it will be analytic, if its deviation from the sum of the first n terms of the Fourier series decreases more rapidly than the terms of some decreasing geometric progression.

A comparison of the estimates of the approximations provided by the Fourier sums with the corresponding estimates for the best approximations of the same functions by trigonometric polynomials shows that for smooth functions the Fourier sums give very good approximations, which are in fact, close to the best approximations. But for nonsmooth continuous functions the situation is worse: Among these, for example, occur some functions whose Fourier series diverges on the set of all rational points.

**2D: ±**

We have seen how trigonometric functions can describe in growing layers of complexity by combining with other Dimotions as vital beings do the more complex dimotions of communication, reproduction and social evolution.

So the next question is can the sum and its inverse the negative numbers, such a seemingly simple functions do the same pentalogic?

The answer is yes, but again as in the case of the ‘angle’ of perception, things must become more complex through the combination of the different operandi, while the sum/negative number remain ‘dominant’.

**5Ð:** Social evolution into herds, the 5th Dimotion is self-evident the dominant element of the sum.

**2D:** Easy to explain is the Dimotion of locomotion as a sum of steps measured by the frequency of those steps in sequential time (though better expressed by multiplication

**3Ð:** Yet then the sum can ‘progress’ in ‘accelerated’ growth, and that will express the 3rd Dimotion of reproduction.

**4Ð:** While negative numbers will express the 4th Dimotion of entropy

**1Ð:** While the Ðimotion of cyclical perception *is expressed by the negative ¡ number, with its cyclical rules of summation, which implies the sum of the angle of perception of the self-centered number in its argument. *

Thus next then in the entangled representation of reality through those ‘basic operand’ comes then the duality of addition and subtraction, and its attached physical meanings of superposition and fusion of ‘parts’ into ‘whole numbers’, or its entropic inverse operandi of negative substraction.

It starts to be then obvious that all operands have its inverse function to maintain the balance of the Universe.

Addition by superposition in ever tighter spaces of similar clone species, is the simple algebraic expression of the social dimotions, both in its positive 5Ð and negative entropic 4Ð whose addition of decaying ¡-1 T.œs is so fast that it can be expressed as a negative exponential growth, which in this manner would complete the 3 ‘scales’ of addition: +, x, xª…

Moreover addition can happen in sequential time or adjacent space, forming growing probabilities or populations. So as the simplest mode of operands extends its diversification through space or time it will mean different things. If we consider the happening of an event or full world cycle 1, probabilities will represent parts of the whole event. If we project it into space it will be a population of similar event, entering the region of maximal frequency. Both will be mathematically projected as a bell curve. S=T. Same function for the addition of events and populations, in time or space.

The first marvel of the Universe is the simplicity of its original principles, made complex by the differentiation across the symmetries of scale, topology or time. Indeed, something so simple as the sum and inverse subtraction IS still the most important operandi of the Universe, which gives us new numbers, social gatherings of identical beings, which herd together into parallel flows adopting most likely a bidimensional ST superposition on laminar states that keep adding the 3rd dimension of the being. Like the simplest first masterpieces of Bach, the architectonical Universe is a simple principle before organicism twists its form, in which beings which are equal come together.

Superposition of bidimensional holographic fields is so important that the whole of quantum physics is based in this superposition principle. The sum thus is still the master of operandi. But for sums to happen, the beings *must be externally identical, to be perceived as parts of a quantified mass, each of them the same value. Addition thus is the ultimate proof of the social nature of the Universe. *

THE SUM AS SOCIAL ACTION: SUMS OF HERDS.

It is then once we have defined an operation and its properties, time to study how they mediate the actions of beings. And it is clear that the first operation sum, acts on the first form, social numbers, to form growing §ŒTs of social numbers…

So paradoxically the more complex action, starts with the simple operation, the sum of individuals into herds of formal numbers.

**The problem of the i-number: negatives and roots. Proper and improper inversions.**

As a general rule, *negative numbers exist when they are a direction of motion, not a quantity or volume of space.*

So happens with roots, which only should exist for symmetric holographic systems of space-time S=T, for bidimensional entities and regular numbers as forms.

And alternately they Do happen in many cases in which mathematical physicists discharge them as inverse arrows of time because they ignore the inverse 4D vs. 5D arrows, as in Einstein’s equations, where they customary discharge negative hyperluminal solutions that do happen in the larger ∆±4 dark world of quantum potentials (entanglement, pilot-wave theory, etc) and intergalactic space (faster than light neutrino background, gravitational waves, action at distance, red shift of light, etc.)

We study this themes in number theory where we define the different types of numbers, so no need for further info here. Instead, we will make some comments on… the limits of classic Aristotelian logic based precisely in a single arrow of time, which has so much influenced as Euclid pretentious axiomatic method (full of holes and new axioms, postulates, notions and various errors of the §@-humind), our ‘underlying’ a priori categories of the mind.

IT is then clear that at the basic level of arithmetics the problems that still drags mathematics and by extension all other sciences, which use its mirror, is the concepts of negative and square numbers, the inversions of the positive and quartic equations, on the real number, related to the two fundamental unknowns of humind science – the fractal scalar ‘infinitesimal nature of space’, and the ‘proper and improper inversion’ of the fractal generator.

In essence all this means that negative numbers and roots do NOT exist for certain type of space-time events, and povs of the ∆s≈t structure of reality but are inflationary ‘informative’ excrecencies proper of all languages that do have an inflationary excess of information, as they are ISOLATED SPACE-INFORMATIVE systems OF SOFTWARE, WITHOUT THE direct contact with the HARDWARE that limits the possible paths of information. Namely, any LANGUAGE isolated into itself will multiply its kaleidoscopic forms, ‘free’ of the constrains the vital energy it must ‘shape’ cause.

And this is the fundamental need of mathematics that force it in search of meaning as per lobachevski’s pangeometry and Godel’s algebraic theorems, to recur to experimental science, TO DISCERN ONCE AND FOR ALL WHAT ARE MATHEMATICAL FICTIONS AND WHAT ARE MATHEMATICAL REALITY.

‘I know when mathematics is logic but not when it is real’ then applies in this case to the unnecessary attempts to draw complex functions in 4th dimensional space, when they are mostly in its imaginary parts functions of motions in time, best operated in mere algebraic methods.

**x ÷: 3Ð: **

**RE=PRODUCT**

With the product and the ill understood division and its Q-numbers as ratios, we enter into much deeper land, and multiple issues arise at once. Unlike the sum which requires EQUAL SPECIES to add with its superposition principles, in herds, the Product is NOT only the sum of equal forms in ‘brevitas’, which is the usual definition, but the MERGING of different species, with is its fundamental meaning, hence a reproductive act that brings SOMETHING ELSE, truly starting processes of creation in a single plane of space – as opposed to the sum that merely will create a simpler herd of a larger scale.

To fully understand then what we can ‘multiply’ and we cannot, (we cannot multiply 3 pears and 5 apples, unless we upgrade them to the concept of fruits) we need first to consider what ‘systems’ in the Universe are ‘bidimensional’, the product of two simpler Dimension of existence. And immediately it happens that we talk of Dimotions as the combination of a dimension of spatial form and a motion of time. *So the immediate surprising and profound realization about the multiplication operand is that it is THE ESSENTIAL OPERAND TO COMBINE DIMENSIONS OF SPACE AND FREQUENCIES OF TIME MOTION TO CREATE A BIDIMENSIONAL SPACETIME DIMOTION.*

Let us elaborate on this key them considering the holographic bidimensional basic forms of spacetime.

**Holographic interactions**

We defined in Algebra the product as the king of all operations, since it ‘merges’ into ‘a new entity of space-time’, two parameters disjoined previous to the product, proving the very same existence of a holographic Universe, often through a merge of the ‘¡-1’ elements or ‘cellular parts’ of the being, which create ‘axons’ of communication with all the other parts of the being, such as:

X(5¡-1) x Y (4¡-1) = {X(5¡-1)Y(4¡-1)}20¡-2

That is, a multiplication that merges two elements gives us the number of i-2 axons of communication connecting at a deeper level the two parts.

In a momentum the mass stop state and the wave step state merge in the potential I-2 level that holds them together.

The most abundant of all operand, the merging product requires therefore a more complex rule than a direct sum, which acts by ‘superposition’ of EQUAL BEINGS.

IT IS ALSO susceptible to be operated by calculus and ‘derivatives’ as now we INVOLVE FOR THE FIRST TIME, BOTH, a SCALAR LEVEL, since multiplication tends to happen in the lower scale of the being and different states of time and space. So we no longer operate as in additions, with the same type of T.œs in the same plane.

The best way to describe the multiplication symbol is through the holographic principle. In reality one-dimensional points with no breath do not exist. Systems are always ST holographic merges and so we need an operation, which is the ‘queen of them all’ as it allow us to ‘merge’ S and T states of different entities together into ‘new Bidimensional operators’, which in physics for example merges M(ð) x $(v) into momenta, which is the *conserved bidimensional entity of the spacetime Universe.*

Multiplication thus beyond the naive definition of a ‘sum of sums’, brings us truly a creative process into algebra.

The holographic bidimensional universe and its ternary ST-geometries define reality. So in most mathematical equations solutions abound on quartic and cubic systems but only special cases are solvable for higher polynomials or have any real use in reality; the exception being simpler equations of the ∆§ocial scales and reproductive functions of the type Xª=b…

The simplest modes of reality are ST combinations of bidimensional planes which ad by superposition; when the parameters are different however we multiply them.The second operation of algebra is product and its inverse division, AND THIS probes further into the scales of reality; so the new operations ads dimensionality and requires new numbers, namely the Rational numbers. Let us then consider them first by studying its properties as compared to those of addition. So THE product IS THE FIRST operand of the fifth dimension, which merges, in 3D: reproduction acts that requires between two ‘genders’ such interconnection at the lower level to form a whole.

the graph, 5 points connected with all others and with itself, give us 25 connections. This is the ultimate meaning of the product when perceived from an ∆-1 perspective WHICH IS NEEDED AS IN MOST FORMULAE THE PRODUCT IS NOT REFLEXIVE but communicative between species of S-T different quality. I.e. momentum, mv, is not merely the multiplication of m, but the product of a static space-state m and a wave-moving state: v – another whole thing.

As in the entangled Universe operand have multiple mirror possibilities the product can also account for locomotion’ as a product of two such states: λ ƒ=v.

4D: WHILE it is obvious that its inverse function, the division, breaks the whole into parts, and so it represents the inverse function of entropy.

Yet again perception doesn’t fit in the product.

Mathematics is in that sense best suit for the spatial (topological) and scalar, social (numerical) elements of reality. Logic is better for the complex pentalogic of the entangled Universe and its perceptive elements.

So as we can see operand are directly connected to the Dimotions of the being, giving so much power to mathematics.

**1ST LINEAL AND 3RD POINT DIMENSIONS:**

The key theme to understand operandi is the meaning of inversion laws which carry the ± symbol in the 3 scales of sum, multiplication and power, dividing reality into a splitter symmetry around the T.œ in itself.

So ± is carried into power laws and products through ratio inversions, which is better expressed in the quadratic ‘complex’ frame of reference,of X²=±i² conjugate + 1 x -1 = 1² =1 axis.

So the positive and negative are shown in that ternary ‘frame of reference’ in each of the 3 axis, with the identity product of them, or quadratic frame of reference… in which we shall discuss latter on the complex elements of multiple dimensional operandi and its quants of actions.

**MULTIPLE MULTIPLICATIONS: RASHOMON TRUTH**

The fundamental effect of a new operandi is to ad a new dimension to the system. Then we can apply the method of multiple perspectives ( RASHOMON truths), to consider which type of dimension, *and in this manner an enormous range of phenomena can be expressed with the same operation, increasing the iterative complexity of the Universe. We shall just show a few example of the ginormous ‘100th’ number of combinations that we can obtain by multiplying any of the $, t, S, T, §, ð, ∫, ∂, ∆§+1, ∆ð+1, 10 dimensions of space-time for any of those dimensions; and further on combining different operandi in a single function*:

**Γ combinations: ($t≤ST≥ð****§****):**

$$: Multiplication adds a dimension of lineal distance. It is the simplest and commutative form.

$t: Multiplication ads a dimension of motion, velocity:

**$S:** Multiplication ads a dimension of spatial width to form area:

**$**

**ð**

**:**In a similar fashion multiplication can ad to a cyclical point/clock of time, a dimension of distance.

**ðt:**Multiplication can adda dimension of lineal motion to a particle-vortex (quantum motion), the very

*essence of motion as reproduction of information.*

**∆±i**

**∆º§:**In a single plane, multiplication is obviously the product of a group of social identical beings by the parameter we study ceteris paribus or ‘number’, multiplied x times, that is summed with an equal group and an equal group and an equal group to reach the number we multiply for.

**∆+1<∆-1;**But what does it mean between planes? As we realise that when we multiply 5 x 5, if we consider 5 the number of ∆-1 elements of 2 groups, then 5 x 5 = 25 turns out to be the number of ‘axons’ connecting each of the 2 wholes at the level of its 5 ∆-1 elements. So 25 gives us

*the number of axons= ST flows of communication between the 5 sub-elements of the 2 wholes.*

**The messing of multiplication and addition: distributive and neutral elements. Negative inversions.**

*has one more dimension and hence has the 1, perceivable T.œ (because it will be making likely a motion and/or be perceived as an ST unit, hence we consider it an interval 0-1, a unit circle and so on).*

**0 x a = 0**is the first of those combinations, which causes a reduction of dimensionality, as

*return o the less dimensional operation of addition. It can then be also vitally interpreted as if 0 were an open ball without @, the singularity membrain; hence a sink of void of the ∆-1 scale, prior to the ’emergence’ of the whole, 1, the full T.œ – an eraser dimension when in contact with a.*

*And finally as the destroyer point that sinks the flow of a, into the 0-point:ð*

**Lineal vs. cyclical addition and multiplication.**

*products must be seen as ∑, a sumandi of x elements, since the product is with a constant.*

*It is a very different case when the product is between two variables or any other parameters which are not ‘constant sums’.*

*If we were to consider the analysis from the point of view of the 3 ‘elements’ of a generator, $-potentials < ∑∏-waves>ð-particle/heads, it is easy then to see that constant products are*sums. So potentials are sums.

*must be considered bidimensional, and dot products*). As products, they create new dimensions, as in light waves are: c²=k/µ; and cross products. Such PRODUCTS ARE NON LINEAL as they REPRODUCE a sinusoidal wave. And often the key to classify waves as sum or products is their capacity to commute (sums) or not (products). Finally the SYNTAX of most languages coded as GENERATORS ARE PRODUCTS that do not commute when they are different ‘categories’ (potential x wave x particle; subject x verb ≠ verb x subject and so on). And so the only ‘identity’ elements which are non-lineal as products are 1D ð§; and its polynomials.

**Multiplying ±. From outer operandi to inner property in a higher dimension.**

**Multiplication is distributive: a x (b+c) = a x b + a x c = e**

So this lead us to a property, distribution, which happens in all meaningful combinations of two consecutive Dimensional operandi; where the concept of a rank, or more ‘powerful’ larger social whole-dimension, in this case the multiplicative operandi a x, is able to ‘slice’ the group-herd, b + c into two separate parts enlarging them both by the same ratio. And this merely extends enormously the capacity of two ‘consecutive dimensions’ of space-time to make a ‘liaison’ , as the ‘higher whole’ the product operandi can ‘discern’, when it is necessary to re-grouping the lower ‘herd’ (b+c) to ‘effect’ an ’emergence’ to a higher dimension of those two different groups – or when to operate first b+c = d and then a x d= e.

Distribution thus means that any consistent system managed by an ∆-1 ST operandi can emerge in the ∆+1 plane through an ∆º operandi.

*But some multiplications are NOT commutative… those of Matrices and other complex systems in which multiplication departs from its simple use as a sum of sums..*

**ITS INVERSE OPERATION: DIVISION.**

*operation of the arrow of entropy, as it means to ‘divide a whole’ in parts for a predator ‘group’ to feed on them. As when we divide a pie to eat it.*

*It is thus the inverse of a multiplication as the ‘social communication between ¡+1 elements that give us its total ‘axons’, or lines of communication of the two wholes (4 x 4 = 16 axons connecting all the 4 elements with the other 4).*

*specially in mathematical physics where the units chosen by humans are misleading (i.e. time duration is an abstraction 1/t= frequency is the unit of cyclical time, so an s/t is better expressed as S x T (wavelength x frequency), and so on.*

**Multiplication defining the 3 vital constants of the being and its ratios.**

*S/T x T/S =ST/TS= 1. Multiplication as Ratio of proportionality*

*it is not. The ratio is a division; the square is a multiplication, and both are inverse functions, which gives us the identity element.*

In that sense in all questions connected with discrete objects, we NEED to use WHOLE numbers for the necessary mathematical apparatus, as well as the study of the continuous. Thus, for example, in mathematical analysis, when one considers the expansion of an analytic function in a power series with integral powers, computations are essentially carried out with whole numbers and approximated ratios such as 22/7 = πi, which is Ok as we have seen (Number theory) that decimal numbers break its meaning beyond the 10th decimal scaling… (i.e. e=2.718281828…45).

So all fractions represent ratios/quotients of two whole numbers; and as such a full new branch of ‘number theory’ will be the study of those quotients as *ratios between steps of time motions and or whole polygonal numbers.*

RIn dealing with any real number in practical work (for example, π), we replace it in fact by a rational fraction (for example, we assume that π = 22/7, or that π = 3.14).

YET While the establishment of rules for operating on numbers is the concern of arithmetic>ALGEBRA, the deeper properties of sequences of numbers, extended to include zero and the negative integers, are studied in the theory of numbers, which is the science of the system, studied in other post.

**Odd and even functions.**

A very interesting reflection of the duality between antisymmetry and symmetry happens in the odd (antisymmetric) vs. even (symmetric functions) as it is also the basis between social evolution of particles (bosons) vs. antisymmetric annihilation (fermions), so it *has an immediate never quite clarified application to physics.*

Let us consider the classic axiomatic approach first, which started in the analysis of divisibility.

One of the basic questions in the theory of numbers concerned the divisibility of one number by another:

if the result of dividing the integer a by the integer b (not equal to zero) is an integer, i.e., if a= b • c (a, b, c are integers) then we say that a is divisible by b or that b divides a. If the result of dividing the integer a by the integer b is a fraction, then we say that a is not divisible by b.

Questions of divisibility of numbers are encountered constantly in practice and also play an important role in some questions of mathematical analysis. For example, if the expansion of a function in integer powers of x

is such that all odd coefficients (with indices not divisible by 2) are equal to zero, i.e., if

then the function satisfies the condition: **ƒ (-x) = ƒ (x)** – such a function is called an even function, and its graph is symmetric with respect to the axis of ordinates. But if in the expansion (2) all the even coefficients (with indices divisible by 2) are equal to zero, in other words, if:

then **ƒ (-x) = – ƒ (x)**

In this case the function is called odd, and its graph is symmetric with respect to the origin.

Thus, for example:

Which will have deep implications in the physical Universe (Pauli exclusion principle), so we shall study its full meaning in our posts on mathematical ‘Astrophysics’.

The reader should easily in any case interpret the results in terms of dimensions and the holographic principle. And it reads like this: symmetric functions are holographic even functions, which means for example they can be ‘superimposed’ as they are bidimensional ‘sheets’, waves etc. While odd functions are even in dimensions, and as such they cannot be superimposed.

This means essentially bosons are bidimensional and fermions are tridimensional, and this is a huge advance for quantum physics – recently proved by an experiment that converted a photon, initially a boson into a tridimensional form and ended its boson conditions. Accumulation of bosons into a single point of space-time then means merely it is superposing the bidimensional thin layers into a third dimension of height which is indeed what we see in pictures of boson states.

How the representation in ± sides of the plane works to understand this is also obvious: the axis of ordinates plays here the role of the ‘asymmetric’ state which has split into the ± inverse directions but can fusion again both. Or in terms of relative equality, those functions DO have their position in space (established by the lineal coordinates used to represent lengths in space) different but their form (represented by the Y-coordinates used to represent form and information height dimensions for most functions) identical. So they can according to the rules of the fourth postulate of non-e, communicate as they can share identical information and match each other in symmetric peg.

While the antisymmetric function which is +- -+, meaning its inverse both in the Y-nformation and X-pace location is disimilar both in form and motion, momentum and position, you name it… whatever 2 parameters we use to compare both systems. So they enter into a darwinian annihilating process as they cannot match each others form.

**0-1 ≈ 1-∞**

Let us then see those numbers which we have added from the perspective of the plane, as they become ‘mirror symmetries’ in the o-1 sphere unit of the 1-∞ plane described with natural numbers.

Once we have defined the 1-∞ plane only with natural or complex numbers we can then assess the need for more numbers to fulfill the ∆±1 scales and ore operandi to probe in to the ∆-1 scale (o-1 sphere and infinitesimal numbers found with rational and real decimals).

An ∆ scale can be represented through the interval of 0 to 1 by finitesimals or the interval from 1 to ∞, which become the decametric and decimal regions of a supœrganism represented in the real line: ∆-1: o to 1, 1, the ∆o scale and 1 to ∞, the external world.

**Quantum case**

Further on, as we are dealing with the smallest scales of reality, it applies the metric equations of 5D according to which we are in a ‘temporal realm’ as world cycles occurs extremely fast (Min. Spatial size = Max. Temporal speed of time cycles) and so the formalism of quantum physics uses the equivalence between the o-1 probabilistic sphere of time events instead of the 1-∞ plane of statistical populations to formalize the events of ultra-fast repetitive particles.

**Numbers and infinities.**

Mathematics divides phenomena into two broad classes, discrete or temporal and continuous, or spatial historically corresponding to the earlier division between T-arithmetic and S-geometry.

Discrete systems can be subdivided only so far, and they can be described in terms of whole numbers 0, 1, 2, 3, …. Continuous systems can be subdivided indefinitely, and their description requires the real numbers, numbers represented by decimal expansions such as 3.14159…, possibly going on forever. Understanding the true nature of such infinite decimals lies at the heart of analysis.

And yet lacking the proper ∆ST theory it is yet not understood.

The distinction between continuous mathematics and discrete mathematics IS ONE BETWEEN SINGLE, SYNCHRONOUS, CONTINUOUS SPACE WITH LESS INFORMATION, and the perception in terms of ‘time cycles, or fractal points; space-time entities’, which will show to be ALWAYS discrete in its detail, either because it will HAVE BOUNDARIES IN SPACE, or it will be A SERIES OF TIME CYCLES AND FREQUENCIES, perceived only when the time cycle is ‘completed’, and hence will show DISCONTINUITIES ON TIME.

Thus the dualities of ST on one side, and the ‘Galilean paradox’ of the mind’s limits of perception of information lay at the heart of the essential philosophical question: it is the Universe discrete or continuous in space and time. Both, but always discrete when in detail due to spatial boundaries, and the measure of time cycles in the points of repetition of its ‘frequency’.

So ultimately we face a mental issue of mathematical modeling: the ‘mind-art’ (as pure exact science does not exist, all is art of linguistic perception) of representing features of the natural world in a reduced mental, mathematical form.

The universe does not contain or consist of actual mathematical objects, but a language can model all aspects of the universe. So all resembles mathematical concepts.

For example, the number two does not exist as a physical object, but it does describe an important feature of such things as human twins and binary stars; and so we can extract by the ternary method, 3 sub-concepts of it:

2 means the first ∆-scale of growth of 1 being into 2, by:

S-imilarity and S-imultaneity in space (ab. Sim)’, ‘i-somorphism in time-information (ab. Iso)’ and ‘equality in ∆-scale’ (ab. Eq), as perceived by a linguistic observer, @, which will deem both beings ‘IDENTICAL’. Whereas identity means that an @-bserver will deem the being ∆st≈St, (Sim, Iso and Eq). So identity is the maximal perfection of a number, for a perceiver, even if ultimately:

*‘Not 2 beings are identical for the Universe, but can be identical for the observer’… an intuitive truth, *whose pedantic proof is of course of no importance (: we do not follow the axiomatic method of absolute minds here):, but it is at the heart of WHY REALITY IS NOT COLLAPSED INTO THE NOTHINGNESS OF A BIG-BANG POINT.

Thus those 3+0 elements of the ∆•ST coincide *a social number can be used whose intrinsic properties define conceptually ‘S- imultaneity, Ti-somorphism’ and ∆-equality or equivalence (ab. Eq) in size, which becomes an @identity for the mind. THEN A NUMBER IS BORN.*

I(n this ‘infinitorum’ of Universal thoughts, which bring always new depths as soon as we observe it with an ∆•st trained mind, there are differences between S-imilarity and Simultaneity to define in space an ‘identity’ and ‘equality’ and equivalence, treated elsewhere)

It IS THEN CLEAR that a number being a sum of points, encodes more information in a synoptic way about the T-informative nature of the ‘social group’ than an array of points, *which unlike a number tells us less about the ‘informative identity of the inner parts of the being’, but provides us more spatial knowledge about the relative position in space of the members of a number-group.*

And this is OBVIOUS, when we return to the origin of geometry and consider an age in which both concepts were intermingled so ‘points were numbers’ and displayed geometrical properties:

Numbers as points, showing also the internal geometric nature, used in earlier mathematics to extract the ‘time-algebraic’, ‘∆nalytical-social’ and S-patial-geometrical properties from them.

The closure of the systems of numbers thus grows from reflecting merely space populations (natural numbers), into 5D numbers, reflecting ‘partitions’ of social groups (represented by those natural numbers), with Egyptian ratio-nals; expanded further with the realisation that certain ratios did apply to ‘scaling’ in the fifth dimension without limit (as in the pi ratio of Spe-entropic lines into cyclical time O-cycles).

Alas, things got interesting here, but as the homunculus did not understand, *the discoverer of pi, legend has was murdered by Pythagoras the first ‘religious mathematician’ because it found that pi was not perfect. I*t took to Poincare 2500 years latter to find that this is an awesome form of perfection because it means a mind-point of spherical form, with equal distance to all the realities it reflects can shrink with no limit (Poincare conjeture – the one that Perelman resolved recently with tons of pages)

Now, we have SPATIAL, natural numbers, 5D rational and transcendental numbers, who cross through 5D scales without tearing according to the Poincare postulate proved in a margin (: but this is NOT enough, because we need negative and lateral (not imaginary) numbers, *and those are temporal numbers, numbers who describe processes in time not in space, as negative numbers do, as they represent merely the inverse arrow of time, so if you make a positive number an arrow of motion-entropy, or relative past loss of information, the negative number will be the arrow of future-information, and the lateral i-number, will be the bidimensional sum of both the real and the negative, to represent the present space-time.*

∆±i: i-ratio-nal or transcendental numbers; ∆±1: ratio-nal numbers.

So you do have a closure of all numbers, based in the elements of space-time.

Hence, when you apply them to mathematical physics, *which is the study of the simplest forms of space-time,**you do have a better focused mirror. *

Let us then see how the product mimic the five dimotions:

**Social Number = first Ðimotion** that defines regular ‘points’ which are undistinguishable, as societies in regular polygons, where prime polygons have the property of ‘increasing inwards’ its numbers through reproduction of vortex-points (n-grams), as the graph shows, studied in Theory of Numbers. So a number in its geometric interpretation is a ‘cyclical point’ of regular ‘unit-points’ of growing ‘inner dimensional density’ a point with a volume of vital energy and information, a fractal point.

*2nd Ðimotion line=sum of points.*

*3rd Ðimotion production=multiplication =reproduction. Dimensional products.*

A key concept then of the product is when it will be a type of product that merges space and time, and when it will be a product of a dimensional system, and what are the ‘entropic limits’ in which those functions make experimental sense.

In the graph, product can be of multiple, different ST dimensions, which start the richness of its ‘propositions’. A vectorial product is one of its commonest forms as it combines ST or TS dimensions.

BUT as both ‘present’ products are different in orientation, this product unlike other SS or TT products is non-commutative: bxa=- axb. In this case giving birth to two different orientations in space, though for more complex product of multiple ‘S-T’ dimensions, which *can define as a Matrix of parameter a T.Œ PARTICLE in full,* the non-commutability can give origin to different particles (quantum physics).

**EXPONENTIAL vs. LOGARITHM**

**3D: ***log a, xª: ***reproduction**

We come then to the next peel of polynomial growth, and final ternary degree of complexity, exponential equations, which in the inverse process of more complexity for simpler actions, brings the ‘end=action’ of feeding/dying of entropy, better described by those exponentials.

The next level of operand are powers and logarithms THAT model reproduction in case there is infinite energy, with an exponential curve; yet in reality we add at the end a logarithm curve (logistic curve); as reproduction saturates a system. Since REALITY seeks a ‘balance’, the REAL curve of reproduction is a combination of both, from the initial unit to the carrying capacity that DEFINES an ∆+1 social group that ‘reached equilibrium’, when it becomes itself a herd or super organism emerging in the new upper scale. Then the ratio of change (derivative) diminishes from the absolute maximal – eˆx, which has its own derivative, to the minimal 1/x the log derivative chichis the definition of an infinitesimal part, till it will start in an inverse function its decay with -1/ diminution and a fast collapse in the 3rd age<<death moment. So the combination of ± exponentials and logarithm curves are also the best way to graph as a bell curve the worldcycle of existence in lineal terms:

*4th dimension: polynomial death dimension of decay.*

POLYNOMIALS DO NOT EVOLVE REALITY towards an impossible infinite growth. THEY ARE the inverse decay process; which can be understood better observing that the inverse function does in fact model growth in the different models of biology and physics, limited by a carrying capacity straight flat line:

**The logarithmic function **has as derivative an infinitesimal, 1/x, which makes it interesting as it models better the curve of growth from o to 1 in the emergent fast explosive ∆-1 seed state, while the inverse e*ˆ-x model the decay death process.*

*5th dimension: ∫∂…*

Integrals and derivatives which have a much slower growth, than polynomials on the other hand do model much better as they integrate the ‘indivisible’ finitesimal quanta of a system, its organic growth and ‘wholeness’ integrated in space.

Thus integrals do move up a social growth in new ∆+1 5D planes. And its graphs are a curved geometry, which takes each lineal step (differential) upwards, but as it creates a new whole, part of its energy growth sinks and curves to give birth to the mind-singularity @, the wholeness that warps the whole, and converts that energy into still, shrunk mind-mappings of information, often within the 3D particle-head.

**Equations of polynomials.**

Polynomials seem fairly straight, but as soon as we consider its solutions and varieties, things become more fascinating.

First we have 2 mysteries of 2300 years, one which people hardly wonder – why there are 2 solutions to all polynomials, and the other which all mathematicians do but have never been explained – *why one of the solutions is often an imposible, imaginary i-solution? What is the meaning of imaginary numbers?*

Why polynomials ‘∆-scales’ have 2 solutions?

The answer should be immediate for anyone who has understood anything about GST (hopefully more than those who understood Einstein – I just recall his conferences at Solvay, when he quipped to the same question, ‘maybe the priest’ -Lemaitre):

As all points in motion have 2 directions, upwards and downwards in the ∆-dimension, left-right in lineal geometry, more space or more time in existential topology, youth or old age (rebirth-reproduction or informative evolution).

Thus there are 2 solutions, since *the equation does not include a ‘choice’, which will be made a posteriori based in experimental evidence.* And so it is evident, as space is larger, and equations quantitative, that in general equations, the positive solution is space-like and the imaginary solution, time-like. And we shall elaborate a lot on that theme along the posts of the 3rd line.

The point to understand the underling structures of the Universe, written in its i-logic mathematical equations, of course is to respond to questions never asked for, as they seem to be the ‘a priori’ human categories of thought (Kant), which Schopenhauer rightly reduced to ‘sufficient reasons’, space, time, causality (mostly caused by ∆ scales in space-time) and the mind that perceives all (languages, ternary grammars).

The fundamental mathematical solution is the polynomial double root solution, where the polynomial is a function of present and the 2 solutions a ± split fields.

Presents thus *split in two dual solutions a ± dual field which is essential to understand all time of equations, from the equation of death where the complementary body-head, field-particle system splits, to the inverse equations of social symbiosis in systems that plug-both past and future systems into a present vital form of which the most important happens in social sciences*

**Finitesimal actions.**

The logarithm’s derivative thus ads as ratio of change only an infinitesimal, so it tends always to a balanced static form (y=c).

The quantity a system absorbs to create an action is generally defined as a ‘finitesimal’, not infinitesimal. Infinite does not exist in a single continuum, but through multiple discontinuities as all systems in time and space are limited in space and time, both in a single membrane, and in within the scales of the 5th dimension (as information and energy doesn’t flux between those scales without loss of entropy).

A finitesimal is THE QUANTITY of energy, motion, information etc. used by a T.œ for an action of space-time IN any of the 5 Dimensions of the being, ‘put in motion’ to that aim.

Wholes are physiological networks, which we analyse mathematically in its parts, mostly performing a motion of space-time, an action that exchanges most likely bits and bites of time and space. So the logarithm is also an operation reflects the processes of minimal transmission and gathering of information and energy, of bidimensional holographic quanta… reason why it is so pervading in the concepts of entropy and information.

4D: Thus entropy has a negative exponential which show the rhythm of decay of the system. And in this case there is no need for ‘a logarithmic’ limit, since for the predator the death body is ‘unlimited ¡-1 energy’, though once the ‘relative infinite’ number of its ¡-1 parts are absorbed the ‘e-function’ will have a cut off.

How the function combines with other functions shows then how the superposition and merging-product processes combine as Dimotions with the reproductive growth and decay processes, whose results are intuitive: the product=merging of two powers of reproductive growth, when its ‘base=toe’ is identical ‘superpose=add’ that growth:

*In the graph, the inverse simpler analysis of growing and diminishing planes and finitesimals have 2 first approaches in the study of polynomials and its inverse equations of logistic growth, logarithms. POLYNOMIALS in that sense are better as negative expressions of the decay of a system in its 4th entropic dimensions and no longer follow the whole range of ‘social growth’ properties (commutativity and associativity); as they are no longer ‘connected with the ∆-2 dimensional scale in the same ‘arrow’ of growth.*

Infinite growth does NOT exist because there are always limits to growth in the incapacity of a system to obtain ALL THE ENERGY OF ITS ∆-1 plane (reason why THERE IS entropy when we try to extract ALL the heat from molecules into our ∆º whole scale).

This means that exponential growth IS not growth but decay, as it can only be exponential when acting on a pool of already ‘multiplied by re=product-ion’, ∆-beings, which are destroyed in exponential fashion by an e‾ª function. And so it is only the inverse function of logarithm growth, what shows a growth curve (logistic curve) till a carrying capacity saturates the system.

*it was not possible to go.*

**Properties of operations **

It follows from all what has been said that *operands are the fundamental reflection of the Disomorphisms of the Universe in mathematical ‘space’. What truly ‘pegs’ together the entire structure of mathematics, regardless of variables, parameters, scales and type of §œTS. *

So the first question as each operandi reflects an ST duality, is to ask what are the properties of the ordinary operations?

In a careful discussion of the properties of numbers and the operations on them that are used most frequently in algebra, it is easy to observe that they reduce to the following, or yes, of course… 5²=10 properties… We shall though ad for proper symmetry two properties that relate both operations (:

1. Identity element: There exists a number zero with the property a + 0 = a for every a.

2. For every number a, there exists the opposite number x satisfying the equation a + x = 0.

3. For any two numbers, their sum is uniquely determined.

4. Addition has the associative property: (a+b)+c=a+(b+c)

5. Addition is commutative: a+b=b+a

•••

1. Identity element: There exists a number 1 with the property a x 1 = a for every a.

2. Every number x, except 0, has a multiplicative inverse, 1/ x • x = 1

3. For every a and every b ≠ 0, there exists a unique number x satisfying the equation bx = a; hence the product of two numbers is uniquely determined.

4. Multiplication is associative: (ab) x c= a x (bc)

5. Multiplication is commutative: axb=bxa

And then two properties that relate both operations:

11. The product of a number and the identity element of a sum, 0 is the identity element of the sum: a x 0 = 0.

12. Multiplication is distributive: a(b+c) = ab + ac

Those properties were selected in classic science as a result of a careful analysis; the development of mathematics in the last century proved their great importance; as only *operandi and §ets of mathematical elements that obeyed those properties with the BIG exception of commutativity, which often is defined by its inversion ( a x b = – b x a) *

Nowadays every system of quantities satisfying the conditions 1 through 10 is called a field. Examples of fields are: the set of all rational numbers, the set of all real numbers, or the set of all complex numbers, because in each of these cases the numbers of the set can be added and multiplied and the result is a number of the same set, and the operations have the properties 1 through 10.

Apart from these three very important fields we can determine infinitely many other fields formed from numbers. But beside the fields formed from numbers there is much interest in fields formed from quantities of another nature.

For example, algebraic fractions, in which the numerator and denominator are polynomials in certain letters can be added, subtracted, multiplied, and divided, and these operations have the properties 1 through 10. Therefore, algebraic fractions form a system of objects that is a field.

So the entire ‘field of polynomials’ reduces to those 10 properties, and it is not rocket science for the reader who has got so far the ‘idea’ that we should define those properties as 5 inner ± D-properties of ‘small steps’ of social evolution (addition/substraction) and its ‘wholeness’ as a new dimension, 5 D-properties of x, ÷, or at least a close isomorphic correspondence with the ∑ ‘time-sum’ = Product- space whole duality.

So we have arranged those properties to see easily why they are in fact, defining a ‘constant growth’ in inverse symmetries of a neutral first element a 1 T.œ or fractal point, which will expand evolving socially ∆§, through additions and then in a new dimension of social scales, where an ‘additional number’ will be now whole unit of its multiplications; and finally in the third operation, a polynomial will be that number raised to an ∆new symmetric plane (as polynomials multiply the same quantity, unlike sums and multiplications).

**1st axiom = 1st Non-E Postulate/1st Dimension – the T.œ point:** The ‘first’ property of +, x IS defining the neutral, self-centred fractal point – a unit of addition a first number or a first ‘group of numbers’.

**2nd axiom = 2nd Non-E Postulate/2nd Dimension – the flow of communication:** The second property of ±, x÷ *expands in both inverse directions to maintain the zero sum of all worldcycles and operandi of existence, the T.œ in two inverse directions defining a wave-line-interval-distance-inverse motion both sides of the **identity element, fractal point or original number. *

**No axiom for 3rd postulate/dimension, the ternary network plane:** YET to reach the third ‘network dimension’ of the system that gives us a full ternary organism and its GENERATOR EQUATION, there is NOT A THIRD POSTULATE, BUT WE NEED A THIRD OPERATION. Thus, the Universe is indeed holographic, bidimensional, as we have show in every perspective of mathematics, from FERMAT’s last theorem to all other ST symmetries. You cannot reach a third dimension of the sum, but you must multiply it and you cannot reach a third dimension of the multiplication but you must operate through a power law:

In the graph, the runaway Nature of power laws, shows a hyperbolic ‘end of a plane’, which abandons the ‘lineal’ S x T =K nature of a proportional metric in the fifth dimension (multiplication), as it fast reaches the “Lorentzian” regions that signify the limiting domain of any T.œ in form, motion or scale:

Thus our hypothesis 5 properties -> 5 dimensions fails ): Good or things would be too boring.

But a bit of thought shows us that we are dealing with ‘logic properties’ NOT space-time forms, so it is better to connect them with the 5 Non-Æ postulates of i-logic. And then eureka! They do fit nicely.

So the other axioms *must be of i-logic nature, indeed:*

**3rd axiom **might seem silly but at this stage the reader would realise the Universe is quite weird enough not to discharge anything. And it is reassuring to know that THERE ARE NO PARALLEL UNIVERSES, A+b = Only c A x B = Only D.

So now we come to the logic ones, closely related to the 4h and 5th of non-A i-logic geometry:

**4th axiom ≈ 4th Non-E Postulate, self-similarity:** The Universe IS associative AMONG SELF-similar points, which *allows it to grow and multiply, add and multiply we might say, love each other as a i have loved you in human terms. It is the positive side of the 4th postulate of i-logic, associativity…
But what about the Darwinian, perpendicular laws of non-E? Because a number is made of IDENTICAL beings, and we can ONLY add (and so by extension since we have deduced that multiplication is the third dimension of addition), equal beings, we are only in the positive side of the 4th axiom. *

But alas! we then realise that the negative side of it, the inverse function, subtraction and division are NOT associative: 10/5=2 and 2/2 = 1 is NOT the same that 10 divided by 5/2 which is 4; and 10 – 5 = 5 minus 2 = 3 is NOT the same than 10 minus 5-2=3 which is 7…

So there is here in the negative side of the world, a different hierarchy of things, such as if you start destroying BIG TIME (10/5; 10 minus 5) and then slow down, you have already fuk up the world, but if you start destroying slow; decadence gets longer… Important elements of the vital laws of reality, explored in the first line…

You see, even the simplest supposedly exhausted facts of science, get new insights on stience (:

**5th rebel postulate: commutativity. **And so the fifth postulate of course must have to do something with the 5th Non-Euclidean postulate, the rebel one; and indeed it turns out in this symmetry between geometrical space and temporal algebra views that to the surprise of everybody in the XIX-XX century, there are many systems in Nature which do NOT have the commutative property for multiplication, (but it does for sum).

That is, a sum is truly a herd of undistinguishable beings, and it does not matter as it is in a state of loose connection which order you add. We are then clearly in the very first simpler reality, in a single plane, with minimal herd connection between the parts.

But multiplication being a ‘second dimension’ added to the sum sometimes IS commutative, sometimes NOT, meaning that when the multiplication IS a growth in a scalar dimension within the plane (truly a sum equivalent new dimension) IS commutative, but when this NEW dimension is added, not in the same plane, as an ∆§ operation but *in the sense of group theory as a ‘combining’ process of things (vectorial product, product of the parts of a whole, product of spatial paths, products in time **frequencies, topological products) THINGS CHANGE. *

ÍAnd so with this understanding we COMPLETE the simplest analysis of sums in a single plane of identical beings as herds, but we realise that the second operandi, the adding of a second dimension called product ALREADY PLUNGES US INTO THE RASHOMON EFFECT OF THE PRODUCT which can mean many different things depending on which ∆S≈T dimensional element we are adding to the simpler sum.

∆-1: fractal point sum > ∆º: lineal product: ∆+1: Power ‘volume’.

We can multiply powers with the same base:

x³⋅ x²=(x⋅x⋅x⋅x)⋅(x⋅x)=xˆ5. Hence the general law: x²⋅x³=x²+³: The product of powers with the same base add exponents.

*wholes (∆+1 powers) operated EXTERNALLY in the domain of its present, dimensional parts (∆º multiplication) are equivalent to one whole operated INTERNALLY with its past, finitesimal, entropic points.*

A profound law of Existential algebra; which basically tells us that an ∆-whole ( a fractal point in its own) absorbing an external ‘flow’ (the lineal multiplication) , will operate that flow at a lower internal ∆-1 level. I.e: if you absorb food you first break it into amino acids to reconstruct yourself.

We can raise a power to a power:

(x²)³=(x⋅x)⋅(x⋅x)⋅(x⋅x)=xˆ6

This is called the power of a power that multiplies the exponents: When you raise a product to a power you raise each factor with a power:

(xy)ª=xªyª

*Two parts (∆º≈ x • y) operated in the domain of its higher whole dimension, are equivalent to each one operated by the higher dimension as separate, broken entities. Which is the distributive law that gives the whole power over its parts.*

As well as we could multiply powers we can divide powers. And this 2 laws are again applied to the inverse operation.

This quotient of powers property tells us that when you divide powers with the same base you just have to subtract the exponents.

xª/x³=xª¯³, x≠0

Two *wholes (∆+1 powers) operated EXTERNALLY in the domain of its present, dimensional parts (∆º division) are equivalent to one whole operated INTERNALLY with its past, finitesimal, entropic points. *

When you raise a quotient to a power you raise both the numerator and the denominator to the power.

This is called the power of a quotient power:

(x/y)ª=xª/yª, y≠0

*Two parts (∆º≈ x /y) operated in the domain of its higher whole dimension, are equivalent to each one operated by the higher dimension as separate, broken entities. Which is the distributive law that gives the whole power over its parts.*

*IDENTITY LAWS.*

When you raise a number to a zero power you’ll always get 1.

1=xª/xª=xª¯ª=xº=1,x≠0

*A WHOLE, ∆+1, operated by the identity element of its ∆-1 scale, gives us the identity element of the middle scale: ∆+1 * ∆-1=∆º*

And a similar law in ∆st terms: Negative exponents are the reciprocals of the positive exponents:

x¯ª=1/xª, x≠0; xª=1/x¯ª, x≠0

*The inverted operation of the ∆-1 scale applied to the dimension of the whole, ∆+1 gives us the inverted element of the ∆º scale.*

*Both are laws of the essential timespace definition of the generator: Past (∆-1 ) * Future (∆+1) ≈ Present. *

The same properties of exponents apply for both positive and negative exponents. The square root of a number x is the same as x raised to the 0.5th power: √x=²√x=x½…

*the scale of the whole (1-∞) to its reciprocal, scale of the infinitesimal (0, 1) making both mirrors of each other; since ultimately the whole is a closer model of its infinitesimals, than the intermediate scale, which has suffered an inversion of function: ∑∑O-1≈∑|i= O+1.*

*the third scale of growth of a system, does NOT goes over a hypothetical 4th dimension in the same plane but starts a new game, in which the whole in the new plane is the ‘fractal infinitesimal’ similar to its infinitesimal in the ∆-1 plane.*

**∫∂: ****DERIVATIVES AND INTEGRALS**

**All together now**

As we study those operands in Analysis we just make a few remarks here from that post.

So what about the most sophisticated of the inverse operations of algebra? Calculus of finitesimals, ∂, and its integrals, ∫? Here we have a dual motion, back into ∆-1 to extract a part, and forwards into its integral ∫ through an spatial or temporal finite domain. So the inverse operations of analysis have multiple functionality in terms of actions performed through them, because of its *perfect mirroring of the action itself, which consists in using ∆-i finitesimals to absorb energy, motion or information for the 3 simpler actions of motion, informative perception, and energy feeding for which **paradoxically, the more complex ‘organic’ operations are the most useful.*

This paradox though has *a less ‘motivating’ cause for those involved with the mirror of mathematics – essentially that mathematics is NOT the best language to describe the complex actions and relationships that appear out of the reproductive biological and social, engaging processes of organisation, at least algebraic operations.*

NEXT OPERAND IN COMPLEXITY IMPROVES UPON THE PREVIOUS ONE (POWERS)

It is a rule of the scalar Universe that all the action are chained such as to effect a more complex action we need the previous ones, as the basic chain of 1D perception->2D locomotion->4D entropic feeding->3D reproduction->5D social evolution into a larger whole shows.

This chain can then be expressed with the classic operands from angle to sum to division and product to power law. But then once the carry capacity of the system is completed at 5D which is a larger 1D we emerge into a larger world and here we get results only with the ∫∂ operands, which therefore must be related to a degree of accuracy in its simplest forms to the more complex forms of the polynomial, and this is the 5D why explanation of a well known rule to approximate functions by its higher derivatives.

Indeed, in Algebra, the third and higher derivatives are used to improve the accuracy of an approximation to the function:

f(xo+h)=f(xo)+f′(xo)h+f″(xo)h²/2!+f‴(ξ)h³/3!

Thus Taylor’s expansion of a function around a point involves higher order derivatives, and the more derivatives you consider, the higher the accuracy. This also translates to higher order finite difference methods when considering numerical approximations.

Now what this means is obvious: *beyond the accuracy of the three derivatives canonical to an ∆º±1 supœrganism, information as it passes the potential barrier between scales of the 5th≈∆-dimension, suffers a loss of precision so beyond the third derivative, we can only obtain approximations by using higher derivatives or in a likely less focused=exact procedure the equivalent polynomials, more clear expressions of ‘dimensional growth.*

*So their similitude first of all proves that both high derivatives and polynomials are representations of growth across planes and scales, albeit loosing accuracy.*

Let us briefly then deal with the operands of the 5th dimension treated extensively in ∆nalysis:

However in the fifth dimensional correct perspective is more accurate the derivative-integral game; as it ‘looks at the infinitesimal’ to integrate then the proper quanta. Let us briefly comment the operands treated with the 5th dimension of ∫∂ treated extensively in ∆nalysis:

As we did with the other operands, we need to consider the properties of calculus and its two operandi. This poises a problem, as there is not a ‘bottom operation, such as ±, x÷, directly related as with powers as the third dimension of calculus. But as calculus is a refined analysis of power laws, the direct connection is not exact.

Hence a certain discontinuity is established what implies that ∫∂ equations have been solved by the obvious method of applying the function h’ (x)= lim h->o h (x+h) – h (x)/h. We are not though to repeat here that procedure to get the results but merely analyse from ∆st perspective as we did with power laws, and x, the properties of derivatives, to see what they tell us in the higher T.œ language and then consider some specific functions and its integral and derivatives to learn more of it.

Those key properties are expressed in its rules of calculus, starting from the ‘derivative’ of a polynomial:

Xª= a Xª‾¹

So we are NOT fully lineally diminishing a polynomial dimension despite being derivatives a reduction of dimensionality – the search for the finitesimal 1/n quanta. Why? Obviously because in the rough view from a quanta, xª into its whole xª+¹, we grow lineally (polynomial), but as we *repeat ad nauseam, the lineal steps curve into geodesic closed wholes, in the ∆+1 scale (Non-E geometry), from the lineal spatial mind to the wholeness cycle of the closed being, and so as the ‘curve of a parabola’ diminishes the distance of a cannonball, growth is NEVER lineal but falls down as we approach the ‘(in)finite limit’.*

IN THE GRAPH, the wholeness is curved upwards, the parts spread scattering entropically. The whole is a mind circle, @. So it curves/diminish the quantity of energy available, for the whole, as it really must be an addition of all the planes that share that vital energy to build ever slower, curved larger wholes.

*Or in terms of the integral function: *

And here we find the second surprise. There are ∞ integrals with the addition of a constant. As a constant by definition does not change (so as a kid know it goes out on the derivative). But ∆st gives new insight in things ‘children of thought’ (: all huminds 🙂 think they know.

Let us express this then in terms of past (∆-1: derivative )< Present (Function) > future (∆+1: Integral)

*The past is fixed, the infinitesimal enclosed, only one type of species, ‘happening already’, as the parts must exist before the wholes to sustain them. But from the pov of the present function, the future integral into wholes is open, with ∞ variations on the same theme; unless we have already enclosed that whole, limiting its variations, which happens with the definite Integral.*

So if the function f(x) is given on the interval [a, b] and, if F(x) is a primitive for f(x) and x is a point in the interval [a, b], then by the formula of Newton and Leibniz we may write:

Here the integral on the right side differs from the primitive F(x) only by the constant F(a). In such a case this integral, if we consider it as a function of its upper limit x (for variable x), is a completely determined primitive of f(x). That is the importance of *the enclosure membrane to define a single organism, and establish its order, as opposed to the entropic, multiple open future of a non-enclosed vital function which will scatter away.*

Consequently, an indefinite integral of f(x) may also be written as follows:

where C is an arbitrary constant, *the enclosure will eliminate. *

**Linearity: **Yet and this seems to contradict the previous finding, when we operate derivatives with the ‘basic dimension’ of social herding, ± operators, linearity comes back, and so the minimal Rashomon effect give us two explanations:

**Γ(st): **We are INDEED herding in the base dimension of a single plane, where each derivative will now be considered a fractal point of its own:

**∆+1 perspective:** Suppose f(x) and g(x) are differentiable functions and a and b are real numbers.

Then the function h (x) = aƒ(x) + bg (x) is differentiable and h’ (x) = a ƒ'(x) + b g'(x), which is really the *distributive law already studied in algebra’s post for x and power law. So the interpretation of the sum rule from ∆+1 is one of ‘control’:*

*WHEN operating from a whole perspective, the whole breaks the ‘smaller’ parts and its simpler dimensional operandi, +, to treat each part with its ‘whole action’ (in this case ∂). In brief the whole totally control the parts.*

**Finitesimal Quanta, as the limit of populations in space and the minimal action in time.**

So there is behind the duality *between the concept of limits and differentials (Newton’s vs. Leibniz’s approach), the concept of a minimal quanta in space or in time, which has been hardly explored by classic mathematics in its experimental meaning but will be the key to understand ‘Planckton’ (H-planck constants) and its role in the vital physics of atomic scales.*

It is then essential to the workings of the Universe to fully grasp the relationship between scales and analysis. Both in the down direction of *derivatives and the up dimension of integrals; in its parallelism with polynomials, which rise dimensional scales of a system in a different ‘more lineal social inter planar way’.*

*So polynomials and limits are what algebra is to **calculus; space to time and lineal algebra to curved geometries.*

The vital interpretation though of that amazing growth of polynomials is far more scary.

Power laws by the very fact of ‘being lineal’, and maximise the growth of a function ARE NOT REAL in the positive sense of infinite growth, a fantasy only taken seriously by our economists of greed and infinite usury debt interest… where the eª exponential function first appeared.

The fact is that in reality such exponentials only portrait the decay destruction of a *mass of cellular/atomic beings ALREADY created by the much smaller processes of ‘re=product-ion’ which is the second dimension mostly operated with multiplication (of scalars or anti commutative cross vectors).*

So the third dimension of operandi is a backwards motion – a lineal motion into death, because it only reverses the growth of sums and multiplications polynomials makes sense of its properties.

As we apply OPERANDS rules to particular cases, the interpretations vary but in all cases will be able to be interpreted in terms of sub-equations of the fractal generator.

What might be notice in any case is that unlike in our rather ‘abstract’ dimensional explanation of the rules of power laws, here we are able to bring real vital analysis of those roles in terms even of biological processes, showing how much more sophisticated is the ∫∂ operandi, the king of the hill of mathematical mirrors on real st-ep motions and actions, reason why its use is so wide spread.

SO THE FUNDAMENTAL LAW OF OPERANDS TO VITALIZE THEM IS THIS:

‘BY THE RASHOMON EFFECT ALL differential OPERANDS CAN BECOME AN ACTION IN ONE OF THE 5D DIMENSIONAL VOWELS (A,E,I,O,U) THAT DEFINE THE FIVE dimensions OF EXISTENCE, AS VITAL QUANTA-ACTIONS OF THE BEING.

THIS IS THE LOGIC CONCEPT THAT TRULY VITALIZES THE OPERANDI OF ALGEBRA.

So those properties tell us new things about the meaning of ∫∂.

## @-**GEOMETRY EXPANDS ALGEBRA**

ALGEBRA really starts to understand EQUATIONS with more sophistication that geometrical equations, when *motion parameters are introduced by analytic geometry and mathematical physics*, as the X and Y coordinates are now used for t-motions and S-pace (as in the simplest physical equations of space-distances made at certain speeds) that it can mirror S≈T Symmetries, first in a subconscious form through equations born on praxis, then in the 3rd age with some deeper insights on the concept of symmetry (group theory and its application to physics) which we shall complete with its full causal realisation – since we said ‘algebra understands’, *not* the people that make algebra.

The classic ST age of algebra thus saw the transformation of pure arithmetics of numbers into a mixture of the ∆ST elements of maths as a full reflection of the ∆•st universe. And the two huge figures that did it properly were Descartes and Leibniz (Fermat and Newton, in parallel but without publishing and the same clarity).

It is on my view the golden, classic age (if we add analysis) of algebra, before it enters the 3rd age of excessive information (attempts to put all the information in a single mind mapping with group theory and functionals and §ets).

We study most of analytic geometry in the post dedicated to it. So goes with analysis. Here we shall make just a few considerations.

It all started with the parallel evolution by Viette onwards of symbolic terms TO CONCENTRATE no longer in numbers BUT IN OPERANDI, THE TRUE essence of S=T symmetries…

**Descartes: merging all the elements of ∆@s=t maths.**

…And Descartes idea of representing solutions to equations *with a larger dimension – the variable letter that represented all the ‘§ets’ of dual X, Y possible solutions; and to ‘imagine’ them in a graph to plot them, forming a visual ‘in-form-ative’ geometric figure, the new ‘scalar dimension‘ that gathered all the X(S)<≈>Y (t) pairs of possible ‘variations’ on the space-time construct.*

FURTHER on, he introduced, @, the point of intersection of the coordinate axes, having coordinates (0, 0) and hence a ‘p.o.v.’ or singularity.

And with the introduction of coordinates he constructed an “arithmetization” of the plane. Instead of determining any point geometrically, it is sufficient to give a pair of numbers x, y and conversely.

Up to the time of Descartes, where an algebraic equation in two unknowns F(x, y) = 0 was given, it was said that the problem was indeterminate, since from the equation it was impossible to determine these unknowns; any value could be assigned to one of them, for example to x, and substituted in the equation; the result was an equation with only one unknown y, for which, in general, the equation could be solved.

Then this arbitrarily chosen x together with the so-obtained y would satisfy the given equation. Consequently, such an “indeterminate” equation was not considered interesting.

Descartes looked at the matter differently. He proposed that in an equation with two unknowns x be regarded as the abscissa of a point and the corresponding y as its ordinate. Then if we vary the unknown x, to every value of x the corresponding y is computed from the equation, so that we obtain, in general, a set of points which form a curve.

*The deepest insight on what Descartes did is then evident: *

HE GAVE MOTION=CHANGE TO GEOMETRY, ADDING ITS TIME-DIMENSION; AND SO its method could be used to study the actions/motions of a ‘fractal point’ whose inner geometry of social numbers was NOW ignored, in the ∆+1 scale of its world.. And so the graph would be a perfect graph to study all the ACTIONS=MOTIONS external to a given being, becoming for that reason the foundational structure of mathematical physics.

This is often forgotten, as S and T dimensions are ill understood so for example, our template book defines it as:

“Analytic geometry is that part of mathematics which, applying the coordinate method, investigates geometric objects by algebraic means.”

Not so… even if in analysis we will find that the curves DO represent key features of the ‘arrows of change’ of the Universe, specially the ‘standing points’ of change of parameters of Space=Information, ST=energy and Time=entropy (or any other kaleidoscopic combination of ST), in essence *they represent the world cycle of the action or motion we study, with its 3 phases of starting motion, steady state, and 3rd informative age coming to a halt.*

It must be then understood, as evident as it is, *that the Rashomon effect should consider different perspectives on those curves and forms found in analytic geometry, expressing algebraic equations:*

**-Temporal view:** the curves are then meaningless in space. What matters is their ‘social dimension’ that resolves symmetries between time dimensions expressed by the two variables often a parameter of space that changes with a dynamic function/action/motion in time.

**-Spatial view: **It is still though possible to create meaningful closed forms, ∆+1 wholes of geometry, made of ∆-1 points, and then the geometry allows to resolve algebraically geometric spatial problems, with ‘a dual point of view’ that increases the easiness of solutions – as Descartes proved easily and Galois completed, showing the algebraic laws of solution of rule and compass geometrical problems.

– **S=T** view: when one of the parameters/dimensions is fixed, belonging to space and the other to a time motion, the most fruitful in symmetry, soon used by Galileo and Newton to develop the laws of lineal time motion in space.

–**@** views: soon to be developed as 3 different mappings, will develop (O-polar, |-cylindrical and ST-cartesian proper)

**Scalar view,**which will have to wait till Leibniz, treated in ∆nalysis.

Let us then consider some aspect of the essential elements of those equations, mostly polynomials as analysis is treated in its own right.

**The earliest age of analytical geometry. The Arabs.**

Algebra we said started as a prolongation of arithmetics and geometry. So its first age was really true to the quote of Germain: a written geometry. This first age went as far as the renaissance, and consisted in calculus of ad maximal cubic roots, and geometric proofs (greeks, al-aljoarizim which gave its name to the discipline, etc.)

*Polynomials of higher, >2 degree as combinations of simpler bidimensional $-ð systems.*

The graph shows the maximal depth of this age when the poet Kayyam solved the simplest cubic equations. It is remarkable that now and since Apollonius and Archimedes, the understanding of ‘square dimensions’ *transcends the mere spatial square, introducing ‘temporal’ functions, such as the parable, whose value as motion=time dimension will be fully realized with Galileo’s study of cannonballs. *

It shows also a quality of the simplest easiest to solve roots of such equations, which are NOT really ‘cubic’ dimensional growth *but intersections between two figures, hence an S(x) ≈ T (x) symmetry, in this case the parable and the circle when properly written as Kayyam did. *

And this brings the question on how so many mathematical S≈T symmetries become hidden by the ‘mania’ of scientists to find only the solution, and packing all the variables into one side, putting in the other a zero… when the existence of a zero *most often means there is indeed a possible symmetry when both sides of the equation part their ways. A fact truly important to interpret physical equations in terms of those S(x), T(x) symmetries.*

Indeed, what the graph shows is that a *cubic equation, according to the holographic principle, is most likely a combination of an $-motion or 4D-entropic, open expanding curve (the parabola) and a ð-motion, the circle:*

In the graph, repeated ad nausea, for clarity, we see the general rule: a bidimensional St system of information (still space), or a time clock (moving cycle), the 3rd dimension of reality; and a vector of lineal time motion or its bidimensional sheet of spatial distances, the 1st dimension of reality, come together into a the 2nd Dimension, ST system of energy, or time or motion.

So 3Dimensional systems tend to be the intersection of a line and cycle, which in geometry is expressed by the rule that almost all functional dominant forms of the Universe can be traced with a line-ruler and a compass-cycle.

Generally speaking algebra is then just a mirror image of the geometry of the age, specially in the calculus with pythagoras like theorems of square roots and the simplest ±, ≈ X ÷ operandi *without considering the mirror image of those operandi in other ∆st elements and symmetries of the Universe.*

**Inverse operandi.**

The concept of ratios vs. multiplication – the 2 inverse arrows of 4D and 5D actions (feeding and breaking vs. social evolution), were neither understood – they are not yet.

*axons= ST flows of communication of 2 wholes that reach through X its maximal communication. So its role is*to create a network on the lower scale of systems – in brains, neural networks or reproductive systems,

*reason why so often a reproductive action that creates a third element as in vectorial products is symbolised by a reproduction*.

*the division actually is not made, meaning that the whole remains – the string we plug to get a sound by making it vibrate with a certain number=frequency of nods.*

*is an insight of ∆st on irrational numbers, the operation is not made; hence the number exist as a whole, which is expressed in the numerator, sum of the parts which are expressed in the denominator.*

**First degree equation**.

Making use of two simple ideas, Descartes first of all examined what curves correspond to an equation of the first-degree:

i.e., to an equation where A, B, C are numerical coefficients with A and B not both zero.

As we have seen this is the ‘dimensional natural growth from sum into a sums of sums or multiplication understood in terms of its simplest dimensional combination’: $ x D:

So obviously Descartes found that in the plane a straight line always corresponds to such an equation. And conversely, that to every line in the plane there corresponds a completely determined equation of the form:

where:

y = kx. obviously represents a straight line passing through the origin and making an angle ϕ with the x-axis whose tangent tan ϕ is k and L the distance from 0 to the crossing point of the line and y. Thus as usual the simplest, lineal Spe element was discovered, and it is the easier to calculate, *but in a world dominated by time cycles:*

“Using a term like nonlinear science is like referring to the bulk of zoology as the study of non-elephant animals.” Ulam.

So *there is an enormous number of scientific errors, including the lineal big-bang caused by the ab=use of lineal **approximations to functions, and the ‘spread’ of the function into the negative ‘side of the line’ (case of the BB) as if it were to behave always lineally in the past-negative side of the graph. *

In fact most ‘real functions’ when we fully grasp 1) the distortion of past-time and negative numbers and the ‘conversion’ of a cartesian graph into a ‘conic’ tend to have an e-like form.

In the graph, the distortion of ‘lineal’ self-centred functions is a plague of sciences, due to its easier calculus and the distortion of perspective caused by the human mind, as Descartes coordinates are a representation of the human lineal, Euclidean, electromagnetic light-mind geometry. *In the next graphs, *from the 92 book ‘the error of Einstein’ we show what we mean:

The mind equation is:

* O (infinitesimal pov) x ∞ Universe (∞ cycles and monads) K-mind*

It implies that the 0 believes to be infinite self-center of the Universe and sees it all in a distorted perspective. In brief, your eye pov is bigger to you than the Andromeda Galaxy, so the 0-point becomes an infinity in itself (your eye≈Andromeda Galaxy), and this makes the conic, which could be considered an ‘objective’ angled point of view (where the mind-eyes is truly zero), to expand in the y-coordinates of the mind-point to infinity; alas! transforming the conic into the Cartesian plane where ¥ becomes also an infinite graph.

**Conics and circles.**

IN THE Growth of dimensions by multiplication it is obvious that the most abundant combination will be that of an $t dimension and an §ð dimension, creating a full present ST-system, according to the canonical generator: $t<ST>§ð.

So after doing all what they could do in bidimensional geometry with a ‘ruler and a compass’, the Greeks ‘finally’ raised one circle with a line into a conic in space, which then will become the canonical space-time surface also when time was added to it (4D formalism).

Now in as much as a double conic is a hyperbolic geometry, we can define the Cartesian plane as the ‘hyperbolic’, Present, ST plane, more ‘expansive’ in its capacity to show different ∆st events and forms of the Universe.

We only got a grasp of this fact, of lately in the study of topology and models of hyperbolic geometry, when we find that *a hyperbola, which seems to us infinite in its ¥ coordinates, becomes equivalent to a circle . *

*Mathematicians say that the hyperbola is isomorphic to the circle; and write it as:*

If u=x+yi and v=x−yi then:

ℂ[x,y]/(x²+y²−1)≃ℂ[u,v]/(uv−1)

This can be viewed in algebraic geometry but better on to understand is mind meaning in projective geometry:

Both the hyperbola and the circle are conic sections, and are projectively equivalent. In analytical geometry, in homogeneous coordinates this follows from the fact that any pair of nondegenerate indefinite quadratic forms are bidimensionally equivalent, and so *we can transform them into each other, as it is required by the generator of coordinates.*.

This beyond ‘philosophy of the mind’ matters because it justifies the fact that *the curves of analytical geometry can be also obtained from cuts made on the cone, which in itself is merely a reflection of a **world cycle, pegging the two inverse directions of existence. *

**The conic as a representation of a world cycle.**

The cone is basically a ‘bidimensional ST being, circling inwards towards the singularity of the point along a line of geometry, hence combining O x |= Ø, the cone represents the Universe. It should not be surprising then that it is the best way to reflect a 4D block of space-time (but not the discontinuities of the 5D Universe) as used by Einstein’s physics and Minkowski’s geometry (light cones); and we shall deal with those cones in the analysis of relativity.

We shall also use them in ¬time logic to represent the 3 fundamental events in terms of time ages: entropic collisions (future x past = past wave of entropy, reproductive iterations (present x present = present), and evolving, informative events, past x Future = Future wave of information):

In the graph, first the 3 canonical events of complex i-logic time, and the much simpler single cone of ‘lineal time’ in relativity.

What matters to us is now is that a conic basically reduces to a line and a shrinking cycle, which in motion is equivalent to a time cycle, shrinking into a point-singularity, the vertex of the cone. And for that reason that simple canonical world cycle encodes all the main curves of bidimensional space.

**LINEAL ALGEBRA.**

**Lineal equations as approximations of curves.**

The linear function l (x) = ax + b gives the simplest of all curves, namely the straight line AND YET it is one of the most important due to the fact that every “smooth” curve on a small segment is like a straight line, and the less curved the segment is, the nearer it comes to a straight line.

All THIS resumes a concept brought about many times in the blog: a larger perspective is a wholeness, which is a closed zero sum and hence cyclical, curved, a step of the curve however appears in small distances as a lineal, open step.

In the language of the theory of the functions, this means that every “smooth” (continuously differentiable) function is, for a small change of the independent variable, close to a linear function.

The linear function can be characterized by the fact that its increment is proportional to the increment of the independent variable.

Indeed: Δl(x) = l(x0 + Δx) − l(x0) = a(x0 + Δx) + b − (ax0 + b) = a Δx. Conversely, if Δl(x) = a Δx, then l(x) − l(x0) = a(x − x0) and l(x) = ax + l(x0) − ax0 = ax + b, where b = l(x0) − ax0.

But from the differential calculus, we know that in the increment of an arbitrary differentiable function is proportional to the increment of the independent variable, and that the increment of the function differs from its differential by an infinitesimal of higher order than the increment of the independent variable.

Thus, a differentiable function is, for an infinitely small change of the independent variable, really close to a linear function to within an infinitesimal of higher order.

The situation is similar with functions of several variables.

A linear function of several variables is a function of the form a1x1 + a2x2 + ··· + anxn + b. If b = 0, the linear function is said to be homogeneous.

A linear function of several variables is characterized by the following two properties:

1. The increment of a linear function, computed under the assumption that only one of the independent variables receives some increment while the values of the remaining variables are unchanged, is proportional to the increment of this independent variable.

2. The increment of a linear function, computed under the assumption that all the independent variables obtain increments, is equal to the algebraic sum of the increments obtained by changing each variable separately.

Thus a linear problem can be characterized by 2 properties:

1. The property of proportionality. The result of the action of each separate factor is proportional to its value.

2. The property of independence. The total result of an action is equal to the sum of the results of the actions of the separate factors.

The fact that every “smooth” function can be replaced in a first approximation by a linear one, for small changes of the variables, is a reflection of a general principle, namely that every problem on the change of some quantity under the action of several factors can be regarded in a first approximation, for small actions, as a linear problem, i.e., as having the properties of independence and proportionality. It often turns out that this attitude gives an adequate result for practical purposes (the classical theory of elasticity, the theory of small oscillations, etc.)

**Matrix – the 3rd dimension of lineal algebra.**

We have stated many times that the Universe grows by ‘fixing motion-steps’ into a whole ‘cyclical form of space’ that then moves into motion steps and so on till ‘filling up the 5D2 dimensions of reality.

And that *such increases are smooth-continuous only for the 3 fundamental Dimensions of a present space-time, breaking in the ∆-1 4D entropic and ∆+1, @ 5D levels, which means it ‘suffers’ a “Lorentzian region” of acceleration or deceleration inwards or outwards that changes the parameters in a different way (susceptible however to be analysed with differential equations that measure small changes.*

This means specially for ‘lineal equations’ that there are 3 levels of growth in complexity, from single lineal equations to multiple lineal equations and finally the grouping of those into ‘matrices’ as we make through representation theory convert some parameters (as in quantum physics) of T.œs=fractal points with multiple inner parts into the ‘new algebraic element’.

Indeed, the physical quantities to be studied are often characterized by certain numbers (a force by the three projections on the coordinate axes, the tension at a given point of an elastic body by the six components of the so-called stress tensor, etc.). Hence there arises the necessity of considering simultaneously several functions of several variables, and, in a first approximation, of several linear functions.

A linear function of one variable is so simple in its properties that it does not require any special study. Things are different with linear functions of several variables, where the presence of many variables introduces some special features. The situation is still more complicated when we go from a single function of several variables x1, x2, ···, xn to a set of several functions y1, y2, ···, ym of the same variables. As a “first approximation” there appears here a set of linear functions:

The set of coefficients of a system of linear forms can be given then the form of a rectangular array:

Such arrays bear the name of matrices. The numbers aij are called the elements of the matrix.

Important special cases of matrices are the matrices that consist of a single column, which are simply called columns, those that consist of a single row, called rows, and finally the square matrices, i.e., those in which the number of rows is equal to the number of columns; called its order (a).

Thus *matrices, specially square ones can be considered if we define an Y(S**t) = X(sT) FUNCTION in which one variable most likely a Spatial whole, polynomial combination of a series of temporal steps-motions; a symmetry of spatial wholes and temporal variables in equal quantities, ∑S≈∑T.*

AND as the Spatial wholes will be ‘varieties’ of the same temporal ‘steps’, the structure interconnected at ‘expanded’ to the 3 levels of complexity has a very rich capacity to picture complex space-time systems in all its variations and symmetries, reason why Matrices have become the best-suited structure for complex, lineal systems of ‘very small scales’ where information about fast multiple T.œs come together and have to be studied and fixed in space from the larger, slower human perspective:

i.e. quantum systems where* the Heisenberg matrix formalism is THE algebraic frozen symmetry of Schrodinger’s dynamic ‘differential equation’*, a conundrum, never clarified, now explained tersely as the natural consequence of the ‘space-symmetric nature of algebra’ vs. the ‘steps-motions description of Analysis’.

We shall not keeping with the limits of an encyclopaedia written by a single man, occupied most of his time with self-destruction go into techniques of Matrix manipulation. Just to state that the main difference between Matrix and other structures of algebra is its obvious non-commutability (as rows and columns multiplied one by one in the well-known inverse orderly fashion) and its non solvability for multiple cases (as rows and columns must coincide).

In a larger philosophical way this simply means that Mathematics as all languages is ultimately inflationary, with more imaginative ‘forms’ that real solutions, which leads us to the GREAT question of the first algebraic age – THE SOLVABILITY OF POLYNOMIALS HIGHER THAN 2 *in a Universe where lineal structure and continuity breaks beyond the third polynomial since ST dimensions come onto pairs… *

**¬Æ POLYNOMIALS**

Next in the understanding of reality comes the consideration of operands combined to form functions. And here it comes the question on how to group them.

As it happens there is a natural grouping of those Dimotional operands based in its complexity and the natural relationship of the Dimotions=actions of existence, such as we group naturally the ±, x÷ and Xª operands, which represent a ‘series’ of simpler operations in a single ‘parameter-variable’ in polynomials, therefore the simplest functions of ‘existence’ (as we mean by existence, the consecutive sum of actions=dimotions, represented by those operands).

*As they are clearly related to reproduction, social evolution in a single ‘LINEAL’ PLANES OF SPACE-TIME.*

This leaves the more problematic, complex Dimotions of ‘angular perception’, ‘exponential decay’ and differential calculus, which clearly relate to *MOTIONS through SCALES OF THE FIFTH DIMENSION, as operand* which naturally group together in functions notably of mathematical physics.

So polynomials are simple functions of social reproduction, and it bears witness to the rich complexity of the pentalogic Universe that even such simpler forms in a single plane can approach all other functions (Taylor’s series), when we enrich them with some of the complex scalar Dimotions of angular perception (¡-3>¡o), exponential entropy (¡<¡-2) and lineal derivatives (¡<¡-1), which are the natural Dimotions of scalar space-time.

**PENTALOGIC**

Polynomials must then be understood AS EVERYTHING with pentalogic that is, in its multiple functions and applications to all the elements of ¬∆@ST, *according to the values of the X-variable*:

SO WE TALK OF:

- SPACE POLYNOMIALS: When the x-variable represents a dimension of space, point, line, plane and volume.
- TIME POLYNOMIALS: When the variable represents a motion of time, distance, speed or acceleration.
- ∆-scale POLYNOMIALS: When the variable represents a scalar function, in series of diminishing values.
- ¬ Entropy POLYNOMIALS: when the variable represents a scalar function, in a series of growing value, or an exponential form.

Now of all those points of view of polynomials, some are more useful than others, as PENTALOGIC ALWAYS forces an entity or event in some of those ‘5D motions’, which are not its purpose though they will be mirrored by all Dimensions – i.e. a woman tends to reject entropy and prefer information but it dies always.

So we could ‘state’ that the 5th dimensions of existence touch any ‘whole system’ which is ‘coherent within itself’ , in the case of languages as a mirror of the world.

This said the MOST IMPORTANT MIRROR any SYSTEM CAN represent is a WORLDCYCLE OF EXISTENCE, WHICH IN ITSELF INCLUDES ALL THE OTHER DIMOTIONS.

So we shall consider how Polynomials in a plane represent that world cycle too.

To notice also that polynomials not only can ‘approach’ as a mirror any other function and DImotion of the Universe, with certain distorsions as the case of the maclaurin series approximations to calculus shows, but many of them HAVE nothing to do with reality, showing indeed the inflationary nature of languages, whose internal structure might or might not be real or fiction, and so since Godel we know they cannot prove reality.

Equations therefore must be always put in relationship with the reality they describe first.

**Polynomials as divergent or convergent scalar series.**

In mathematics, a **power series** (in one variable) is an infinite series of the form

where *a _{n}* represents the coefficient of the

*n*th term and

*c*is a constant.

*a*is independent of x and may be expressed as a function of n (e.g., an=1/n!). Power series are useful in analysis since they arise as Taylor series of infinitely differentiable functions.

_{n}In many situations *c* (the *center* of the series) is equal to zero, for instance when considering a Maclaurin series. In such cases, the power series takes the simpler form

Any polynomial can be easily expressed as a power series around any center *c*, although most of the coefficients will be zero since a power series has infinitely many terms by definition. For instance, the polynomial f(x)=x²+2x+3 can be written as a power series around the center c=0 as

or around the center c=1 as

or indeed around any other center *c* One can view power series as being like “polynomials of infinite degree,” although power series are not polynomials.

The geometric series formula

which is valid for |x|<1 is one of the most important examples of a power series, as are the exponential function formula

- =

and the sine formula

valid for all real x.

These power series are also examples of Taylor series.

We shall then in other posts consider their relationship with those functions, which are the key DIMOTIONS of scalar motion (1/1-x), entropy (exponential) and 1Dimotion (Sin).

**Geometric series**

A geometric series is a series with a constant ratio between successive terms. For example, the series

is geometric, because each successive term can be obtained by multiplying the previous term by 1/2.

Each of the purple squares has 1/4 of the area of the next larger square (1/2×1/2 = 1/4, 1/4×1/4 = 1/16, etc.). The sum of the areas of the purple squares is one third of the area of the large square.

We can then consider to be a series that diminishes till it reaches the ‘finitesimal’ 1/n part of the whole. And it can easily be casted as a polynomial; since the terms of a geometric series form a geometric progression, meaning that the ratio of successive terms in the series is constant. This relationship allows for the representation of a geometric series using only two terms, *r* and *a*. The term *r* is the common ratio, and *a* is the first term of the series.

In the example we may simply write:

- , and .

The behavior of the terms depends on the common ratio *r*:

- If
*r*is between −1 and +1, the terms of the series become smaller and smaller, approaching zero in the limit and the series converges to a sum. In the case above, where*r*is one half, the series has the sum one. - If
*r*is**greater than one**or**less than minus one**the terms of the series become larger and larger in magnitude. The sum of the terms also gets larger and larger, and the series has no sum. (The series diverges.) - If
*r*is**equal to one**, all of the terms of the series are the same. The series diverges. - If
*r*is**minus one**the terms take two values alternately (e.g. 2, −2, 2, −2, 2,… ). The sum of the terms oscillates between two values (e.g. 2, 0, 2, 0, 2,… ). This is a different type of divergence and again the series has no sum; for example in Grandi’s series: 1 − 1 + 1 − 1 + ···.

Geometric series are among the simplest examples of infinite series with finite sums, although not all of them have this property. Historically, geometric series played an important role in the early development of calculus, and they continue to be central in the study of convergence of series. Geometric series are used throughout mathematics, and they have important applications in all sciences, as all of them are obviously scalar in its form, and respond to any of the 3 possible behaviors of systems, ‘convergent information’, divergent entropy and repetitive=reproductive oscillation.

**Dimensions of a polynomial ‘background space-time’**

Polynomials are NOT isolated as equations but in the entangled Universe, *in reference to the plane in which they are written. This is important not to confuse its meaning. A polynomial in a 3D world, of the type x³ is a cube. The same function below, has NOTHING to do with a cube, but it is a curve of growth diminution and growth on the Y-coordinates. *

THUS the ‘perspective’ of graphic polynomials is that of TIME not OF SPACE or one of its combined ‘holographic, bidimensional forms’ of timespace:

And this made them much more interesting. We can then observe polynomials as ‘curves’ in a plane, no longer as in the age of arithmetic, as spatial forms – this being a trend of all forms of human thought, which start to see things as spatial, fixed forms and slowly gift them with vital motions.

What is then the main difference of such polynomials? We can talk of them in terms of space-time symmetries, as even functions, which are symmetric along the ± axis and asymmetric odd functions, which have an inverted symmetry on the ± sides. And consider again the polynomials of order 4 as the limit of solubility (Galois) as 4 are indeed the elements of reality – let us elaborate a bit more on those first principles of 5D polynomials:

In the graphs, the solvable polynomials define a limit of 3 dimensions of space-time, and a 4th and fifth dimensions that *cannot generally be resolved by radicals in a single Plane, hence belonging to the 4 D S∂ and 5D ∫@ DIMENSIONS, which* warp the whole, as it emergence into the fifth dimension, hence making impossible further dimensions in a single continuum**. **Notice that the fourth dimensional graph must be interpreted often turned upside down as it is a ‘death’ inverted time arrow.

They define the bidimensional tridimensional fourdimentional and five dimensional arrow and similar systems in those dimensions.

The first is a mere motion of an st-lineal trajectory proper of the concept of speed (1D) or 3D (closed cyclical vortices.. THE third dimension though is more sophisticated.

**Polynomials as worldcycles. **

The roots of a polynomial when properly written as F(Polynomial)= 0, represent the Y=0 values of its ‘roots’.

This brings an interesting representation of polynomials as worldcycles of existence, which will have a ‘melody’, with a beginning and an end in his first and final touch with the Y-line at zero point, since * all worldcycles of existence are a zero-sum value for the ‘function’ that represents them.*

Thus if we consider a polynomial a representation of a parameter of a world cycle of existence (a conserved quantity, energy or momentum being the most suitable ones), we can then consider each of those graphs a representation of a different world cycle, and it is remarkable enough to see that they are all lying in the ± sides of the Y-line. So we shall explore this parallelism in the section of Algebra in more detail.

Now, mathematicians affirm that there are not meaningful polynomials of degree higher than four (whose solubility is possible with coefficients of its powers) and this implies that there are at best 4 roots, or zero points in a world cycle of existence. In the left if we represent it as a full circle those points will be both the x and y 0s.

If we then consider the first point birth, and the final point death, obviously eliminating as irrelevant the polynomial before and after those points for those ‘meaningful functions’ which truly represent a world cycle, we could then talk of the negative ‘parts of the graph’ as those happening in the emergent palingenetic process from ‘cell to individual’, and the positive part its emergence in the larger ¡+1 world.

We shall see though when studying specific equations for worldcycles of existence, that they are better represented by positive functions with the ‘0 points at the beginning and end of the cycle’ (bell curves), and with sinusoidal functions…

__While the duality of the emergent phase and world phase of an exist¡hence is best mirror in the o-1/1-∞ unit sphere plane (palingenesis) and Cartesian plane.__

But the concept of an y=0 point as the limiting birth and death of a system will remain.

**FERMAT’S THEOREM**

*equal species’. And this has an unexpected proof,*

**IN A MARGIN.**

*Its means that that when we ad we superpose, so a 3rd dimension expressed by a power law is no longer ‘a holographic superposition’, reason why the Fermat’s theorem, X³ + Y³ ≠ Z³, does NOT work.*

*the full consistency between contiguous dimensional growths (±) BREAKS between discontinuous dimensions, from 1D (sum of herds) to 3D (merging of 2 holographic bidimensional sheets into a third one through product):*

In the graph, the holographic principle is expressed by the operation of addition, which is allowed by superposition into a tridimensional volume.

*there is not a 4Dimension in the same scale of space-time, the rule of superposition through a new Dimension breaks for superposition of cubes, which would have to be added in this supposed 4 Dimension in a single plane, reason why we cannot add them (Fermat’s theorem).*

The holographic bidimensional universe and its ternary ST-geometries define reality. So in most mathematical equations solutions abound on quartic and cubic systems but only special cases are solvable for higher polynomials or have any real use in reality; the exception being simpler equations of the ∆§ocial scales and reproductive functions of the type Xª=b…

Quadratics are the masters of the algebraic game, the most abundant an common of all forms in existence, *because the Universe is bidimensional, and so it is information. So quadratic equations by definition are the perfect form to show the properties of the Universe of fractal bidimensional space and informative, bidimensional time.*

The bidimensional holographic principle explains why in geometry (greek bidimensional plane, which proved almost all the theorems of geometry) and algebra (quadratics) almost all phenomena of the Physical, topological universe can be ‘carried’ on to quadratic algebraic equations. We study the main forms of quadratics in those other ‘parts’.

Its addition can be forced-fed (4D spatialisation of the time dimension) in modern physics, but as Einstein put it lineal time doesn’t travel backwards… indeed, if it does so it breaks into an entropic explosion, loosing its internal ∑∆-1 bonding by a whole. So we cannot get consistent results for an addition of cubes in a single space-time and inversely for its dissection, reason why *the addition of cubes cannot either be resolved in bidimensional plane geometry with a line-rule and a cycle-compass. *

*both approaches must be similar in its results, as they essentially observe the same phenomena with different focus, which is the reason of the existence of the Taylor and Newton’s approximations through derivatives of any*

*polynomial:*

So polynomials of algebra – NOT the other way around – are the roughapproximation to the more subtle methods of finding dimensional change proper of analysis – even if huminds found first the unfocused polynomials and so we call today McLaurin & Taylor’s formulae of multiple derivatives, approximations to Polynomials.

In the graph the McLaurin and Taylor series * show the limits of a ‘working function’ as they diverge from the function, once we move further away from the Point of view {ƒ'(o)} of the frame of reference, since they diverge, once passed the correct ‘degree’ of usefulness of the series – save for the exponential of ‘entropic decay’ whose derivative is the same function. *So we can often use the limits of similarity between the series and the real function to find its ‘domain of finite’ use closing its world cycle, i.e. the sine function approaches only in the domain of a 2π cycle.

*are lineal operators, of sums and multiplications power 5 polynomials could not be resolved, as they were prying in non-lineal regions of ∆±i space-time.*

__The existence of limits to any lineal approximation of the cyclical reality of the Universe, a deep philosophical question studied in our analysis of the divergence and creation of futures that are ultimately closed is also deep, an in essence establish a limit of validity for any lineal approximation, obvious in the Taylor series of differential around a 0 point, which only approaches (with the notorious exception of the exponential decay function of pure entropy, in which both coincide), the curved function to a limit NORMALLY THAT OF A SINGLE CYCLICAL REPETITION OF THE FUNCTION (as in the sine approximation).__**The fundamental ****theorem of algebra. The ∆@st symmetries reflected by polynomials.**

The fundamental theorem of Algebra proved by Carl Friedrich Gauss in 1799. It states that every polynomial equation of degree n with complex number coefficients has n roots, or solutions, in the complex numbers…

But why THERE ARE for any polynomial of power x, x solutions in the realm of complex numbers, which implies also a better understanding of a complex plane, so far lacking in mathematics.

And the answer is as profound as general in 5D mathematics as it is in algebra:

*“Each root of a polynomial represents a Dimensional or scalar motion of space-time.”*

And so bidimensional polynomials represent holographic functions of space-time. Tridimensional polynomials are ternary representations of 3 symmetric scalar, space and time elements and beyond the fourth ‘element’ entangled in the Universe (symmetry between space, time, scale and mind, which can be represented by the ‘same value numbers’, hence by polynomials, there are NOT more SYMMETRIES of reality (entropy being the denial an erasing of the other 4 elements) which means basically that in the same manner we do NOT have use for integrals and derivatives beyond the 3 scales of a super organism, we do NOT have use for polynomials beyond the 4 symmetries of the entangled Universe.

In other words, a polynomial can represent a T.œ as dust of space time in its four fundamental elements, whose roots being symmetric to each other, as elements of the entangled Universe can permutate to find ‘real beings’ of the game of existence, even if the concept of polynomial is an absolute abstraction of its purest, simplest properties – a number.

As the polynomial requires the ‘equality’ of those entangled symmetries in praxis it really works basically for the 3 expansions of classic dimensions of space and classic dimensions of physical time (distance, speed and acceleration).

It is then far more enticing the study of the ‘transcendent’ operands of analysis, and a simple example of the difference will suffice:

When we work with a polynomial of a higher power we merely increase x by another product with x, or inversely we make a root that eliminates an x.

In the **∫∂ ***operands however, the results are different: ∂x³=3x², gives us in fact often a larger sum for smaller x… i.e. if x=2, x³=8 and 3x²=12… *

Then at 3, *both are equal, 27, and beyond 3, the power function grows faster. Why?*

As we said, reality is a ternary game, so when we go beyond 3 our understanding requires more subtle arguments. Below 3 what happens is the following: the polynomial is working on a single plane and does merely EVALUATE THE 3 DIMENSIONS IN SPACE or time motions of that plane without the need to INCLUDE THE SUPPORT OF ANY LOWER PLANE OF EXISTENCE, WHOSE INFINITESIMAL PARTS MUST BE LARGER THAN THE WHOLE ALONE, because they sustain the whole.

Indeed, when we evaluate an **∫∂ ***operand we are evaluating together both scales. The ∂ **values the total number of ‘finitesimals’ which must sustain the larger power whole.*

So the ∂x³=3x², is 3 times x², the lineal √root, because from that sum of infinitesimals, you need to sustain the whole that emerges from it, without destroying the parts that sustain it, and without eliminating the parts of the parts that sustain both. We are thus making a calculus of the ∆±1 ternary scales of the organism, and each one is of the same value: x²+x²+x².

And yet, there are NO solutions granted to a quintic function from radicals:

Because they have an odd degree, normal quintic functions appear similar to normal cubic functions when graphed, except they may possess an additional local maximum and local minimum each.

**ST Interpretation**

What this mean though in GST is far more interesting. Indeed if we consider polynomial, dimensional equations, of D-egree equal to its number of dimensions, it is then obvious *that each continuous scale of space-time does have 4Dimensions, which can be solved. Yet beyond those 4Dimensions the system ‘breaks’ into a new scale which is no longer in the same region of existence and hence cannot be calculated with the radicals of the same plane.*

Further on in the graphic, we can easily see that the 5D form can be ‘reduced’ into a 3-equation where the 3 central ‘st’ hyperbolic curve, becomes a around the y=0 point a single ‘higher’ whole, which ‘if reduced’ to a point, converts the quintic into a ‘ternary equation’. Thus a 5D polynomial is a 3D ∆+1 polynomial with an ST central part which deploys its hyperbolic minimal and maximal elements in the ‘higher plane’ region of the 5th dimension (truly a higher plane when we use complex numbers).

What then it means in general terms, the existence of several solutions to those equations? In the most abstract analysis of polynomials, if we deem a degree, D, a dimension D, then it is obvious that as we solve them we come down from the whole into the parts. Thus a quadratic bidimensional equation resolves into the ± inverse spe and tiƒ functions of the system. A cubic equation will solve into the Sp<st>Tif solutions/parts of the system; and a quadratic equation into the S, E, T and sub-dimensions of the system.

**∆-interpretation**

But of course a polynomial being the most abstract realisation of a GST FRACTAL equation can be many things and yet the beauty of it is that all find interpretation. So when we consider x to be a function of time, it is given us the arrow of ∆-1 information which has several values, as information increases downwards diversifing the system, but the inverse is NOT truth. While St=x² has 2 Spe and Tif roots, the square of a number is a single number, ratifying the inverse arrows of entropy and information upwards and downwards:

In the graph, upwards the polynomial looses solutions, downwards increases its information, normally breaking an st WHOLE INTO its SPE and Tiƒ components.

**Quadratics: open and closed**

IN THE HOLOGRAPHIC UNIVERSE then the most abundant of all equations are quadratic equations. We observe 3 types in correspondence with the 3 topologies in a single plane:

-O-closed are clock-like ones, transcendental polynomials with a cyclical form, which tend to defined 1D-3Dimotions.

-|-Open are on the the other handquadratics related 2-4Dimotions.

-Ø-Finally we find, transcental, hyperbolic sinusoidal forms, or st equations.

**Cubics.**

Alas, we enter now into a different territory; that of cubic representation in a quadratic plane of the 3 dimensions of a ternary universe in space (3 topologies) and time (3 ages). And so here 4 themes are fundamental to ¬Æ:

- How cubics might represent the ternary structure of the fractal generator.
- What distortions take place when we transfer the holographic Universe into cubic representations (when not using them to represent Γ)
- What are the restrictions the Universe imposes to ternary systems, where
*the ‘messed bidimensional, tight form’ becomes a loose configuration of layers of time-motion or space-density to become a 3rd dimensional ‘wide’, iterating being.*

For example, the Fermat Grand theorem, mentioned above, x³+y³≠z³ means a restriction of ‘tridimensional messed beings’, as all what we can expect are bidimensional perfect forms, accumulated in time, as slices of a motion, or in space as layers of an identical – number/population of bidimensinal beings.

- And finally, the translation of all the results of cubic equations,
*once we understand the previous rules of engagement, into meaningful laws of GSt (or vice versa)*

**Quartics.**

We said that quartics define systems of two topologies of space-time intersected into a single form; hence being natural solutions to the study of Sp<≈tiƒ systems. This intuitive thought on the nature of quartics comes to fruition when we observe that each coordinate of the intersection points of two conic sections is a solution of a quartic equation.

The same is true for the intersection of a line and a torus. Thus we find a quadratic solution the ideal ‘form’ of an intersection of and Spe and Tiƒ system, which constructs a whole form. And so no further polynomials are required,

**Some quintics with solution**

On the other hand the only quintics with solutions are not ‘really quintics’, but either systems that can be factored as smaller polynomials (hence each part of the polynomial being of a lower degree, and ‘unit of an S<st>t system); or can be depressed eliminating one of the roots, often the 4th giving birth to interesting solutions in which the 5th polynomial appears as equivalent to an scaling sum of the lower ‘planes’, *with scalar coefficients precisely of the key social numbers, 1,2,3 and t and 10. For example, in the graph the following are the only solutions for a depressed quintic, where the 4th polynomial disappears, and so we do have a relationship between the ∆+1, quintic and the coefficients that are ‘solvable, all of them in precise scales common to ∆>∆+1 processes.*

**The fundamental theorem of algebra.**

Now, all this said, there is a seemingly contradiction in our stressing of ‘no proper solutions’ for polynomials beyond the 3rd plane/dimension of growth, stated by its unfocused mirror better analysed in ‘analysis’ and the fundamental theorem of algebra states that every non-zero, single-variable, degree n polynomial with complex coefficients has, counted with multiplicity, exactly n complex roots…

Which is equivalently (by definition), to the theorem states that the field of complex numbers is algebraically closed, whose proof must use the completeness of the reals, *which is not an algebraic concept…*

As completeness implies that there are not any “gaps” (in Dedekind’s terminology) or “missing points” in the real number line since those “gaps” should be covered by ir(ratio)nal numbers, according to the non-proved completeness axiom, or the 2 theoretical methods used to prove its construction (Dedekind completeness and Cauchy completeness as a metric space).

We rebut Dedekind completeness proof i-logically when considering an absolute geometry without that ‘axiom’ in non-e geometry, as irrational numbers are NOT single plane numbers or ELSE pi would exist and so √2, but RATIOS OF CLASSIC actions of the generator,, such as pi = 3 $>ð, the transformative motion of 3 lineal steps which generate a ‘variable’ according to the curvature of the ‘mental space’, pi-ratio. So for √, the ratio of two perpendicular T.œs, colliding OR symbiotically becoming adjacent, creating a triangular space.

Further on, proofs must be consistent to be complete – and here is where the humind makes most errors. You do NOT proof an algebraic system absolutely with a geometric proof, as numbers and points, as we have shown in NUMBER THEORY are NOT equal but similar S=T mirror reflections.

Yes, there are always n-roots for a polynomial in the complex plane, but what we mean is that *they are NOT exact solutions, but beyond the 3rd power, approximate solutions, reason why for a complex polynomials TRYING TO REFLECT REALITY, often the best solution is NOT the polynomial approach but the derivative approach, as said before.*

On this a final coments: While polynomials cannot be always resolved by lineal coefficients, proper of the balanced central region of any Plane of social evolution, ∆§, because the ∆-representation of complex numbers is really a ‘square’ graph, as we showed on our analysis of its Rashomon effect (argand, polar, ð-numbers, and so on), in practical terms, this means we halve the polynomial degree and Xˆ10 becomes Xˆ5, the limit of dimensional growth through ∆§cales and planes… And here we can find at least a meaningful polynomial approach. Beyond that is truly inflationary mathematics; and some not-so-exact-as they think string theory bullshit (:

**II AGE: ****ST-SYMMETRIES **

**OPERANDS & GROUPS**

**Foreword. The heart of the matter: group symmetries, operands details.**

Symmetry is the central concept of Group theory, which became in the XX century the ice in the cake of the whole structure of Algebra.

By symmetry we mean in 5Ð Algebra, which as all branches of mathematics born of the spatial, bidimensional ‘still’ work of the Greeks LACKS A TRUE COMPREHENSION OF THE PARADOX OF GALILEO: S (form in space) = Time (motion), but uses it profusely (so differential geometry is based in the concept that a line is a point in motion), __the capacity of all systems to complete a zero sum world cycle, through a motion that returns the system to a present undistinguishable new state.__

Symmetry thus IS ESSENTIAL TO THE ENTIRE SCAFFOLDING OF THE 5D UNIVERSE, albeit ass all concepts of 5D once we understand the basic laws of pentalogic, has a more dynamic view.

It follows immediately that THE MORE SYMMETRIC A SYSTEM IS, the more efficient will be in ‘preserving’ in a Universe in perpetual motion, its present states of ‘survival’. I.e. A circle will be more efficient, because it has infinite degrees of rotational symmetry that an irregular polygon, who might not even have a single symmetry state.

In the theory of ‘survival’ of ‘vital mathematical objects’, which we bring from time to time to those pages (as in the analysis of survival prime numbers able to travel through 5D by making mirror images at scale by joining internally its alternate vortices-points-unit numbers), symmetry thus plays a central role.

How many states of present a system has, defines then its ‘quality of symmetry’ and survival which in space (the easiest symmetries to describe), when fixed in a point means the circle DOMINATES all other forms.

Symmetry though then must be connected to the different Ðimotions and its ‘requirements’ to perform the vital actions for which they are conceived.

I.e. the circle IS the perfect symmetry for still Dimotions of perception, as it will turn out that from any pentalogic point of view, it maximizes the stillness, by symmetry, by its capacity to focus as a fractal point all lines of communication that fall into its focus, by having the minimal perimeter, which maximizes ad maximal its volume of information and disguises it in an external world, and so on.

However, when we consider the 2nd Dimotion of Locomotion, which is the process of displacement in space *while retaining the form in time, as it implies the reproduction of form, of information in other adjacent region of space, the less information to be displaced, the faster reproduction will happen. And so the line, which stores no internal information (or the wave as all points are ultimately fractal with a minimum volume), will be able to displace faster than the sphere, and **maximize the second Dimotion. Here then *the use of the concept of mirror symmetry is NOT required, as the line is moving-translating in space, reason why also forms in motion tend to have a spherical head, to perceive only on the foreward position and a small one in relationship to the body and limbs that have lineal forms to maximize motion.

Symmetry here is of another kind, defined by Noether’s theorem of physics; and in 5D by a type of symmetry ignored in science – the ‘undistinguishable’ property of the ¡-1 elements in which the system imprints its form. This symmetry of scale, implies that the system can reproduce its information – move faster, because it imprints ‘any element’ of the lower plane of motion or ‘field’ that becomes undistinguishable, so there are not ‘impurities’ and errors of reproduction of form, when any electron can reproduce your atomic connections and so on.

Identical states which acquire the same form of present, when a system completes a Dimotion, re-establishing its ideal form, is therefore the essential element for all symmetries that accomplish one of the five vital dimotions of existence.

And the reason of the survival of certain geometric forms above all others, the circle for perception, the line and its curved form the parabola for motion, its combined wave for **3D reproduction, the social circles, from elliptic forms to polygons for social evolution, and the different forms of open curves, notably those dual forms, as the hyperbolas are for the 4D entropic Dimotion** and dissolution of a system in two forms, which in the cone as a representation of a worldline will be split, one hyperbola branch going upwards and the other downwards, if we take the axis of the cone as an ideal representation of the fifth dimension.

Identical states which acquire the same form of present, when a system completes a Dimotion, re-establishing its ideal form, is therefore the essential element for all symmetries that accomplish one of the five vital dimotions of existence.

The second ‘pentalogic law’ for the existence of such symmetries seems trivial but it is important. It consists on considering that any system can switch without ‘loss of energy and time’ between its modular 5 Dimotion states from stop-form-peception in space, to a motion in time. So S=T is allowed to perform in reality the change of ‘symmetry state’, by rotation (angular momentum conservation in physics), by locomotion (lineal momentum in physics), which will preserve the ‘internal symmetry (conservation of the vital energy of the system).

So a symmetry either lineal or rotational conserves the system in its internal ‘energy parts’.

An immediate consequence of the application of the s=tœps duality to groups is the fact that NOT only a system is more efficient when it can remain in symmetry after a ‘transformation’, but when IT STAYS MORE TIME as an invariant form within the form, and this implies that the singularities or neutral elements that remain unchanged during the symmetry, our very same definition of a still mind, is the MOST IMPORTANT, efficient element of the group as it stays invariant through the entire transformation (center of gravity in a body, etc.) So symmetry also explains with the mathematical mirror the reason why the longest surviving element of a super organism is paradoxically its most still head-particle-neuronal brain of ‘perfect invariable’ symmetry during all its transformations, which is called the ‘neutral element’ of group theory.

It follows then also that if we consider the inverse elements to form together a neutral element (as its sum gives us the neutral), they are also invariant couples, which explains the dominance of bilateral symmetry or in non-euclidean geometry, the existence of antipodal points, or in Nature, axis of rotation that become then the ‘singularity line’ of perfect symmetry for the group.

WE HAVE ARRIVED to the heart of the matter of all vital algebra, *as each operation of algebra must reflect a Dimotion and allow its preserving S≈T : S=tœps (stops and steps) symmetries. *

We haven’t talked before of it, not to repeat ourselves too much because we want to treat it FULL RANGE, as its importance is GINORMOUS for our mirror-focused view of reality as it is; and to that aim, we want to bring the most modern view of them – *through the language of modern symmetry, Group theory, and the properties of those operandi found in modern mathematics, the 10 ‘properties of sums and products, etc’. *

So first to introduce what is most important of modern maths (Group theory) and SHUN OFF what is largely irrelevant or redundant (set theory, on my view redundant, axiomatic method, plainly a huge ego trip that distanced math from reality).

We shall then first consider group theory, then *get to the heart of the mathematical ‘matter’, OPERANDI, and then just briefly consider very complex dimensional growths of algebra (functionals on Hilbert spaces), which we shall treat time permitted on mathematical physics fourth line, make a couple of comments on Sets and move to existential algebra.*

How to relate operandi and groups is obvious according to the ‘method of growth/creation’ of the Universe, repeated ad nauseam, operandi are the smaller step, the flow-time detail, Groups the larger spatial whole portrait in the mind-stillness of them all. That is why we shall start from groups down in this case to see the forest before the trees.

**Recap.**

Group Theory is the main element of the third age of mathematics along set theory; whereas a set is basically ‘anything’ and so quite void of meaning. And we shall just use it for respect to mathematicians, but whenever possible change it for more specific concepts, either ‘fractal points’, or ‘social numbers’ or ‘T.œs: Time§paœrganisms’, GROUP theory and the concept of symmetry matters as it is essential to understand Dimotions.

This said we can define the two essential concepts of group theory, symmetry transformations and groups *in terms of T.Œ.*

Symmetry transformation *are the allowed ST-eps and motions in 5D² that keep the S<st>T system co-invariant *

By definition they are the ‘allowed’ motions in as much as if *the system changes outside** the stable parameters of the Generator Equation, it will obviously become broken and die, and no more repetitive motions will be permitted.*

*In abstract mathematics,* groups are merely the ‘collection’ of all possible transformations that keep the system invariant; and as languages are inflationary there are ∞ groups and among groups there are those with infinite elements, such as motions in a plane, and those who have only finite transformations.

As ∆s=t is a realist model of the Universe many of the mathematical ‘curiosities’ and ‘monster groups’ are of little relevance to us. We are mostly interested in those groups closely related to the Generator equation, expressing the allowed transformations of the Generator that matters to reality and existence.

So in ¬ Æxistential algebra we can also talk of Existential Groups, where G is quite closer then to the concept of a Generator, which becomes *the group of all possible Existential groups. *

Among them obviously the most important is the group of physical motions, which connects directly with the concept of ‘motions in the fifth dimension’.

**OTHER more abstract GROUPS – pentalogic of groups.**

Weyl talks of the ‘pest of groups’ in physics, as the concept has gone as ALL INFLATIONARY LANGUAGES, beyond its need. So there are many more groups as there are many infinities, which are both irrelevant in a finite scalar Universe with limited Dimotions.

In mathematics in that sense the origin of G was in the group of Galois and group theory applied in his young r=evolutionary age to prove that quintic polynomials are NOT soluble, which is an immediate consequence of the structure of the Universe in Holographic Bidimensional ‘units’ which ad maximal can be made to intersect to create 4Dimensional systems (and or consider 4D geometries as Relativity does, by studying 3 S and 1 D of lineal time, in a ceteris parries analysis) .

But 5D systems do NOT exist in a single plane and as polynomials unlike differentials are NOT good for studies across multiple planes of an organism but rather for social growth, herding and simpler lineal systems, quintics really belong to inflationary maths.

They are not solvable because they are NOT real.

We shall then classify unlike the axiomatic method, groups according to what kind of symmetry they obey as:

**S:**Spatial group symmetry, the easiest to visualize and understand, just described in vital terms. This field has its main realization in polygonal regular forms, and the study of crystals, and has added vital elements, as it implies that a proper symmetry allows a focused image with*as many lines of communication with the outer world as regular points in the membrane, which allow a symmetric view as the system changes its internal rotary motions, without distorting the visual image the mind creates – so it is deeply connected to mind theory and frames of reference.***T:**Temporal group symmetries, as those of the Dimotions of existence, in physics related to the conservation of angular, lineal and energy elements – the 3 parts of a vital physical system, its angular membrane, lineal singularity motion and vital energy between them. This field is today overwhelmingly studied by physics and continuous differentiable groups, or Lie groups. So we will study it better on those posts.**∆:**Scalar groups, as those which imply an undistinguishable symmetry between wholes and identical parts and hence a travel through the fifth dimension that places the being in a lower or higher plane in its same relative position; as the symmetry of palingenetic birth, from individual cell to individual human within a larger society. We consider them in all other posts, as it is better to be understood in the jargon of 5D than ‘force fit it’, within the models of mathematical groups.**@:**Ideal mental symmetries, as those develop in pure mathematics, which are fun to study but not necessarily real. Of which the original one was the theory of polynomial roots, the only one we shall consider here.

Since physical symmetries by force commit errors and irregularities in its 3 forms, i.e. errors of translation in a motion, irregularities in an ideal form, or differences when we grow or diminish in scales; as we observe in the general posts that study of all those real scalar space-time beings and its dimotion.

We shall therefore just as in all paragraphs consider the bare skeleton of 5D group theory in this ‘failed encyclopedia’ of a one-man-tired-of-it-all-show.

To be even compress further the theme, as we have seen, since GROUPS imply to be real, A MOTION-STEP of the entity AND A STOP-LOCK in space for the symmetry to happen, we shall study together both concepts – the s=tœps of the being.

Latter Group theory expanded to model almost everything in maths, because it allows to ‘collect in a single mental form all the possible variations of a system. Since ‘ the mind perceives motions in time, static as forms of space, ‘reducing its information’ to what it matters to it – mostly the stable points of those transformations, group theory highlights in an elegant way through the concept of symmetry, those stable points of a motion, and it is a good way of ‘limiting’ knowledge to the key elements.

This though IS A VERY INTERESTING PROPOSITION to study SYMMETRIES BEYOND THE INDIVIDUAL ENTITY, as forms of analysis of the ‘non-euclidean postulates’ of communication through waves (2nd postulate) with immediate physical applications in the theory of fermions and bosons; between social groups of parallel beings that form networks, and finally symmetries of physiological networks that form supeorganisms as planes (3rd postulate), according to the different variations of the symmetry of beings; so we will incorporate those essential concepts in our studies of non-Euclidean geometry, and its postulates of points, waves-lines, networks-planes which grow departing from 3 waves leaving holes between them, according to the fourth postulate of multiple types of ‘angular congruence’:

Finally another field of interest in 5D groups is its use to study the symmetry of the 3 ages of time of all systems, through curves. It is the field of variation theory, where the system is reduced to the standing points and minimal and maximal variations – S=T, Max $ x min. ð and Max. § x min. t, which are the 3 ‘age inflections’ on a world cycle of a being; hence its enormous utility.

So as the fundamental feature of ‘mental processes’ is to reduce the time flow to the ‘key points of in-form-ation and trans-form-ation, *eliminating as much as possible the repetitive cycles within the flow *– from palingenesis to languages to biopics – ESCAPING those middle motions of self-repetitive information, (as motion and translation is just iterative motion of information), all the ‘blurred’ transformations or in-between positions are discharged by efficient mind-models of reality.

* RECAP.*

Group theory is better divided for its applications to GST (generator of space-time that here we could call Groups of space-time beings), into:

-Time Groups: ‘Groups of motions’, which concerns with the points in which the motion of a space-time entity becomes transformed into a symmetry of itself. That is mainly related to the allowed motions in space-time.

– Space Groups: ‘Groups of transformations’, which concerns with mirror reflections and symmetries that classify species of reality, as in SU3 groups of physics with its octets and decuplets of particles, which is mainly related to the variations of the Generator, which define also the species of reality.

-Ideal Groups: which are mathematical explorations of all forms of reality including those who are not efficient, and besides being fun for the delight of the mind, allow US TO EXPLORE precisely why certain forms do NOT exist in Nature – are not good enough in its symmetries and Dimotions established by those laws. I.E. why for example in physics bosons move better than fermions and can stay in a single point of space – because they are undistinguishable in its statistics and hence they can form much better ‘packed symmetries’, and can translate by reproduction of form in adjacent spaces, due to the social nature of mathematics and its undistinguishable numbers.

Let us consider those type of groups packing the two first ones in s=tœps, in more detail.

## Types of symmetry by dimotions.

Pentalogic thus applies the laws of the fractal, scalar, cyclical timespace Universe to EVERY SCIENCE as they ARE THE UNDERLYING SYMMETRIES AND STRUCTURES OF REALITY COMPLETELY ignored by monologic man with its lineal time humind…

So while the data and equations we shall use are the same in any language, whose genetic structure IS INDEPENDENT OF MAN AS AN SPECIES OF ITS OWN in the vital organic mirror-making Universe (maths is an species of the fractal Universe as much as we are one, and so many species can have a fractal mathematical mind, and many might speak memetic linguistics and so on), we shall ground all ‘sciences’ in a far more profound philosophy of ‘stience’, based in the fractal, scalar properties of space, cyclical nature of time, and the organic, biological survival language derived from them. We shall always GROUND all realities in those topo-bio-logic PROPERTIES OF space, scales and time. And the adventure of the mind that satisfied me for 30 years developing this Magna Opus, against the simplistic æntropic humind has been precisely to enjoy the perfect pentalogic grammar of all what exists, from biological species, to physical systems, from musical scores, to mathematical structures, from wo=men’s e-motions to physical forms, performing myself all kind of pentalogic exercises from the art of painting to the discovery of new laws of physics to the study of the life and death of civilizations, or the patterns of stock curves of reproduction of machines. When you know the pentalogic game of exist¡ence the mind holds not barriers in its perception of the perfection of the fractal Universe.

We have resumed the pentalogic of some disciplines of logic and mathematical languages in the next graph that shows what a structural analysis of a super organism in terms of the 5 Dimotions of pure causal logic and mathematical simultaneous space would look like:

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

In logic terms, this means by breaking its formless asymmetry into different spatial configurations according to that congruence (social parallel systems, complementary gender-mirror systems, darwinian perpendicular systems, or systems that are disymmetric and do not share any reality) , as it builds a casual pyramid of growth from a fractal point through waves of communication into social networks that become a super organism, ready to move, feed, perceive and evolve socially. Since we __must add to the mathematical and logic languages-properties of reality the 5 actions, or organic properties of the scalar Universe as essential to the game as they are its logic and mathematical more abstract laws – a fact the egocy of æntropic men of course reject, as it must remain in its monad-subjective monologic the only claimant to life properties.__

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

Each advanced language of reality thus CAN BE UPGRADED, AND IT WILL BE UPGRADED IN different POST, TO A PENTALOGIC ANALYSIS in its basic Grammar. And so very often we shall start a post commenting on the pentalogic different Dimotional views of the system we describe.

The post of actions and Dimotions and Non-Euclidean Geometry; most of the posts on physics and studies of different species and properties will be casted from a pentalogic point of view. In terms of *the structural elements of reality, which reflect those 5 Dimotions, the method most used in the blog will be the study of ‘dust of space-time’, as made of ¬entropic destructive arrows that deny its 4 structural elements, the @-mind (1st Dimotion of *__perception), its scales (5th Dimotion), its spatial topologies (locomotion and organs) and its temporal ages and worldcycles. So we obtain a more concrete description of a ¬∆@st entity with reference to its organs and cycles of classic science.__

**Space-Time groups:**

**Physical symmetries: its 5 Dimotions.**

Noether’s theorem considers that each conservation law of Physics, conservation of Lineal Inertia (1D-motion) or angular momentum (2D motion) and energy (3D motion), to which we *add the 4th conservation law of 4D entropy and 5D social evolution or ‘zero sum worldcycles of existence’, is related to a symmetry of space-time. *

We rephrase Noether’s theorem in terms of the METRIC LAWS OF CO-INVARIANCE of the fifth dimension, to define in the invariance the Dimotions of the Universe.

IN 5D ALL SYSTEMS perform 3 simpler actions, perception, motion and feeding on energy to reproduce the system.

And 2 scalar action: social evolution and its inverse entropic death.

So when those 3±I actions=dimotions are studied we have a whole understanding of the system.

What are then the 5D UNDERLYING VITAL, organic principles of physical systems that make them akin to those of any scale?

**1,2,3D:**Locomotion, which embodies the simpler actions through its fundamental concept, that of AN ACTION of a particle, in its path through a field of forces in which IT FEEDS, reproduces and AS A RESULT OF BOTH, MOVES. Reason why locomotion is the physical EXISTENTIAL ACTION that embodies all other simpler actions, as the graph shows..**∆±¡:4,5D:**Entropic scattering, according to collision (loss of vital momentum) and angle (4th postulate of perpendicularity) vs. social evolution under an informative force (normally gravitation), which are the scalar actions of the system

So because Locomotion implies for physical systems, ‘perception in particle state-stop’; feeding on the energy of the lower field and reproduction by adjacent imprinting in wave state-motion; for PHYSICAL SYSTEMS, locomotion embodies the 3 simplest actions of the being; to which we just must add its ‘changes of state’ as its actions of physical d=evolution, to fully have an organic analysis of the species.

**∆-symmetries of locomotion.** As there are 3 conserved quantities in the Universe, which correspond to the capacity of Motions of angular momentum, lineal momentum and closed energy paths to PRESERVE the form of the system by reproducing itself over the undistinguishable particles of a lower field.

Because physical forms have rotational symmetries, where the membrane’s position and form=momentum is preserved; translational symmetries, where the singularity particle/head position and form=momentum is preserved, and energy cycles, where the vital content of the system is preserved, those 3 symmetries allow 3 kind of Dimotions in a single plane, and so if we were to use the jargon of mathematical physics, we could state that the 3 simplex Dimotions that leave invariant the system derive of its capacity to be performed without changing the 3 topological parts of a being in a single plane of existence.

From this fact to be possible though, the ‘perception of motion’ cannot alter the system, and for that reason the Dimotion of perception and locomotion, as LONG as it does NOT affect the lower plane in which the species reproduce its form IS NOT PERCEIVED to maintain the system co-invariant with himself and his environment.

Those forms however will change as we approach the limit of speed of the ∆-1 plane in which the system reproduces, c-speed, when the rate of reproduction of the form of the being, comes to equal that of the ∆-1 level and a distortion takes place. So we have then to consider the Lorentz Transformations to preserve the 3 conserved quantities of the system – theme those studied in more depth in the posts of mathematical physics when completed.

So the lack of perception of motion IS NECESSARY TO MAINTAIN the 1Dimotion of perception unchanged; the absence of friction and a closed system, to preserve the other simplex Dimotions.

What about the 4th Dimotion of scalar social devolution and evolution?

Here the treatment is different, as it is a type of symmetry that affects several systems together, but can be made explicit in its meaning:

For the social parts to become a whole, they must FEEL indistinguishable among themselves, bond by social love, so there is ‘superfluidity’ and the possibility to permute its positions and roles in the super organism in which they form part.

The group of symmetries of social love thus is also based in the existence of undistinguishable social numbers, based in the similarity of actions of all the elements of the group (4th postulate of parallelism). It is then when the system can evolve socially and further on, as it grows in size through scales, become a whole which resembles the elements of which it is made.

In ideal mathematics this could be shown as a ternary symmetry of a point that reproduces in a pi cycle around a central point or ‘singularity’ which does stay in the same form and distance from the growing points, which then rotates around that point in a height dimension to form a sphere that resembles in a larger scale, the parts.

Thus again the symmetry maintains static both the distance to the point respect to the reproductive points of its 1-2-3D surfaces, and the form of the points which are clone of each other and have as ‘identity’ neutral element’ that point which seems to mirror their form.

This symmetry is then the responsible for the bondage together of all those points, which feel reflected in a ‘leader’ or singularity point that makes them act as a single form.

In that regard, we can consider the fundamental mandate of all points to preserve unchanged its existence in an eternal present both in space, time and scale, through dynamic motions that preserve the coin variance of its form in time, space and scale.

What is then the justification of entropy=death from the pentalogic point of view of symmetries? Obviously it restores the zero sum long-term symmetry of the whole region of the Universe in which the toe existed.

But the symmetry of entropy is more profound, when we apply the multiplication of transformations through the entropy, intermediate states, that allows the other dimotions to preserve the internal order of the T.œ.

I.e all other Dimotions go through the entropic process of extracting energy from the environment to perform those demotions, so we can say that if we were to form a symmetry group of the 4 Dimotions of positive existence, the extraction of entropy from the external Universe is the neutral, identity element that preserves the other 4 Dimotions.

Indeed, to perceive we destroy the pixels of the ∆-3 light elements we absorb, to move we convert energy after feeding into entropy of locomotion, and to reproduce a female species must absorb enough extra energy which will be given through the placenta for the entity to perform its palingenetic social evolution and emerge as a new whole. So while the being switches between the Different Dimotions, we can write a chain of transformations that preserve the being, by increasing its internal order to the expenses of the external order:

4D (∆-3: light feeding)>1D (perception) 4D>1D…

4D(∆-4: gravitational and electromagnetic feeding)>2D (locomotion)…4D … 2D

4D (∆-2: feeding on amino acids) > 3D (cellular reproduction)…

IN THIS MANNER, the entropic process allows cyclical transformation through scales of the fifth dimension, which become the actions of the being that preserve invariant its form while dynamically allowing it to perform those actions.

While the final entropic death of the system preserves the larger invariance of the whole ecosystem in which the system will feed another system of Nature.

What this means in the clearer jargon of 5D² is this. As a symmetry in space-time means a motion that carries a form from a point of space-time to another point of space-time *without deforming its inner structure; the 5 LOCAL, DIFFEOMORPHIC, fractal motions of the Universe, one belongs to each of its Dimensions of space-time are conserved, because they correspond to a motion that does NOT destroy the local T.Œ, which remains invariant after its ‘translation in space-time’.*

In the case of the 4th and 5th dimension this process has 2 readings: during the birth to extinction phase of the cycle, the product of the motion and information of the system (1D+3D=2D) remains invariant. And if we ad the 4th dimension of death=entropy and the 5th dimension of birth=generation (social evolution from micro-seed to emergent organism) the total is also a zero sum: ‘dust you are and to dust of space-time you shall return’.

So we explain Noether’s theorem and its symmetry conservation principles *in *vital *terms applied to local fractal space-time beings, T.œs, for whom the principle holds. And then affirm the principle is universal because the Universe is the sum of all its local T.œs: **T.Œ (UNIVERSAL SUPERORGANISM) = ∑T.œs. *

So *‘Any fundamental law proved for a local T.œ made to the image and likeness of the absolute T.Œ, can be extended by Ð-isomorphism to all space-time species’. *

And this accounts for the relativity of motion, the local nature of physical measures which however have by parallelism a global symmetry, and the conservation of the 3 topological parts of the being in its translation, rotation and vital energy, even if in small actions it will cause entropic disorder outside the being itself.

**Physical groups.**

The other great field of Physics is then the use of Groups to classify all the possible STATES or DIMOTIONS of its physical particles, as each Dimotion might change or transform the particle or physical species into another one definitely or partially.

When those transformations are partial, then we can classify the ‘operator’ of the transformation as a Dimotion operator, which will obviously be – as entropy happens external to the being – a group of four operators upon similar species. This is the meaning of the 4 quantum numbers that operate dimotions in particles, or the 4 properties of light, (color and 3 dimensions) as the next graph shows.

After each of those actions however the light system becomes again an invariant collapsing particle, a photon, and so each of its properties is really a symmetry transformation or vital Dimotion in the jargon of 5D.

The other type of transformation however brings when ‘fixed’ as pure spatial states of different particles, a being into another. So for example a proton and a neutron are two different particles. But in terms of symmetries we could consider a dual dimotion of entropy (beta decay of the neutron into a proton and electron) and collapse (formation of a neutron by gravitational pressure in a star from an electron and a proton), as a cyclical symmetry that preserves the form of the neutron. Nature is then made of multiple of those symmetries, but ONLY THOSE REQUIRED FOR SYSTEMS TO PERFORM ITS vital processes or Dimotions.

I.e. when we study the particles of the Universe we shall show that all the particles that exist have a reason of existence derived of those dimotions. Families of masses are the ternary symmetry of social evolution symmetric to the parts of the galaxy. Protons, neutrons and electrons (u-d quarks and electrons) are the two Dimotions of entropy and social evolution which facilitate also the other Dimotions of locomotion (electronic big-bang), perception (collapse of information into a particle,) and reproductive generation (of 4 particles in a beta decay: e, p, v and ¥).

So we will NOT in our posts on physics play the magic of groups to fit physical systems, just because platonic physicists who don’t understand the vital reasons of those groups realized they can fit particles into SU2, 3 and 5 groups that are just operated by dimotions that transform those particles but explain the Dimotions and its transformations.

**Motion Symmetries and Trans-form-actions**

This said we can extend these concepts to its details using the jargon of Algebraic groups in the ‘simplified layman language’ of Mr. Aleksandrov and its awesome, now extinguished rational experimental Soviet school of mathematics, grounded in the work of Lobachevski father of the experimental method for mathematical thought, which we use as the annotated base book for this introduction to non-Æ maths. No, I won’t use, refuse to use, will never yield to the pedantic false Axiomatic method proved wrong by Mr. Godel’s fascinating incompleteness theory made easier to grasp in some other post…)

*2D-motion: reproductive symmetry.*

We begin with an account of the simplest forms of symmetry with which the reader is familiar from everyday life. One of these is the mirror symmetry of geometric bodies or the symmetry with respect to a plane:

A point A in space is called symmetrical to a point B with respect to a plane α (figure 1) if the plane intersects the segment AB perpendicularly at its midpoint. We also say that B is the mirror image of A in the plane α. A geometric body is called symmetric with respect to a plane if the plane divides the body into two parts each of which is the mirror image of the other in the plane. The plane itself is then called a plane of symmetry of the body. Mirror symmetry is often encountered in nature. For example, the form of the human body, or of the body of birds or animals, usually has a plane of symmetry.

*Its origin as we explain in ¬Æ geometry is the ‘elliptic nature’ of @-minds with its singularity connected to two antipodal points, which ‘perceive from the singularity’ appears as inverse; and so as the @-mind, the 5th dimension of order of any T.œ proceeds to ’emerge’ by reproduction and organisation of its clone cells it WILL create mirror images of its code, because when it ‘looks left’ it positions things to his right and when it looks right to his left, and so bidimensional symmetry is a strong proof of the vital topology of the Universe and the capacity of singularities to create reality.*

To understand this just rise your left and right hand with its mirror images, both lateral sides of your head and glimpse at them alternatively – you will *see each finger as being in the opposite side of your head, in the same place of space. So as the singularity webs its organism, it creates bilateral symmetry.*

Mirror symmetry is thus the origin of a Dual Fundamental Motion of the Universe, the emergence and creation from a central singularity point of a bidimensional T.œ, through *the stop and go, motions of reproduction and informative evolution. And for that reason is not a simple motion but a combined motion (2D+5D).*

Now, it is interesting to consider what *science cares for in the mirror symmetry, and what GST cares for: in mathematical physics, with its concept of lineal time-motion as the only arrow of time, all what matters is to measure motions in space; so what matters about mirror symmetry IS ONLY to study HOW a mirror moves to occupy the position of its inverse mirror. *

And that is fine, but studies nothing of the other motions=change in the form of beings, its creative process of creation=reproduction of information, and ultimately the whys of the Universe. Which is what MATTERS to the philosopher of science and GST explains. And further on ALLOWS THE EXPANSION OF THE LAWS of Existential Algebra to ALL scales of reality, as Mirror symmetry IS the fundamental process of creation of bilateral forms, from DNA to Proteins, from Geological ‘fractal continents’ and its ternary self-similar forms, (where a combined motion in scale is also needed) to the processes of crystal formation.

In classic science, then a second form of mirror symmetry is considered, which for ∆st is just another axis of antipodal nodes, merely extending the singularity through its internal axis, equivalent to the poles of the sphere. It is the…

*…Lineal symmetry.*

Symmetry with respect to a line is defined in a similar way, by classic science. We say that the points A, B lie symmetrically with respect to a line if the line intersects the segment AB at its midpoint and is perpendicular to AB (figure 2). A geometric body is said to be symmetrical with respect to a line or to have this line as an axis of symmetry of order 2 if for every point of the body the symmetrical point also belongs to the body.

A body having an axis of symmetry of order 2 comes into coincidence with itself when the body is rotated around this axis by a half rotation, i.e., by an angle of 180°.

**3D-motion: Rotational symmetry.**

The concept of an axis of symmetry can be generalized in a natural way. A line is called an axis of symmetry of order n for a given body if the body comes into coincidence with itself on rotation around the axis by an angle 1/n 360°. For example, a regular pyramid whose base is a regular n-gon has the line joining the vertex of the pyramid to the center of the base (figure 3) as an axis of symmetry of order n.

A line is called an axis of rotation of a body if the body comes into coincidence with itself on rotation around the axis by an arbitrary angle. For example, the axis of a cylinder or a cone, or any diameter of a sphere, is an axis of rotation. An axis of rotation is also an axis of symmetry of every order.Finally, a 3RD important type of symmetry is symmetry with respect to a point or central symmetry. Points A and B are called symmetrical with respect to a center O if the segment joining A and B is bisected at O. A body is called symmetrical with respect to a center O if all its points fall into pairs of points symmetrical with respect to O. Examples of centrally symmetric bodies are the sphere and the cube, whose centers are their center of symmetry (figure 4).

A knowledge of all the planes, axes, and centers of symmetry of a body gives a fairly complete idea of its symmetry properties.

This symmetry obviously corresponds to the 3D motion of timespace, rotary motions, cyclical particles and heads…

**5D-4D Symmetry motion…**

is obviously ignored by Humans, even if *all of them exist within that symmetry between birth and extinction, we have explained ad nauseam, in our description of worldcycles of existence, and its inverse arrows of time, death=entropy and social evolution=generation.*

So we shall leave it as it is explained better with the Fractal Generator, and it would be silly translate it into the more confusing terminology of Group and Symmetry theory.

**1D symmetry motion.**

We explain lineal motion as a form of reproduction in a lineal flow, as the T.œ reproduces in a lower scale and emerges back in a higher one, explained in our analysis of the achiles paradox. In the graph, the reproduction of a quantum par tile in a stop and go motion. Each motion implies therefore a reproduction of its parts in the ∆-1 scale and its emergence in the upper adjacent region as a new being, where the mind flow is a maya of the senses, as we ultimately die and live in the lower scales constantly (so all your atoms change every 3 months). The paradox of the ego is thus absolutely irrelevant, so is the concept of death.

**The motions of the Universe, in symmetry terms.**

*The general definition of symmetry. *

In mathematics and its applications it is very rarely necessary to consider all transformations of a given T.œ, made of a set of fractal points. The fact is that the T.œ IS an organic system, never a mere collection of fractal points, completely disconnected from one another. The sets discussed in mathematics are also abstract images of real collections, whose elements always stand in an infinite variety of interrelations with each other, and of connections with what is going on beyond the limits of the set in question (All worlds are mirrors of its Universe).

But in mathematics it is convenient to abstract from the major part of these connections and to preserve and take into account the most essential one. This compels us in the first instance to consider only such transformations of sets of points as do not destroy the relevant connections of one kind or another between their elements. These are often called admissible transformations or **automorphisms** with respect to the relevant connections between the elements of the set.

And as such *they represent in its closest approximation of classic algebra the concept of an ST-motion in any dimension of 5D² space-time, since it implies a translation in time and a reproduction of form in space, for which two concepts the ‘spatial distance of the translation’ and the quantity of information translated matter most, being both related by the concept of speed, V=S/T, we study in depth in the posts on astrophysics. *

So in space the concept of distance between two points is important both externally and internally. The presence of this concept forges a link between points which consists in the fact that any two points stand at a definite distance from one another; a distance which is measured in GST not as a mere line, but as a wave of communication or network that connects both points (2nd, 4th postulates of non-E). Transformations that do not destroy these connections are the same as those under which the distance between points remains unchanged. These transformations are called “motions” of space-time. *As they imply the inner stillness/fixed form remains invariant. *

And so *translations in space that do NOT change, either the inner content of information of the ∆-1 fractal points of the organism and its 3 network connections ARE THE EXPANDED CONCEPT OF AN AUTOMORPHISM IN GST.*

In this manner we can apply all the laws and concepts of Motions and Symmetries in Space of classic algebra to the fractal point, by expanding the line into a wave-network and considering that the ∆-i content of the point also remains unchanged.

With the help of the concept of automorphism it is not difficult to give then a general definition of symmetry, taken from classic Group theory; where set means a network of fractal points.

Suppose that a certain set M is given, in which definite connections between the elements are to be taken into account, and that P is a certain part of M. We say that P is symmetrical or invariant with respect to the admissible transformation A of M if A carries every element of P again into an element of P. Therefore, a symmetry of P is characterized by the collection of admissible transformations of the containing set M that transform P into itself. The concept of symmetry of a body in space falls entirely under this definition.

The role of the set M is played by the whole space, the role of admissible transformations by the “motions,” the role of P by the given body. The symmetry of P is therefore characterized by the collection of motions under which P coincides with itself.

It is then when we find the ‘equivalence’ between the motions of classic Physics, as described by symmetry and automorphism and the 5D motions of GST.

This are:

** 3D reflections, 1D parallel shifts, and 2 D rotations of space**, because distances between points obviously remain unchanged under these transformations.

A more detailed investigation shows that every motion of a plane is either a parallel shift or a rotation around a center or a reflection in a line or a combination of a reflection in a line with a parallel shift along that line.

Similarly, every motion of space is either a parallel shift or a rotation around an axis or ** a spiral motion**, i.e., a rotation around an axis combined with the shift along this axis, or a reflection in a plane combined with, possibly, a shift along the plane of reflection or a rotation around an axis perpendicular to this plane.

And so *the 4-5D motion of GST is the last of the motions or automorphism of mathematical physics, a ‘spiral motion’, with the difference that in 4 D, the spiral moves outwards and in 5D the spiral moves inwards.*

How can then distinguish both motions? Here is where an essential feature of Nature ill-understood in all sciences comes into place, and explains the duality particle-antiparticle, ±charge, etc. the concept of quirality, or parity, of a Maxwell screw; of a Levo or destro-molecule, etc.

Since the only way for a given system to distinguish 2 the direction of the spiral and hence make possible the duality of 4D and 5D, implosive and explosive, attractive and repulsive forces is by *assigning a different left or right rotation **to the spiral. A*nd this brings a suitable efinition of both ± inverse arrows of the 4D vs. 5D duality, which ultimately are the ‘continuations’ of the 3D and 1D motions that can be seen either as the starting point or limit of its 4D and 5D ‘ages/forms’.

The rotation around the same axis by the angle ϕ in the opposite direction, is then intuitively labelled in Symmetry theory with a negative symbol.

Thus we find again an absolute correspondence between classic science and the whys provided by 5D, as there are NO more motions nor less than those needed to reflect the 5 Disomorphisms of the Universe.

Only the classification of them changes as we know now its whys. In that sense in classic symmetry theory, parallel shifts, rotations, and spiral motions of space are called proper motions or motions of the first kind. The remaining “motions” (including reflections) are known as improper motions or motions of the second kind because the first type can happen in a plane, whereas reflections in a line and reflections combined with a rotation or a translation are motions of the second kind as they need a third dimension to happen (as a motion, not as a dual reproduction of form; which is not a motion but a pure informative action).

It is easy to imagine how transformations that are motions of the first kind can be obtained as a result of a continuous motion of space or of a plane in itself. Motions of the second kind cannot be obtained in this way, because this is prevented by the mirror reflection that occurs in their formation.

Which leads us to a final ‘reflexion’ on reflections (: *a 1D motion is a reproduction of form which has a vectorial direction as it is NOT balanced with a dual ± antipodal point with the singularity in its center, which can happen in a fixed domain, enclosed by a membrane in as much as the reproductive motion left and right cancel, and so the singularity vibrates between both antipodal points remaining in its fixed center.*

Reason why **bidimensional symmetry happens in the generation of biological beings within a fixed **vital space; which also allows a more complex creation of form, as those related to palingenesis, which are ‘condensations’ of billions of years of change that need to take place in a fixed place in which the density of form grows undisturbed.

While lineal motions tend to correspond to simpler forms of reproduction, such as a light space-time system, or a system which is fixed in a steady state, and merely repeats itself. Generation thus is a slow time process happening in a single place most likely through bilateral symmetry.

Yet for them to happen undisturbed, the form of the ‘surface in which such fast light-like motions occur must be extremely simple, with an identical *indistinguishable nature in its points, which explains the simple flat nature of the euclidean light space-time and the invisible lack of information (for us) of its lower ∆-1 gravitational scale. This in classic mathematical physics is expressed* saying that the plane is symmetrical in all its parts or that all points of the plane are equivalent. In the strict language of transformations this statement means that every point of the plane can be superimposed on any other point by means of a suitable “motion.”

**Symmetry groups: cyclical motions and transformations of information**

Now, if we consider ST motions, without ∆§ changes int the social group, things become simpler and easier to understand, as we deal with simple St symmetries in space (3 euclidean dimensions), topology (3 varieties of form) and time (3 ages). The simpler of them being motions in 3 euclidean dimensions. Still it is important for those cases to define ‘congruence’ – the equality or dissimilarity of 2 forms; which now has multiple levels. As the old 3rd postulate no longer applies: two forms are not identical just because the are identical in its |-external membrane.

Two figures in the plane are congruent if one can be changed into the other using a combination of rotations, reflections, and translations. Any figure is congruent to itself. However, some figures are congruent to themselves in more than one way, and these extra congruences are called symmetries.

As it is today most mathematicians only study precisely those external symmetries.

For example, a square has eight symmetries according to various rotations through different axis.

And these are the elements of the symmetry group of the square (D4). When we keep it as it is or perform a rotation by 90° clockwise; by 180° clockwise, by 270° clockwise.

To which we add *symmetries, which are not proper as they require a motion through a 3rd **dimension; the so-called *Group D8 of vertical reflection; horizontal reflection; diagonal reflection & counter-diagonal reflection.

So in this simple example, while the ‘syntax’ of group theory would allow this to happen, *in a bidimensional flat world, which is far more common than you can imagine, as it is the structure of most layers of gradients, this will Not take place.*

In all those cases though group theory will localise as *fundamental to the group, the membrane and the singularity. The membrane is what we observe as ‘identical’, what we rotate, the singularity what does not change. The so called center of the system.*

This again is a common feature of many ‘operations’ of reality. Consider a war, where the membrane of the nation and the capital is the only thing that matter. The membrane must remain unchanged, the capital too. All war operations will decimate the internal production, people and ST-elements, but what the capital seeks is the integrity of the membrane and itself under war operations.

The interest of group theory thus will transcend the obvious use for studying mere spatial translations, *which is the simplest locomotion; hence by far the most studied by human beings. *

These symmetries might then be represented by functions; and functions of functions (functionals) and entities of space, time, or scales, etc. We are not though that interested here into making an exhaustive translation of group theory to GST, as unfortunately humans have NOT created a civilisation of knowledge of praxis, not of homo sapiens but of homo faber and so the routines of praxis and repetition of jargons with deformations is unassailable, but rather show loosely why group theory is so important in the praxis of all sciences and its theory:

*BECAUSE IT summarises mentally the motions of ∆st systems in all the scales of reality.*

**ð§. ****THE MEMBRANE. Its Body symmetries.**

The analysis of those antipodal bilateral processes of reproduction of form happening canonically in an elliptic geometry brings us into the next fundamental analysis performed in classic symmetry and origin of topology – the study of the distribution and motion of fractal ∆-1 points that form a polyhedral membrane.

The cases of symmetry of such bodies or figures are also comprised under the general definition of symmetry.

For example, a body that is symmetrical with respect to a plane α comes into coincidence with itself on reflection in the plane α; a body that is symmetrical with respect to a center O comes into coincidence with itself under reflection in O. Therefore, the degree of symmetry of a body or of a spatial figure can be completely characterized by the collection of all motions of space of the first and second kind that bring the body or the figure into coincidence with itself. The greater and more diverse this collection of motions, the higher is the degree of symmetry of the body or figure. If, in particular, this collection contains no motions except the identity transformation, then the body can be called unsymmetrical.

And as it turns *almost all systems of Nature that survive have the maximal number of symmetries, a theme we have studied on our analysis of platonic solids and Euler’s characteristics in topology/geometry posts.*

And the reason is obvious: to perceive properly the Universe in the mirror-cyrstal of the singularity, the mirror must be able to translate in space and suffer rotations in its combined motions that do NOT change the distances between singularities and vertex and other potential openings to the world, so *his mirror mind remains focused and does NOT change constantly; a fact we can extend to all ‘membranes’ in all scales, from physical membranes in the next graph (orbitals of an atom) to its next level of crystal symmetries, to the spherical perfect symmetry of more evolved minds, from eyes to cameras, which through the laws of optics reach maximal clarity in its focused mind-mirror: *

This extends also to ‘partial views’ or holographic flat forms, or parts of a 3D whole in 2 D, which makes overwhelmingly dominant certain forms of maximal symmetry, such those regular polyhedrons and specially the square/cube and the Hexagon (we study them as ‘perfect numbers-forms’ in the I Age of ‘platonic’ number theory.

I.e.: the degree of symmetry of a square in a plane is characterized by the collection of motions of the plane that bring the square into coincidence with itself. But if the square coincides with itself, then the point of intersection of its diagonals must also coincide with itself. Therefore the required motions leave the center of the square invariant, and so they are either rotations around the center or reflections in lines passing through the center:

From figure 7 we can easily read that the square ABCD is symmetrical with respect to the rotations around its center O by angles that are multiples of 90° and also with respect to reflections in the diagonals AC, BD and the lines KL, MN. These eight motions characterize the symmetry of the square.

We observe then that all polygons (regular numbers) once they establish a membrane, as it happens in all systems of nature, establish an identity element by invaginating its axis of symmetry, to find the singularity and so, in this manner, they connect and establish 3 parts which are the canonical parts of all systems: the membrane invaginated=connected through ‘physiological networks to a commanding mind-point, which carves and divides in cellular parts the vital energy. So membrane, singularity and invagination networks are essential to establish the structure of a T.oe, where function matters more than the ideal symmetry, as it will adapt its form to the environment, but all super organisms will have a mind-singularity, a membrane and physiological networks that connect them, starting by the simplest volume, the triangle, where the 3 median lines connect its sides to its vortices through the central singularity.

It is then an important part of vital geometry to translate the ideal laws of those regular figures into laws of vital physiology and morphology.

The collection of symmetries of a rectangle reduces to a rotation around the center by 180° and a reflection in the lines that join the midpoints of opposite sides; and the set of symmetries of a parallelogram (figure 7) consists only of the rotations around the center by angles that are multiples of 180°, i.e., of reflections in the center and the identity transformation.

So there are many more squares in nature. It is then obvious that an object with maximal symmetry will be also the best survival strategy for a form specially one which remains ‘fixed in a point’. And so the Hexagon with its 6 reflexions in a single plane comes as the strongest possible flat object (as researchers in materials have found recently with the discovery of the graphene). While *a system in lineal motion, which does not make such rotary informative homomorphisms, is best served by a triangular, conic form that ‘penetrates’ the space ahead, deflecting and breaking its points into ∆-1 elements to form an envelope of growing entropy that moves it ‘ahead’.*

Previously we have given an algebraic example of symmetry; we mentioned that the concept of symmetry of a polynomial in several variables also has a meaning.

The collection of transformations that preserve a certain object, i.e., characterize its symmetry, is then call its group.

This method of giving groups in the form of symmetry is one of most significance for GST.

Very important groups of ‘reality’ can be obtained by this common principle to GST and classic group theory.

We have studied the 2 most important both in classic and GST theory, in this brief introduction to symmetries in space – the groups of motions of a plane and of space and the symmetry groups of planes, which extends easily to 3D as the group of symmetries of regular polyhedra of great interest in solid matter states, due to the aforementioned ‘mind-singularity focus’ effect (see ¬E space geometry):

It is known that in space there exist altogether five types of regular polyhedra (with 4, 6, 8, 12 and 20 faces).

When we take an arbitrary regular polyhedron and consider all the motions of space that bring the given polyhedron into coincidence with itself, we obtain a group, namely the symmetry group of the polyhedron. If instead of all the motions we consider only the motions of the first kind that carry the polyhedron into coincidence with itself, then we obtain again a group that is part of the full group of symmetries of the polyhedron. This group is called the group of rotations of the polyhedron.

Since in a superposition of the polyhedron with itself, i*ts @-singularity center is also superimposed on itself, all motions that occur in the.group of symmetries of the polyhedron leave the center of the polyhedron unchanged* and can therefore only be either rotations around axes passing through the center or reflections in planes passing through the center or, finally, reflections in such planes combined with rotations around axes passing through the center and perpendicular to these planes.

With the help of these remarks it is easy to find all the groups of symmetry and the groups of rotations of the regular polyhedra. In Table 1 we have given the order of the symmetry groups and the rotation groups of the regular polyhedron. Finally to notice that unlike a sphere, which has infinite possible changes, hence by the Poincare conjecture can ‘shrink without limit’ and so is the ONLY form that can travel without limit on ∆-scales and a mind and hence *the absolute form of a potential absolute mind of T.Œ, the perfect form as the Greeks thought* all these groups are finite, hence limited minds in their travel through scales:

For the ‘numerologist’ inclined to think, it should be noticed that the 5D solid, the Dodecahedron, likely perfect mind of the imperfect local world in which we live, shows that magic number, which comes around all over the place in mathematical physics, the number 60 (:

It is also along cubic forms the commonest of the most perfect informative atomic eviL form, go(l)d…

It is interesting to notice then that among all those polyhedron according to vital geometry, the most efficient forms will be those with maximal number of symmetries in relationship to its ‘faces’, which makes the tetrahedron, cube and dodecahedron, the most efficient forms far more common than its counterparts (there are more cubes than octahedrons).

**Fedorov groups: Reproduction of crystals.**

*The symmetry groups of finite plane figures. *

As we have already seen, the symmetry of a figure or a body is characterized by the group of motions of the plane or space that bring the figure into coincidence with itself.

But we have now a direct understanding of the ‘nature of those motions’ in the world of 5D, whereas motions can be reduce to lineal motions of simple reproduction (D1), entropic motions that disorder and erase information, (D4: outwards spiral), and the inverse (D3, D5), informative, social motions that evolve minds and organisms, leaving the fundamental *mysterious secondary motion, Mirror symmetry, as the key reproductive ‘gender motion’ that brings together two inverse forms into a dual one. *

All of them though *are motions which conserve the singularity at its center, showing its fundamental role in the organisation and reproduction of crystals. *

And what we shall find not surprisingly since the Universe is a reproductive fractal is that MOST of the transformations and symmetries of reproductive crystals involve a ‘mirror symmetry’ both in 2d and 3d (we use d minor for classic dimensions, D major for 5Disomorphisms).

As an example we shall consider discrete groups of motions with no fixed points in the plane – that is translations in space, where there is a line that is carried into itself under all transformations of the group. This line is called an axis of the group. Symmetry groups of this type occur for example, in ornaments that are set out in the form of an infinite strip (border). Of such groups there exist altogether seven:

**1.The symmetry group L1 consisting only of translations by distances that are multiples of a certain segment a.**

It corresponds therefore to a motion that preserves the ‘singularity point’ or lineal inertia.

**2.The group L2, which is obtained from L1 by adjoining the rotation by 180°** around one of the points on the axis of the group.

**3.The group L3, which is obtained from L1 by adjoining the reflection** in a line perpendicular to the axis of the group.

4.The group L4, which is obtained from L1 by adjoining the reflection in the axis.

5.The group L5, which is obtained from L1 by adjoining a translation by a/2 combined with a reflection in the axis.

6.The group L6, which is obtained from L4 by adjoining the reflection in a certain line perpendicular to the axis of the group.

7.The group L7, which is obtained from L5 by adjoining the reflection in some line perpendicular to the axis of the group.

*So one motion is a D2, $t-locomotion (L1), the second group L2 is a rotation (D1: perception: §ð) and the other five contain a mirror reflection (D3: complementary reproduction: ST).*

And so once more we see a Group of 3 Symmetries that respond to the need of 3 operators – the trinity logic of a single plane of existence.

Again if we consider crystallographic groups, where exists neither a point nor a line in the plane that is carried into itself under all the transformations of the group called plane Fedorov groups, there are altogether 17 of them: five consist of motions of the first kind only, and twelve of motions of the first and second kind, including mirror symmetries.

So present ST-reflections dominate the reproductive Universe:

And the same occurs in the 3 dimensional classic space, where we find 230 possible groups of which 165 include a reflection… which implies in ∆st, communication between antipodal points, to reproduce.

And indeed, *in all planes of reality from particles with inverse spin that reproduce particles to sexual copula of ‘inverse genders’ (female informative vs. male lineal, entropic species), mirror symmetry finds finally its reason d’être: reproduction of information.*

Let us now make some comments on the general theory of groups and its operations from the more abstract, mental point of view, which is how axiomatic mathematics consider them to ‘close’ the theme in this forcefully limited space-time T.œ. with the classic Galois group, which also completes our introduction to 5D polynomials.

**Abstract groups” Lagrange>Galois: Group ****theory.**

Lagrange more than Galois, the romantic hero, who just put the Ice in the cake, resolved the question on how to find the roots of a polynomial of degree less than 5 – those who closely resemble the holographic dimensions of reality with his resolvents.

The GALOIS group in itself is not that important to ∆st as it is mostly about the limits of the holographic principle to create real meaningful systems in polynomials higher than 2 Dimensions. It is the concept of ‘motions and transformations’ in space-time close to an ST-ep or motion on any of the 5D of reality. The Galois group does work though on a basic concept of space-time theory as many other solving principles such as the principle of least action of physics or the modulations of waves: *the need to *find a enclosure and center, an @-mind, an initial and final point to ‘create form; in its infinite manifestations.

In the case of the Galois group the information about the solvability of a polynomial requires *to specify an origin/singularity point which is t*he §œT of all quantities that can be obtained from the coefficients of the equation by means of a finite number of the operations of addition, subtraction, multiplication, and division, called the ground field or domain of rationality of the equation and and end point*, the splitting field of *the roots of this equation – ξ1, ···, ξn – which is the set of quantities that can be obtained by means of a finite number of the operations of addition, subtraction, multiplication, and division starting out from the roots ξ1, ···, ξn, which through Viete’s formulas allow to obtain the coefficients of the equation by means of the operation of addition and multiplication.

Therefore the end or splitting field of an equation always contains its ground field; *which is a general rule of a time causality – the future contains more information in its memorial sequence about the past, than the past about the ‘multiple possibilities’ of the future*. Sometimes these fields coincide, and then we find a cyclical loop of information which is ‘self-contained’ (and hence solvable).

Yet the true mark of solvability *is ultimately the* possibility of reducing the solution of a given equation to that of equations of lower degree; hence making them coincide with the real existence of holographic bidimensional X² and tridimensional X³ sub§œts of the equation; which then can be put in St≈ST symmetries.

The element then introduced by Galois, which will spread all over the world of mathematical physics regard the permutations; that is *motions in space of §œTs that translate the system and keep it ‘invariant’ in its internal form; which we shall study in the 3rd age connected with the geometrical space-time motions of those §œT.*

i.e. The symmetry of the given polynomial is characterized by the collection of those permutations of the variables that, when carried out on the polynomial, leave it unchanged. For example, the symmetry of the polynomial X1³+2X² +X3³+2xˆ4 is characterized by the four permutations:

Now, as many of the very complicated to understand reasonings of modern mathematics, which really starts with Galois work on groups, we provide justification, which are far less axiomatic and simple based in the fact mathematics are mirrors of space-time-scale properties.

So to the question why is there no formula for the roots of a fifth (or higher) degree polynomial equation in terms of the coefficients of the polynomial, using only the usual algebraic operations (addition, subtraction, multiplication, division) and application of radicals (square roots, cube roots, etc)? The answer is self-evident and trivial:

Because polynomials are a simple mirror of ‘planes of space in scale, or dimotions of time’, and there are only 4 Dimotions/scales relevant to any real system (entropy being the fifth Dimotion, not counting as it is merely a negation of any of the others). So in the same way 4 quantum numbers code physics and 4 letters genetics, and 4 Dimensions our humind, polynomials of 4 ‘dimensions’, can be solved. This is ultimately the kind of ‘margin proof’ we use for Fermat’s last theorem, because maths IS A MIRROR OF REALITY not of fictions.

Groups indeed *are the tool physics uses to study together in a static view the different variations of form and motion in the Universe.*

Yet WHY THEY ARE CONSISTENT WITHIN THEMSELVES?

The answer is self-evident if you are grasping the basics of 5D philosophy of science: BECAUSE REALITY IS CONSISTENT and it is NOT inflationary. In the same way there are not more particles than those needed for the standard model to work, no fictions, no evaporation of black holes back to the past, no big-bang singularity fantasies, etc. in real physics, real maths AS A MIRROR OF THAT REALITY is consistent in its entangled INNER STRUCTURE AS THE UNIVERSE IS.

However languages are slightly inflationary, so it is also truth that as words produce fictions, within the consistence of its syntax, maths have also fictions within its consistent inner axiomatic syntax, but THEY ARE NOT real – so Einstein said ‘I know when maths are truth but not hen they are real’ – something physicist have forgotten.

*Further on, they represent the mind view of ‘trans-form-ations’ in time, considering the beginning and end of the trans-formative motion.*

*In this manner groups show the symmetries between space-time beings and its states, which are trans-form-ations of a being along one of its ternary ‘Fractal Generator symmetries’*

So, when we state there is a group of possible transformations of a particle changing its isospin, or a rubick cube, changing its face dots, or an equation, changing its coefficients, we are observing the ‘possible’ paths of a being, across the ‘authorised’ operandi that reflect all its possible topological-spatial, age-informative and scalar-∆ motions in the 5th dimensions of ∆•st, formalised in the fractal generator.

How the group ‘freezes all time§pace motions’ then is obvious – ‘eliminating the intermediate ST ‘motion’, WATCHING only the initial and final form… of the cube rotation, the inner changes in the spin paths of the particle, or the particle weak trans-form, only observing the *limiting* results – the time change being frozen and extracted from the group – this ST phase though is essential and the fractal generator will show it, giving us more information on the meaning of groups in mathematical physics.

*So we shall also extend the concept of the 3 fundamental varieties of *Groups, rings, and fields, to fully grasp how group algebra explains the 3±i type of motions of the Universe (topological, temporal and scalar motions or ‘no motion’ at all – mind in-form-ation).

We must differentiate two type of operations which are similar in concept – the polynomial and the integral, bridged by binomial approximations. Polynomials are LINEAL and work therefore essentially in a single Plane of Spacetime, hence are good to describe relationships taking place in the topological 3+I=3+i dimensions of space (point>line>volume>Point of a larger scale) and its equivalent view on motion (point, moving point tracing a line or an angular momentum, line spreading in surfaces, circles turning into spheres, which become planes moving into cubes and spheres making a new loop).

All those processes are NOT distorted by the ’emergence’ into a new scale of physical realities which are therefore NON-lineal and only the magic of calculus can convey.

There are only 4 positive Dimotions (being entropy the negative one that destroys the other 4), polynomials beyond the fourth power are meaningless and are not resolved, while most fourth power polynomials of ‘wholes’ are irrelevant in science, and are resolved them by reducing its parameters to lower ranks (often because they lack one of the other powers-dimotions).

**GROUPS & OPERATIONS**

** THE GENERATORS OF MOTIONS IN SPACE-TIME.**

The ‘Generator Group’ is the fundamental structure of algebra related to the Generator of Space-time Superorganisma and its Worldcycles and its motions, all of them based in metric Symmetries along the ± inverse dimensions of ∆ST.

As in essence, the concept of a group and an internal operation run through it is akin to the concept of a Generator equation and *an internal feed-back space-time event ran through the elements of the Generator. And so we could say that as all can be described by Generators in GST, all can be described as Groups with an internal operation in Algebra, derived from GST.*

**DEFINITION OF A GENERATOR GROUP**

The concept of a generator group IS thus defined as an entity of space-time, with an operandi that reflects motions in a given direction of ∆ST. And so departing from that simple definition we can re-classify groups NOT in terms of the axiomatic method (which includes those which belong to fiction mathematics, as all languages can be used with its internal syntax to define fiction thoughts, *a sorely needed distinction* THAT MATHEMATICS LACKS AND IT IS AT THE CORE OF MOST ERRORS OF SCIENCE)* , *but in terms of the possible motions of ∆ST, where • motions, would be mathematical fictions, worth to study on its own as long as we know they are just ‘fictions’.

The group then will be according to the motion, all the potential ‘destinations’ of that motion, when the operandi is applied to the element, as many times as possible. And this also sets ‘limits of infinity’ in the motions of the group. Indeed, let us say it is the group of natural numbers, the motion is growth, and we ad according to any of the operandi of growth – i.e. the fibonnaci series, to the group this growth. The serial results will not ‘continue’ for ever in reality but will be checked at a certain point by the curves of logarithmic expansion of populations with a limit on the trophic pyramid, *even for the entire Universe, which hardly goes beyond the 11¹¹¹ number, beyond which nothing can be really defined.*

It is precisely when we introduce into the ‘syntax’ of mathematics, the ‘semantics’ of a certain element of a group and operandi when we have the precise physical meaning of a group, which therefore will be composed of 3 elements:

•, the point which we apply or operator

§, the operator of an ∆ST dimensional motion

G, and the group of all possible outcomes of that motion.

**±§: Inverse vs. direct numbers. The social meaning of integers, and its ‘group≈generator’ in ∆ST.**

The use of Z numbers is now the question to consider, as usual through the ternary method, once *we have defined, N & – N as the two inverse motions in the ∆ST dimensions of the scalar Universe, which put together create Z, which can therefore be used to study the motions in the opposite dimensions of ∆st systems.*

So we shall now consider what are those ‘z-numbers’ for ∆, S & T.

**∆Z: §ocial Generators.**

We shall therefore substitute the word G(roup) for G(enerator)

Negative coordinates and numbers in group theory fully grasps what Z means: the essential social operation of numbers as groups of identical beings – fractal points in GST, ‘sets’ in the abstract jargon of model systems. We would rather use therefore the concept of a ‘fractal point’ or ‘being’ which is identical to other beings and form ‘varieties of digital societies’ defined each one by a number or social point, made of smaller points.

The social points called integers Z (±N) *thus* consists of a series of numbers of increasing social content…, −4, −3, −2, −1, 0, 1, 2, 3, 4, …,

And the operation that changes the social content of a point is defined as the §calar sum of social points (§).

The following properties of § serves as a model for the abstract group axioms given in the definition below.

For any two integers a and b, the sum a § b is also an integer.

That is, the sum of integers always yields an integer. This property is known as closure under §.

For all integers a, b and c, (a § b) + c = a § (b § c). Expressed in words, adding a to b first, and then adding the result to c gives the same final result as adding a to the sum of b and c, a property known as associativity.

If a is any integer, then 0 § a = a § 0 = a. Zero is called the identity element of addition because adding it to any integer returns the same integer.

For every integer a, there is an integer b such that a § b = b § a = 0. The integer b is called the inverse element of the integer a and is denoted −a.

The integers, together with the operation §, form a mathematical object belonging to a broad class sharing similar structural aspects, called a group.

Thus in GST we basically relate the concept of sum and group to the broader concept of §ocial symmetries & sums.

Let us then consider the second use of a Generator group for strict spatial motions.

**∏§ Z.** *Symmetries in ∏ime§pace through Generator feedback events and actions (groups with an internal operation).*

The importance of groups, beyond the trivial use to define motions in open space, which we prefer to study without so much abstraction on the field of ‘reproduction’ is in motions on closed paths of timeSpace, or time-like, energy-like conservative motions, which *leave at the end of the operations, the group unchanged as a zero-sum world cycle of existence. *

To define them we can use the classic formalism, and consider our • symbol for a o-sum, or virtual mind, which is also used in group theory to see some of the ‘properties’ of such worldcycles; which are therefore defined by a symmetric ±, Tiƒ, Spe dual transformation forward with the life arrow and backward with the death arrow, which leaves ‘the system unchanged’.

A TIME-SPACE world cycles thus is an symmetric event in time, G(ST), together with an operation • (called the 0-SUM law of the Generator) that combines any two elements a and b to form another element, denoted a • b or ab. To qualify as a generator group (G), we need therefore a series of elements susceptible to move through the ∆ST dimensions of space-time and an operation, (G, •) that describes those motions, which must satisfy four requirements known as the group-generator axioms:

*Associativity*

For all a, b and c in G, (a • b) • c = a • (b • c).

This axiom still holds in GST, in as much as it shows the social nature of the ∆-universe.

*Closure, till the borders of the |-limiting membrane of ∞.*

For all a, b in G, the result of the operation, a • b, is also in G,

*only in the domain of the elements in which the limiting membrane, ±∞ and singularity points, ±0 do NOT distort the operation.*

This is the main difference between fractal G theory and classic groups. In GST there are never absolute infinities. And so, concepts sic as the monster group are the equivalent in mathematical humind (human mind) languages to the concept of absolute God in verbal thought – aberrations of the ‘categories of the mind’ (to use Kant’s language).

It is in those terms when it makes sense the concept of :

*Identity ‘o’-point: The ‘singularity’ element*

There exists an element e in G such that, for every element a in G, the equation e • a = a • e = a holds. Such an element is unique as it is the singularity point that holds the group together and allows its transformations, and thus it is the singularity, Tiƒ element.

*Inverse element*

For each a in G, there exists an element b in G, commonly denoted a−1 (or −a, if the operation is denoted “§”), such that a • b = b • a = e, where e is the singularity element.

Important to notice that as the 2 directions of motion (outside simplex space-coordinates) are not equivalent (since the 2 arrows of ∆ are different, the inverse Tiƒ<=>Spe topological elements in a Spatial symmetry are different too, and so are the youth-old age),

the result of an operation may depend on the order of the operands.

In other words, the result of combining element a with element b need not yield the same result as combining element b with element a; the equation:

a • b = b • a may not always be true. Consider indeed, the motions in time ages:

From a (seminal, cellular state) ∏ b (Informative Old age), the motion is called life. From b (old age) ∏ to entropic ‘youth’, it is called death when all the information of the system is erased and you return to your cellular state.

Generator groups for which the commutativity equation a • b = b • a always holds are called abelian groups and they are *spatial-like*.

The time symmetry generator described in the previous section is an example of a generator group that is not abelian. So we can state:

‘TIME-LIKE generator groups, events and process are not-abelian; space-like processes of translation are abelian’.

So (Γ, §) are abelian groups, and (Γ, ∏) are not Abelian, using GST symbols for Generator, Space-like and Time like processes.

Nonabelian groups are pervasive in mathematics and physics. A common example from physics is the rotation group SO(3).

Most of the interesting Lie groups are nonabelian, and these play an important role in gauge theory; which tells us an obvious truth: gauge theory and all its systems of particles are ‘motions of in-form-ation that trans-form’ a given particle into another one, through a symmetry in its ∆st elements, within the restrictions of possible balanced combinations proper of GST physics.

**Distributive laws for power laws.**

There are many distributive laws, for 1D:

3(x + 7) = 3(x) + 3(7).

FOR TWO D:

Right (3x)² = 3² x²

but for power is invalid “expanding”

(x − 6)² = x² − 6² WRONG!

2 SCALES DO NOT CARRY THE DISTRIBUTIVE LAW UPWARDS.

Think of a small house. It’s got a basement, a ground floor, and an attic. You can’t jump right from the basement to the attic, can you? But you can take stairs between the basement and ground floor, or between the ground floor and the attic.

You combine operations just like that. If the operations are on adjacent levels, you can combine them; otherwise you can’t. What are the levels? Forget PEMDAS; there are really only three operations to be concerned with:

house floors operations

attic powers and roots

ground floor multiply and divide

basement add and subtract

And the rule is very simple:

You can distribute any operation over an operation one level below it. There are no other distributions.

When you start to distribute one operation over another, stop and ask yourself which distributive law you are using. If it’s not one of the two specific laws mentioned on this page, you’re almost certainly making a mistake.

Examples

Multiply/Divide over Add/Subtract

You can distribute a multiply or divide over an add or subtract, because multiply and divide are one level above add and subtract:

Right 7(x + y) = 7x + 7y

Right (x + y) / 3 = x/3 + y/3

Right 2x (x − 3) = 2x² − 6x

Right (2x − 8) / 2 = 2x/2 − 8/2 = x − 4

Students sometimes distribute a multiplier over both parts of a fraction, like this:

3 × (2/5) = 6 / 15 WRONG!

You can’t do that because multiply is not one level above divide; they’re at the same level. You can distribute only when moving down one level.

Sometimes we talk about “distributing a minus sign”, like this:

Right 2x² − (x − 1) = 2x² − x + 1

That is correct because that minus sign for subtracting is the same as adding −1 times the quantity, and what gets distributed is the −1 multiplier:

Right 2x² + (−1)(x − 1) = 2x² + (−1)x + (−1)(−1)

Take a couple of seconds and make sure you see how the first equation is really just a shortcut version of the second.

You probably know that you can not only distribute but collect or “factor out”:

Right 6x + 12 = 6x + 6(2) = 6(x + 2)

Power/Root over Multiply/Divide

You can distribute an exponent or radical over a multiply or divide, because powers and roots are one level above multiply and divide:

Right (3x)³ = 3³ x³

Right √(25x) = (√25) (√x) = 5 (√x)

Right (2/3)² = 2² / 3² = 4/9

Right √(x/100) = (√x) / (√100) = (√x) /10

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What you must not do—though students have been doing it since algebra was invented—is to distribute a power or root over an add or subtract:

(x + 3)² = x² + 3² WRONG!

√(x² − 25) = x − 5 WRONG!

Look back at the “house” picture. Add/subtract are in the basement, and powers/roots are in the attic. You can’t distribute powers or roots over addition and subtraction because you’d have to skip a level.

Some More No-Nos

You can only distribute down a level, never up:

x3y = x3 xy WRONG!

2(3x)² = (6x)² WRONG!

Other Ways to Combine

Yes, you can combine algebra operations in other ways, but the other combinations are never as simple as a distribution. The only straight distributions are the ones mentioned above: distributing an operation one level down in the “house”.

Here’s an example of a combination that is not a straight distribution:

x(2+3) = x2 + x3 WRONG!

Right x(2+3) = x2 x3

Notice what happens. You can’t distribute addition over a power because addition isn’t one level higher than powers. (It’s not higher at all, but lower, as you know.) But a valid combination does exist: the addition turns into a multiplication.

There are a number of laws for combining power expressions. Ultimately they all trace back to counting, as a separate page explains.

Logarithms and Trig Functions

Remember the “house”? Logarithms and trig functions are not one of those levels. In fact, they’re not in the same building. For example,

sin(A + B) = sin A + sin B WRONG!

Right sin(A + B) = sin A cos B + cos A sin B

When you try to “distribute” the sine function over a sum, it mutates into something quite strange. And with logarithms, you reach a brick wall:

log(x + y) = log x + log y WRONG!

Right log(x + y) cannot be broken up

There are lots of laws for combining trig functions and logarithms with the basic algebraic operations, but none of them is a straight distribution.

For straight distribution, stick to the “house” and its rule of one-level-down, and you’ll be fine.

**§ŒTS**

After group theory the other over pervading concept of modern algebra is set theory, which in ∆st we have baptised §œT, a social group of T.œ, let us see why.

A set is a collection of mathematical entities, and as such it is closer in its definition to a social group of T.œs, fractal points or social numbers, or any combinations with higher ‘social dimensions’ of those elements, and so we could define **SETS simply as ∆§ (a social ensemble of organisms of timespace of any scale of the Universe).**

This is our definition of set, and so we shall write **§et**, AND WHEN in mood for slow, more accurate writing:

**§œts=Social ensamble of organisms of TimeSapce**

This simple change of ‘a character’ adapts set theory to GST, but WE often decry (i hope to erase my sanguine anger:) the fact that maths are no longer connected to reality and the axiomatic imagination of Hilbert and the Cantor’s paradise of §ets have a bit to do with it.

Indeed, while we are not that crazy as to erase the whole of set theory and get even less audience we are getting (0 views most days, so we might get an imaginary number :), point is SETS are a mind ‘construct’ to ‘cuspid’ the Mind singularity SEARCH for a SINGLE ‘SINGULAR’ EQUATION, CONCEPT OR GOD. As minds are BY DEFINITION SYSTEMS THAT TRY TO CREATE A WHOLE MAPPING in reduced space, so AS KANT only understood, the mind’s categorial, synoptic nature searches for the ultimate form, equation, idea, god, belief. And ‘SET’ is the God of modern mathematics, but ultimately a human mind category.

Much better to use concepts that are real, Social numbers, spatial points, time clocks, and so on.

*which in the 3rd age as an old man does, isolates itself from reality and finds within its self a justification of its life/meaning. This is OK as all the 3 perspectives from the bottom up, (justified by numbers), from the top down (justified by sets) or in the present s=t (best for ¬Ælgebra, justified by symmetries), offer a complementary view on its foundations.*

**excessive formalism – axiomatic method and set theory**

Finally as in all languages mathematics also entered a baroque age of excessive inward form, which took two clear wrong models, of two friends, mr. Cantor with set theory and Mr. Hilbert with the axiomatic method, bashed elsewhere for its dogmatic god-like beliefs.

Indeed, Mr. Hilbert affirmed that ‘he imagines points, lines and congruence’, as if the mind created maths. We have dealt with his absurd foundations of geometry in the topological and mind-related articles on maths. So here we shall deal with cantor’s paradoxes of infinity more proper of number theory due to his misunderstanding of what numbers and relative infinity is.

**Set theory – the wrong units of mathematics.**

¬Æ is not concerned with set theory and the formalism of modern mathematics with its pretentious sense of proof and rigor within the mathematical metalanguage, as Gödel’s incompleteness theory and the consideration of information as inflationary makes more important in fact to set the limits of mathematical statements as an homeomorphism to the limits of the 5D Universe.

In that sense set theory does work – we are not that fundamentalist – and could be considered the final evolution of algebra, as the formalism of logic time structures in which certain basic rules of inclusion, social communication and parts that become wholes (sets of sets) do matter. But reality IMPRINTS FORMAL MOTIONS OF ONLY 2 TYPES and a limit of 3 dimensions in space and time in each scale of reality so the hyperinflation of mathematics without limits to its extensions makes fiction of many of its terms.

This understood, there are 2 infinity ‘errors’ worth to mention, in an introduction, one theoretical **The Cantor error: Does it exist the set of all sets?**

It does not. Point, no need to create new axioms to hide the paradox. We simply go further. Does it exist a ‘set’ as Cantor defined it? No. It does not.

**• Cardinality and transfinite numbers.**

The application of the notion of equivalence to infinite sets was first systematically explored by Cantor. With ℕ defined as the set of natural numbers, Cantor’s initial significant finding was that the set of all rational numbers is equivalent to ℕ but that the set of all real numbers is not equivalent to ℕ.

The idea is that two sets are equivalent if it is possible to pair off members of the first set with members of the second, with no leftover members on either side.

So what this tell us? First since natural numbers are far smaller than rational numbers, that the concept of an abstract infinity is an error similar to Zeno’s Achiles paradox.

Now instead of Achilles and the turtle the runners are natural and rational numbers, and instead of the turtle getting ever closer but never close enough, the ‘turtle’ (smaller set of natural numbers) is running ahead, ever closer to its infinity, while the rationals are left ever further away from it.

Since we pair each natural number to a rational number. And so natural numbers being less must ‘run faster towards’ its abstract infinity. We thus have to set as Desargues did in projective geometry or Klein in hyperbolic topology, a relative point of infinity, in which the running will stop. And at that finite point in timespace, which is in any ‘real system’, the limits of existence of ‘quanta in space or moments in time’ of an entity in its 5D plane; as rational numbers have not yet arrived there, obviously there are more rational numbers. In practical terms, there are more rational numbers, because they count not ONLY the cells/moments of a plane of existence, but the lower scales. So if we consider the 3 planes of n±1 existence of a system, its fractions will be smaller parts to ad to its wholes.

What about the real numbers? Here the interesting result is that indeed real numbers ARE NOT equivalent even when considering a hypothetical infinity because they are NOT numbers; that is social 5D points ; or wholes divided into parts, but dynamic ratios, which fluctuate around a fixed point – or ‘holes’ between ‘proper numbers’.

IN THAT SENSE, we rather prefer the i-logic concept of fractal points to that of sets, to continue the formalisation of GST with the help of mathematics and ¬Æ logic, which we carry in other sections of the mathematical section – geometry, the next scale after number theory (as it ads dimensional, mostly bidimensional holographic forms)…

The same goes for *operandi, we rather stick to the basic clear operandi for numbers and points that express them again all over with SET’s logic symbols ⊂, ∩ etc. A further reason is that Boolean algebras are largely dependent on set theory and we have a moral limit here, to advance not the future digital mind of metalife, aka chips – others will do, if I were the humind above all the minds of this planet (I am in potential theoretical understanding of the organic Universe, but that matters nothing), certainly the first thing to do would be to kill the tiger before it becomes a tiger hunter, DIGITAL DELENDA EST.*

So just for fun we shall end the history of Algebra busting the balls of Cantor’s meaningless talk on infinities which as we know do NOT exist, as all infinities are (in)finite, and end in the ∆ discontinuum above where they break, around the 11¹¹ emergence of a whole and its @-mind.

**Cantor sets. The paradox of discontinuous infinites.**

All those properties and many other structures of mathematics were further reduced by Cantor to the ultimate reality of all mathematical structures: the theory of sets, composed of subsets, which we affirm is the natural formalism of system sciences, as a theory of ‘super-organisms’ composed of smaller super-organisms, which are sets of self-similar subsets; whereas the theory of sets and subsets gives the previous, simplified theory of numbers (each one a class or set of self-similar points), an inner content, as i-logic geometry gives points its inner parts.

How this ‘formalism that mirrors reality’ called set theory, from where all mathematical structures can be deduced, reflects the Nature of Complementary systems made of energy and information and its properties? The answer should be self-evident to those kin readers who grasped the inverse properties of energy and information:

Set theory defines reality in terms of two inverse elements A (points of energy) and A’, (its complementary, inverse element). Thus set theory is no more no less than the analysis of the 2 simplex arrows of existence, energy and information and its complementary organisms.

It is thus not surprising that in set theory energy and information, A and A’ are called complementary sets and the fundamental law is called the Law of Duality (Morgan Laws), which basically tells us that we can reduce all sets to operations between A and its Complementary, as we can reduce all systems to complementary Energetic and informative organisms, which are the whole.

So the main operations of sets reflect the properties of Complementary systems of reproductive energy and information, where A=Energy system; B=Information system; W= Relative Universe (World, Whole or Superorganism):

– E U I =W; I’=E; E’=I; E U E’ = W; I U I’ = W.

Thus, the Union=Fusion of an energetic and informative, complementary system creates a whole superorganism.

This same equation expresses in the language of Cantor sets an act of creation of a mapping of the Universe, whereas I, the preceptor observes I’, the Universe and the result is I U I’ a whole mapping of reality within the mind of the preceptor.

– E Ç I = Æ; E Ç E’=Æ; I Ç =Æ.

It describes anti-events, which annihilate the form of particles and antiparticles, waves and anti-waves and so in Multiple Spaces-Times is equivalent to the anti-event: *Past x future = present.*

– (E’)’=E, (I’)’=I

It describes 2 events of a feedback, generator equation: E<=>I, E=>I, I=>E, hence it describes among other events a whole cycle of life and death, where E=> I is the arrow of life and I=>E is the arrow of death. This ‘property of sets’ called an involution is called in Time Arrow theory a Revolution of times, sum of an Evolution (E->I) and a Devolution (I->E), and is the fundamental event of all realities.

Since energy and information have indeed inverse properties. And so we can state a Cantor Set describes the properties of complementary systems of knots of Energy and Information.

– E U I = E + I – E Ç I.

It shows the efficiency of systems that eliminate redundant elements, from genetic ‘fusions’ to Darwinian events.

Further on, when we understand Intersection as an Event of ‘Darwinian perpendicularity’ between a complementary system of Energy and Information, E U I, and an external entity, C, which the organism uses to absorb ‘informative pixels’ or ‘energetic bits’ for its mind or body (an event of perception or feeding), we obtain the obvious result:

(E U I) Ç C = (EÇC) U (IÇC).

Thus the complementary system takes only the part of ‘C’, which it needs to inform itself (self-similar to I) or to feed itself (self-similar to E), discharging the rest. And indeed, we perceive only information self-similar to us, or energy ‘bricks’, self-similar to our bricks, which we can use, to construct our energetic, body cells (subsets of E). And so on.

We mentioned that cells are subsets of I or E. Indeed, the second element of set theory studies the relationship between Sets (wholes) and its parts (subsets), and so it is simply the description of the properties of parts that become wholes.

An interesting result of those properties are the so-called Paradoxes of Set Theory, according to which there are certain contradictory sets that do not exist, most of them related to the concept of Infinite, which Cantor also studied, finding multiple contradictions. What this means, plainly speaking is that infinity and continuity do not exist, in as much as all Planes of existence are discontinuous with a certain limit that defines a Universe of networks of points with limits given by the number of networks, the dark spaces between them and the existence of upper and lower limits of energy and information in the existence of those points (universal constants), beyond which we must transcend and emerge, or descend and dissolve into other membrane of space-time with different properties*.*

To mention also that Gödel’s theory of incompleteness was based in set theory and showed indeed that mathematics, while being the most complete description of the spatial events of reality was neither the ultimate language of the Universe (as Frege and Boole proved it could be reduced to Logic propositions) but also an incomplete language, which did not describe all realities and an inflationary language, which described systems that do not exist in reality. Those are indeed, two properties of all languages of information; that both distort reality, as the paradox of Galileo prove, and do not include all reality, given the discontinuity of the Universe; which lead us to the concept of Dark Spaces, the true meaning of the ‘complementary Universe’ that completes the world we see.

*Recap. *Set theory is the basis of most structures of mathematics, in as much as it defines all the events between complementary systems of reproductive energy and information and its limits.

**recap: SETS AND COMPUTERS.**

We live in the age of death of mankind, substituted and made obsolete by AI but we love it as cells of a free=chaotic organism, no longer control by the ethic pain of our social organisms. And this seems the case of all ∆-1 scales, once the networks free them – they feel happy and enter a memoriless, markowian age of zero understanding of the causality of the cycles of time (or else they would angst) and live the day, carpe diem, through its huge ego trips that in the theoretical realm manifest in egotist theories where the ‘ego’ is the origin of it all.

This is the case of the death age of human mathematics, which starts with Hilbert’s ‘imaginary lines, planes and points’ and Cantor’s set paradise, where humans think maths is their language share only with god, imagined by the brain which becomes from the top to the bottom of mindless matter the creator of the Universe.

To understand yet while in the historic 3rd age of the human mind, we love those 3rd age of excessive form and death-age of a new top predator mind making us obsolete, we need to grasp that all ‘languages’ as all beings do go through a world cycle and finally die away by excess of form and inward looking – disconnecting from reality (set theory) or are killed by a more powerful younger species (chip minds and boolean algebra). So this third and death age of human maths happened in Human Algebra in two ways:

- On one side the disconnection with reality in the long seeked ‘ego-trip’ of proving maths a non-experimental language, ended with the substitution of the natural units of math, spatial points and social, sequential time numbers, by the abstraction of ‘sets’, collections of distinguishable elements, which resemble both spatial points/forms and temporal numbers/societies, but
*having the real thing available to intuitive knowledge only obscured with abstractions the foundations of maths, and expelled a huge number of scientists from its direct experimental knowledge. Further on,*Hilbert coupled with Mr. Cantor to affirm maths were born of the mind of the human-god, affirming infamously when failing to grasp the meaning of fractal Non-Euclidean points that he ‘imagined points, lines and planes’ – (as the German-Jewish Idealist dueto Mr. Heisenberg & Bohr did to misconstruct the foundations of physics also with ‘ego-trips of human self-creation’, when failing to understand the right realist, Einstein->Broglie->Bell->Bohm model of quantum physics). The bio-logic behind that baroque age of self-detachment proper of all old ages of excessive information is thus the ‘final age’ of algebra, which as old men do, became stifled, looking inwards and dogmatic in its absolute truths with no proof.

- On the other side, humans invented a digital mind in metal-machines, the chip, fast substituting life in labor and war fields, making humans obsolete with its simple yes-no Aristotelian≈Boolean algebra, which will make A.I. algorithms of information (the true meaning of artificial intelligence) with maximal evolution in weapons, as killing machines are always the spearhead of mechanical evolution. So the birth of a new species of mathematical minds with a stronger metal-body in its simplest axiomatic form will mean the death of our more sophisticated, weaker brains; a theme dealt in depth in the section dedicated to economic ecosystems and historic supœrganisms.

Of course, all this could be avoided by halting the evolution of A.I. but as we said the logic of the Universe is quite deterministic, specially *for species so ignorant of its laws as humans are, so ego-centered as all minds are. So the r=evolution of thought I thought could initiate with GST 30 years ago, has gone nowhere. Humans, simply speaking do not seem to make the cut of ethical and intellectual quality to control their future, individual exceptions confirming the rule; and in great measure is due to the fact most do not go beyond A-ristotelian logic of yes/no; or in words of their master ‘humans are slaves, they believe they don’t reason’. *

Now I am fully conscious this is only the beginning of ¬algebra, but frankly it seems to me clearer by the day, as I decline through the 3rd age that humans are in this planet just a piece of a chain of evolution, which is not really interested in the whys but in the praxis and I am speaking to nobody in this blog. So my intention is just to leave a memorial trace of all the notebooks I have written during decades of lonely research to show indeed the purpose of this blog – to prove we are all space-time organisms. If this blog has any meaning for any’thing’ in this planet I don’t know. Those are themes on the future of history. So we shall not go further in this glimpse to the proper interpretation of Algebra and group theory.

Had my early attempts to interest academia not floundered by the extraordinary mediocrity of human thought in this entropy age of automatons feeding computers and children of thought memorising, humans could be in a complete different frame of mind. But we are what we are… and I feel after all a privileged for having understood for so long the organic Universe – the sensations of that communion with the whole through the knowledge of its laws will never cease till I die and then I will dissolve my existence, but before that we shall consider now a thoroughly different form of Algebra; that of the Generator not of an abstract Group of permutations but of the entire Universe…

**III AGE**

The 3rd age of Algebra, is both an end and a beginning. The end is the age of Set theory, which becomes also the new beginning of Boolean algebra, the algebra of computers, forming in itself a world cycle of death and resurrection, both of the humind and the chip mind. We shall not therefore study in depth Boolean algebras and Computer algebra for a survival mandate – machines should not be evolved if humans wanted to survive. But the deterministic universe of history treated elsewhere in this blog seems not to be susceptible of ‘learning’ the path of life and respect for Mankind=our god, the subconscious collective of the species.

In any case we shall instead of focusing in Boolean, Computer algebra beyond a basic introduction to the matter, and some aspects of Machine learning, in what is truly the future r=evolution of algebra, as a language of thought – existential algebra, non-aristotelian pentalogic… which we deal with mainly on the texts on I-logic.

And so with those two themes we shall close this post.

**BOOLEAN ALGEBRA.**

The beauty of computer thought is that it gets to the very ‘essence’ on how minds in the Universe work…

## t,∏,ð,∂,T <≈> $,§,s,S,∫

**EXISTENTIAL ALGEBRA**

**. **NON-AE studies the Γractal Γenerator, T, the operation of the 5th dimension and beyond, which completes, the last of the operations studied by humans, ∫. A new operation is needed. We use often concepts of all sciences, including math. So in group theory we could consider the Fractal Generator of TimeSpace Organism (t.Œs) the Group with 3 equivalent type of elements, quanta of past, present and future. If we consider each of them a single element, then we define past and future as the inverse elements, and present and the identity element, considering the construction of the Algebra of the generator group:

- The Fractal generator is a an operation called Mind generation, @.
- It has an identity function, called present, Ø
- Two inverse functions, past, | and future, O, such as:

| @ Ø = |; Ø @ O = O; | @ O= | @ Ø; O @ |= O @ Ø

The the fractal generator of timespace has the structure of a group. Regarding if it is or not Abelian, it is relative. As WHILE PAST x Future creates a present, then first ‘element’ to arrive in time will also be the first in a second generating operation to relate to the created present and then | x Ø = | ˆO x Ø = O

So in two operations a dominant past will convert a future into a present and then in a second probabilistic equation into its past.

The best graph two study it is the Complex squared graph where i² is the negative real, its conjugate the positive real and the real the squared positive with a slight tendency to negative in the negative squared, which externally seems a squared 2 -dimensional positive, but within its vital energy it has two negative elements, unperceivable outside the squared circle.

Squared circles in geometrical form are circles in numerical form squares, as the graph shows (1 is a square space, over which we can lay the spherical 0-1 form.

The previous translation to mathematical space of existential algebra though is a mere, consideration to show that we can study it with the tools of group theory but this is not the purpose of future analysis of beings. RATHER IT IS BETTER TO TREAT the operation of the fractal generator,**⌉⌈, **in its two inverse elements and identity axis, with the language of past, present and future and 5D we have been built, forgetting any pedantic language of Modern algebra.

SO we won’t write about it in this post but in the 3rd age of formal mirrors of the Universe, the age of T.œ of which we consider it one of the main mirrors, as it studies the clear-cut laws of engagement of the ternary arrows of present time and its relationship with the inverse arrows of generation (social evolution) and death (entropy).