1. Quadrature of a circle of the projective universe

© А.I.Imaev, Y.К.Imaeva, «Quadrature of a circle of the projective universe». – Yekaterinburg: publishing house «Socrat», 2013. – 136 p., il. ISBN 978-5-906361-05-9

Quadrature of a circle of the projective universe

Galileo: «Nature is like a book written in the mathematical language».

Cosmologist Max Tegmark: «Our Universe is mathematics itself, it’s a gigantic mathematical structure», «The more we study physics, the more mathematical patterns we find», «Universe is a mathematical object, it exists in the platonic sense, but not in the space-and-time; vice versa, space and time exist inside it». «It changes our point of view. It means that mathematics is not a language that we’ve invented but a deep structure which we gradually discover like archeologists; it is an abstract and constant object, when we remove coats, we find a code». «This theory is closer to truth, since if the reality is a mathematical object, we are able to understand it» [1].

Since the aftermath of Einstein the search for the Unified Field Theory has been going on. Einstein wanted to construct pure physics of the field, where one could reject the notion of ‘substance’. He was looking for a unified principle which governed all processes and interrelations in the Universe. The most ancient of exisiting problems – the problem of the quadrature of a circle – is able to present a simple explanation of the Universe organization, to crack open the curtain over the Theory of Everything.

In order to disclose this thought we shall use the words of Stever Hawking, who is one of the most famous physicist of our time, about the cookbook of the Universe. What ingredients shall we need? Hawking, being a famous cosmologist and a scientific expositor, suggests three ingredients which are necessary for ‘cooking’ the Universe: mass, energy and space. Taking into account the most famous formula in the world Е=mc² about connection between energy and mass, Hawking leaves two ingredients: energy and space. Therefore, if there are these two ‘ingredients’, the Universe will objectively exist.

But what is energy? Nowadays the notion of ‘energy’ is widely used both by scientific and unscientific sources of knowledge. Sometimes people try to explain odd and strange phenomena with the help of some unusual energies. But, for example, the Bible, being the main source of spiritual knowledge, uses the expression ‘the Strength of Mind’ instead of ‘energy’. The Strength of Mind is a living force as a reason for all processes which take place in the world. In opposition to this, in a contemporary scientific sense by the term ‘energy’ one can mean a non-living force. «A classical Greek word ἐνέργεια means «action, activity, force, power». In 1807 Thomas Young was the first to use the term ‘energy’ in the modern understanding of this word instead of the notion ‘living force.

Energy is a general quantitative measure of movement and interrelation of all types of matter. Other definitions of energy are unscientific. The notion of energy is to connect all natural phenomena. But if energy is a measure, we shall further demonstrate that while having this definition of energy, it is born by space and is a consequence of the existence of space. Therefore, there remains only one thing which gives rise to the Universe – it is space.

But how can we define what space is? Space is not only defined, but also described and understood through geometry. Geometry can be different, but if there is space, it will definitely have geometry. If there is geometry, there is a proportion, i.e., numerical ratios. Proportion leads to patterns, to the structure. So, what is space? Space is a geometrical and numerical structure. Geometry and a number give rise to physics.

This idea is not new, moreover, this idea and this conclusion let us perceive the world from scientific point of view. The question is to find such a geometrical and numerical structure which would be appropriate to natural phenomena observed and would definitely describe them and joined everything into one Unified theory.

Let us demonstrate that this geometrical and numerical structure is the quadrature of a circle.

Mathematical essence.


                                                 1                                2                            3

Quadrature of a circle is a square matrix of all sizes superposed on the transcendental fractionality of a circular line. 1 – a circular line is divided into 4 parts, 2 – a circular line is divided into 12 parts and so on. This division can be considered as a polygon which is close to a circular line.

As is known, perimeter of a circle is calculated according to the formula 2πr, where r – radius of a circle, π – the transcedental number. The transcedental number is perpetual: π = 3,141592653 ….. – fraction digits are lost in perpetuity, they change one another without any remarkable periodicity. π-number is an infinite fraction.

π-fraction, which is a part of the formula 2πr of the perimeter of a circle, does not let calculate the exact final value for the perimeter of a circle. In order to calculate the perimeter of a circle one should use the mathematical notion of the limit – a certain number which a sequence of numbers goes to. By ‘the perimeter of a circle’ we shall mean the limit which the variable perimeter of a regular polygon, which is inscribed into this circle, goes to (approaches), when the number of its sides can be endlessly doubled. If one inscribes a square, an octagon and so on into a circle and doubles the number of sides, one will receive a polygon whose number of sides will go to perpetuity and the side length will go to zero.


A polygon with a large number of sides approaches a circular line.

The mathematical essence of the quadrature of a circle consists in the fact that a polygon which approaches a circular line will never become a circular bar which has π-number in it.

While analyzing the existing mathematical apparatus it is necessary to state that there is not any other tool for perceiving and describing the world except quadrature. The notion of a vector, differential and integral calculus, notion of the limit, of a complex number – all these things are advanced methods of quadrature. We are able to describe a curvilinear object only through its quadrature. In other words, if we understand how to construct the quadrature on an object, then it means that we understand how to study this object.

Physics of quadrature of a circle.

A square contains an important component of the space structure – angle of 90°. Looking back, we can see that there are no right angles in nature. Nature does not create straight lines, everything is constructed with the help of circular lines. The only object of nature which contains the angle of 90° is the light.

The light is an electromagnetic wave. It is well-known that an electric wave and a magnetic wave are perpendicular to each other.


Electric and magnetic components of the light are perpendicular.

In the third volume of «Elementary physics»under the editorship of the academician G.S. Landsberg it is written: «strength Е and induction В of an electric and a magnetic fields in a wave are perpendicular to each other and to the wave direction.

The study of any wave directions demonstrates that for any direction the analogous position of vectors Е and В will be correct: both of them are perpendicular to the wave direction; it means that their vibrations will take place perpendicularly to this direction, i.e., the electromagnetic wave is transverse.


The electromagnetic wave is transverse

Transverseness of vibrations is absolutely common property of any electromagnetic wave which does not depend either on the choice of the wave direction or on the type of the radiating element. The same common property is mutual perpendicularity of fields Е and В in an electromagnetic wave».

It is also well-known that the light is straightforward.


Photographer Lars van de Goor,

source: www.larsvandegoor.com

According to the law of reflection, if the light faces any obstacles on its way, it reflects from them and coordinates its movement with the perpendicular to the surface of this obstacle. The law of reflection is correct for the electromagnetic emission with any length of a wave.


Reflection of the light. The angle of incidence equals to the angle of reflection. Both angles are calculated from the perpendicular to the surface of the obstacle.

Therefore, because of the straightforwardness the electromagnetic field of the reality is the quadrature in space of the physical vacuum. The existing natural electromagnetic field is demonstrated on the basis of the inner geometrical matrix. The quadrature of the physical vacuum can be called ‘the matrix of life’.

It will not be great exaggeration to say that for some centuries there has already been a domain of science which studies the quadrature of the physical vacuum, — i.e., spectral analysis. As is well-known, spectral analysis is based on the fact that atoms of every element radiate or absorb light at some certain frequencies. Frequencies, vibrations, fluctuations – all these things are geometrically connected with the length of a wave, and the length of a wave is determined by the quadrature. Every chemical element has its own unique linear spectrum. According to this spectrum one can perform the remote evaluation of the chemical composition. And according to the spectrum of heavenly bodies one can make a conclusion about the content of their matter.

Quadrature of the physical vacuum. Lengths of electromagnetic waves.
length λ demonstrates quadrature of the physical vacuum

Our eyes are able to perceive electromagnetic waves in the range from 440 to 700 nm. The visible range contains the spectrum of the rainbow starting from violet with smaller length of a wave to red with larger length of a wave. Waves, which are beyond the limits of this range, are called ultraviolet and infrared, respectively.

Spectral analysis. Atoms of every chemical element radiate or absorb light at some certain frequencies.


Spectral line


Absorption line


Radiation line

Taking into account the abovementioned transfer of the quadrature to the physics of the world order, we shall refer to words of Anatoli Vasilievich Rykov who is the author of the book «Vacuum and Universe Matter»: «Nature does not need in four interactions: it uses a unified scenario of acting forces, the basis of which is electromagnetism».

Also we would like to note that any law of proportionality which describes some natural phenomena is a method of the quadrature, and that, consequently, the quadrature lies in the basis of the observed phenomena. No law of proportionality can exist outside the quadrature.

Physical vacuum in terms of the quadrature of a circle.

Physical vacuum in terms of the quadrature of a circle is situated where an ideal circular line is. An ideal circular line is situated outside the quadrature, since there is only a circle as a polygon with a large number of sides in the quadrature.

In the place where an ideal circular line is there ‘is nothing’, and since a geometrically circular line (circles of all possible sizes) is everywhere, consequently, the physical vacuum is everywhere.

While describing an ideal circular line, the quadrature results in the transcedental number π. At the same time it turns out that the maximum fractionality, i.e. fractionality of the transcedental number, of space will be in the place of the physical vacuum. Here, the maximum fractionality will be present not in the physical vacuum itself, but «along the edge» of a circular line, where the fractionality film appears, the loop of the number π. The quadrature and the vacuum itself are divided by the infinitesimal number – i.e., the difference between the infinitesimal quadrature of a polygon and the number π itself.

Externalized forms of matter cannot be constructed in the infinitesimal fractionality. A circular line of natural objects which is visible to us is structurized, i.e., it is a polygon with limited finite number of sides.

 11  12
 Koch curve  Fractal «Koch snowflake»..

Fractionality of space creates a circular line. A circle, being a polygon, is matter.

Einstein wrote that laws which describe the field are structural laws. He defined ‘a substance’ as ‘a huge concentration of energy in a comparatively small space». These polygons of matter describe a substance as stationary waves with a certain whole number of knots.


Polygons of the quadrature which create the structure in the physical vacuum – stationary waves of energy which create the substance.

Quadrature of a circle and the relativity theory.

The famous Einstein’s question «Did the God have the choice when he was creating the Universe?» is a question about the geometrical and mathematical predeterminacy of the structure of the Universe. The relativity theory is constructed on the basis of the quadrature of a circle.

According to Einstein, when matter and energy are present, space will be flat and straight. The mediator of gravity is the structure of space and the mechanism with the help of which gravity acts is the space warp. Objects which have mass create distortion of space-and-time.

In addition, from the point of view of the relativity theory, «light velocity does not depend on the movement of a radiating source. Material bodies cannot have velocity which is more than light velocity. Light velocity is the upper limit for velocities of all material bodies». Quadrature of a circle of space explains why no object can move faster than light velocity.

In terms of the quadrature of a circle all objects which have mass are always composed of curved surfaces and are constructed in fractionality. All bodies which have mass are parts of rotating systems. Einstein wrote: «Our world is non-Euclidean. Its geometrical nature is determined by masses and their velocities» [2].

The light is in the inner Euclidean space. Straightforwardness is always «inside» in relation to space-and-time of an observer. Light is always inside and it is a constructor of the curved space, therefore, no matter how fast the curved surface moves or changes, it cannot outrun straightforward light which is inside. Movement along the curved surface cannot be faster than movement along straight lines which connect different points of this curved surface.

Stability of the light velocity lies in the basis of the relativity theory. Space in the relativity theory is able to stretch and shorten, time is able to slow and speed up from the point of view of different observers. Quadrature of a circle explains this by a difference of fractionality which is distortion. Mass and time are distorted while light is straightforward.

According to the quadrature of a circle of the space structure, any object which has mass by definition acquires curved geometry both of its construction and its movement, and it cannot be faster than movement along a straightforward line.

An observer can think that he or she moves along a straightforward line himself, but in fact it means that the radius of a circle of his movement is very large, it is much larger than a section of the way of movement.

«According to the relativity theory there is no substantial difference between mass and energy. Energy has got mass and mass is energy». Since a light ray which is visible to us is also a wave – space deformation, change of state of a substance, change of the quadrature, i.e, a light ray possesses energy and energy has got mass. Consequently, in some cases deviation of the visible light from straightforward movement will be obvious, but this fact does not cancel the law of quadrature as an inner construction of space.

Quadrature of a circle agrees with gravitational theories both of Newton and Einstein, there is no necessity in their conflict. It demonstrates correctness of the formula of the Newtonian gravitational law and explains geometry of Einstein’s gravitation.

Despite the fact that, from the point of view of an observer, it may seem that light must logically move diagonally or along a curved line in relation to an observer, anyway light moves straightforward,  by quanta, from the inside, and mass and time are distorted here.


Quadrature of a circle and quantum mechanics

Quadrature of a circle of the space structure explains the main principle of quantum mechanics, i.e., discreteness of energy and matter. Quadrature of a circle leads us to understanding geometry of a light quantum.


An initial quantum of the geometrical organization of the Universe

Geometry of a light quantum is the angle of 90° which is restricted by a circular line with fractionality of π-number. Fractionality of π-number means that light is not able to go out of limits of a circular line. It clarifies the wave-corpuscle duality which is a mystery of modern physics. The number π creates a particle which is the length of a wave.

Because of the transcedence of π-number the square matrix of the Universe is discrete and is divided into parts by a circular line which is strictly ordered. Light is simultaneously both a wave and a particle. Discreteness of the square matrix in time looks like a wave.

We shall see the confirmation of this in the following definitions: “Energy of a light quantum of a uniform ray decreases in proportion to the increase of the length of a wave. Energy of photons of the red flash is twice less than energy of photons of the violet flash”. Energy of a photon in the quantum theory is proportional to its frequency and inversely proportional to the length of a wave in the wave theory. Energy of a photon Е = hn, where h – the Planck constant, n – frequency of light. Energy is constructed by the geometry of space.

“Interference phenomenon which was discovered by Davisson and Germer actually demonstrated that electrons are like waves and that mass has got wave representation” [3].


Propagation of an electromagnetic wave.

Mass and charge are discrete. The irrationality loop creates particles in the physical vacuum.

Reduced Planck constant ħ=h/2π

The geometry of a light quantum demonstrates that the angle is 90°, light is inside, and a circular line is external in relation to light. It follows that a straightforward geometrical matrix of the Universe structure is hidden inside material forms and outside we are able to see a circular line in construction of natural objects.

Thus, quadrature of a circle combines “two fundamental pillars which underpin the building of modern physics”, i.e., the quantum form of π-number and structural polygons of the smooth geometrical model of the relativity theory.

Quadrature of a circle and the holographic principle of reality.

The holographic principles stipulates that all 3-dimensional objects can be encoded in two dimensions. According to this principle, there is a encoded information on the edge of the Universe, some kind of “a sheet of information which describes the whole Universe from the inside”.

When we speak of the quadrature of a circle of the space structure, we are to add that this “sheet of information”, and more precisely an information loop is situated around every object in the Universe which has got mass and lifetime. Further we shall demonstrate that it is fractionality of a loop which determines both mass and lifetime of an object.

Both the holographic principle and the quadrature of a circle demonstrate that space-and-time continuum of the observed reality can be described without going out of limits of three dimensions which are realized by us.

Quadrature of a circle finds two fundamental axes – the electromagnetic cross – in the construction basis of the 3-dimensional reality. 3-dimensionality is created by distortion and fractionality, the simplest examples for this is the surface of a sphere. A sphere has got volume, but this volume in nature is constructed discretely through inserted curved surfaces, the distance between which is arbitrarily small. This is ‘an onion’ of the universe, “when we remove its coats, we find the code”, and the simple definition of which is the quadrature of a circle.

The same way a person sees the world – we see the world as being flat, we see not a volume but a perspective, to be more exact, we realize the volume through perspective and we are not able to see a volumetric thing simultaneously from all sides.

Quadrature of a circle and fractal geometry of nature

The explanation of the fact why all objects which possess mass are constructed in fractionality and have curved surfaces is presented by fractal geometry. Fractals have fractional dimensionalities which characterize irregularity and distortion. All objects which are constructed fractionally also have fractional dimensionalities.

“Usually when you look at a surface, you can see all its complexity and it seems to you as very non-mathematical object. It is necessary to think how to get what you see and it succeeds through endless repeating”, “fractals surround us in nature every now and then” [4].

Geometry of nature is complex not by a basic geometrical form but by a multiple inscription of basic forms which creates spirals.

The thing that seems very complex from the geometrical point of view – geometry of a forest, a shore, clouds, human lungs, — in fact can be achieved through fractal construction, i.e., through multiple repeating of the same operation which consists of very simple geometrical forms.

The holographic principle and fractal geometry are united by a main common provision: a small part contains a general image.

“You can develop formulas which will describe clouds, flowers, trees. These things are all around us. Nature is full of self-similar objects” [4].

When one performs fractal construction of a 3-dimensional space, its dimensionality becomes less than three. The existing 3-dimensional reality has got the dimensionality more than two and less than three.

According to the relativity theory, “division into time and space has no objective sense, as time is not absolute any longer”. Space in all directions has got fractional dimensionality, i.e., it is seen in time. Time appears in the form of overcoming discreteness while changing the state of a substance. Here the restriction of light velocity in the vacuum lies.

Time and mass are connected and are defined by the fractionality of space. Time determines the velocity of processes which is connected with mass.


Source of the illustration: [4].

From the point of view of fractal geometry and quadrature of a circle, dimensionality of an ideal circular line equals to zero. This confirms perceiving the physical vacuum as an ideal circular line.

“Fractal geometry suggests us a method of detailed and very exact description of the world we live in, including wildlife” [4].

There must be one principle which possesses the possibility of infinite number of variations in order to explain all possible natural phenomena. All infinite difference of fractionality which is mathematically presupposed in the quadrature of a circle constitutes the world that we realize.

Within frameworks of polygons of the quadrature irrational numbers, i.e., infinite fractionalities, represent themselves in the thing that is created by infinite variations. For example, it is well-known that every snowflake has got a certain geometrical form — a hexagon -  in its basis. Snowflakes are six-pointed and there are no snowflakes of any other structure. At the same time there are no two similar snowflakes. All of them are individual like fingerprints of a person’s hand. It is supposed that one cubic meter of snow has about 350 million of snowflakes, every of which is unique.


A snowflake is composed through inscription of hexagons. Fractionality of a circular line creates infinite variations of snowflakes within the framework of a hexagon.

The photo of water from the book of the Japanese researcher Masaru Emoto “The Messages from Water”.


Quadrature of a circle and string theory

The main principal novelty of the string theory is that in this theory the notion of a particle is replaced by the notion of a string. The theory suggests that in the basis of the whole substance there are extensions, i.e., thin invisible strings. Pulsating strings which lie in the basis of the universe and string loops ideally agree with the idea of the quadrature of a circle as the geometrical theory of Everything.

Also according to the string theory, “different modes of vibrations of fundamental strings lead to different masses and constants of interaction”, “properties of elementary “particles” – their masses and constants of different interactions – are exactly defined by resonant modes of vibrations which are realized through inner strings of these particles” [3].

According to the quadrature of a circle, strings do not need to vibrate. The quadrature of a circle contains and takes into account arbitrarily small possibilities of changing a substance. Thus, a substance is constructed along lines of the quadrature of a circle, depending on the certain state of the system.

Quadrature of a circle of the space structure is able to answer all questions “why are observed phenomena and their laws as they are?”.

According to the string theory, vibrations of string loops are resonant, “along the length of a string there is exactly a whole number of maximums and minimums”, these are, without doubt, polygons of the quadrature.

“In order to explain the real world it is necessary to understand the mechanism of symmetry distortion”, — says David Gross, a physicist, a Nobel Prize winner and a specialist in the sphere of the string theory.

Inequality in the world is created by an infinite small parameter — the transcedence veil — which is present everywhere. Quadrature of a circle is the difference of fractionality, the constant flow of forms, the thing that we call energy and different types of matter. The constant flow of forms is caused by the irrationality which is present everywhere in the space structure itself.

The string theory offers the implementation of new dimensionalities for describing interactions which exist in nature. “Additional interactions require additional dimensions”. But while describing physical processes we shall not need the introduction of new dimensions, if we take completeness of the Universe as a basis.

Completeness of the Universe

“A fractal is possible to construct when one divides a geometrical figure into smaller fragments”.

Theoretical physics suggests the mathematical apparatus for describing physical processes. But another point of view is also possible – treating the Universe as a mathematical object allows us to understand ongoing processes completely, basing on the existing geometrical and mathematical structures of space.

The Universe as a mathematical object cannot become this object without initial completeness of everything. The Unified theory of the field is impossible without inner completeness of everything, as the notion of the field presupposes the interconnection of everything that is inside it. Einstein wrote: “lines of force or, in other words, the field helps us calculate the force in any point of space”.

In 1980 the book of David Bohm “Wholeness and the Implicate Order” was published where he introduced his theory of the Universe structure which is based on performed experiments.

Bohm stipulates that everything in the Universe is interconnected at the underlying level which he called ‘concealed’ or ‘implicate inner order’. David Bohm wrote: “The Universe with all its elements, including us, is in fact a gigantic complete system where everything is interdependent which is not always evident. Everything that can be touched and that exists in the world separately — rocks, oceans, forests, animals and people, — is a visible level of the universe. However, all these things and phenomena only seem to be separated, in fact they are interconnected at the underlying level of higher completeness — the concealed order which is not available for our perceptive organs”.

Completeness of the Universe is confirmed by newest scientific experiments which result in new directions in physics. The principle of non-locality in the quantum mechanics stipulates that two particles can show themselves as parts of a single whole, without depending on the distance between them.

Gregg Braden, a geologist, an explorer of the Earth, an engineer in the sphere of the space electronics, the author of popular books, wrote: “In 70s Bohm offered a clearer metaphor which lets picture the world as a single whole which is presented in the form of separated parts. Pondering over interconnections of the Universe, he was more and more sure that the world was like a gigantic space hologram.

In a hologram any part of an object contains the whole object in a diminished view. From Bohm’s point of view, all things that we can observe in the environment are views of something much more realistic which is happening at the underlying level of the universe in the sphere of the hidden and genuine being. According to this approach, “things up are the same as down”, “things inside are the same as outside”. In other words, any system consists of systems of a smaller scale, which are identical to it in their essence.


A good example of the hologram is the elegant simplicity of a human body. It does not matter where the DNA-molecule is taken from — from a hair, a finger of a hand or a drop of blood, it will contain the genetic code of the whole organism. The genetic model of a human is always found without any changes in it» [5].

Although physics in its speculations refers to the Universe, each new discovery makes us realize that it is more correct to discuss only its visible part and introduced notions of the dark energy and dark matter which, according to calculations, occupy almost the whole space of the Universe and confirm this idea. The Theory of Everything must take this into account.

While observing only a part of the Universe, we feel the necessity to explain how objects which are visible to us are interconnected. For this purpose we introduce notions of forces and energies, through which everything is interconnected. This leads to some contradictions, since in fact these forces and energies are relative. Speaking of the organization of the Universe, we have to take into account that all things, excluding geometry and mathematics, that we use for its descriptions, will be of a figurative character. In fact some figures are more fixed in our notion of the existing reality, as, for example, an image of the independent material particle, and some figures are less fixed. Energy transmission is the movement of not a substance itself but its state, which is characteristic of all wave states. «The observed movement of a wave is the movement of some state of a substance, but not of a substance itself».

Quadrature of a circle geometrically explains the initial completeness of everything. Then the emptiness of the physical vacuum at the same time can be considered as the initial completeness.

The completeness of the Universe is confirmed by the first law of thermodynamics – conservation of energy and matter, nothing is taken from nowhere and nothing disappears in nowhere. Matter and energy can change the form and transform into each other, but their quantity will remain constant and we can consider this as being a direct definition of the inner completeness of everything.

The analogy of the initial completeness is the mystery of the rainbow. Each colour is already present in the white colour. Besides, all differentiations of the material world are present in the initial completeness. Newton wrote: «colours are not born anew, they become visible due to division» [3].


Quadrature of a circle and the issue of the origin of the Universe

The issue of the origin of the Universe in the contemporary scientific understanding is mostly the issue of the origin of the matter, i.e., of the mass.

According to the relativity theory, an object which has got mass creates distortion of the empty physical vacuum. In places where the vacuum is distorted there is mass, i.e., matter. If the mass of an object distorts space-and-time, from the point of view of treating reality as a mathematical object the following adverse idea will be logically correct: distorted geometry of space-and-time leads to the mass.

Appearance of the mass lies not in the existence of special particles but in the geometrical organization of the Universe.

Einstein wrote: «Our theory becomes more and more abstract. But our constant final aim is to understand reality better and better» [2].

The initial completeness of everything is empty and non-distorted. There are no notions of time and mass in the initial completeness.

Making analogy with the Big Bang, one can say that space cracked when there appeared a transcendent fraction in the initial emptiness. After that, raising or explosion took place, i.e., space distortion.

Here one must discuss the general initial coefficient of constructing space. Distortion of completeness leads to the appearance of matter. Matter is therefore created along lines of abruption of the completeness. Numerical and geometrical laws of the observed world depend on lines of abruption of the completeness. It means that the number which creates abruption is the general numerical coefficient.


A photo of a separate atom (shadow of a separate atom).

For the first time it was made in 2012, Center for Quantum Dynamics, Griffith University, Brisbane.

Source of the illustration:


The number π demonstrates the transcendent distortion of space which creates mass in the projective Universe taking into account the continuity of a fraction. The continuity of the transcendence of the number π is «Higgs field» – i.e., the thing that adds mass to particles. Higgs boson is not a particle, it is something that «covers» particles.

Mass in an attribute of any particle in the distorted universe which is connected by the fractionality of irrational numbers in its geometry. Particles «sink in» the fractionality. Thus, the biggest travel speed and the absence of mass are attributes of particles which do not have π-number in them and which move along paths which do not contain the number π. These are light particles – photons. It is well-known that photons are massless particles and that light is straightforward.

We would like to note that it is not the physical vacuum that has got mass, it is a circular line which appears in it. The larger the fractionality of a circular line is, the larger the mass is.

The less the fractionality is, the less the mass is. Different levels of fractionality determine the experimental registration of a set of different particles, masses of which differ in a hundred times.

The completeness decreases the mass. For example, it is well-known that a whole brick weighs less than the same brick when it is broken into pieces.

The sacred spring of the Theory of Everything

A Nobel Prize winner, an academician G.I. Alferov:

«Choose the way of a scientist and you’ll be closer to the God».

Finally, we shall point out the source where we have taken the idea of the quadrature of a circle of the projective Universe from. The geometrical code of nature is kept in the Bible, the most ancient book on the Earth.

Any theory by its definition cannot be called ‘the Theory of Everything’, unless it includes a huge layer of spiritual knowledge, i.e., knowledge which provides a frame for the existence of the civilized society and the science as it is. Besides, the Theory of Everything cannot but include the notion of consciousness, otherwise, it will be not the Theory of Everything but a theory of a part of visible reality.

Isaiah said: «And the vision of all is become unto you as the words of a book that is sealed, which men deliver to one that is learned, saying, Read this, I pray thee: and he saith, I cannot; for it is sealed» [Is.29:11].

In the Book of Revelation it is also written about the existence of a sealed book: «And I saw the right hand of him that sat on the throne a book written within and on the backside, sealed with seven seals. And I saw a strong angel proclaiming with a loud voice, Who is worthy to open the book, and to loose the seals thereof?» [Rev.5:1-2]. By «outside» the Bible means ikons and parables. Philip the Apostle: «The truth did not come into the world adamical, but it came in the form of symbols and ikons» [the acanonical Gospel of Philip, art.61, Nag Hammadi library].

«Inside» the Bible there is a geometrical and a numerical code of nature. The Bible is written in «Spirit and Truth» [Gospel of John 4:23]. The Book of Revelation (the Apocalypse) starts with words: «I was in the Spirit» [1:10]. Those who are «in the Spirit» are open for perceiving the inner structure of reality. «And he said to them: you are allowed to learn secrets of the Kingdom of God, and everything can be outside in parables» [Gospel of Mark 4:11].

According to the Gospel of John, the Universe appeared through the Message of the Creator: «Ad initium there was Logos and God had Logos. Everything through Him started to be and without Him nothing started to be what started to be» [Gospel of John 1:1,3]. How is the visible Universe created through a immaterial Logos of the Creator? The connection between the visible and the invisible can be represented only through geometry and mathematics.

The Creator’s Logos is a geometrical structure, as, on the one hand, it is immaterial and, on the other hand, it describes the visible Universe distinctly and clearly. «Visible came out of invisible» [Paul the Apostle, Jew.11:3].

In order to combine scientific and spiritual knowledge it is more correct to discuss not the matter but a projection. The Bible describes reality as a projection of Spirit and also mathematical and geometrical principles of its construction, i.e., it describes the presence of a certain structure of reality which is considered to be a projection of Spirit.

In order to disclose the quadrature of a circle as a geometrical structure of reality which is concealed in the Bible, we shall compare the first line of the first biblical book – the Book of Genesis of the Old Testament and last lines of the last biblical book – the Book of Revelation of the New Testament.

The Book of Genesis: «At the first God made the Heaven and the Earth, and the Earth was waste and without form; and it was dark on the face of the deep; and the Spirit of God was moving on the face of the waters»  [Gen.1:1-2]. These lines are supplemented by Solomon’s words concerning the creation of the world: «When He set a compass upon the face of the depth, when he strengthened the fountains of the deep» [Sol.8:27-28, the Old Testament]. The key notion here is the word ‘the depth’, which is used in the Bible for a lot of times.

Why is a circular line made ‘upon the face of the depth’? ‘The depth’ in the Bible means the infinite fractionality of a circular line which is described now by the transcendence number π which is included into the formula of a circle 2πr.

There are two approaches to history. The first approach suggests that modern scientific  achievements are unique for the human history. Another approach presupposes periodicity or, in other words, frequency of events. The Bible responds to the question concerning the uniqueness of human achievements through words of Ecclesiast: «Is there [any] thing of which it may be said, See, this [is] new; it hath been already of old time, which was before us» [Ecclesiastes, 1:10].

A circle is described in the Book of Revelations in the form of description of the holy city, «which comes down from Heaven», — light space: «And I heard a great voice out of heaven saying: Behold, the tabernacle of God is with men, and He will dwell with them; And the city lieth foursquare, and the length is as large as the breadth» [Rev.21:3,16]. ‘A foursquare whose length is as large as the breadth’ is a square and in a more general case, when we speak of equality of all sides, it is a rhombus. Diagonals of a rhombus decussate. Therefore, a ‘foursquare’ with equal sides describes the geometry of light in the best way possible.

Thus, the geometrical structure of the world which combines a circular line as ‘dark on the face of the deep’ and a quadrangle with equal length and width as the light which engenders is presented in the Bible through the quadrature of a circle.

Apart from this evident but metaphorical description, the definition of the quadrature of a circle is presented ion first lines of the Bible when the first day of Creation is described: «And God saw the light, that it was good: and God divided the light from the darkness» [Gen.1:4]. This is a simple definition, which combines both spiritual and mathematical and physical sense of the first day of Creation, becomes clear, taking into account the mathematical essence of the quadrature of a circle. A polygon which approaches a line of a circle will never become a circular line which contains the number π, therefore, the light is divided from the darkness, a polygon is divided from an ideal circular line, an electromagnetic matrix is divided from the physical vacuum by the transcedent number. «And the light shines in the darkness, and the darkness has not mastered it» [Gospel of John, 1:5]. The darkness cannot master the light, as the light and the darkness are different structures in their essence.

The quadrature of a circle as the geometrical alphabet of the Universe can be found also in the following words of Jesus Christ: «Again, the kingdom of Heaven is like unto a net, that was cast into the sea» [Matt.13:47].

Polygons of the quadrature, stationary waves with the whole number of knots which constitute the structure of the matter in the physical vacuum are described in the Bible as «cornerstones», «spiritual stones» of the Universe. Book of Job: «Where wast thou when I laid the foundations of the earth? Declare, if thou hast understanding. Whereupon are the foundations thereof fastened, or who laid the corner stone thereof» [Job38:4, 6].

Paul the Apostle: «And did all drink the same spiritual drink: for they drank of that spiritual Rock that followed them: and that Rock was Christ» [1 Cor.10:4]. «That spiritual Rock» means consequent polygons of the quadrature which are in the basis of the world projection.

Jesus Christ: «Show me the stone which the builders have rejected. That one is the cornerstone.» [Gospel of Thomas, 66], «a stone which the builders rejected, this one has become the chief cornerstone» [Gospel of Mark 12:10]. These words mean that it is the structure which is a key to determine the organization of the Universe in any of its scopes.

In the Book of Revelation there is a basic formula of reality. «And the city lieth foursquare, and the length is as large as the breadth: and he measured the city with the reed, twelve thousand furlongs. The length and the breadth and the height of it are equal» [Rev.21:16]. We shall assume that ‘equal height and breadth of a foursquare’ is с, then a foursquare-square itself is с². Let us denote the length, width and height of the city which is stated in the citation as а, b and d. If а=b=d, as it is stated in the citation, then abd is the cube volume. The projection of reality appears on the basis of the fact that a, b , d and с are not equal to one another, and they exist only in fractions of irrationalities. Their values constitute the structure of visible space. The formula of reality which follows from this abstract from the Book of Revelation is,
and it can be represented through Newtonian and Coulomb’s laws.
Discrete transformation of a straight 2-dimensional basis, i.e., of an electromagnetic matrix through time, creates distorted 3-dimensional space which is visible to us. Here, one can compare the structure of our space to a sieve or a bolter. Existing natural objects which are created through time have got stratified structure.

The quadrature of a circle presents explanation for one more secret of modern physics – the reason of the effect of the second law of thermodynamics. From the point of view of treating the Universe as a mathematical object, the second law of thermodynamics is explained through the irrational fractionality of the space structure which aims to bring everything to the condition of the uniform fractionality.

The effect of the second law of thermodynamics is called ‘corruption’ of space in the Bible.

The Theory of Everything which is described in the Bible discloses the way of the spiritual formation of a human which has got a strict reasonable explanation – physics of the world order. Loops over the physical vacuum are biblical ‘deep’, on the basis of which a human being is born due to the structure and the Spirit, through light. It is space of the transcedent fractionality, i.e., of ‘corruption’. Corruption of the material world, the influence of loops of irrational fractionality of the space structure does not leave us other possible correct ways except the way of spiritual formation which engenders and defeats the death. The Book of Job: «Therefore snares are round about thee, and sudden fear troubleth thee; or darkness, that thou canst not see; and abundance of waters cover thee.» [Job 22:10-11]. The word «waters» in the Bible, starting from the very first line of the Bible, means the environment of a human, i.e., space-and-time. Paul the Apostle: «So when this corruptible shall have put on incorruption, and this mortal shall have put on immortality, then shall be brought to pass the saying that is written, Death is swallowed up in victory» [1 Cor.15:54].

Quadrature of Great Pyramids

Being some kind of models of the quadrature of a circle, there are Great Pyramids everywhere on the Earth. Pyramids are located around the globe, both on the land and on the ocean bottom, being the evidence of the high level of development of previous civilizations. After a lot of milleniums it is rather difficult to evaluate real dimensions of some pyramids. But the evidence that has been achieved by researchers demonstrate that the ratio of the perimeter of the base of pyramids – Giza Plateau in Egypt – and their doubled height equals to  the approximate value of π. Thus, π is implied in the structure of pyramids.

Pyramids are inscribed into the geometry of the environment to a high precision, and due to its own geometry they arrange and rectify space. This explains the effect of pyramids – ‘corruption’ or the influence of fractionality of π is stopped in pyramids. The pyramidal effect means conservation of perishable goods, suspension of decay and fermentation processes.

But not only processes which take place in these pyramidal constructions are interesting, also the fact that these processes take place only due to a peculiar geometrical form of a pyramids is quite captivating. The geometrical form imposes the origin of certain processes inside it. The space geometry determines its properties.

The ancient knowledge of the mysterious nation – Maya – describes the world through the quadrature of a circle.

There remained not many texts from the Maya, but in one of them, in the book «Popol Vuh» («Book of the Community», ‘Book of Counsel’, or it is often translated as «Book of the People») there are lines which describe the appearance of the Universe through the quadrature of a circle: «And here we shall set forth the revelation, the declaration, and the narration of all that was hidden; … The true book written a long time ago exists but its observation is concealed from those who looks for and thinks. Magestic were its appearance and narration of the fact how everything appeared: heaven and earth; how its four corners and four main points were established and recognized; how it was divided and how the heaven was divided. And here was the rope for measuring delivered; and it was set in heavens and on the earth, in four corners, in four main points, as it was called by the Creator and the Maker, Tepeu and Gucumatz of life and all things» [6]. The image of the ‘rope’ demonstrates the geometry of the world order.

Mayan calendar is based on the space geometry. The Maya created the more exact calendar than that that is used by the world nowadays. Exactness of their calendar means that they understood perfectly what our world and its organization are like. Being transformed into our chronology and having received the worldwide fame, the date 21.12.2012 of the Mayan calendar means the change of epochs. A new epoch is accompanied by the new knowledge, new discoveries which change the life of people and our understanding of the world.

The same image of a rope which goes through the whole Universe is being used at present by scientific expositors who try to explain the search for a solution of the geometrization problem, a part of which is The Poincare conjecture, about the form of the Universe. The image is in the following: if one represents this ‘rope’ as a loop, then will it be possible to retract this loop or will it twine around a «hole»?. Consequently, if it is possible to retract this rope, the Universe has got a spherical form; if not, then it has got the form of a torus with one hole or the form of a torus with two holes. The Poincare conjecture is famous for the fact that it was resolved by the Russian mathematician Grigori Perelman who rejected the prestigeous mathematical award and also a million of dollars which were awarded to him for finding the solution. Perhaps, after introducing solutions for these extremely difficult problems to the academic community, Grigori Perelman realized much more in the issue of the geometrization of the Universe than is kept in these mathematical theorems. The Bible which keeps knowledge of the world which comes from Jesus Christ and prophets alleges that the problem of the geometrization of the Universe cannot be resolved without taking into account the cornerstone, i.e., ‘a foursquare’ in the Book of Revelation.

Quadrature of a circle and «The Secret Doctrine» of Blavatsky

Blavatsky: «Neglecting science is steeping into darkness ».

«The Secret Doctrine» of Helena Petrovna Blavatsky, a theosopher of the 19th century, is an attempt to transfer the ancient doctrine which keeps the Theory of Everything and which is extant only in the metaphorical form. We shall present only some interesting phrases of Helena Petrovna which disclose this idea.

«Outstanding scholars will have to turn their mind to studying the symbolic global language with its numerical and geometrical keys».

«There is a «general abstraction», i.e., the thing that we call «a constant coefficient» or «a general function» in mathematics, which is included not in a particular construction but in all constructions; in other words, as a general ratio of 1 to 31415».

«The whole Cosmos must inevitably exist in a single energy source», «light, warmth, sound, coupling, etc.; it is the «spirit» of electricity which is the Life of the Universe».

The symbolic global language includes a very ancient symbol – the Egyptian ankh. The ankh looks like a cross with one curved end. In ancient frescos this cross is held by pharaohs in their hands, it can be seen on statues and stone figures of gods, it is made in the form of decorations. This symbol-hieroglyph in the form of ankh means «life». If one succeeds in reading the Egyptian ankh, it means that one has learnt main structural components of the geometrical and mathematical structure of the Universe.

The curtain of the transcedent fractionality, 180° of the radian measure π, creates initial division into two parts, into a particle and an antiparticle as a necessary condition for represenation of the form out of the physical vacuum through electromagnetic interaction, i.e., a straightforward matrix.


The Egyptian ankh is the beginning. In other words, it is the geometry of the first particle which has got mass and which is visible from the physical vacuum.

Here, notions «positive» and «negative» are just the case of agreement. But what’s the point of existence of charges? The point is in division the whole into two parts. From the point of view of treating reality as a mathematical object, the thing that we call the efficiency of an electrical charge is the restoration of the distorted symmetry of the geometrical structure. For example, ions bundle into molecules under the influence of electric forces. Atoms become ions when they start to have abundance or lack of electrons, i.e., they face the distortion of the symmetry of the structure.

If the statement that «The Secret Doctrine» of Blavatsky was a reference book for Einstein is correct, then it is clear that this book helped Einstein understand the main idea of the relativity theory.

In contrast, we shall use the words of Blavatsky who was commenting Vishnu’s words concerning the circle of the Universe and formulation of the relativity principle.

«The Secret Doctrine»: «There is no better definition for a natural symbol and the evident nature of a Deity, which has got its circle everywhere (and it is endless) and therefore it has also got its central point everywhere; in other words, in every point of the Universe».

The relativity principle: «The fundamental principle of the special theory of relativity which stipulates that all observers who move with constant speed experience the effect of the same physical laws and, consequently, every observer who moves with constant speed has the right to stipulate that he is at rest».

Taking into account also the equivalence principle which is a basis of the general theory of relativity, according to which an accelerating observer has got the right to allege that he is at rest, «while thinking that the force which influences him is determined by the gravitational field», every observer, therefore, has got the right to allege that the whole world turns around him and that he is in the central point of the Universe.



As a conclusion we shall present examples of the action of the quadrature in different spheres of the world:

• Light waves of different lengths correspond to different colours. The quadrature of a colour is around us.

• Sand which is scattered on the vibrating metal surface takes contours of some certain figures of different kinds. Probably, mysterious circles in fields have got the same reasons of construction.

• A blast wave is the change of differentiation on the border of structures, along the perimeter of a circle.

• Water surface tension (borders of structures).

• The conventional quadrature of the Earth – meridians and parallels. The quadrature of the celestial sphere: bending and straight ascending – straight continuation of the earth quadrature which is projected at the sky – an imaginary mesh for measurements and description.

• The key word in the process of transferring information in the quantum science is ‘coherence’. The initial completeness by its definition has got coherence in relation to any condition and it allows to be the beginning of any process – creation of projections of any world states.

• Why is it so hard to create a computer which will be analogous to the human brain? The human brain as a fractal contains the image of completeness. The brain analyzes the world, basing on the completeness and deciphering particularities in the visible. A computer creates completeness out of parts, which is impossible to be realized in full in terms of the existing reality. The space structure creates the infinite number of variations; it means that all variations are impossible to be downloaded in the form of a program. Consideration of all possible variations is possible only from the whole thing to the particular thing, which happens in nature.

• Electric ignitions during the process of impregnation and at every heart beat. One can describe the heart rate as the quadrature of a heart. «In a healthy heart the prescribed rate goes through a muscle as an electric wave. These waves follow the exact sequence and control all heart beats».

• Quadrature of the optical vision. A human glance means very detailed  instant «photos» of the reality, the duration of which is about 100 ms. Some time is necessary for perceiving these «photos». If one moves very fast, he or she has got less time to perceive «photos» and contours of the landscape tend to coalesce. Insects which move faster have got less detailed «photos» of the reality but they make these ‘photos’ more frequently.

Using the law of light reflection, optics is able to explain why we see our reflection and reflection of objects which surround us when we look into a mirror.


Law of light reflection.

Here, if we have a point which is reflected in a mirror and if we construct a point which is symmetrical to it in relation to the mirrored surface, then, according to the law of reflection, rays which are reflected from the mirrored surface will «go out» of this symmetrical point  that we have constructed behind the mirror.


The symmetrical point which has been constructed by us is called ‘an imaginary point’ in optics, since when we look into a regular mirror, there is no such point in fact. It is just the situation when rays which have been reflected from the mirror strike our eyes, our brain completes the construction of the reflected beam and continues it behind the mirror until it crosses an imaginary image in the point M’.

The same way we see real forms of the projective space in the existing reality. A human brain possesses geometrical vision, i.e., an ability to construct space geometrically. When we look into a mirror, we see an imaginary image behind the mirrored surface. When we look at the world which surrounds us, we see real forms of the projective space through the curtain of irrationality through the electromagnetic quadrature. Vision is restricted by the optical range. The optical range is a part of the electromagnetic quadrature of the physical vacuum.

Naturally, a human brain is able to understand the Euclidean geometry, i.e., geometry of light – to understand the angle of 90°. Through the angle of 90°, through the light a human being sees, realizes and constructs the world. Apart from the Euclidean geometry there are other types of geometry, i.e., geometry of convoluted spaces, but these types of geometry are not natural for understanding a human being. If, according to Plato, «A Deity, while constructing the Universe, applies geometry», then a human being applies geometry in order to see, realize and describe the Universe. A human being geometrizes space, applies the square grid to it mentally or in drawings, but convoluted objects are realized through quadratures of curves and for this purpose there are differentials and integrals. Geometry of a convoluted thing, of darkness, of something that we call ‘matter’, is not natural for human understanding. The Book of Job: «He setteth an end to darkness, and searcheth out all perfection: the stones of darkness, and the shadow of death» [Job28:3]. A human being perceives the world only through the quadrature of a circle, he or she perceives a circle through a square – he or she sees the matter through the light.


In the end we would like to discuss the Bible once again: Can you create a book which  will exist throughout some milleniums and will be timely for every day throughout all these milleniums? What is able to live in science for thousands of years? Geometry and mathematics are. What is changing in science? Our interpretation of observed phenomena. Our theories change in course of time. Thus, what’s the point of truth? The truth results in geometry, mathematics and spiritual knowledge. Having combined them all, we shall get the Theory of Everything, i.e., the theory which is as old as the hills; it is always well-known, always timely and always keeps the secret – the depth, and when we dive into it, we apprehend the world.




[1] Prof. Max Tegmark, Massachusetts Institute of Technology. Quarry for the citation: BBC film «What is Reality».

[2] А. Einstein, L. Infeld «The Evolution of Physics: The Growth of Ideas From Early Concepts to Relativity and Quanta». – Saint-Petersburg.: Amfora, 2008.

[3] Brian Greene «The Elegant Universe (Superstrings, Hidden Dimensions and the Quest for the Ultimate Theory)».

[4] the film of the studio NOVA «Hunting the Hidden Dimension», 2008.

[5] Gregg Braden «The Divine Matrix: bridging time, space, miracles and belief».

[6] Jose Arguelles «The Mayan Factor».


More detailed description of the biblical code is presented in the book «Quadrature of a circle of the projective universe».