Ecology of knowledge. Informative: Fractal geometry discovered by Benoit Mandelbrot describes the ordered chaos of nature and demonstrates the principle of infinite nesting of self-similar structures into each other based on simple mathematical relationships. A fractal (from Latin fractus, “broken, broken”) is an infinitely self-similar geometric figure, each fragment of which is repeated as the scale decreases.
Fractal geometry, discovered by Benoit Mandelbrot, describes the ordered chaos of nature and demonstrates the principle of infinite nesting of self-similar structures into each other based on simple mathematical relationships. A fractal (from Latin fractus, “broken, broken”) is an infinitely self-similar geometric figure, each fragment of which is repeated as the scale decreases.
Is the Universe really infinite or just very big? Does the Universe have a center? Does she have boundaries? They do not exist, just as a fractal has no center or boundaries. Imagine that everything around is a fractal. And we are also part of this fractal. Infinite self-similarity.
The Universe expanding around us is not the only one; we may be surrounded by billions of other universes. Perhaps our world is only part of the Multiworld - a hypothetical set of all possible parallel universes. There are hypotheses that the Multiverse universes may have different laws of physics and different quantities spatial dimensions.
Most scientists recognize that the Universe has a fractal structure: planetary systems are united into galaxies, galaxies into clusters, clusters into superclusters, and so on. Previously, scientists believed that the distribution of matter could be considered continuous, starting with objects about 200 million light years in size. Data on more than 900 thousand galaxies and quasars showed that there is no continuity even at a scale of 300 million light years.
The findings contradict the fundamentals of the theory Big Bang, according to which in the first moments after the birth of the Universe, matter was distributed evenly and continuously.
A number of scientists believe that in the time that has passed since the Big Bang, under the influence of gravity, fractal structures on a universal scale could not have time to form.
Today there is no single mathematical model or theory that can describe every aspect of the Universe. The theory of infinite nesting of matter - fractal theory - is an alternative philosophical and cosmological theory that is not included in the standard academic fields of science. Currently, there is no theory of the fractal universe. According to researchers, based on Einstein's theory of relativity, the creation of such a theory is possible. If academic science accepts that matter in the Universe is distributed in the form of a fractal, it will require a revision of almost all existing models Universe.
Fractals embody the principle of repetition - copies, which are found in abundance in nature. This geometric shapes, which look the same at any degree of zoom. Fractal geometry is not “pure” geometric theory. This is a concept, a new look at well-known things, a restructuring of perception that forces the researcher to see the world in a new way.
Aristotle, Descartes and Leibniz argued that matter is divisible indefinitely. In every particle, no matter how small it may be, “there are cities inhabited by people, cultivated fields, and the sun, moon and other stars shine, like ours,” the Greek philosopher Anaxagoras stated in his work on homoeomerism in the 5th century BC .
The main postulate of the legendary “Emerald Tablet” of Hermes Trismegistus states: “What is below is similar to what is above.” This principle was accepted as an axiom by followers of Hermetic philosophy, who argued for an analogy between the micro and macro worlds.
The sacred teachings of all ancient civilizations are permeated by the idea of the existence of a harmonious Universe. The Egyptian goddess of truth and order, Maat, represented the embodiment of the principle of the natural order of things. The Greeks, who studied with the Egyptians, associated the word “cosmos” with civilization, translated as “embroidery” and expressing the harmony and beauty of “self-similarity”. If we look at these objects at different scales, the same elements are constantly revealed. All of them can be described in the form of mathematical equations.
Principles sacred geometry, which is based on fractals, “Platonic solids”, the Golden Ratio spiral, the number Phi, in equally inherent in humans, flowers, and stars. Everything that exists in the real world is a fractal: circulatory system, crowns and leaves of trees, clouds and oxygen molecule.
Research related to fractals is changing the usual ideas about the world around us. Fractals force us to reconsider our views on geometric properties objects. Fractals describe real world sometimes even better than traditional physics or mathematics.
We cannot describe a rock, a piece of landscape, the surface of the sea, a rock, or the boundaries of an island with straight lines, circles, and triangles. This is where fractals come to our aid. With the help of fractals, these structures can be modeled and created, which is used in various computer programs.
When we look closely at a fractal shape, we see the same structure regardless of the degree of magnification. This similarity can be seen in nature, looking at mountains, clouds, and coastlines at different angles. Nature is an unbroken web.
Fractal geometry is the geometry of nature. Nature itself takes advantage of her achievements and examples of this can be found everywhere: from the spirals of a shell and daisy flowers to the symmetry of a hexagonal honeycomb. “Self-similarity” can be found when studying the shapes of molecules or galaxies. All objects in the Universe interpenetrate each other.
Fractal geometry predetermines the shapes of the molecules and crystals that make up our bodies and the Cosmos. In fact, it is the key to understanding the Universe.
Fractal structure is genetic code Universe. published
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SECOND CHAPTER OF THE BOOK "THE THEORY OF RELATIVITY AS AN ACCIDENT"
THE RELATIVITY PRINCIPLE AND THE NONLINEAR WORLD
“There is not a single concept about which I would be sure that it will remain unshakable. I'm not even sure I'm on the right track at all."
A. Einstein
SHINE LIGHT ON THE PRINCIPLE OF RELATIVITY AND YOU WILL UNDERSTAND...
It is well known that all the consequences of the special theory of relativity can be obtained from two postulates: Lorentz transformations and the ABSOLUTEITY of the speed of light (that is, its independence from the speed of the source and receiver).
In other words, the principle of relativity (PR) is redundant - it is not needed by the “theory of relativity”. And the theory of relativity itself, in accordance with its origin, should be called the “theory of absoluteness”...
Is it a coincidence that software can be removed from it without harm to the service station? Nothing happens by chance in science. A. Einstein considered his concept to be the theory of relativity. That is, the result of generalizing Galileo’s software from mechanical forms of motion to any other, including EMF. In fact, as is now becoming completely clear, the essence of the principle of relativity is its linearity, which reflects the invariance of processes with respect to inertial reference systems, which corresponds to the usual linear symmetry that exists between systems uniformly moving relative to each other. What processes are we talking about? Of course, about linear, reversible, that is, single-level ones. For the material world, this is primarily simple mechanical movement. However, the linear essence cannot be extended to purely nonlinear phenomena, including interlevel motion (transformation of matter into free EMF and vice versa).
Multilevel and interlevel movements correspond to their own fractal symmetry; it is nonlinear and has little in common with the principle of relativity, although it completely replaces it hierarchically organized world.
Why, in this case, are the consequences of Einstein’s “theory of absoluteness” true, and where is the limit of their applicability?..
SHED LIGHT ON THE PRINCIPLE OF RELATIVITY AND YOU WILL UNDERSTAND
The purpose of this chapter is to show that the principle of relativity (PR) cannot be extended to the phenomena of our world that lie beyond the boundaries of the material world and its mechanics. First of all, this implies that Einstein’s theory of relativity (TR) was derived from an incorrect axiom - from the extension of relativity to phenomena covering the material world and EMF.
Let's start with obvious things that do not require deep penetration into the subtleties of the theory. Does software really not work where light, EMF, appears? Yes it is.
According to Einstein, software can be formulated as follows:
All identical physical phenomena under the same initial conditions proceed in the same way in all inertial systems ah countdown.
However, the author of TO implies that, for example, a reference system moving at the speed of light is not included in the number of inertial systems. The question is, why? Yes, apparently because the speed of light is prohibited for material objects. So be it! But this speed is allowed (by nature!) for “photons” that do not have rest mass. In this case, can we consider the moving frame of reference, combined with the photon, to be inertial (according to Einstein)? Again, no, although I would really like to: all material bodies without exception in such a reference system move at the same speed - the speed of light and have, accordingly, zero length, infinite mass and density, and stopped time.
That is, the world of free EMF does not want to include the material world as an objective reality, subject to the principle of relativity. The underlying reasons for this EMF behavior will be clarified in the subsection “Software and system asymmetry of the hierarchical world,” but for now we note that from the point of view of the material world, the world of black holes is also characterized infinite densities and stopped time. And this, of course, is not accidental. We just live in a hierarchically organized world, managing not to notice it.
It seems to us, living in the material world, that the principle of relativity is universal - it works, even in the presence of EMF: it is enough just to postulate the absoluteness of the speed of light in a vacuum and derive the Lorentz transformations from these two postulates, replacing the good old Galilean transformations with them.
Without dwelling on the criticism of this approach, we note the following: the software in this case turns out to be somehow one-sided, clearly skewed. From the point of view of objects of the real world, software exists and functions, but from the point of view of objects of free EMF, it does not exist, it is a fiction. Is it correct to consider such an expansion of the powers of software as its generalization to all phenomena of the material world and free EMF in the aggregate? Or is this still a semi-generalization of it? What to do with the “third world” - the world of black holes (BH), which clearly represents a third form of existence of matter, still little known to us? How else is it necessary to generalize the software in order to correctly take into account this post-material form of organization of matter, unknown to the author THAT?
Don't try to find the answer to additional questions- They are not here. But the answer to the main question is clear. There is no correct generalization of software to phenomena that lie outside the material world. Yes, some consequences of TO are in good agreement with experimental data, but these consequences, as will be clear from further analysis, are not at all the result of generalizing Galileo’s TO to new forms of motion (levels of organization of matter), including free EMF.
Light is a delicate matter. It does not obey the principle of relativity. Concluding the introductory part of the chapter devoted to software, we will give a curious example when literally the same phenomenon associated with the emission of light “proceeds” differently IN THE SAME REFERENCE SYSTEM.
Imagine a stationary frame of reference and two observers in it - at points A and B. Let a source of green light move at a constant, fairly high speed relative to it - in the direction from point A to point B. Then at the same time the observer at point A will see a source of red light, and at point B - blue. That is, the same phenomenon IN THE SAME REFERENCE SYSTEM will look very different.
Yes, this is just a consequence of the Doppler effect, but it should not conflict with the software. But let’s forgive the Doppler effect for this little prank: then it will help us understand the main thing - what the “theory of relativity” is.
THE THEORY OF RELATIVITY AND DIFFERENT FORMS OF MATTER
It became commonplace the fact that the theory of relativity, including general relativity (Einstein’s theory of gravity, as it is now more streamlined to say), is not able to adequately describe and explain the phenomena occurring on the border of the material world and the world of “black holes” (BM and BH).
Indeed, from the point of view of this theory, a black hole is a kind of singularity, that is, a region (point?) where everything is infinite: the mass of a material body collapsing there, because this body has reached the maximum speed (the speed of light), density and even the time during which it occurs this collapse.
That is, on the one hand, TO prohibits material bodies from behaving in a similar way (after all, for this it is necessary to spend and endless energy), and, on the other hand, instructs them to behave exactly this way (at the VM-BH boundary). It is generally accepted that overcoming such “small difficulties” lies on the path to creating a “theory of quantum gravity.”
To a lesser extent, it is understood that the theory of relativity does not adequately describe the phenomena at the boundary between matter and free EMF. How does this inadequacy manifest itself? But is modern quantum electrodynamics, which adequately describes free EMF, integral part or a special case of THAT? Of course not.
Moreover, they are incompatible, “unseamable”: the quantum nature of the first (albeit inconsistent, not brought to logical completion) and the continual nature of the second exclude the possibility of their unification: they are alternative! This means that at least one of them does not adequately reflect reality. Can you guess which one?
Einstein, well aware that one of the two theories ( quantum mechanics or the theory of relativity) must give way to another, categorically did not accept and sharply criticized the following aspects quantum theory:
- the probabilistic nature of the description of free EMF (which, as is becoming increasingly clear, is associated with the fundamental unobservability of the phenomena of free EMF from the point of view of an observer from the material world - the BEB between them interferes - but which does not affect objective nature such description);
- nonlocality free field, a special kind of connection between free field objects, for which there is no speed limit (which in itself violates the postulates of TO), a phenomenon that was later experimentally confirmed many times, that is, a fundamental reality that is completely inexplicable from the point of view of TO;
- quantum leaps – destruction, collapse wave function(and, therefore, the collapse of all information about the previous state of the system - “pure synergetics”!) - the complete destruction of Einstein’s ideas about determinism in our (his) world.
As you know, time put everything in its place: quantum mechanics turned out to be right, and, therefore, the theory of relativity was wrong.
But what does the principle of relativity, which this chapter is devoted to, have to do with it? Despite the fact that the theory of relativity is unable to describe the levels of organization of matter that lie below the material world (free EMF) and above it (post-material world of BH).
But this means that TO is a theory of a purely material world, and the principle of relativity for the material world was formulated by Galileo - several centuries before Einstein.
The conflict is that Einstein sincerely believed that he had generalized Galileo’s software to all phenomena of our world, including, at a minimum, phenomena associated with free EMF. As we have just seen, this is an unfortunate misconception.
RELATIVITY AND NONLINEARITY
Let's try to understand the origins of this misconception of the author of TO. As it became clear from the previous discussions, problems with TO arise directly where the material world borders on the free EMF or the world of black holes. Let us note that in both cases there is significant nonlinearity (although by no means a singularity, as represented by the theory of relativity).
In both cases, these are large entropy barriers (LEBs), separating the fundamental levels of matter organization from each other (free EMF - VM and VM - BH world). In both cases, matter overcoming the BEB is associated with an abrupt change in the properties of this matter, which in the first case is described as the collapse of the wave function (destruction of the free EMF structure, albeit not observed by us), and in the second - as gravitational collapse(with the destruction of the structure of a collapsing material object).
In both cases, therefore, there is a destruction of the previous structure of matter, or, what is the same, a collapse of information (“forgetting” one’s previous state). It is these two essential nonlinearities, which we called BEB, that exclude the possibility of extending the principle of relativity to different, albeit adjacent, fundamental levels of the organization of matter.
Indeed, phenomena at a lower level are fundamentally unobservable from a higher level, and vice versa. The only thing available to us is to observe the finished result of the collapse, if we're talking about about the absorption of a photon by matter (or about the generation of a particle-antiparticle pair by a quantum) and the observation of a collapse process infinite in time - if we are talking about the absorption of a material object by a black hole.
In other words, there is no equality, no parity between the two fundamental levels of organization of matter - they are neighboring levels of the world hierarchy (one is higher than the other), and there is no symmetry between them (the BEB is nonlinear, introducing irreversibility into their “relationships”). The principle of relativity is a consequence of one of the symmetries of our world - the invariance of the equations of mechanics under Galilean transformations.
Where symmetry and linearity are violated (BEB, interlevel phenomena), the principle of relativity cannot be preserved: in any modification it must remain linear, because it is designed to reflect the properties of symmetry.
When it was discovered that Maxwell's equations were non-invariant under Galileo's transformations, by and large, there were two correct paths further development physicists:
- abandon attempts to extend Galileo’s software, as a purely linear apparatus, to nonlinear phenomena associated with different fundamental levels of organization of matter (light - matter);
- try to find a fundamentally new form of symmetry - nonlinear, interlevel - and after that decide which principle corresponds to the new type of symmetry (clearly, not the principle of relativity).
However, as you know, physics has gone its own way. At first, Hertz tried to rearrange Maxwell's equations in such a way as to return them to the “womb of Galileo's transformations” - the attempt failed.
Then Poincaré obtained almost all the relations that would later be called the theory of relativity by performing a group-theoretic analysis of Maxwell's equations (that is, based on symmetry considerations). By default, Poincare, therefore, proceeded from the fact that the two studied levels of organization of matter (EMF and matter) are symmetrical with each other, equivalent, equal.
This would be a completely rigorous physical and mathematical approach if it were not for the false initial premise: symmetry was attributed to multi-level phenomena associated with the nonlinear transformation of matter.
Physics of the early twentieth century, however, did not accept Poincaré's results. True, not at all for fundamental reasons, but to the extent that that physics had not even heard of group theory, and considerations of symmetry were not considered anything else worthy of attention.
Einstein, who introduced physical analysis specific situations, was more fortunate: he was immediately noticed and understood, although not all of his results were then perceived as true. Moreover, they cannot be considered as such in our time - the time of nonlinear science.
THE RELATIVITY PRINCIPLE AND SYSTEM ASYMMETRY OF THE HIERARCHICAL WORLD
In any case, both Poincaré and Einstein proceeded from the false premise of symmetry between two neighboring (but not equivalent!) hierarchical levels of matter organization: free EMF and matter.
What is their “inequality”?
Free EMF is a lower level of organization of matter than matter. What does this mean? It is well known from the theory of complex systems that an element (lower level of organization) can freely exist outside the system (higher level of organization), while a system - a union of elements - without these elements does not make sense: it functions due to the “movement” of elements, operating with them.
The situation is similar with the relationship between free EMF and matter: free (!) EMF - by definition, is not associated with material objects, is “detached” from them and exists independently. And this is understandable, because free EMF is the lower level of matter organization in relation to matter - it is allowed to do this.
Matter, as a higher level of organization of matter, is unthinkable apart from electromagnetic radiation: a substance “functions” by absorbing and, most importantly, emitting. Any material object is a source of electromagnetic radiation with a spectrum that is completely determined by its temperature.
All this is significant confirmation that free EMF and matter are, respectively, the lower and upper levels of matter organization in relation to each other.
No less important is the “quantitative aspect”: all photons “weigh” less than the elementary “building block” of the material world - the proton: photons have a smaller measure of matter, being representatives of a lower level of its organization.
But some important conclusions follow from this:
- these levels are unequal;
- the relationship between them is nonlinear
character;
- that is, between them there is a nonlinear interlevel, large (for fundamental levels) entropy barrier - BEB;
- nonlinearity between levels of organization of matter in the hierarchical world causes asymmetry between them (indeed, all types of symmetry known to physics are linear);
- the asymmetry of the hierarchical world - a kind of “arrow (spear) of evolution” - is directed “from bottom to top”, that is, opposite to the direction of the “arrow of time”, reflecting the realities of VNTD (the second law of thermodynamics);
- this asymmetry is a manifestation of the most fundamental property of our open nonlinear world, which has not yet been formulated; this property is the VALVE OF SPACE – MATTER;
- The VENTILITY of our universe is a consequence of the fact that it organizationally represents a continuous discretium of space-matter (NDSM);
- THE VALVE OF SPACE – MATTER manifests itself in our world as the ability of matter to evolve – to self-complicate;
- finally, a rather trivial, but important final conclusion for us: Galileo’s principle of relativity is a reflection of the invariance (symmetry) of our world in its single-level representation, which does not go beyond the boundaries of the material world (Newtonian mechanics);
- according to all of the above, the principle of relativity (PR) cannot be generalized, extended to essentially nonlinear, initially asymmetric interlevel relationships such as “free EMF - matter” or, even more so, “matter - the world of black holes (BH)”;
- the construction of a special theory of relativity based on such a generalization of software is fundamentally incorrect, although it looks outwardly beautiful; incorrect precisely for reasons of symmetry - asymmetry (linearity - nonlinearity) of our world, its space-matter.
SPECIAL THEORY OF RELATIVITY AND THE DOPPLER EFFECT
However, despite everything that has just been said, the special theory of relativity (SRT) “works”, as they say, gives results. How to explain this fact? The explanation of the first level, lying on the surface, has already been given - long before us: with the same success with which Einstein obtained the conclusions of SRT from his two postulates (PO + the absoluteness of the speed of light), its consequences can be obtained without using software, but based on ready-made Lorentz transformations, which, as is known, are older than SRT. Of course, it looks more artificial, “tight,” less impressive, but no less effective. However, an incorrect message in both cases (for the same reasons) cannot give a completely correct result. Let us explain what is meant by this.
Yes, the special theory of relativity (SRT) is not a generalization of the principle of relativity. But SRT is still, undoubtedly, a kind of generalization! This is precisely what allows it to explain many experimental facts that do not fit into classical physical models.
What does SRT generalize? Why can’t its final results (and not just its initial positions) be considered completely correct?
We will consider these and related issues in more detail in the next chapter, but here we will focus on a thesis statement of the main points.
Attention! The so-called “special theory of relativity” is - try to be genuinely surprised here - an incomplete generalization of the Doppler effect - well known phenomenon, associated, in particular, with electromagnetic field, emitted by a moving material object.
It turns out that the Doppler effect can be generalized to phenomena of a purely material, gravitational nature (“gravitational” Doppler effect). If the “electromagnetic” Doppler effect is associated with the “weighting” of photons emitted by an object moving towards the observer (that is, an increase in the energy of the photon, a decrease in its wavelength), then the “gravitational” Doppler effect explains the facts of weighting (increase in mass) and a decrease in the length of the material itself object moving towards the observer, which generalizes the known The ripple effect to the real level.
Moreover, it is precisely this increase in mass and reduction in the length of a material object that serves as the true cause of the “electromagnetic” Doppler effect, which we have known about since school years - the effect of shortening the wavelength (increasing quantum energy) of electromagnetic radiation emitted by a material object moving towards observer.
True, here there is a need to overcome the newly emerged asymmetry (in relation to objects moving away from the observer), but this difficulty is easily overcome. This is what you can verify by reading the next chapter, “The Doppler Gravitational Effect or the Restoration of Symmetry.”
THE RELATIVITY PRINCIPLE AND FRACTAL SYMMETRY
"Despite great literature about symmetry... it is very difficult to find out the position of symmetry in the system of sciences... We will not find in this literature an exact specific indication of what the phenomena of symmetry are in natural processes»
IN AND. Vernadsky
Before moving on to the question of what the “theory of relativity” actually generalizes (see the next chapter “Doppler’s gravitational effect or restoration of symmetry”), let us once again return to Galileo’s principle of relativity. We have just realized that the generalization of Galileo’s principle of relativity to nonlinear interlevel phenomena of an asymmetrically structured hierarchical world is incorrect. First of all, this concerns nonlinear relations of the “matter – free EMF” type, which the theory of relativity considers to be its area of competence. Without any reason, as we have seen, for, by generalizing the linear principle of relativity to nonlinear phenomena, TO performs an incorrect operation.
It would be appropriate to recall once again that Galileo himself was, perhaps, the first to realize the impossibility of nonlinear symmetry: he examined the so-called scale symmetry and came to the conclusion that it was impossible. Indeed, if you increase the linear dimensions, for example, of an architectural structure (or a living organism) N times, then its volume and mass will increase in proportion to the cube of N, and the strength associated with the cross-sectional area of the load-bearing structures (bones of a living organism) will increase only in proportion to the square of N This means that such large-scale symmetry inevitably leads to the destruction of the object. Something similar happens with free EMF when the photon energy exceeds the threshold: it collapses, forming a particle-antiparticle material pair.
Moving from one fundamental level of organization of matter to another, we will try to implement the same large-scale (that is, nonlinear) symmetry, but not with the geometric dimensions of objects (they are different levels are generally incommensurable - the geometry of a material object and the geometry of a free field wave are different entities), but with their structures.
Let us make a guess: if nonlinear scale symmetry is possible in the world at all, then this is a symmetry between the structures of physical objects, and not between their geometric dimensions.
What is the structure of the fundamental level of organization of space-matter and how does it transform during the transition of matter from one fundamental level to another?
As already mentioned, everyone fundamental view movement and the corresponding fundamental level of organization of space-matter is characterized by its fundamental length - the minimum possible wavelength of the corresponding field. It is the fundamental length that is the quantitative basis of the structure of space-matter at a given level. Quite conventionally, such a structure can be represented as a network with a cell size equal to: f.
In fact, naturally, there is no material network, since the structure is an essential, ideal concept, with its shadow - the wavelength - appearing to us only when we remember the universality wave properties matter. The structure is a kind of information matrix of prohibitions and permissions for a certain form of motion of matter (wave in essence).
It was also said earlier that the fundamental lengths (and, therefore, structures) of the levels under consideration differ by approximately 60 orders of magnitude, and about the incommensurability - the irrationality of their relationship. But all this does not prevent us from drawing the main conclusion from everything said above: between the structure of free EMF and the structure of the material world there is a scale symmetry with a scale factor equal to approximately 10^60. This symmetry is not invented by us, it is given by nature itself. But what kind of symmetry is this if it can neither be seen nor touched?
It turns out that it is possible! There are no entities that do not appear to us in the form of natural objects, processes, and phenomena. The scale similarity of structures appears to us in the form of nonlinear interlevel FRACTAL SYMMETRY.
Since the theme of fractal symmetry runs throughout this series, we will limit ourselves to the main points of understanding:
1. Nonlinear symmetry, as a concept, overcomes the framework of geometry (and today’s mathematics in general) and falls only under a broader general scientific definition of symmetry, such as:
The first object is fractally symmetrical with respect to the second if it, moving in any permitted way in space and time (that is, evolving), can become structurally identical to the second object, conditionally motionless in space and time.
Using this definition, it can be argued that any cell biological organism fractally symmetrical to the entire organism (experiments in animal cloning fully confirm this).
2. Fractal symmetry is a systemic concept, it is symmetry in an essentially asymmetric, hierarchically structured world. This is a symmetry between a system and its element, impossible in principle if we operate with linear concepts such as the principle of relativity, but inevitably realized in a nonlinearly evolving world, the space-matter of which has valve properties (see the example of “cell-organism” in paragraph 1).
3. Fractal symmetry, being a generalization of all experience systems thinking person, is not something completely unexpected. This concept grew naturally from the results that preceded it:
The idea of a monad - a “mentally active substance” that perceives and reflects other monads and the whole world as a whole (G.V. Leibniz);
The paradox of set theory, according to which a part and a whole can be of equal power (G. Cantor);
The experimentally proven phenomenon of the abrupt emergence of a new biological species as a result of a simultaneous (symmetrical) change in any characteristic in the majority of individuals (V.L. Komarov);
The theory of nomogenesis, which recognizes (but does not explain) the presence in a living organism of internal information content, which is in contact in some still unclear way with cosmic information content (L.S. Berg);
Concepts of isomorphism - similarities between systems (phenomena, processes) of different levels (L. von Bertalanffy);
Understanding that in the violation of a certain type of symmetry one can see the hidden manifestation of a symmetry of another, higher type or a hierarchy of symmetries linked to each other (N.V. Ovchinnikov);
The law of indistinguishability of the part and the whole (the law of conservation of information) (G.V. Chefranov);
Properties of fractal self-similarity (scaling) - the fractal repeating itself at different scale levels (B. Mandelbrot);
Hypotheses about the holographicity of the brain and any complex system up to the Universe as a whole (K. Pribram, D. Bohm, M. Talbot);
- “philosophy of instability”, according to which, in a state far from equilibrium, each part of the system “sees” the entire system, obeying its order, acting coherently with its other elements (I.R. Prigogine);
Discovery of a number of periodically repeating adaptive reactions of the body to increasing loads (L.H. Garkavi, G.B. Kvakina, M.A. Ukolova);
Ideas about coevolution as the coexistence of tempoworlds - systems complex structures, developing at different rates (E.N. Knyazeva, S.P. Kurdyumov);
The periodic system of space-matter, proposed in the second book of this series;
And many other well-known and significant scientific results, including periodic table chemical elements D.I.Mendeleev and N.I.Vavilov’s law of homological series.
Concluding this chapter, however, let us once again dwell on the fractal symmetry between the fundamental levels of organization of space-matter. How specifically and clearly does it manifest itself?
Most clearly - in the method of transformation of matter, overcoming the BEB from the bottom up. To overcome this barrier, matter must undergo collapse: whether we are talking about the transformation of free EMF into matter or matter into a post-substance state of a black hole. In the first case, this is a collapse of the wave function, in the second, a gravitational collapse, but both of them are accompanied by a collapse of information about the previous state of matter.
It is also characteristic that in order for this transformation to occur, matter lower level must:
- either overcome the quantitative barrier (the photon must have a critical energy sufficient for the birth of a material particle-antiparticle pair; a neutron star must have a mass greater than the critical one);
- or be directly absorbed material object a higher level of organization (respectively, matter or a black hole).
Another manifestation of fractal symmetry between fundamental levels is the “violation” of the law of conservation of energy (mass) at the boundary of two levels – free matter and EMF.
Formally speaking, symmetry (the conservation law) is indeed violated: the photon (EMF) does not have a rest mass - it arises only in the process of interlevel transformation of matter.
But this is exactly the case when the violation of one type of symmetry calls to life another type of symmetry - a higher level - nonlinear interlevel fractal symmetry (FS). And the law of conservation of mass (energy) must be rethought as the law of interlevel transformation of matter. It is necessary to understand that the famous “relativistic” formula (E=mc;) is formal expression precisely this inter-level fractal law, where both energy and mass are understood precisely (and exclusively) as a quantitative measure of matter at the corresponding level of organization.
Since the number of fundamental levels of matter organization is unlimited, this interlevel formula is just a particularity that needs generalization.
Without duplicating the considerations set out in other chapters devoted to this and related issues, we nevertheless note that, as follows from the results presented further in the chapter “Light and Matter”, the energy of the EMF and the mass of the material world, between which the law of transformation of matter establishes correspondence , differ not only formally (“formally” they are almost identical), but also in essence - significantly to a greater extent than science had imagined before. We are talking about the independence of the free EMF from the material world of the gravitational field (GP): the absence of rest mass in the free EMF is not a mere formality, it entails a complete absence of interaction between the free EMF and the GP of the material world.
The fundamental levels of matter organization are separated – connected by BEB-FS.
Finally, we note that the concept of FS allows us to rethink our understanding of facts relating to fundamental levels of organization of space-matter that are more distant in relation to the material world and make certain predictions about their properties and features that still need confirmation.
Continuation of the book.
Symmetry is the ideathrough which a personfor centuries triedto stitch and create order, beautyand perfection.
Term "symmetry" translated from Greek means congruentlyregularity, proportionality, uniformity in arrangementparts. Ancient philosophers considered symmetry, order and certainty to be the essence of beauty. The Concise Oxford Dictionary defines symmetry as beauty, conditioned by the proportionality of parts of the body or any whole, balance, similarity, harmony, consistency. However, it does not cover the full depth and breadth of this concept.
We encounter symmetry everywhere - in nature, technology, science, art. It exists not only in the macroworld, but is also inherent in the micro- and megaworld. Symmetry, understood in the broadest sense, is opposed to chaos, disorder; it is observed wherever there is at least some order. In this sense, not only objects of nature are symmetrical (snowflakes, leaves, fish, insects, the human body, etc.), but also such ordered phenomena as regular shift day and night, seasons, the cycle of water and other substances in nature, etc. The idea of symmetry can be expressed in words such as balance, harmony, perfection.
For humans, symmetry has an attractive force. We love to look at the manifestation of symmetry in nature: symmetrical crystals, snowflakes, flowers that are almost symmetrical. Architects, artists, poets and musicians have known the laws of symmetry since ancient times. Geometric patterns are constructed strictly symmetrically; Classical architecture is dominated by straight lines, angles, circles, equality of columns, windows, arches and vaults. Of course, symmetry in art is not literal. The laws of symmetry of a work of art do not imply uniformity of forms, but a deep consistency of elements.
The concept of symmetry runs through the entire centuries-old history of human creativity. The laws of nature that govern the infinite variety of phenomena are also subject to sym-
metrics. Symmetry can be found almost everywhere if you know where and how to look for it. All the diversity of the world around us is subject to amazing manifestations of symmetry. J. Newman wrote about this very successfully: “Symmetry establishes a funny and surprising affinity between objects, phenomena and creations that outwardly seem to be unrelated to anything: earthly magnetism, the female veil, polarized light, natural selection, group theory, invariants and transformations , the working habits of bees in the hive, the structure of the space, the designs of vases, quantum mechanics, flower petals, the interference pattern of X-rays, cell division, equilibrium configurations of crystals, snowflakes, music, the theory of relativity..." (Quoted from the book: Tarasov L.V. This amazingly symmetrical world, 1982.)
A strict mathematical idea of symmetry was formed relatively recently - in the 19th century. The modern approach to symmetry assumes the immutability of an object in relation to any operations or transformations performed on it. The modern definition of symmetry is formulated as follows: symmetrical is an object (object) that can besomehow change, resulting in an object that matcheswith the original one. According to the definition, first of all there must exist the object is the bearer of symmetry. It is, of course, different for different manifestations of symmetry. These are material objects or properties. Objects must have some signs - properties, processes, relationships, phenomena, which do not change under symmetry operations. Changes in these objects must also occur, but not just any, but only those that transform it into identical to myself. And finally, there must be a property of the object that does not change, i.e. remains invariant.
Let us emphasize that invariance does not exist on its own, not in general, but only in relation to certain transformations, and changes (transformations) are of interest insofar as something is preserved. In other words, without change there is no point in considering conservation, just as without conservation there is no interest in change. Thus, symmetry expresses the preservation of something despite some changesdisagreement or maintaining something despite change. Sim-
metric presupposes the immutability not only of the object itself, but also of any of its properties in relation to transformations performed on the object.
The immutability of certain objects can be observed in relation to various operations - rotations, translations, mutual replacement of parts, reflections, etc. In this regard, different types of symmetry are distinguished.
ROTARY SYMMETRY. An object is said to have rotational symmetry if it aligns with itself when rotated through an angle of 2t. /P, Where P can be equal to 2, 3, 4, etc. to infinity. The axis of symmetry is called the axis nth order axis.
TRANSPORTABLE (TRANSLATIONAL) SYMMETRY. Such symmetry is spoken of when, when moving a figure along a straight line for some distance A or a distance that is a multiple of this value, it is combined with itself. The straight line along which the translation is carried out is called transfer axis, and the distance a - by elementary transfer or period. Associated with this type of symmetry is the concept of periodic structures or lattices, which can be both flat and spatial.
MIRROR SYMMETRY. Mirror symmetrical is an object consisting of two halves that are mirror counterparts to each other. A three-dimensional object transforms into itself when reflected in a mirror plane, which is called plane of symmetry.
SYMMETRY OF SIMILARITY are peculiar analogues of previous symmetries with the only difference that they are associated with a simultaneous decrease or increase in similar parts of the figure and the distances between them. The simplest example of such symmetry are nesting dolls.
Sometimes figures can have different types of symmetry. For example, some letters have rotational and mirror symmetry: Ж, Н, Ф, О, X.
The so-called geometric symmetries are listed above. There are many other types of symmetry that are abstract in nature. For example, SWITCH SYMMETRY, which consists in the fact that if identical particles are swapped, then no changes occur; HEREDITY is also a certain symmetry.
GAUGE SYMMETRIES involve changes in scale. For example, it is known that when a body rises to a certain height, the energy expended depends only on the difference between the initial and final heights, but does not depend on the absolute height. They say that there is a symmetry of the height reference point, and it is classified as a gauge symmetry. All fundamental interactions are of a gauge nature and are described by gauge symmetries. This fact reflects the unity of all fundamental interactions. Gauge invariance allows us to answer the question: “Why and why does this kind of interaction exist in nature?” This is due to the fact that the requirement of gauge invariance gives rise to a specific type of interaction. Therefore, the form of the interaction is no longer postulated, but is derived as a result of gauge invariance.
A unified theory of all physical interactions is built on this principle. It is interesting to note that this principle goes far beyond physics and can become powerful. regulatoryprinciple in solving social and economic problemsth character. It seems that principles such as social justice, equality, a sustainable standard of living for the population and others can be brought into line with some symmetry.
In inanimate nature, symmetry primarily arises in such a natural phenomenon as crystals, of which almost all solids are composed. It is this that determines their properties. The most obvious example of the beauty and perfection of crystals is the well-known snowflake. All snowflakes, despite the variety of their shapes, have mirror and rotational symmetry of the 6th order. It has been proven that all crystals can have rotational symmetry of the 2nd, 3rd, 4th and 6th orders. The symmetry of the crystal is associated with the presence crystal lattice - spatial lattice of atoms. This shows that symmetry limits the possibilities of structural options.
Physical laws and phenomena are also subject to the laws of symmetry. R. Feynman wrote that “the whole variety of laws of physics is permeated by several general principles, which are somehow contained in each law. Examples of such principles are some properties of symmetry” (Feynman, 1987).
There are several symmetries of physical laws:
Physical laws are unchanging, invariant with respect to transfers in space, which is due uniformity aboutwanderings. This means that when a device is transferred from one point in space to another, its properties, operating features and experimental results will not change.
Physical laws invariant with respect to rotation in space. It's called isotropy of space. For example, whether the installation is turned to the north or east, the results of the experiment will be the same.
The symmetry of physical laws is determined by homogeneity of time, those. They invariant with respect to enosesin time. Thus, the homogeneity of space and time are properties of symmetry.
The principle of relativity of natural laws - This is also symmetry with respect to the transition from one inertial reference system to another. This symmetry establishes the equivalence of all inertial reference systems.
No physical phenomena change when rearrangement of two ideally identical particles(for example, electrons or protons) - permutation symmetry.
Another type of symmetry of physical laws is invariantity in relation to specular reflection. This means that two physical installations, one of which is built as a mirror image of the other, will function identically. Note that this symmetry is broken under certain interactions.
Symmetry properties are among the most fundamental properties physical systems. However, not all laws of nature are invariant to any transformations. For example, more geometricThe Chinese principle of similarity does not apply to physical laws. Even G. Galileo guessed that the laws of nature are asymmetrical with respect to changes in scale. R. Feynman gives an example with a model of a cathedral, which is made of matches. If it is increased to its natural size, the structure will collapse under its own weight. From the point of view of modern physics, the lack of symmetry of physical laws regarding similarity transformation is explained by the fact that the order of the sizes of atoms has an absolute value that is the same for the entire Universe. Laws of classical
physicists stop working in the microworld, and the laws of quantum mechanics take their place. This is already a manifestation of asymmetry, i.e. asymmetry.
There is a deep connection between symmetry and conservation laws. At the beginning of the 20th century. E. Noether formulated a theorem according to which if the properties of a system do not change due to any transformation over it, then this corresponds to a certain conservation law - Noether's theorem. Since the independence of properties from a transformation means the presence of symmetry in a system with respect to a given transformation, Noether’s theorem can be formulated as a statement that the presence of symmetry in a system determines the existence of a conserved physical quantity for it. So, for example, the law of conservation of momentum is a consequence of the homogeneity of space, and the law of conservation of energy is a consequence of the homogeneity of time. Conservation laws, acting in the most various areas and in various specific situations, express what is common to all situations, which is ultimately associated with the corresponding principles of symmetry. Thus, symmetry is associated with conservation and highlights various invariants - some kind of “reference points”. We can say that symmetry brings order to our world. In the world around us, “everything flows, everything changes,” it is filled with interactions and transformations, randomness and uncertainty are present everywhere. But at the same time the laws of the world reveal symmetry: energy is conserved, summer is followed by winter, etc. Selection symmetrythere is something in common both in objects and in phenomena, emphasizing that despite the fact that the world is diverse, but at the same time he one, because in various natural phenomena there are blackyou are a community.
In the world of living nature, all the main types of geometric symmetries also appear. The specific structure of plants and animals is determined by the characteristics of the habitat to which they adapt and the characteristics of their way of life. Any tree has a base and a top, a “top” and a “bottom” that perform different functions. The significance of the difference between the upper and lower parts, as well as the direction of gravity, determines verticalorientation the rotary axis of the “wood cone” and planes of symmetry. The leaves are characterized by mirror symmetry. This
The same symmetry is also found in flowers, but in them mirror symmetry more often appears in combination with rotational symmetry. There are also frequent cases of figurative symmetry (acacia branches, rowan trees). It's interesting that in in the floral world, the most common is rotational symmetry of the 5th order, which is fundamentalbut is impossible in periodic structures of inanimate nature. Academician N. Belov explains this fact by the fact that the 5th order axis is a kind of instrument of the struggle for existence, “insurance against petrification, crystallization, the first step of which would be their capture by the grid” (quoted from the book:). Indeed, a living organism does not have a crystalline structure in the sense that even its individual organs do not have a spatial lattice. However, ordered structures are represented very widely in it.
In the world of fish, insects, birds, mammals, it is characteristic bilateral symmetry(bilateral translated from Latin - “twice lateral”) - this is what mirror symmetry is called in biology. This is due to the fact that, unlike plants, which do not change their place of residence, movement in space is important for animals: they do not have symmetry regarding the direction in which they move, i.e. the back and front of the animal are asymmetrical. The plane of symmetry in animals, in addition to the vector of direction of movement, is determined, as in plants, by the direction of gravity. This plane divides the animal into two halves - right and left. The same applies to humans.
Symmetry of similarity manifests itself in nature in everything that grows. The tree trunk has an elongated conical shape. The branches are usually arranged around the trunk in a line similar to a screw, but it gradually tapers towards the apex. That's an example simsimilarity geometry with a helical axis of symmetry. Every living organism repeats itself in a similar way. Nature discovers bydobie as its global genetic program. Similarity rules living nature as a whole. Geometric similarity is considered a general principle of spatial organization of living structures. A birch leaf is like another birch leaf, etc.
There is another remarkable symmetry - self-similarity or scale invariance (scaling), which has the most direct relation to nature. When building models that describe the world around us, we are accustomed to using such
well-known geometric concepts such as line, circle, sphere, square, cube and others. But in fact, the world is structured according to more complex laws. It turned out that it is not always possible to limit ourselves to such simple concepts, i.e. The world cannot always be studied using only a “ruler and compass.” Euclid's geometry is not able to describe the shape of clouds, mountains, trees, or the seashore. The fact is that clouds are not spheres, mountains are not cones, etc. Nature shows us a completely different level of complexity than we used to think. In natural structures, as a rule, the number of different scales is infinite.
Mathematicians have developed mathematical concepts that go beyond traditional geometry, the ideas of which, as they are now beginning to understand, allow us to understand the essence of nature more and more deeply. One such striking example is fractalnew geometry, the central concept of which is the concept "fractal". This word is translated into Russian as "frombroken object with fractional dimension."
There are many different definitions of fractal. First of all, the mathematical concept of a fractal identifies objects that have structures of various scales, reflecting the hierarchical principle of organization. Fractals have the property self-similarity: their appearance does not change significantly when viewed through a microscope with different magnifications, i.e. a fractal looks almost the same no matter what scale it is observed at. In other words, a fractal consists of similar elements of different scales and, in fact, is a pattern that repeats when scales change. A small fragment of such an object is similar to another, larger fragment, or even to the structure as a whole. That's why they say that a fractal is a structureround, consisting of parts that are similar to the whole. Fractals to some extent reflect the principle of Eastern wisdom: “one in all, and all in one.”
The main feature of fractals is that they have fractional dimension, which is a consequence of scale invariance. WITH mathematical point vision of geometric objects, including fractals, can be considered as a set of points embedded in space. For example, the set of points forming a line in Euclidean space has the dimension D = 1, and the set of points forming a surface in three-dimensional space has the dimension D = 2. The ball has the dimension D = 3. Their characteristic feature is that the length of the line, surface area or volume are proportional, respectively, to the linear scale to the first, second or third power, i.e. their dimension coincides with the dimension of the space in which they are embedded. However, there are objects for which this is not the case. Such objects, in particular, include fractals, the dimension of which is expressed as a fractional number 1< DJ < 3, где Df- fractal dimension. In Fig. Figure 2.1 shows one such typical example, demonstrating that the curve can have a dimensionDf > 1, so-called Koch curve.
It is constructed as follows. The original segment of unit length is divided into three equal parts. Then the constructions shown in Fig. 2.1. As a result, in the first generation (n = 1) we obtain a broken curve consisting of four links 1/3 in length. The length of the entire curve in this generation is £(1/3) = 4/3. The next generation (n = 2) is obtained by the same operation on each straight link of the first generation. Here we get a curve consisting of N= 4 2 = 16 links, each length 5 = Z" 2 = 1/9. The entire length is L(l/9) = (4/3) 2 = 16/9. And so on. At the nth step, the length of the straight link 6 = 3~ P . The number of generations can be represented as P= - 1п^/1пЗ, and the length of the entire broken line L(5) = (4/3)" = ex P ln£/ln3 = 6 1 ~ D f, D f = W/W = 1.2628. Number of segments N(6) = 4 P = 4~ 1p/1p3 and can be written as N(5) = 5~ Df , Where Df - fractal dimension of the Koch curve. Thus, the Koch curve is a fractal with fractal dimension Df = In 4/3. In a similar way, you can build many varieties of other fractals. It is also possible to construct objects for which it is necessary to enter not one, but several dimensions. Sometimes such objects are called mathematicallywith fractals, which, unlike natural or physical fractals, have ideal self-similarity. For physical fractals (real-life objects), self-similarity or scale invariance is satisfied approximately(or, as they say, on average).
An example of a fractal object often found in nature is a coastline. In Fig. 2.2 shows the southern
Rice. 2.2. Coast of southern Norway
part of the coast of Norway, which has the appearance of a strongly indented line. It can be shown that it is impossible to measure the length of such a line using the usual methods of Euclidean geometry. But fractal geometry is well suited for this purpose. It turned out that the length coastline well described by the formula L(5) = a8 l ~ Df, Where 5 - scale used for measurement (for example, some kind of compass), A - number of scale units. For the coast of Norway it is DJ ~ 1.52, for the UK coastline - DJ ~ 1.3. In nature, fractal structures are common: cloud outlines, smoke, trees, coastlines and river beds, cracks in materials, bronchi of the lungs, porous sponges, lichen-like branching structures, powder surfaces, arteries and cilia covering the intestinal walls, and many others. which, at first glance, do not have any regularities in their structure. But the lack of order in them is illusory.
Outwardly, they look like jagged, “shaggy” or “holey” objects, representing something intermediate between points, lines, surfaces and bodies.
The introduction of the concept of fractal and fractal geometry makes it possible to identify previously hidden patterns in the structure and properties of natural objects that have a disordered structure, to classify and study their properties. When we look at a fractal object, it appears disordered to us nom. When we zoom in or out we will see the same thing again. This is a manifestation of the property of symmetry -mas staff invariance, or scaling. This is what determines their unusual properties. Due to their self-similarity, fractals have a surprisingly attractive beauty,
which is not found in other objects. They can describe many processes that have not yet been described due to their fractional dimension and self-similarity. It is even believed that the fractal world is much closer to the real one, since many natural objects demonstrate the properties of fractals. Apparently, it’s not in vain that they say that nature loves fractals. Such an amazing similarity between the real world and the fractal world is due, first of all, to the fact that the properties of the physical world change slowly with changes in scale. Sand on the shore has many properties in common with those of pebbles. A small stream is in many ways similar to big river . Such invariance relative to scale - characteristic fractals. In living nature, the appearance and internal structure
are specified in the genotype algorithmically. A tree branch is similar to the tree itself, since it is built using the same algorithm. This applies to the circulatory system of animals, humans, and the complex leaves of some plants. Various fractal sets can also be obtained using simple (elementary) transformations, for example, like P X 2 P +1 = x"+ s, where c is some, P= 1,2,3.... Many numbers obtained using this formula, at certain values of c, also have the properties of fractals. By displaying them on a plane or in three-dimensional space, amazingly beautiful images are obtained (see, for example, Fig. 2.3 and Fig. 2.4).
It is interesting to note that fractal mathematics can be used to analyze changes in prices and wages, error statistics in telephone exchanges, word frequencies in printed texts, etc.
Let us emphasize that symmetry in living nature is never absolute; there is always some degree of asymmetry. Although we encounter symmetry almost everywhere, we often notice not it, but its violation. Asymmetry - the other side of symmetry. Asymmetry is asymmetry, i.e. lack (violation) of symmetry.
Rice. 2.4. "Eye of the Seahorse"
Symmetry and asymmetry are two polar opposites of the objective world. At different levels of development of matter there is symmetry - relative order, then asymmetry is a tendency to disrupt rest, movement, and development.
Asymmetry is already present at the level of elementary particles and manifests itself in the absolute predominance of particles over antiparticles in our Universe. Famous physicist F. Dyson wrote: "Discoveries last decades in the field of particle physics force us to pay special attention to the concept of symmetry breaking. The development of the Universe from the moment of its origin looks like a continuous sequence of symmetry violations. At the moment of its emergence in a grandiose explosion, the Universe was symmetrical and homogeneous. As it cools, one symmetry after another is broken, which creates the possibility for the existence of an ever-increasing variety of structures. The phenomenon of life fits naturally into this picture. Life is also a violation of symmetry" (quoted from the article: I. Akopyan // Knowledge is power. 1989).
Molecular asymmetry was discovered by L. Pasteur, who was the first to distinguish “right-handed” and “left-handed” molecules of tartaric acid: right-handed molecules are like a right-handed screw, and left-handed ones are like a left-handed one. Chemists call such molecules stereoisomers. Stereoisomer molecules have the same atomic composition, same sizes, the same structure - at the same time they are distinguishable, since they are mirror asymmetrical, those. the object turns out to be non-identical with its mirror double.
Therefore, here the concepts of “right-left” are conditional. It is now well known that molecules of organic substancesforming the basis of living matter, have an asymmetricalcharacter, those. they enter into the composition of living matter only either as rights, or how left-handed molecules. Thus, each substance can be part of living matter only if it has a very specific type of symmetry. For example, the molecules of all amino acids in any living organism can only be left-handed, sugars - only right-handed. This property of living matter and its waste products is called dissymmetry. It is completely fundamental. Although right-handed and left-handed molecules are indistinguishable in chemical properties, living matter not only them distinguishes but also makes a choice. It rejects and does not use molecules that do not have the structure it needs. How this happens is not yet clear. Molecules of opposite symmetry are poison for her. If a living creature found itself in conditions where all food was composed of molecules of opposite symmetry that did not correspond to the dissymmetry of this organism, then it would die of starvation. In inanimate matter there are equal numbers of right- and left-handed molecules.
Dissymmetry is the only property due to whichwe can distinguish a substance of biogenic origin fromnonliving matter. We cannot answer the question of what life is, but we have a way to distinguish living from non-living. Thus, asymmetry can be seen as the dividing line between living and inanimate nature. Inanimate matter is characterized by the predominance of symmetry; during the transition from inanimate to living matter, asymmetry predominates already at the microlevel. In living nature, asymmetry can be seen everywhere. This was very aptly noted in the novel “Life and Fate” by V. Grossman: “In a large million Russian village huts there are not and cannot be two indistinguishably similar. All living things are unique. The identity of two people, two rosehip bushes is unthinkable... Life stalls there, where violence seeks to erase its originality and characteristics.”
Symmetry and asymmetry form a unity; they are interrelated with each other, like two sides of the same coin. It is impossible to imagine a completely symmetrical world, just as it is impossible to think about a world completely devoid of symmetry. Symmetry underlies things and phenomena, expressing something common, characteristic
common to different objects, while asymmetry is associated with the individual embodiment of this commonality in a specific object.
Based on the principle of symmetry method of analogies, involving the search general properties in various objects. Based on analogies, physical models of various objects and phenomena are created. Analogies between processes allow them to be described by general equations. The principles of symmetry underlie the theory of relativity, quantum mechanics, solid state physics, atomic and nuclear physics, particle physics. A method for solving problems based on symmetry considerations has been developed.
The principles of symmetry express the most general properties of nature; they have more general character than the laws of motion. Therefore, testing the principles of symmetry has always been of interest to physicists, and the search for new symmetries is one of the tasks of physics in general. The search for new properties of symmetry is at the same time a search for new conservation laws. Our ideas about symmetry are established by generalizing experimental data. Some symmetries turn out to be only approximate. On the other hand, by generalizing experience, we discover new conservation laws and, consequently, new principles of symmetry.
There is a point of view according to which there are three stages in our knowledge of the world: level of phenomena or events, lawnew to nature And principles of symmetry, Climbing them, we get to know nature more deeply and further, understand it better. Level phenomena the most basic. This is everything that happens in the world: the movement of bodies, collisions of particles, absorption and emission of light and many other phenomena. At first glance, it seems that there is nothing in common between them. However, upon closer examination we find that between phenomena havethere are certain relationships, which they call laws. In principle, if we had complete information about all phenomena and events in the world, then we would not need laws. On the other hand, if we knew all the laws or one comprehensive law of nature, then the invariance properties of these laws would not provide anything new. But, unfortunately, we do not even know most of the laws of nature. Therefore, knowledge of the properties of symmetry, as E. Wigner wrote, “consists in giving a structure to the laws of nature or establishing an internal connection between them, just as laws establish a structure or relationship in the world of phenomena” (Wigner, 1971). Therefore they say that if laws govern phenomena That the principles of symmetry arelaws of physical laws. Thus, it can be said that symmetry characterizes the era synthesis, when disparate knowledge merges into a single, holistic picture.
Identification of various symmetries in nature, and sometimes postulating them, has become one of the methods for theoretical study of the micro-, macro- and mega-world. The laws of nature allowpredict phenomena, and the principles of symmetry - discoverNature laws. For example, Maxwell's equations in electrodynamics are derived from the symmetry between electrical and magnetic phenomena. D. Maxwell proceeded from the belief that the interactions of electric and magnetic fields should be symmetrical, and therefore introduced an additional term into his equations that took this circumstance into account. Confidence in the symmetry of the laws of nature led him to the conclusion about the existence of electromagnetic waves. It can also be said that A. Einstein’s idea, which led him to the creation of the theory of relativity, was based on confidence in the deep symmetry of nature, which should simultaneously embrace mechanical, electromagnetic and all other phenomena.
O. Moroz in his book “In Search of Harmony” wrote that physicists chase symmetry just as travelers chase an elusive mirage in the desert. A beautiful, alluring picture appears on the horizon, but as soon as you try to get closer to it, it disappears, leaving a feeling of bitterness.
For centuries, symmetry has remained a subject that has fascinated philosophers, astronomers, mathematicians, artists, architects and physicists. The ancient Greeks were completely obsessed with it - and even today we tend to encounter symmetry in everything from furniture arrangement to haircuts.
Just keep in mind that once you realize this, you'll probably feel an overwhelming urge to look for symmetry in everything you see.
(Total 10 photos)
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1. Broccoli Romanesco
Perhaps you saw Romanesco broccoli in the store and thought it was another example of a genetically modified product. But in fact, this is another example of the fractal symmetry of nature. Each broccoli floret has a logarithmic spiral pattern. Romanesco is similar in appearance to broccoli, and in taste and consistency - to cauliflower. It is rich in carotenoids, as well as vitamins C and K, which makes it not only beautiful, but also healthy food.
For thousands of years, people have marveled at the perfect hexagonal shape of honeycombs and asked themselves how bees could instinctively create a shape that humans could only reproduce with a compass and ruler. How and why bees have passionate desire create hexagons? Mathematicians believe this is an ideal shape that allows them to store the maximum amount of honey possible using the minimum amount of wax. Either way, it's all a product of nature, and it's damn impressive.
3. Sunflowers
Sunflowers boast radial symmetry and an interesting type of symmetry known as the Fibonacci sequence. Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (each number is determined by the sum of the two previous numbers). If we took our time and counted the number of seeds in a sunflower, we would find that the number of spirals grows according to the principles of the Fibonacci sequence. There are many plants in nature (including Romanesco broccoli) whose petals, seeds and leaves correspond to this sequence, which is why it is so difficult to find a clover with four leaves.
But why do sunflowers and other plants observe mathematical rules? Like the hexagons in a hive, it's all a matter of efficiency.
4. Nautilus Shell
In addition to plants, some animals, such as the Nautilus, follow the Fibonacci sequence. The shell of the Nautilus twists into a Fibonacci spiral. The shell tries to maintain the same proportional shape, which allows it to maintain it throughout its life (unlike humans, who change proportions throughout life). Not all Nautiluses have a Fibonacci shell, but they all follow a logarithmic spiral.
Before you envy the math clams, remember that they don’t do this on purpose, it’s just that this form is the most rational for them.
5. Animals
Most animals have bilateral symmetry, which means they can be split into two identical halves. Even people have bilateral symmetry, and some scientists believe that human symmetry is the most important factor, which influences the perception of our beauty. In other words, if you have a one-sided face, you can only hope that it is compensated by other good qualities.
Some go to complete symmetry in an effort to attract a mate, such as the peacock. Darwin was positively annoyed by the bird, and wrote in a letter that "The sight of the tail feathers of a peacock, whenever I look at it, makes me sick!" To Darwin, the tail seemed cumbersome and made no evolutionary sense, as it did not fit with his theory of “survival of the fittest.” He was furious until he came up with the theory of sexual selection, which states that animals evolve certain features to increase their chances of mating. Therefore, peacocks have various adaptations to attract a partner.
There are about 5,000 types of spiders, and they all create an almost perfect circular web with radial supporting threads almost equal distance and spiral fabric for catching prey. Scientists aren't sure why spiders love geometry so much, as tests have shown that a round cloth won't lure food any better than a canvas irregular shape. Scientists suggest that radial symmetry evenly distributes the impact force when the victim is caught in the net, resulting in fewer breaks.
Give a couple of tricksters a board, mowers, and the safety of darkness, and you'll see that people create symmetrical shapes, too. Due to the complexity of the design and incredible symmetry of crop circles, even after the creators of the circles confessed and demonstrated their skills, many people still believe that they were made by space aliens.
As the circles become more complex, their artificial origin becomes increasingly clear. It's illogical to assume that aliens will make their messages increasingly difficult when we couldn't even decipher the first ones.
Regardless of how they came to be, crop circles are a pleasure to look at, mainly because their geometry is impressive.
Even tiny formations like snowflakes are governed by the laws of symmetry, since most snowflakes have hexagonal symmetry. This occurs in part because of the way water molecules line up when they solidify (crystallize). Water molecules become solid by forming weak hydrogen bonds, they align in an orderly arrangement that balances the forces of attraction and repulsion, forming the hexagonal shape of a snowflake. But at the same time, each snowflake is symmetrical, but not one snowflake is similar to the other. This happens because as each snowflake falls from the sky, it experiences unique atmospheric conditions that cause its crystals to arrange themselves in a certain way.
9. Milky Way Galaxy
As we have already seen, symmetry and mathematical models exist almost everywhere, but are these laws of nature limited to our planet? Obviously not. A new section at the edge of the Milky Way Galaxy has recently been discovered, and astronomers believe the galaxy is an almost perfect mirror reflection myself.
10. Sun-Moon Symmetry
Considering that the Sun has a diameter of 1.4 million km and the Moon is 3,474 km in diameter, it seems almost impossible that the Moon can block sunlight and provide us with about five solar eclipses every two years. How does this work? Coincidentally, while the Sun is about 400 times wider than the Moon, the Sun is also 400 times farther away. Symmetry ensures that the Sun and Moon are the same size when viewed from Earth, so the Moon can obscure the Sun. Of course, the distance from the Earth to the Sun can increase, which is why we sometimes see rings and partial eclipses. But every one or two years a fine alignment occurs, and we witness a spectacular event known as complete solar eclipse. Astronomers don't know how common this symmetry is among other planets, but they think it's quite rare. However, we should not assume that we are special, as it is all a matter of chance. For example, every year the Moon moves about 4 cm away from the Earth, meaning that billions of years ago every solar eclipse would have been a total eclipse. If things continue like this, total eclipses will eventually disappear, and this will be accompanied by the disappearance of annular eclipses. It turns out that we are simply in the right place in right time to see this phenomenon.
Frosty patterns on the window, the intricate and unique shape of snowflakes, sparkling lightning in the night sky fascinate and captivate with their extraordinary beauty. However, few people know that all this is complex fractal structures.
Infinitely self-similar figures, each fragment of which is repeated as the scale decreases, are called fractals. Vascular system human, animal alveolar system, gyri sea shores, clouds in the sky, outlines of trees, antennas on the roofs of houses, cell membrane and star galaxies- all this amazing product of the chaotic movement of the world is fractals.
The first examples of self-similar sets with unusual properties appeared in the 19th century. The term "fractals", which comes from the Latin word "fractus" - fractional, broken, was introduced by Benoit Mandelbrot in 1975. Thus, a fractal is a structure consisting of parts similar to the whole. It is the property of self-similarity that sharply distinguishes fractals from objects of classical geometry.
Simultaneously with the publication of the book “Fractal Geometry of Nature” (1977), fractals gained worldwide fame and popularity.
T The term "fractal" is not mathematical concept and in connection with this does not have a strictly generally accepted mathematical definition. Moreover, the term fractal is used to refer to any shapes that have any of the following properties:
Non-trivial structure on all scales. This property distinguishes fractals of such regular figures as a circle, ellipse, graph smooth function and so on.
Increase the scale of the fractal does not lead to a simplification of its structure, that is, on all scales we see an equally complex picture, while when considering a regular figure on a large scale, it becomes similar to a fragment of a straight line.
Self-similarity or approximate self-similarity.
Metric or fractional metric dimension, significantly superior to topological.
Construction is possible only with the help of a recursive procedure, that is, defining an object or action through oneself.
Thus, fractals can be divided into regular and irregular. The first ones are mathematical abstraction, that is, a figment of the imagination. For example, the Koch snowflake or the Sierpinski triangle. The second type of fractal is the result natural forces or human activity. N Regular fractals, unlike regular ones, retain the ability to self-similarity within limited limits.
Every day fractals find more and more greater application in science and technology - they describe the real world as well as possible. We can give examples of fractal objects forever; they surround us everywhere. Fractal like natural object represents shining example eternal continuous movement, formation and development.
Fractals have found wide application in computer graphics to construct images of natural objects, for example, trees, bushes, mountain ranges, sea surfaces, etc. The use of fractals in decentralized networks has become effective and successful. For example, the IP address assignment system in the Netsukuku network uses the principle of fractal information compression to compactly store information about network nodes. Due to this, each node of the Netsukuku network stores only 4 KB of information about the state of neighboring nodes; moreover, any new node connects to the common network without the need for central regulation of the distribution of IP addresses, which, for example, is actively used on the Internet. Thus, the principle of fractal information compression ensures the most stable operation of the entire network.
The use of fractal geometry in the design of “fractal antennas” is very promising.
Currently, fractals have become actively used in nanotechnology. Fractals have become especially popular among traders. With their help, economists analyze the exchange rate of stock exchanges, financial and trading markets.In petrochemistry, fractals are used to create porous materials. In biology, fractals are used to model the development of populations, as well as to describe systems of internal organs.Even in literature, fractals have found their niche. Among works of art works with a textual, structural and semantic fractal nature were found.
/BDE mathematics/
Many Julias (in honor of French mathematician Gaston Julia (1893-1978), who, together with Pierre Fatou, was the first to study fractals.His work was popularized by Benoit Mandelbrot in the 1970s)
Geometric fractals
The history of fractals in the 19th century began precisely with the study of geometric fractals. Fractals clearly reflect the property of self-similarity. The most obvious examples of geometric fractals are:
Koch curve - non-self-intersecting continuous curve infinite length. This curve is not tangent at any point.
Cantor set- loose uncountable perfect set.
Menger sponge is an analogue of the Cantor set with the only difference that this fractal is constructed in three-dimensional space.
Triangle or Sierpinski carpetis also an analogue of the Cantor set on the plane.
Weierstrass and van der Waerden fractalsrepresent a non-differentiable continuous function.
Trajectory of Brownian particlesis also not differentiable.
Peano curve is a continuous curve that passes through all points of the square.
Tree of Pythagoras.
Consider the triadic Koch curve.
To construct a curve, there is a simple recursive procedure for forming a fract of curves on a plane. First of all, you need to set an arbitrary polyline with finite number units, the so-called generator. Next, each link is replaced by a generating element, or rather a broken line, similar to a generator. As a result of this replacement, a new generation of the Koch curve is formed. In the first generation, the curve consists of four straight links, the length of each of which is 1/3. To obtain the third generation of the curve, the same algorithm is performed - each link is replaced by a reduced generating element. Thus, to obtain each subsequent generation, all links of the previous one are replaced by a reduced generatrix of elements. Then, the nth generation curve for any finite n is called a prefractal. In the case when n tends to infinity, the Koch curve becomes a fractal object.
Let's turn to another method of constructing a fractal object. To create it, you need to change the rules of construction: let the forming element be two equal to the segment, connected at right angles. In the zeroth generation, we replace the unit segment with a generating element so that the angle is on top. That is, with such a replacement, the middle of the link shifts. Subsequent generations are built according to the rule: the first link on the left is replaced with a formative element in such a way that the middle of the link is shifted to the left of the direction of movement. Next, the replacement of links alternates. The limiting fractal curve constructed according to this rule is called the Harter-Haithway dragon.
In computer graphics, geometric fractals are used to simulate images of trees, bushes, mountain ranges, and coastlines. 2D geometric fractals are widely used to create 3D textures.
After graduating from university, Mandelbrot moved to the USA, where he graduated from the University of California Institute of Technology. On his return to France, he received doctorate at the University of Paris in 1952. In 1958, Mandelbrot finally settled in the United States, where he began working at the IBM research center in Yorktown.
He worked in the fields of linguistics, game theory, economics, aeronautics, geography, physiology, astronomy, and physics.
Fractal (lat. fractus - crushed) is a term introduced by Benoit Mandelbrot in 1975. There is still no strict mathematical definition of fractal sets. ABOUT n was able to generalize and systematize “unpleasant” sets and construct a beautiful and intuitive understandable theory. He discovered the wonderful world of fractals, the beauty and depth of which sometimes amaze the imagination and delight scientists, artists, philosophers... Mandelbrot’s work was stimulated by advanced computer technologies, which made it possible to generate, visualize and explore various sets.
Japanese physicist Yasunari Watanaba created computer program, drawing beautiful fractal patterns. A 12-month calendar was presented at the international conference "Mathematics and Art" in Suzdal.
