The concept of symmetry is found in many areas of human life, culture and art, as well as in the field of scientific knowledge. But what is symmetry? Translated from ancient Greek, this means proportionality, immutability, correspondence. Speaking about symmetry, we often mean proportionality, orderliness, harmonious beauty in the arrangement of elements of a certain group or components of an object.

In physics, symmetries in equations that describe the behavior of a system help simplify the solution by finding conserved quantities.

In chemistry, symmetry in the arrangement of molecules explains a number of properties of crystallography, spectroscopy or quantum chemistry.

In biology, symmetry refers to the forms of a living organism or identical parts of the body that are regularly located relative to the center or axis of symmetry. Symmetry in nature is never absolute; it necessarily contains some asymmetry, i.e. such parts may not match with 100% accuracy.

Symmetry can often be found in the symbols of the world's religions and in repeated patterns of social interactions.

What is symmetry in mathematics

In mathematics, symmetry and its properties are described by group theory. Symmetry in geometry is the ability of figures to be displayed while maintaining properties and shape.

In a broad sense, a figure F has symmetry if there is a linear transformation that takes this figure into itself.

In a narrower sense, symmetry in mathematics is a mirror reflection relative to a line c on a plane or relative to a plane c in space.

What is an axis of symmetry

A transformation of space relative to a plane c or line c is considered symmetrical if each point B goes into point B" so that the segment B B" turns out to be perpendicular to this plane or line and is divided in half by it. In this case, the plane c is called the plane of symmetry, the straight line c is the axis of symmetry. Geometric figures, such as regular polygons, can have several axes of symmetry, while a circle and a ball have an infinite number of such axes.

The simplest types of spatial symmetry include:

• specular (generated by reflections);
• axial;
• central;
• transfer symmetry.

What is axial symmetry

Symmetry about an axis or line of intersection of planes is called axial. It assumes that if you draw a perpendicular through each point of the axis of symmetry, then you can always find 2 symmetrical points on it, located at the same distance from the axis. In regular polygons, the axes of symmetry can be their diagonals or midlines. In a circle, the axis of symmetry is its diagonals.

What is central symmetry

Symmetry about a point is called central. In this case, at an equal distance from the point on both sides there are other points, geometric figures, straight or curved lines. When connecting symmetrical points of a line passing through a point of symmetry, they will be located at the ends of this line, and its middle will be precisely the point of symmetry. And if you rotate this straight line, fixing the symmetry point, then the symmetrical points will describe the curves in such a way that each point of one curved line will be symmetrical to the same point of another curved line.

In geometry, a property of geometric figures. Two points lying on the same perpendicular to a given plane (or line) on opposite sides and at the same distance from it are called symmetrical with respect to this plane (or line). A figure (flat or spatial) is symmetrical with respect to a straight line (axis of symmetry) or plane (plane of symmetry) if its points in pairs have the specified property. A figure is symmetrical with respect to a point (center of symmetry) if its points lie in pairs on straight lines passing through the center of symmetry, on opposite sides and at equal distances from it.

Definition of symmetry

The concept of “symmetry” (Greek symmetria - proportionality), according to one of the greatest mathematicians of the twentieth century. Hermann Weyl (1885 - 1955), "is the idea through which man has throughout the centuries tried to comprehend and create order, beauty and perfection." Usually the word “symmetry” means harmony of proportions - something balanced, not limited by spatial objects (for example, in music, poetry, etc.). On the other hand, this concept also has a purely geometric meaning, consisting in the natural repetition in space of equal figures or their parts. As E.S. Fedorov wrote (1901), “symmetry is the property of geometric figures to repeat their parts, or, to be more precise, their property in different positions to come into alignment with the original position.”

However, speaking about symmetrical figures, one should distinguish between two types of equality: congruent (Greek congruens - combined) and enantiomorphic - mirror equal (Greek enantios - opposite, morphe - form). In the first case, we mean figures or their parts, the equality of which can be revealed by simple combination - overlapping each other, i.e. “own” movement, transferring the left (L) figure (for example, the left screw, hand) to the left, the right (R) - to the right, in which all points of one figure coincide with the corresponding points of the other. In the second case, equality is revealed through reflection - a movement that transforms an object into its mirror image (left to right and vice versa).

In this case, all points of the spatial figure become pairwise symmetrical relative to the plane. As a result of such transformations (movements), the object is combined with itself, i.e. transforms into itself. In other words, it is invariant with respect to this transformation, and therefore symmetric. The transformation itself, which reveals the symmetry of an object, called the symmetry transformation, preserves unchanged the metric properties of the parts of the object, and therefore the distance between any pair of their points. Thus, objects can be considered symmetrically equal if all points of one of them are translated into the corresponding points of another according to a single rule.

Symmetry concept

Symmetry is a concept that reflects the order existing in nature, proportionality and proportionality between the elements of any system or object of nature, orderliness, balance of the system, stability, i.e. some element of harmony.

Millennia passed before humanity, in the course of its social and production activities, realized the need to express in certain concepts the two tendencies it had established primarily in nature: the presence of strict orderliness, proportionality, balance and their violation. People have long paid attention to the correct shape of crystals, the geometric rigor of the structure of honeycombs, the sequence and repeatability of the arrangement of branches and leaves on trees, petals, flowers, plant seeds, and reflected this orderliness in their practical activities, thinking and art.

The concept of “symmetry” was used in two meanings. In one sense, symmetrical meant something proportional; symmetry shows the way many parts are coordinated, with the help of which they are combined into a whole. The second meaning of this word is balance.

Symmetry is one of the most fundamental and one of the most general patterns of the universe: inanimate, living nature and society. We encounter symmetry everywhere. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human knowledge; it is widely used by all areas of modern science without exception.

Over thousands of years, in the course of social practice and knowledge of the laws of objective reality, humanity has accumulated numerous data indicating the presence of two tendencies in the world around us: on the one hand, towards strict orderliness and harmony, and on the other, towards their violation. People have long paid attention to the correct shape of crystals, flowers, honeycombs and other natural objects and reproduced this proportionality in works of art, in the objects they created, through the concept of symmetry.

Objects and phenomena of living nature have symmetry. It not only pleases the eye and inspires poets of all times and peoples, but allows living organisms to better adapt to their environment and simply survive.

In living nature, the vast majority of living organisms exhibit various types of symmetries (shape, similarity, relative location). Moreover, organisms of different anatomical structures can have the same type of external symmetry.

The principle of symmetry states that if space is homogeneous, the transfer of a system as a whole in space does not change the properties of the system. If all directions in space are equivalent, then the principle of symmetry allows the rotation of the system as a whole in space. The principle of symmetry is respected if the origin of time is changed. In accordance with the principle, it is possible to make a transition to another reference system moving relative to this system at a constant speed. The inanimate world is very symmetrical. Often, symmetry breaking in quantum particle physics is a manifestation of an even deeper symmetry. Asymmetry is a structure-forming and creative principle of life. In living cells, functionally significant biomolecules are asymmetrical: proteins consist of levorotatory amino acids (L-form), and nucleic acids contain, in addition to heterocyclic bases, dextrorotatory carbohydrates - sugars (D-form), in addition, DNA itself is the basis of heredity is a right-handed double helix.

Principles of symmetry

The principles of symmetry underlie the theory of relativity, quantum mechanics, solid state physics, atomic and nuclear physics, and particle physics. These principles are most clearly expressed in the invariance properties of the laws of nature. We are talking not only about physical laws, but also others, for example, biological ones. An example of a biological law of conservation is the law of inheritance. It is based on the invariance of biological properties with respect to the transition from one generation to another. It is quite obvious that without conservation laws (physical, biological and others), our world simply could not exist.

Aspects without which symmetry is impossible:

1) the object is the bearer of symmetry; things, processes, geometric figures, mathematical expressions, living organisms, etc. can act as symmetrical objects. 2) some features - quantities, properties, relationships, processes, phenomena - of an object, which remain unchanged during symmetry transformations; they are called invariants or invariants. 3) changes (of an object) that leave the object identical to itself according to invariant characteristics; such changes are called symmetry transformations; 4) the property of an object to transform, according to selected characteristics, into itself after its corresponding changes.

Thus, symmetry expresses the preservation of something despite some changes or the preservation of something despite a change. Symmetry presupposes the invariability 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.

Types of symmetry

1)ROTARY SYMMETRY. An object is said to have rotational symmetry if it aligns with itself when rotated through an angle of 2?/n, where n can be 2, 3, 4, etc. to infinity. The axis of symmetry is called the axis of the nth order.

2)TRANSPORTABLE (TRANSLATIONAL) SYMMETRY. Such symmetry is said to occur when, when moving a figure along a straight line to some distance a or a distance that is a multiple of this value, it coincides with itself. The straight line along which the transfer is carried out is called the transfer axis, and the distance a is called the 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.

3) 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 the plane of symmetry. It is enough to look at the real world around us to be convinced of the paramount importance of mirror symmetry with the corresponding symmetrical element—the plane of symmetry. In fact, the shape of all objects that move along the earth’s surface or walk, swim, fly, roll near it, usually has one more or less well-defined plane of symmetry. Everything that develops or moves only in the vertical direction is characterized by cone symmetry, that is, it has many planes of symmetry intersecting along the vertical axis. Both are explained by the action of gravity, the symmetry of which is modeled by a cone.

4) SYMMETRY OF SIMILARITY They are unique analogues of previous symmetries with the only difference being 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, Х. The so-called geometric symmetries are listed above.

There are many other types of symmetries 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 are associated with a change in scale. 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.

Scientific and practical conference

Municipal educational institution "Secondary school No. 23"

city ​​of Vologda

section: natural science

design and research work

TYPES OF SYMMETRY

The work was completed by an 8th grade student

Kreneva Margarita

Head: higher mathematics teacher

year 2014

Project structure:

1. Introduction.

2. Goals and objectives of the project.

3. Types of symmetry:

3.1. Central symmetry;

3.2. Axial symmetry;

3.3. Mirror symmetry (symmetry about a plane);

3.4. Rotational symmetry;

3.5. Portable symmetry.

4. Conclusions.

Symmetry is the idea through which man has tried for centuries to comprehend and create order, beauty and perfection.

G. Weil

Introduction.

The topic of my work was chosen after studying the section “Axial and central symmetry” in the course “8th grade Geometry”. I was very interested in this topic. I wanted to know: what types of symmetry exist, how they differ from each other, what are the principles for constructing symmetrical figures in each type.

Goal of the work : Introduction to different types of symmetry.

Tasks:

Study the literature on this issue.

Summarize and systematize the studied material.

Prepare a presentation.

In ancient times, the word “SYMMETRY” was used to mean “harmony”, “beauty”. Translated from Greek, this word means “proportionality, proportionality, sameness in the arrangement of parts of something on opposite sides of a point, straight line or plane.

There are two groups of symmetries.

The first group includes symmetry of positions, shapes, structures. This is the symmetry that can be directly seen. It can be called geometric symmetry.

The second group characterizes the symmetry of physical phenomena and laws of nature. This symmetry lies at the very basis of the natural scientific picture of the world: it can be called physical symmetry.

I'll stop studyinggeometric symmetry .

In turn, there are also several types of geometric symmetry: central, axial, mirror (symmetry relative to the plane), radial (or rotary), portable and others. Today I will look at 5 types of symmetry.

Central symmetry

Two points A and A 1 are called symmetrical with respect to point O if they lie on a straight line passing through point O and are on opposite sides of it at the same distance. Point O is called the center of symmetry.

The figure is said to be symmetrical about the pointABOUT , if for each point of the figure there is a point symmetrical to it relative to the pointABOUT also belongs to this figure. DotABOUT called the center of symmetry of a figure, the figure is said to have central symmetry.

Examples of figures with central symmetry are a circle and a parallelogram.

The figures shown on the slide are symmetrical relative to a certain point

2. Axial symmetry

Two pointsX And Y are called symmetrical about a straight linet , if this line passes through the middle of the segment XY and is perpendicular to it. It should also be said that each point is a straight linet is considered symmetrical to itself.

Straightt – axis of symmetry.

The figure is said to be symmetrical about a straight linet, if for each point of the figure there is a point symmetrical to it relative to the straight linet also belongs to this figure.

Straighttcalled the axis of symmetry of a figure, the figure is said to have axial symmetry.

An undeveloped angle, isosceles and equilateral triangles, a rectangle and a rhombus have axial symmetry.letters (see presentation).

Mirror symmetry (symmetry about a plane)

Two points P 1 And P are called symmetrical relative to the plane a if they lie on a straight line perpendicular to the plane a and are at the same distance from it

Mirror symmetry well known to every person. It connects any object and its reflection in a flat mirror. They say that one figure is mirror symmetrical to another.

On a plane, a figure with countless axes of symmetry was a circle. In space, a ball has countless planes of symmetry.

But if a circle is one of a kind, then in the three-dimensional world there is a whole series of bodies with an infinite number of planes of symmetry: a straight cylinder with a circle at the base, a cone with a circular base, a ball.

It is easy to establish that every symmetrical plane figure can be aligned with itself using a mirror. It is surprising that such complex figures as a five-pointed star or an equilateral pentagon are also symmetrical. As this follows from the number of axes, they are distinguished by high symmetry. And vice versa: it is not so easy to understand why such a seemingly regular figure, like an oblique parallelogram, is asymmetrical.

4. P rotational symmetry (or radial symmetry)

Rotational symmetry - this is symmetry, the preservation of the shape of an objectwhen rotating around a certain axis through an angle equal to 360°/n(or a multiple of this value), wheren= 2, 3, 4, … The indicated axis is called the rotary axisn-th order.

Atn=2 all points of the figure are rotated through an angle of 180 0 ( 360 0 /2 = 180 0 ) around the axis, while the shape of the figure is preserved, i.e. each point of the figure goes to a point of the same figure (the figure transforms into itself). The axis is called the second-order axis.

Figure 2 shows a third-order axis, Figure 3 - 4th order, Figure 4 - 5th order.

An object can have more than one rotation axis: Fig. 1 - 3 axes of rotation, Fig. 2 - 4 axes, Fig. 3 - 5 axes, Fig. 4 – only 1 axis

The well-known letters “I” and “F” have rotational symmetry. If you rotate the letter “I” 180° around an axis perpendicular to the plane of the letter and passing through its center, the letter will align with itself. In other words, the letter “I” is symmetrical with respect to a rotation of 180°, 180°= 360°: 2,n=2, which means it has second-order symmetry.

Note that the letter “F” also has second-order rotational symmetry.

In addition, the letter has a center of symmetry, and the letter F has an axis of symmetry

Let's return to examples from life: a glass, a cone-shaped pound of ice cream, a piece of wire, a pipe.

If we take a closer look at these bodies, we will notice that all of them, in one way or another, consist of a circle, through an infinite number of symmetry axes there are countless symmetry planes. Most of these bodies (they are called bodies of rotation) also have, of course, a center of symmetry (the center of a circle), through which at least one rotational axis of symmetry passes.

For example, the axis of the ice cream cone is clearly visible. It runs from the middle of the circle (sticking out of the ice cream!) to the sharp end of the funnel cone. We perceive the totality of symmetry elements of a body as a kind of symmetry measure. The ball, without a doubt, in terms of symmetry, is an unsurpassed embodiment of perfection, an ideal. The ancient Greeks perceived it as the most perfect body, and the circle, naturally, as the most perfect flat figure.

To describe the symmetry of a particular object, it is necessary to indicate all the rotation axes and their order, as well as all planes of symmetry.

Consider, for example, a geometric body composed of two identical regular quadrangular pyramids.

It has one rotary axis of the 4th order (axis AB), four rotary axes of the 2nd order (axes CE,DF, MP, NQ), five planes of symmetry (planesCDEF, AFBD, ACBE, AMBP, ANBQ).

5 . Portable symmetry

Another type of symmetry isportable With symmetry.

Such symmetry is spoken of when, when moving a figure along a straight line to some distance “a” or a distance that is a multiple of this value, it coincides with itself The straight line along which the transfer occurs is called the transfer axis, and the distance “a” is called the elementary transfer, period or symmetry step.

A

A periodically repeating pattern on a long strip is called a border. In practice, borders are found in various forms (wall painting, cast iron, plaster bas-reliefs or ceramics). Borders are used by painters and artists when decorating a room. To make these ornaments, a stencil is made. We move the stencil, turning it over or not, tracing the outline, repeating the pattern, and we get an ornament (visual demonstration).

The border is easy to build using a stencil (the starting element), moving or turning it over and repeating the pattern. The figure shows five types of stencils:A ) asymmetrical;b, c ) having one axis of symmetry: horizontal or vertical;G ) centrally symmetrical;d ) having two axes of symmetry: vertical and horizontal.

To construct borders, the following transformations are used:

A ) parallel transfer;b ) symmetry about the vertical axis;V ) central symmetry;G ) symmetry about the horizontal axis.

You can build sockets in the same way. To do this, the circle is divided inton equal sectors, in one of them a sample pattern is made and then the latter is sequentially repeated in the remaining parts of the circle, rotating the pattern each time by an angle of 360°/n .

A clear example of the use of axial and portable symmetry is the fence shown in the photograph.

Conclusion: Thus, there are different types of symmetry, symmetrical points in each of these types of symmetry are constructed according to certain laws. In life, we encounter one type of symmetry everywhere, and often in the objects that surround us, several types of symmetry can be noted at once. This creates order, beauty and perfection in the world around us.

LITERATURE:

Handbook of Elementary Mathematics. M.Ya. Vygodsky. – Publishing house “Nauka”. – Moscow 1971 – 416 pages.

Modern dictionary of foreign words. - M.: Russian language, 1993.

History of mathematics in schoolIX - Xclasses. G.I. Glaser. – Publishing house “Prosveshcheniye”. – Moscow 1983 – 351 pages.

Visual geometry 5th – 6th grades. I.F. Sharygin, L.N. Erganzhieva. – Publishing house “Drofa”, Moscow 2005. – 189 pages

Encyclopedia for children. Biology. S. Ismailova. – Avanta+ Publishing House. – Moscow 1997 – 704 pages.

Urmantsev Yu.A. Symmetry of nature and the nature of symmetry - M.: Mysl arxitekt / arhkomp2. htm, , ru.wikipedia.org/wiki/

from Greek symmetria - proportionality) - uniform, similar arrangement of elements of the form of some artificial object; in the broad sense of the word - invariance (constancy) of the structure, shape of a material object (system of objects) relative to its transformation, due to which symmetry is associated with the preservation of certain quantities characterizing a given object (system), for example, energy, momentum, etc. (Noether's theorem in theoretical physics). (See also Syngonies, Crystals, Crystallography).

Excellent definition

Incomplete definition ↓

Symmetry

The ordering of the whole is, according to Plato, the transformation of the whole into harmony, and a certain structure of harmony is symmetry, proportion, rhythm.

a) Plato did not give a sufficiently clear and developed definition of symmetry, although this concept is very important for aesthetics. His statements about symmetry (Phileb, 23c - 27d), unfortunately, are too general. They boil down to approximately the following: imagine some empty background on which nothing is drawn. Let's draw a figure on this background - a circle, square, triangle, rectangle, etc. Such a figure is indicated using a straight or curved line. Let us further assume that we do not consider the background we have taken and the drawn figure separately from each other, but as something whole. This representation is correct, because the figure has somehow occupied and subjugated a certain part of the background. What kind of figure is this, what specific appearance does it have? Its appearance can be beautiful or ugly, proportionate or disproportionate, symmetrical and asymmetrical. Did we give the figure the exact look we wanted, or did we fail? Our aesthetic sense will tell you whether this figure is good or bad, whether it is slender or not slender, beautiful or ugly, etc. This is the simplest and universal reasoning that must be kept in mind in order to understand the content of Plato’s difficult dialogue “Philebus” .

Instead of talking about the background, Plato introduces the concept of the infinite. Of course, Plato’s words that the infinite “can” be as big and as small as you like will not immediately become clear, that it is empty and contains nothing in itself. So, our background is the Platonic infinite. Next, on our background we draw a certain figure, i.e. we limit some part of the background. Plato calls this figure with a not very clear term - “limit”. The limit in this case is simply the limitation of a known part of the background. But our drawing, which limited part of the background from the rest of the background, created exactly a certain figure. Plato calls this figure with a not entirely clear term - “confusion” of the infinite and the limit. This is not any kind of confusion of any different objects. This term can be compared to how a drawing of a figure is perceived when this figure, standing out against some background, actually “blends” with this background, but it is clear that this concept of “blending” is specific. Even more difficult and incomprehensible is Plato’s term, which he uses to designate exactly what kind of figure we got, that is, what kind of idea we wanted to embody in the drawing, whether the idea, for example, of a triangle or the idea of ​​a circle, or even any specific idea. Plato called this the "cause of confusion." The word “cause” here is either unfortunate, or we simply failed to translate the corresponding Greek term. It is clear, however, that this figure is completely definite. This is not a figure at all, but a triangle, rectangle, circle, etc. Is this the figure that we wanted to draw? Here a new stage in understanding the drawing appears, which Plato calls three terms at once: “symmetry”, “truth” and “beauty”. Of course, the figure we get is either symmetrical or asymmetrical, or it corresponds to our idea and therefore is true, or we made a mistake in something when drawing, and then it is not true, and it is either beautiful or ugly. This is also clear. But the too general nature of these terms and the absence of any discussion about their interdependence make them not entirely clear, which is why there was a lot of controversy about this in the comments of ancient authors on Plato’s Philebus. Consequently, symmetry according to Plato’s Philebus presupposes at least four different concepts - the infinite, the limit, the mixing of both and the reasons for this mixing. And, besides, even in this case, the concept of symmetry is not yet very clearly distinguished from the concept of truth and beauty. If we keep in mind Plato’s love for the architectonics of concepts and their schematism, the division of beauty, truth and symmetry is nothing more than a repetition of the original dialectic of the infinite, limit and confusion at the highest level. The most interesting and closest to our understanding of aesthetics is the discussion of pleasure, or enjoyment, and rationality. Pleasure, or enjoyment, is something limitless, since it, taken by itself, is insatiable, eternally strives, as if blindly, and has no limit. Rationality, mind, or intellect, on the contrary, is always based on a certain system, on certain precise distinctions, on abstinence from pleasures, and therefore is a firm and definite principle, a “limit.” If by beauty Plato understands the synthesis of pleasure and intelligence, that is, as if the inner side of the proportionality of symmetry, then he obviously foresees the later very widespread European teachings about the combination of pleasure and intelligence in beauty. The true concept of beauty always includes not only pleasure, but also reasonable ideology. Plato's doctrine of symmetry turns out to be not so naive and general; it to some extent reflects both the real aesthetic reality and its real perception.

b) We proceeded from the fact that aesthetic and all other terminology was developed by Plato gradually, sometimes with great effort, and often took on unclear and confusing forms. However, it is impossible to study Plato’s aesthetics on the basis of only some materials from the Phileb. It is necessary to pay attention to the use of the term “symmetry” in other dialogues.

For example, the following is interesting in the “Laws” (Legg., II 668 a): “After all, what is equal is equal and what is symmetrical (symmetron) is symmetrical, not because it pleases or suits someone’s taste, but the criterion here is primarily truth, and not something else.” In this case, “symmetry” already presupposes “truth,” so that, at least on this point, we were correct in our guess regarding the place of “symmetry” in the Philebus. Adjacent to the Philebus is the judgment in the Laws (Legg., VI 773 a): “What is equal and proportionate in relation to virtue is infinitely higher than what is excessive (acratoy).” These examples also show that it was not for nothing that Plato placed his “symmetry” in such a general area as the area of ​​\u200b\u200bthe creative mixture of the limit and the infinite. These two texts very weakly emphasize the structural side of symmetry, so “proportionality” here can be understood in the broadest sense. Just as “truth” and “beauty” have some kind of correspondence (i.e., mutual correspondence between the limit and the infinite), symmetry is the same correspondence.

About the structural nature of symmetry we read: “The temple of Poseidon himself was one stage in length, three plephras in width, and in proportion (symmetron) to that in height” (Critias, 116 d). What symmetry means here is unclear to us. But it is clear that some kind of structural correspondence is meant. The same kind of structural principle can be encountered in the Sophist, where it talks about the distortion of objects formed as a result of perspective:

“If they [artists] create true symmetry of beautiful objects, then you know that the higher appears smaller than the lower, and the lower appears larger, due to the fact that the former are visible to us from afar, and the latter close... Do not they also separate under such circumstances, artists are with the truth, when they give to the images they decorate not really beautiful “dimensions” (tas oysas simmetrias), but seemingly so” (Soph., 235 e - 236 a). Here “symmetry” only hints at structure, but in reality it means (as it is translated) precisely “dimensions” or (if we also translate the prefix of this word) “the totality of sizes.”

Let us cite a text that refers to being composed of units of length, but without any structural relationship between these lengths: “Being equal, it will be of the same measures [i.e. e. “from the same number of units of measure”], with what it will be equal to... If it is more or less, compared to what it is commensurate with (xymmetron), then in relation to the lesser it will have more measures [larger in size], and in relation to the larger it will have fewer measures [smaller in size]... With what it is incommensurable (me symmetron), in relation to that it will once have smaller measures, another time larger” ( Parm., 140 b). By “symmetry,” obviously, here we mean simply mathematical commensurability, i.e., the possibility of finding a single measure of measurement.

c) To characterize the term “symmetry,” the text from Plato’s dialogue “Theaetetus” (147d-148a) is important. This text presents significant difficulties from a purely philological side. Its idea boils down to the fact that Plato brings to the fore when studying symmetry rectangles, where the sides are measured by a certain rational number, and the diagonals by an irrational number. The relationship between the side and diagonal of each such rectangle creates a special kind of symmetry, on the basis of which, as studied by modern architectural theorists, ancient masters erected temple buildings of the classical period.

The discussion about symmetry from Theaetetus has not gone without response in modern art criticism literature. Namely, D. Hambidge, in his doctrine of dynamic symmetry in architecture,3 refers precisely to this place in Plato’s Theaetetus, although he does not subject it to a special analysis. It is based on a large amount of art history and natural science material and, among other things, on the analysis of all the main architectural elements of the Parthenon (as well as other Greek temples)4. If we keep in mind the terminology of Theaetetus, then the name of the symmetry considered by this author as “dynamic” should be considered very successful.

The discussion about symmetry in the Theaetetus in its essence does not go beyond the Philebus, but only concretizes it. The unification of “limit” and “limitless” in an artistic image is achieved in “Theaetetus” with the help of geometric construction. Geometry in the dialogue “Theaetetus” here serves as the bodily and practical principle with the help of which Plato makes his abstract constructions. With the help of geometry, Plato tries to translate the practice of ancient fine art (in this case architecture) into scientific language.

In Plato's concept of symmetry there is a rather significant discrepancy with the usual understanding in Western European aesthetics. This discrepancy is most noticeable due to the too large scope of this concept in Plato. Now symmetry is represented mainly as the presence of mutually equivalent parts located around a certain center or axis. Plato's concept of symmetry was reduced to the presence of mutually equivalent parts with a very expanded understanding of the “center” or “axis”. Here we think of not only numerical and geometric relations, but also relations of any spheres of existence and life in general.

Most of all, of course, “symmetry” is thought of by Plato (like all other aesthetic forms) in relation to the soul and the cosmos. As we will see, it is already characteristic of all elementary figures from which Plato’s cosmos is built (Tim., 69 b), but it is especially fixed on the living body and soul and in the relationship between soul and body (Tim., 87 c). We can say that symmetry here has the same broad meaning as in pre-Socratic aesthetics, but only in it the creative moment is emphasized, completely dissolved in the cosmological and physical idea of ​​the world among the pre-Socratics.

Excellent definition

Incomplete definition ↓