What is symmetry and when did it originate. What does symmetry mean in different sciences? Examples of symmetrical equilibrium

Definition of the word "Symmetry" according to TSB:

Symmetry - Symmetry (from Greek symmetria - proportionality)
in mathematics
1) symmetry (in the narrow sense), or reflection (mirror) with respect to the &alpha plane. in space (with respect to the straight line a on the plane), is the transformation of the space (plane), in which each point M passes into a point M such that the segment MM is perpendicular to the plane &alpha. (straight a) and divides it in half.
The .alpha. plane. (line a) is called the plane (axis) C.
Reflection is an example of an orthogonal transformation that changes orientation (as opposed to proper motion). Any orthogonal transformation can be carried out by sequential execution of a finite number of reflections - this fact plays an essential role in the study of S. geometric shapes.
2) Symmetry (in broad sense) is a property of a geometric figure Ф, characterizing a certain regularity of the form Ф, its invariability under the action of movements and reflections. More precisely, a figure Φ has S. (symmetric) if there exists a nonidentical orthogonal transformation that maps this figure into itself. The set of all orthogonal transformations that combine the figure Ф with itself is a group called the symmetry group of this figure (sometimes these transformations themselves are called symmetries).
Thus, a plane figure that transforms into itself upon reflection is symmetrical with respect to a straight line—the S. axis (Fig. 1). the symmetry group consists of two elements. If the figure Φ on the plane is such that the rotations with respect to some point O through an angle of 360°/n, n is an integer &ge. 2, translate it into itself, then F has S. of the nth order with respect to the point O, the center of S.
An example of such figures are regular polygons (Fig. 2). group S. here - the so-called. cyclic group of order n. A circle has a S. of infinite order (because it is combined with itself by turning through any angle).
The simplest types of spatial S., in addition to S. generated by reflections, are central S., axial S. and S. of transfer.
a) In the case of central symmetry (inversion) with respect to the point O, the figure Ф coincides with itself after successive reflections from three mutually perpendicular planes, in other words, the point O is the middle of the segment connecting the symmetrical points Ф (Fig. 3). b) In the case of axial symmetry, or S. with respect to a straight line of the nth order, the figure is superimposed on itself by rotation around a certain straight line (S. axis) through an angle of 360 ° / n. For example, a cube has a straight line AB with a C. axis of the third order, and a straight line CD with a C. axis of the fourth order (Fig. 3). in general, regular and semiregular polyhedra are symmetrical with respect to a series of lines.
The location, number, and order of axes of crystallography play an important role in crystallography (see Crystal symmetry). axial C. Straight line AB, called the mirror-rotary axis C. of order 2k, is the C. axis of order k (Fig. 4). A mirror-axial line of order 2 is equivalent to a central line. d) In the case of translation symmetry, the figure is superimposed on itself by translation along some straight line (transfer axis) on some segment. For example, a figure with a single translation axis has an infinite number of S. planes (since any translation can be carried out by two successive reflections from planes perpendicular to the translation axis) (Fig. 5). Figures with several transfer axes play an important role in the study of crystal lattices.
S. has become widespread in art as one of the types of harmonious composition. It is characteristic of works of architecture (being an indispensable quality, if not of the entire structure as a whole, then of its parts and details - plan, facade, columns, capitals, etc.) and decorative and applied art. S. is used as the main technique for constructing borders and ornaments (flat figures, respectively, having one or more S. of transfer in combination with reflections) (Fig. 6, 7).
S. combinations generated by reflections and rotations (exhausting all types of S. geometric figures), as well as transfers, are of interest and are the subject of research in various fields of natural science. For example, helical S., carried out by rotation through a certain angle around an axis, supplemented by a transfer along the same axis, is observed in the arrangement of leaves in plants (Fig. 8) (for more details, see Symmetry in Biology). C. the configuration of molecules, which affects their physical and chemical characteristics, is important in the theoretical analysis of the structure of compounds, their properties, and behavior in various reactions (see Symmetry in chemistry). Finally, in the physical sciences in general, in addition to the already indicated geometric symmetry of crystals and lattices, the concept of symmetry in the general sense (see below) is of great importance. So, the symmetry of the physical space-time, expressed in its homogeneity and isotropy (see Relativity theory), allows you to establish the so-called. Conservation laws. generalized S. plays an essential role in the formation of atomic spectra and in the classification elementary particles(see Symmetry in physics).
3) Symmetry (in the general sense) means the invariance of the structure of a mathematical (or physical) object with respect to its transformations. For example, the S. laws of the theory of relativity is determined by their invariance with respect to Lorentz transformations. The definition of a set of transformations that leave all the structural relations of an object unchanged, i.e., the definition of the group G of its automorphisms, has become the guiding principle of modern mathematics and physics, which makes it possible to penetrate deeply into the internal structure of the object as a whole and its parts.
Since such an object can be represented by elements of some space P endowed with a corresponding structure characteristic of it, transformations of the object are transformations of P. a representation of the group G in the transformation group P (or simply in P) is obtained, and the study of the C. of the object is reduced to the study of the action of G on P and the search for invariants of this action. In the same way, the S. of the physical laws that govern the object under study and are usually described by equations that are satisfied by the elements of the space P, is determined by the action of G on such equations.
So, for example, if some equation is linear on the linear space P and remains invariant under transformations of some group G, then each element g from G corresponds to a linear transformation T g in the linear space R of solutions of this equation. Compliance g
&rarr. T g is a linear representation of G and knowledge of all such representations of it allows us to establish various properties of solutions, and also helps to find in many cases (from “symmetry considerations”) the solutions themselves. This, in particular, explains the necessity for mathematics and physics of a developed theory of linear representations of groups. Specific examples see Art. Symmetry in physics.
Lit .: Shubnikov A.V., Symmetry. (Laws of symmetry and their application in science, technology and applied arts), M. - L., 1940. Kokster G. S. M., Introduction to geometry, per. from English., M., 1966. Weil G., Symmetry, trans. from English., M., 1968. Wigner E., Etudes on symmetry, trans. from English, M., 1971.
M. I. Voitsekhovsky.
Rice. 1. A flat figure symmetrical with respect to the straight line AB. point M is converted to M&rsquo. upon reflection (mirror) relative to AB.
Rice. 2. A star-shaped regular polygon with symmetry of the eighth order about its center.
Rice. 3. A cube having line AB as a third-order symmetry axis, line CD as a fourth-order symmetry axis, point O as a center of symmetry. The points M and M of the cube are symmetrical both about the axes AB and CD and about the center O.
Rice. 4. A polyhedron with mirror-axial symmetry. straight line AB is a mirror-rotary axis of the fourth order.
Rice. 5. Figures with translation symmetry: the upper figure also has an infinite number of vertical axes of symmetry (second order), i.e. reflection planes
Rice. 6. A border superimposed on itself either by transfer to a certain segment along the horizontal axis, or by reflection (mirror) about the same axis and transfer along it to a segment twice as small.
Rice. 7. Ornament. the transfer axis is any straight line connecting the centers of any two curls.
Rice. 8. A figure with helical symmetry, which is carried out by translation along the vertical axis, supplemented by rotation around it by 90 °.

Symmetry is in physics. If the laws that establish relationships between the quantities that characterize a physical system, or determine the change in these quantities over time, do not change under certain operations (transformations) that the system can be subjected to, then these laws are said to have S. (or are invariant) with respect to data transformations. Mathematically, S. transformations constitute a group.
Experience shows that physical laws are symmetrical with respect to the following most general transformations.
Continuous transformations
1) Transfer (shift) of the system as a whole in space. This and subsequent spatio-temporal transformations can be understood in two senses: as an active transformation - a real transfer of a physical system relative to a chosen reference system, or as a passive transformation - a parallel transfer of a reference system. S. physical laws with respect to shifts in space means the equivalence of all points in space, that is, the absence of any selected points in space (homogeneity of space).
2) Rotation of the system as a whole in space. S. physical laws with respect to this transformation means the equivalence of all directions in space (the isotropy of space).
3) Changing the origin of time (time shift). S. regarding this transformation means that physical laws do not change with time.
4) Transition to a reference system moving relative to this system with a constant (in direction and magnitude) speed. S. with respect to this transformation means, in particular, the equivalence of all inertial frames of reference (see Relativity theory).
5) Gauge transformations. The laws describing the interactions of particles with some kind of charge (electric charge, baryon charge, lepton charge, hypercharge) are symmetric with respect to gauge transformations of the first kind. These transformations mean that the wave functions of all particles can be simultaneously multiplied by an arbitrary phase factor:

The concept of symmetry is found in many areas human life, culture and art, and in the field of scientific knowledge. But what is symmetry? Translated from the ancient Greek language, this is proportionality, immutability, correspondence. Speaking of 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 describing 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 regularly located relative to the center or axis of symmetry of the form of a living organism or the same parts of the body. Symmetry in nature is not 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 world religions and in repetitive 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 display, while maintaining properties and shape.

In a broad sense, a figure F is symmetric 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 straight line c on a plane or relative to a plane c in space.

What is an axis of symmetry

A transformation of space with respect to a plane c or a straight line c is considered symmetric if, in addition, each point B goes to a point B "so that the segment B B" is perpendicular to this plane or straight line and divides it in half. In this case, the plane c is called the plane of symmetry, the straight line c is called the axis of symmetry. Geometric figures, such as regular polygons, can have several axes of symmetry, and the circle and the ball have an infinite number of such axes.

The simplest types of spatial symmetry include:

• mirror (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 a perpendicular is drawn through each point of the symmetry axis, then on it one can always find 2 symmetrical points located at the same distance from the axis. In regular polygons, the axes of symmetry can be their diagonals or midlines. In a circle of an axis of symmetry - 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 shapes, straight or curved lines. When connecting symmetrical points to a straight line passing through a point of symmetry, they will be located at the ends of this line, and just the point of symmetry will be its midpoint. And if you rotate this straight line, fixing the point of symmetry, then the symmetrical points will describe the curves so that each point of one curved line will be symmetrical to the same point of the other curved line.

A balanced composition seems right. It looks stable and aesthetically pleasing. Although some of its elements may stand out as focal points, no part draws the eye enough to overwhelm the rest. All elements are combined with each other, smoothly connecting with each other and forming a single whole.

Unbalanced composition causes tension. When a design is disharmonious, its individual elements dominate the whole, and the composition becomes less than the sum of its parts. Sometimes such disharmony can make sense, but more often than not, balance, order, and rhythm are the best solution.

It is easy to understand what balance is from the point of view of physics - we feel it all the time: if something is not balanced, it is unstable. Surely as a child you swung on a swing-board - you are on one end, your friend is on the other. If you weighed about the same, it was easy for you to balance on them.

The following picture illustrates the balance: two people of the same weight are at an equal distance from the fulcrum on which the swing is balanced.

Seesaw in symmetrical balance

The person on the right end of the board swings it clockwise, while the person on the left end swings it counterclockwise. They apply the same force in opposite directions, so the sum is zero.

But if one person were much heavier, the balance would disappear.

Lack of balance

This picture seems wrong because we know that the piece on the left is too small to balance the piece on the right, and the right end of the board must be touching the ground.

But if you move the larger piece to the center of the board, the picture will look more believable:

Seesaw in asymmetrical balance

The weight of the larger figure is offset by the fact that it is located closer to the fulcrum on which the swing is balanced. If you've ever been on a swing like this, or at least seen others doing it, then you know what's going on.

Compositional balance in design is based on the same principles. The physical mass is replaced by a visual one, and the direction in which the force of gravity acts on it is replaced by a visual direction:

1. Visual Mass is the perceived mass of a visual element, a measure of how much attention a given page element draws.

2. Visual direction is the perceived direction of the visual force in which we think an object would move if it could move under the influence of physical strength acting on it.

There are no tools to measure these forces, and no formulas to calculate visual balance: to determine if a composition is balanced, you rely only on your eyes.

Why is visual balance important?

Visual balance is just as important as physical balance: an unbalanced composition makes the viewer feel uncomfortable. Look at the second seesaw illustration: it doesn't seem right because we know the seesaw has to touch the ground.

From a marketing perspective, visual mass is a measure of the visual interest that an area or element on a page generates. When a landing page is visually balanced, every part of it creates some interest, and a balanced design keeps the viewer's attention.

In the absence of visual balance, the visitor may not see some design elements - most likely, he will not look at areas that are inferior to others in visual interest, so that the information associated with them will go unnoticed.

If you want users to know everything you intend to tell them, consider developing a balanced design.

Four types of balance

There are several ways to achieve compositional balance. The pictures in the section above illustrate two of them: the first is an example of a symmetrical balance, and the second is an example of an asymmetric one. The other two types are radial and mosaic.

Symmetrical balance is achieved when objects of equal visual mass are placed at an equal distance from the fulcrum or axis at the center. Symmetrical balance evokes a sense of formality (which is why it is sometimes called formal balance) and elegance. A wedding invitation is an example of a composition that you most likely want to make symmetrical.

The disadvantage of symmetrical balance is that it is static and sometimes seems boring: if half of the composition is a mirror image of the other half, then at least one half will be quite predictable.

2. Asymmetric balance

Asymmetrical balance is achieved when objects on opposite sides of the center have the same visual mass. In this case, on one half there may be a dominant element, balanced by several less important focal points on the other half. Thus, a visually heavy element (red circle) on one side is balanced by a row of lighter elements on the other (blue stripes).

Asymmetric balance is more dynamic and interesting. It evokes a sense of modernity, movement, life and energy. Asymmetric balance is harder to achieve because the relationships between elements are more complex, but on the other hand it leaves more room for creativity.

Radial balance is achieved when elements radiate from a common center. The rays of the sun or the circles on the water after a stone has fallen into it are examples of radial equilibrium. Maintaining the focal point (fulcrum) is easy because it is always in the center.

The rays diverge from the center and lead to it, making it the most noticeable part of the composition.

Mosaic equilibrium (or crystallographic balance) is a balanced chaos, as in the paintings of Jackson Pollock. Such a composition does not have pronounced focal points, and all elements are equally important. The lack of hierarchy, at first glance, creates visual noise, but, nevertheless, somehow all the elements fit together and form a single whole.

Symmetry and asymmetry

Both symmetry and asymmetry can be used in a composition, no matter what type of equilibrium it is: you can use objects with a symmetrical shape to create an asymmetrical composition, and vice versa.

Symmetry is generally considered beautiful and harmonious. However, it can also seem static and boring. Asymmetry usually appears more interesting and dynamic, although not always beautiful.

Symmetry

Mirror symmetry(or bilateral symmetry) occurs when two halves of the composition, located on opposite sides of the central axis, are mirror images of each other. Most likely, when you hear the word "symmetry", you imagine exactly this.

The direction and orientation of the axis can be anything, although it is often either vertical or horizontal. Many natural forms that grow or move parallel to the earth's surface are mirror-symmetric. Her examples are butterfly wings and human faces.

If the two halves of the composition reflect each other absolutely exactly, such symmetry is called pure. In most cases, the reflections are not completely identical, and the halves are slightly different from each other. This is incomplete symmetry - in life it is much more common than pure symmetry.

Circular symmetry(or radial symmetry) occurs when objects are arranged around a common center. Their number and the angle at which they are located relative to the center can be any - symmetry is preserved as long as there is a common center. Natural forms that grow or move perpendicular to the earth's surface are circularly symmetrical, such as the petals of a sunflower. Alternation without reflection can be used to show motivation, speed, or dynamic action: imagine the spinning wheels of a moving car.

Translational symmetry(or crystallographic symmetry) occurs when elements repeat at regular intervals. An example of this symmetry is repeated fence slats. Translational symmetry can occur in any direction and at any distance, as long as the direction is the same. Natural forms acquire this symmetry through reproduction. With translational symmetry, you can create rhythm, movement, speed, or dynamic action.

A butterfly is an example of mirror symmetry, a fence slat is translational, a sunflower is circular.

Symmetrical forms are most often perceived as figures against a background. The visual mass of a symmetrical figure will be greater than that of an asymmetrical figure of similar size and shape. Symmetry creates balance on its own, but it can be too stable and too calm, uninteresting.

Asymmetrical shapes don't have the same balance as symmetrical ones, but you can balance the whole composition asymmetrically. Asymmetry often occurs in natural forms: you are right-handed or left-handed, tree branches grow in different directions, clouds take on random shapes.

Asymmetry leads to more complex relationships between the elements of a space and is therefore considered more interesting than symmetry, which means it can be used to draw attention.

The space around asymmetrical shapes is more active: patterns are often unpredictable, and overall you have more freedom to express yourself. back side asymmetry in that it is more difficult to make it balanced.

You can combine symmetry and asymmetry and achieve good results - create a symmetrical balance of asymmetrical shapes and vice versa, break up a symmetrical shape with a random label to make it more interesting. Collide symmetry and asymmetry in the composition so that its elements attract more attention.

Principles of Gestalt Psychology

Design principles do not emerge from nothing: they follow from the psychology of our perception of the visual environment. Many design principles grow out of the principles of Gestalt psychology and also build on each other.

So, one of the principles of Gestalt psychology concerns precisely symmetry and order and can be applied to compositional balance. However, this is perhaps the only principle applicable to it.

Other principles of Gestalt psychology, such as focal points and simplicity, add up to the visual mass, and the good continuation factor, common destiny factor, and parallelism set the visual direction. Symmetrical forms are most often perceived as figures against a background.

Examples of different approaches to web design

It's time for real examples. The landing pages below are grouped into four types of balance. Perhaps you will perceive the design of these pages differently, and that's good: critical thinking is more important than unconditional acceptance.

Examples of symmetrical equilibrium

The Helen & Hard website design is symmetrical. The About Us page in the screenshot below and all other pages on this site are balanced in a similar way:

Screenshot of Helen & Hard's "About Us" page

All elements located on opposite sides of the vertical axis located in the center of the page mirror each other. Logo, navigation bar, round photos, title, three columns of text - centered.

However, the symmetry is not perfect: for example, columns contain different amounts of text. By the way, look at the top of the page. Both the logo and the navigation bar are centered, but visually they don't appear to be centered. Maybe the logo should have been centered on the ampersand, or at least the area next to it.

The three menu text links located on the right side of the navigation bar have more letters than the links on the left side - it seems that the center should be between About and People. Maybe if these elements were not really centered, but visually centered, the whole composition would look more balanced.

The Tilde homepage is another example of symmetrical balance design. Like on Helen & Hard, everything is arranged around a vertical axis running down the center of the page: navigation, text, people in photos.

Screenshot of Tilde homepage

As in the case of Helen & Hard, the symmetry is not perfect: first, the centered lines of text cannot be a reflection of the photograph from below, and secondly, a couple of elements stand out from the general row - the “Meet the Team” arrow points to the right, and the text at the bottom of the page ends with another right arrow. Both arrows are calls to action and both break the symmetry, drawing additional attention to themselves. In addition, the color of both arrows contrast with the background, which also attracts the eye.

Examples of asymmetric equilibrium

Carrie Voldengen's homepage showcases an asymmetrical balance around a dominant symmetrical shape. Looking at the composition as a whole, you can see several forms that are separate from each other:

Screenshot of the Carrie Voldengen website

Most of the page is occupied by a rectangle consisting of a grid of smaller rectangular images. The grille itself is symmetrical in both the vertical and horizontal axes and appears to be very solid and stable - you could even say that it is too balanced and looks immovable.

The block of text on the right breaks the symmetry. The lattice is contrasted with text and a round logo in the upper left corner of the page. These two elements have approximately equal visual mass acting on the grating from different sides. The distance to the imaginary fulcrum is about the same as the mass. The block of text on the right is larger and darker, but the round blue logo adds weight to its area and even matches the top left corner of the grid in color. The text at the bottom of the grid seems to hang from it, but it is light enough not to disturb the compositional balance.

Notice how the white space also seems to be balanced. The voids on the left, top and bottom, as well as on the right under the text - balance each other. There is more white space on the left side of the page than on the right side, but there is extra space on the right side at the top and bottom.

The images in the header of the Hirondelle USA page change from one to another. The screenshot below was taken specifically to demonstrate the asymmetric compositional balance.

Screenshot of Hirondelle USA

The column in the photo is shifted slightly to the right of the center and creates a noticeable vertical line, since we know that the column is a very heavy object. The railing on the left creates a strong connection to the left edge of the screen and feels solid enough too.

The text above the railing seems to rest on it; in addition, on the right it is visually balanced by a photograph of a boy. It may seem that the railing seems to be hanging from the column, upsetting the balance, but the presence of the boy and the darker background behind him balance the composition, and the light text restores the balance as a whole.

Examples of Radial Equilibrium

The homepage of Vlog.it demonstrates radial balance, as seen in the screenshot. Everything but the object in the upper right is organized around the center, and the three rings of images rotate around the center circle.

Screenshot of the Vlog.it homepage

However, the screenshot does not show how the page loads: a line is drawn from the lower left corner of the screen to its center - and from that moment on, everything that appears on the page rotates around the center or radiates from it, like circles on water.

The small circle in the upper right corner adds translational symmetry and asymmetry, increasing visual interest in the composition.

There are no circles on Opera's Shiny Demos homepage, but all the text links radiate from a common center, and it's easy to imagine the whole structure rotating around one of the central squares, or maybe one of the corners:

Screenshot of Opera's Shiny Demos homepage

The name Shiny Demos in the top left and the Opera logo in the bottom right balance each other out and also seem to emanate from the same center as the text links.

This is good example that it is not necessary to use circles to achieve radial balance.

Examples of mosaic equilibrium

You might think that mosaic balance is the least used on websites, especially after the paintings of Jackson Pollock were cited as an example. But mosaic equilibrium is much more common than it seems.

A prime example is the home page of Rabbit's Tale. Letters scattered across the screen definitely create a sense of chaos, but the compositional balance is there.

Screenshot of Rabbit's Tale homepage

Almost equal in size areas of color and space, located on both sides, on the right and on the left, balance each other. The rabbit in the center serves as a fulcrum. Each element does not attract attention on its own.

It is difficult to figure out which specific elements balance each other, but in general there is a balance. Maybe the visual mass on the right side is a little bigger, but not enough to upset the balance.

Sites with a lot of content, such as news portals or magazine sites, also exhibit tiled balance. Here is a screenshot of The Onion home page:

Screenshot of The Onion homepage

There are a lot of elements, their arrangement is not symmetrical, the size of the text columns is not the same, and it is difficult to understand what balances what. Blocks contain different amounts of content, and hence their sizes vary. Objects are not located around some common center.

Blocks of varying sizes and densities create a somewhat cluttered feel. Since the site is updated every day, the structure of this chaos is constantly changing. But overall, the balance is maintained.

Conclusion

Design principles draw heavily from Gestalt psychology and perceptual theory and are based on how we perceive and interpret our visual environment. For example, one of the reasons we notice focal points is because they contrast with the elements around them.

Symmetries can be exact or approximate.

Symmetry in geometry

Geometric symmetry is the most well-known type of symmetry for many people. A geometric object is said to be symmetrical if, after it has been transformed geometrically, it retains some of its original properties. For example, a circle rotated around its center will have the same shape and size as the original circle. Therefore, the circle is called symmetric with respect to rotation (has axial symmetry). The kinds of symmetries possible for a geometric object depend on the set of available geometric transformations and what properties of the object must remain unchanged after the transformation.

Types of geometric symmetries:

Mirror symmetry

In physics, invariance under a rotation group is called isotropy of space(all directions in space are equal) and is expressed in the invariance of physical laws, in particular, the equations of motion, with respect to rotations. Noether's theorem relates this invariance to the presence of a conserved quantity (the integral of motion), the angular momentum.

Symmetry about a point

Sliding symmetry

Symmetries in physics

In theoretical physics, the behavior of a physical system is described by some equations. If these equations have any symmetries, then it is often possible to simplify their solution by finding conserved quantities (integrals of motion). So, already in classical mechanics, Noether's theorem is formulated, which associates a conserved quantity with each type of continuous symmetry. From it, for example, it follows that the invariance of the equations of motion of the body over time leads to the law of conservation of energy; invariance with respect to shifts in space - to the law of conservation of momentum; invariance under rotations - to the law of conservation of angular momentum.

supersymmetry

Transfer in a flat four-dimensional space-time does not change the physical laws. In field theory, translational symmetry, according to Noether's theorem, corresponds to the conservation of the energy-momentum tensor. In particular, purely temporal translations follow the law of conservation of energy, while purely spatial shifts follow the law of conservation of momentum.

Symmetries in biology

Symmetry in biology- this is a natural arrangement of similar (identical, equal in size) parts of the body or forms of a living organism, a set of living organisms relative to the center or axis of symmetry. The type of symmetry determines not only the general structure of the body, but also the possibility of developing animal organ systems. The body structure of many multicellular organisms reflects certain forms of symmetry. If the body of an animal can be mentally divided into two halves, right and left, then this form of symmetry is called bilateral. This type of symmetry is characteristic of the vast majority of species, as well as humans. If the body of an animal can be mentally divided not by one, but by several planes of symmetry into equal parts, then such an animal is called radially symmetrical. This type of symmetry is much less common.

Asymmetry- lack of symmetry. Sometimes the term is used to describe organisms that lack symmetry in the first place, as opposed to dissymmetry- secondary loss of symmetry or its individual elements.

The concepts of symmetry and asymmetry are reversed. The more symmetrical an organism is, the less asymmetric it is, and vice versa. A small number of organisms are completely asymmetric. In this case, one should distinguish between the variability of shape (for example, in an amoeba) and the lack of symmetry. In nature and, in particular, in living nature, symmetry is not absolute and always contains some degree of asymmetry. For example, symmetrical plant leaves do not exactly match when folded in half.

Biological objects have the following types of symmetry:

• spherical symmetry of rotations in three-dimensional space through arbitrary angles.
• axial symmetry (radial symmetry, rotational symmetry of an indefinite order) - symmetry with respect to rotations through an arbitrary angle around an axis.
  • rotational symmetry of the nth order - symmetry with respect to rotations through an angle of 360 ° / n around any axis.
• bilateral (bilateral) symmetry - symmetry with respect to the plane of symmetry (mirror reflection symmetry).
• translational symmetry - symmetry with respect to shifts of space in any direction for a certain distance (its special case in animals is metamerism (biology)).
• triaxial asymmetry - lack of symmetry along all three spatial axes.

Radial symmetry

Usually two or more planes of symmetry pass through the axis of symmetry. These planes intersect in a straight line - the axis of symmetry. If the animal will rotate around this axis by a certain degree, then it will be displayed on itself (coincide with itself). There can be several such axes of symmetry (polyaxon symmetry) or one (monaxon symmetry). Polyaxon symmetry is common among protists (such as radiolarians).

As a rule, in multicellular animals, the two ends (poles) of a single axis of symmetry are not equivalent (for example, in jellyfish, the mouth is on one pole (oral), and the top of the bell is on the opposite (aboral). Such symmetry (a variant of radial symmetry) in comparative anatomy is called In a 2D projection, radial symmetry can be preserved if the axis of symmetry is directed perpendicular to the projection plane.In other words, the preservation of radial symmetry depends on the viewing angle.

Radial symmetry is characteristic of many cnidarians, as well as most echinoderms. Among them there is the so-called pentasymmetry, based on five planes of symmetry. In echinoderms, radial symmetry is secondary: their larvae are bilaterally symmetrical, while in adult animals, external radial symmetry is violated by the presence of a madrepore plate.

In addition to typical radial symmetry, there is two-beam radial symmetry (two planes of symmetry, for example, in ctenophores). If there is only one plane of symmetry, then the symmetry is bilateral (animals from the group Bilateria).

A crystallographic point symmetry group is a point symmetry group that describes the macrosymmetry of a crystal. Since only 1, 2, 3, 4, and 6 orders of axes (rotational and improper rotation) are admissible in crystals, only 32 of the entire infinite number of point symmetry groups are crystallographic.

Anisotropy (from other Greek. ἄνισος - unequal and τρόπος - direction) - the difference in the properties of the medium (for example, physical: elasticity, electrical conductivity, thermal conductivity, refractive index, speed of sound or light, etc.) in different directions within this medium; as opposed to

Symmetry I Symmetry (from Greek symmetria - proportionality)

in mathematics

1) symmetry (in the narrow sense), or reflection (mirror) relative to the plane α in space (relative to the straight line a on the plane), is the transformation of the space (plane), in which each point M goes to the point M" such that the segment MM" perpendicular to the plane α (straight a) and cut it in half. Plane α (straight a) is called the plane (axis) C.

Reflection is an example of an orthogonal transform (See Orthogonal Transform) that changes orientation (See Orientation) (as opposed to proper motion). Any orthogonal transformation can be carried out by sequentially performing a finite number of reflections - this fact plays an essential role in the study of the symmetry of geometric figures.

2) Symmetry (in a broad sense) - a property of a geometric figure F, which characterizes some regularity of the form F, its invariance under the action of movements and reflections. More precisely, the figure F has a S. (symmetric) if there exists a nonidentical orthogonal transformation that maps this figure into itself. The set of all orthogonal transformations that combine a figure F with itself, is a group (See group) called the symmetry group of this figure (sometimes these transformations themselves are called symmetries).

So, a flat figure that transforms into itself upon reflection is symmetrical with respect to the straight line - the C axis. ( rice. one ); here the symmetry group consists of two elements. If the figure F on the plane is such that rotations about any point O through an angle of 360 ° / n, n- an integer ≥ 2, translate it into itself, then F has S. n-th order with respect to the point O- center C. An example of such figures are regular polygons ( rice. 2 ); group S. here - the so-called. cyclic group n-th order. A circle has a S. of infinite order (because it is combined with itself by turning through any angle).

The simplest types of spatial S., in addition to S. generated by reflections, are central S., axial S. and S. of transfer.

a) In the case of central symmetry (inversion) about the point O, the figure Ф is combined with itself after successive reflections from three mutually perpendicular planes, in other words, the point O is the middle of the segment connecting the symmetrical points Ф ( rice. 3 ). b) In case axial symmetry, or S. relative to a straight line n th order, the figure is superimposed on itself by rotation around some straight line (N-axis) at an angle of 360 ° / n. For example, a cube has a line AB axis C. of the third order, and a straight line CD- C. axis of the fourth order ( rice. 3 ); in general, regular and semiregular polyhedra are symmetrical with respect to a series of lines. The location, number, and order of axes of crystallization play an important role in crystallography (see Crystal Symmetry). k around a straight line AB and reflection in a plane perpendicular to it, has a mirror-axial C. Straight line AB, is called the mirror-rotary axis C. of order 2 k, is the C axis of the order k (rice. 4 ). A mirror-axial line of order 2 is equivalent to a central line. d) In the case of translation symmetry, the figure is superimposed on itself by translation along some straight line (transfer axis) on some segment. For example, a figure with a single translation axis has an infinite number of S. planes (since any translation can be carried out by two successive reflections from planes perpendicular to the translation axis) ( rice. 5 ). Figures having several transfer axes play an important role in the study of crystal lattices.

S. has become widespread in art as one of the types of harmonious composition (see composition). It is characteristic of works of architecture (being an indispensable quality, if not of the entire structure as a whole, then of its parts and details - plan, facade, columns, capitals, etc.) and decorative and applied art. S. is also used as the main technique for constructing borders and ornaments (flat figures, respectively, having one or more S. transfer in combination with reflections) ( rice. 6 , 7 ).

S. combinations generated by reflections and rotations (exhausting all types of S. geometric figures), as well as transfers, are of interest and are the subject of research in various fields of natural science. For example, helical S., carried out by rotation through a certain angle around an axis, supplemented by a transfer along the same axis, is observed in the arrangement of leaves in plants ( rice. eight ) (for more details, see the article Symmetry in biology). C. the configuration of molecules, which affects their physical and chemical characteristics, is important in the theoretical analysis of the structure of compounds, their properties, and behavior in various reactions (see Symmetry in chemistry). Finally, in the physical sciences in general, in addition to the already indicated geometric symmetry of crystals and lattices, the concept of symmetry in the general sense acquires great importance (see below). So, the symmetry of the physical space-time, expressed in its homogeneity and isotropy (see Relativity theory), allows you to establish the so-called. conservation laws; generalized symmetry plays an essential role in the formation of atomic spectra and in the classification of elementary particles (see Symmetry in physics).

3) Symmetry (in the general sense) means the invariance of the structure of a mathematical (or physical) object with respect to its transformations. For example, the S. laws of the theory of relativity is determined by their invariance with respect to Lorentz transformations (See Lorentz transformations). Definition of a set of transformations that leave all the structural relations of the object unchanged, i.e., the definition of a group G its automorphisms, has become the guiding principle of modern mathematics and physics, allowing deep insight into internal structure object as a whole and its parts.

Since such an object can be represented by elements of some space R, endowed with an appropriate characteristic structure for it, insofar as the transformations of an object are transformations R. That. get a representation of the group G in transformation group R(or just in R), and the study of S. of the object is reduced to the study of the action G on the R and finding invariants of this action. In the same way, the laws of physics that govern the object under study and are usually described by equations that are satisfied by the elements of space R, is determined by the action G to such equations.

So, for example, if some equation is linear on a linear space R and remains invariant under transformations of some group G, then each element g from G corresponds to a linear transformation Tg in linear space R solutions of this equation. Conformity g → Tg is a linear representation G and knowledge of all such representations of it allows us to establish various properties of solutions, and also helps to find in many cases (from "symmetry considerations") the solutions themselves. This, in particular, explains the necessity for mathematics and physics of a developed theory of linear representations of groups. For specific examples, see Art. Symmetry in physics.

Lit.: Shubnikov A.V., Symmetry. (Laws of symmetry and their application in science, technology and applied art), M. - L., 1940; Kokster G. S. M., Introduction to geometry, trans. from English, M., 1966; Weil G., Symmetry, trans. from English, M., 1968; Wigner E., Etudes on Symmetry, trans. from English, M., 1971.

M. I. Voitsekhovsky.

Rice. 3. A cube having line AB as a third-order symmetry axis, line CD as a fourth-order symmetry axis, point O as a center of symmetry. The points M and M" of the cube are symmetrical both about the axes AB and CD, and about the center O.