How to draw a symmetrical object. Axial symmetry rule

Since ancient times, man has developed ideas about beauty. All creations of nature are beautiful. People are beautiful in their own way, animals and plants are amazing. The sight is pleasing to the eye gemstone or a salt crystal, it’s hard not to admire a snowflake or a butterfly. But why does this happen? It seems to us that the appearance of objects is correct and complete, the right and left halves of which look the same, as if in a mirror image.

Apparently, people of art were the first to think about the essence of beauty. Ancient sculptors who studied the structure human body, back in the 5th century BC. The concept of “symmetry” began to be used. This word is of Greek origin and means harmony, proportionality and similarity in the arrangement of the constituent parts. Plato argued that only that which is symmetrical and proportionate can be beautiful.

In geometry and mathematics, three types of symmetry are considered: axial symmetry (relative to a straight line), central (relative to a point) and mirror symmetry (relative to a plane).

If each of the points of an object has its own exact mapping within it relative to its center, there is central symmetry. Its example is such geometric bodies as a cylinder, a sphere, a regular prism, etc.

Axial symmetry points relative to a line provides that this line intersects the middle of the segment connecting the points and is perpendicular to it. Examples of the bisector of an undeveloped angle isosceles triangle, any straight line drawn through the center of the circle, etc. If axial symmetry is characteristic, the definition of mirror points can be visualized by simply bending it along the axis and putting equal halves “face to face.” The desired points will touch each other.

With mirror symmetry, the points of an object are located equally relative to the plane that passes through its center.

Nature is wise and rational, therefore almost all of its creations have a harmonious structure. This applies to both living beings and inanimate objects. The structure of most life forms is characterized by one of three types of symmetry: bilateral, radial or spherical.

Most often, axial can be observed in plants developing perpendicular to the soil surface. In this case, symmetry is the result of rotation of identical elements around a common axis located in the center. The angle and frequency of their location may be different. Examples are trees: spruce, maple and others. In some animals, axial symmetry also occurs, but this is less common. Of course, nature is rarely characterized by mathematical precision, but the similarity of the elements of an organism is still striking.

Biologists often consider not axial symmetry, but bilateral (bilateral) symmetry. An example of this is the wings of a butterfly or dragonfly, plant leaves, flower petals, etc. In each case, the right and left parts of the living object are equal and are mirror images of each other.

Spherical symmetry is characteristic of the fruits of many plants, some fish, mollusks and viruses. Examples of radial symmetry are some types of worms and echinoderms.

In human eyes, asymmetry is most often associated with irregularity or inferiority. Therefore, in most creations of human hands, symmetry and harmony can be traced.

I . Symmetry in mathematics :

Basic concepts and definitions.

Axial symmetry (definitions, construction plan, examples)

Central symmetry (definitions, construction plan, whenmeasures)

Summary table (all properties, features)

II . Applications of symmetry:

1) in mathematics

2) in chemistry

3) in biology, botany and zoology

4) in art, literature and architecture

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1. Basic concepts of symmetry and its types.

The concept of symmetry r goes back through the entire history of mankind. It is found already at the origins of human knowledge. It arose in connection with the study of a living organism, namely man. And it was used by sculptors back in the 5th century BC. e. The word “symmetry” is Greek and means “proportionality, proportionality, sameness in the arrangement of parts.” It is widely used by all areas of modern science without exception. Many great people have thought about this pattern. For example, L.N. Tolstoy said: “Standing in front of a black board and drawing different figures on it with chalk, I was suddenly struck by the thought: why is symmetry clear to the eye? What is symmetry? This is an innate feeling, I answered myself. What is it based on?” The symmetry is truly pleasing to the eye. Who hasn’t admired the symmetry of nature’s creations: leaves, flowers, birds, animals; or human creations: buildings, technology, everything that surrounds us since childhood, everything that strives for beauty and harmony. Hermann Weyl said: “Symmetry is the idea through which man throughout the ages has tried to comprehend and create order, beauty and perfection.” Hermann Weyl is a German mathematician. His activities span the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what criteria one can determine the presence or, conversely, absence of symmetry in a given case. Thus, a mathematically rigorous concept was formed relatively recently - at the beginning of the twentieth century. It's quite complicated. Let us turn and once again remember the definitions that were given to us in the textbook.

2. Axial symmetry.

2.1 Basic definitions

Definition. Two points A and A 1 are called symmetrical with respect to line a if this line passes through the middle of segment AA 1 and is perpendicular to it. Each point of a line a is considered symmetrical to itself.

Definition. The figure is said to be symmetrical about a straight line A, if for each point of the figure there is a point symmetrical to it relative to the straight line A also belongs to this figure. Straight A called the axis of symmetry of the figure. The figure is also said to have axial symmetry.

2.2 Construction plan

And so, to construct a symmetrical figure relative to a straight line, from each point we draw a perpendicular to this straight line and extend it to the same distance, mark the resulting point. We do this with each point and get symmetrical vertices of a new figure. Then we connect them in series and get a symmetrical figure of a given relative axis.

2.3 Examples of figures with axial symmetry.

3. Central symmetry

3.1 Basic definitions

Definition. Two points A and A 1 are called symmetrical with respect to point O if O is the middle of the segment AA 1. Point O is considered symmetrical to itself.

Definition. A figure is said to be symmetrical with respect to point O if, for each point of the figure, a point symmetrical with respect to point O also belongs to this figure.

3.2 Construction plan

Construction of a triangle symmetrical to the given one relative to the center O.

To construct a point symmetrical to a point A relative to the point ABOUT, it is enough to draw a straight line OA(Fig. 46 ) and on the other side of the point ABOUT set aside a segment equal to the segment OA. In other words , points A and ; In and ; C and symmetrical about some point O. In Fig. 46 a triangle is constructed that is symmetrical to a triangle ABC relative to the point ABOUT. These triangles are equal.

Construction of symmetrical points relative to the center.

In the figure, points M and M 1, N and N 1 are symmetrical relative to point O, but points P and Q are not symmetrical relative to this point.

In general, figures that are symmetrical about a certain point are equal .

3.3 Examples

Let us give examples of figures that have central symmetry. The simplest figures with central symmetry are the circle and parallelogram.

Point O is called the center of symmetry of the figure. In such cases, the figure has central symmetry. The center of symmetry of a circle is the center of the circle, and the center of symmetry of a parallelogram is the point of intersection of its diagonals.

A straight line also has central symmetry, but unlike a circle and a parallelogram, which have only one center of symmetry (point O in the figure), a straight line has an infinite number of them - any point on the straight line is its center of symmetry.

The pictures show an angle symmetrical relative to the vertex, a segment symmetrical to another segment relative to the center A and a quadrilateral symmetrical about its vertex M.

An example of a figure that does not have a center of symmetry is a triangle.

4. Lesson summary

Let us summarize the knowledge gained. Today in class we learned about two main types of symmetry: central and axial. Let's look at the screen and systematize the knowledge gained.

Summary table

	Axial symmetry	Central symmetry
Peculiarity	All points of the figure must be symmetrical relative to some straight line.	All points of the figure must be symmetrical relative to the point chosen as the center of symmetry.
Properties	1. Symmetrical points lie on perpendiculars to a line. 3. Straight lines turn into straight lines, angles into equal angles. 4. The sizes and shapes of the figures are preserved.	1. Symmetrical points lie on a line passing through the center and a given point of the figure. 2. The distance from a point to a straight line is equal to the distance from a straight line to a symmetrical point. 3. The sizes and shapes of the figures are preserved.

II. Application of symmetry

Mathematics	In algebra lessons we studied the graphs of the functions y=x and y=x The pictures show various pictures depicted using the branches of parabolas. (a) Octahedron, (b) rhombic dodecahedron, (c) hexagonal octahedron.
Russian language	The printed letters of the Russian alphabet also have different types of symmetries. There are “symmetrical” words in the Russian language - palindromes, which can be read equally in both directions.	A D L M P T F W– vertical axis V E Z K S E Y - horizontal axis F N O X- both vertical and horizontal B G I Y R U C CH SCHY- no axis Radar hut Alla Anna
Literature	Sentences can also be palindromic. Bryusov wrote a poem “The Voice of the Moon”, in which each line is a palindrome. Look at the quadruples of A.S. Pushkin “ Bronze Horseman" If we draw a line after the second line we can notice elements of axial symmetry	And the rose fell on Azor's paw. I come with the sword of the judge. (Derzhavin) "Search for a taxi" "Argentina beckons the Negro" “The Argentine appreciates the black man,” “Lesha found a bug on the shelf.” The Neva is dressed in granite; Bridges hung over the waters; Dark green gardens Islands covered it...

Biology

The human body is built on the principle of bilateral symmetry. Most of us view the brain as a single structure; in reality, it is divided into two halves. These two parts - two hemispheres - fit tightly to each other. In full accordance with the general symmetry of the human body, each hemisphere is an almost exact mirror image of the other Control of the basic movements of the human body and its sensory functions is evenly distributed between the two hemispheres of the brain. The left hemisphere controls the right side of the brain, and the right hemisphere controls the left side.

Botany

A flower is considered symmetrical when each perianth consists of an equal number of parts. Flowers having paired parts are considered flowers with double symmetry, etc. Triple symmetry is common for monocotyledonous plants, fivefold - for dicotyledonous plants. A characteristic feature of the structure of plants and their development is spirality. Pay attention to the leaf arrangement of the shoots - this is also a peculiar type of spiral - a helical one. Even Goethe, who was not only a great poet, but also a natural scientist, considered helicity one of characteristic features of all organisms, a manifestation of the innermost essence of life. The tendrils of plants twist in a spiral, the growth of tissues in tree trunks occurs in a spiral, the seeds in a sunflower are arranged in a spiral, and spiral movements are observed during the growth of roots and shoots.

A characteristic feature of the structure of plants and their development is spirality. Look at the pine cone. The scales on its surface are arranged strictly regularly - along two spirals that intersect approximately at a right angle. The number of such spirals in pine cones is 8 and 13 or 13 and 21.

Zoology

Symmetry in animals means correspondence in size, shape and outline, as well as the relative arrangement of body parts located on opposite sides of the dividing line. With radial or radial symmetry, the body has the shape of a short or long cylinder or vessel with a central axis, from which parts of the body extend radially. These are coelenterates, echinoderms, and starfish. With bilateral symmetry, there are three axes of symmetry, but only one pair of symmetrical sides. Because the other two sides - abdominal and dorsal - are not similar to each other. This type of symmetry is characteristic of most animals, including insects, fish, amphibians, reptiles, birds, and mammals.

Axial symmetry

	Various types symmetry physical phenomena: symmetry of electric and magnetic fields (Fig. 1) The distribution is symmetrical in mutually perpendicular planes electromagnetic waves(Fig. 2)	Fig.1 Fig.2
Art	Mirror symmetry can often be observed in works of art. Mirror" symmetry is widely found in works of art of primitive civilizations and in ancient paintings. Medieval religious paintings are also characterized by this type of symmetry. One of Raphael’s best early works, “The Betrothal of Mary,” was created in 1504. Under a sunny blue sky lies a valley topped by a white stone temple. In the foreground is the betrothal ceremony. The High Priest brings Mary and Joseph's hands together. Behind Mary is a group of girls, behind Joseph is a group of young men. Both parts of the symmetrical composition are held together by the counter-movement of the characters. For modern tastes, the composition of such a painting is boring, since the symmetry is too obvious.

Chemistry	A water molecule has a plane of symmetry (straight vertical line). DNA molecules (deoxyribonucleic acid) play an extremely important role in the world of living nature. This is a double-chain high-molecular polymer, the monomer of which is nucleotides. DNA molecules have a double helix structure built on the principle of complementarity.
Architeculture	Man has long used symmetry in architecture. The ancient architects made especially brilliant use of symmetry in architectural structures. Moreover, the ancient Greek architects were convinced that in their works they were guided by the laws that govern nature. By choosing symmetrical forms, the artist thereby expressed his understanding of natural harmony as stability and balance. The city of Oslo, the capital of Norway, has an expressive ensemble of nature and art. This is Frogner - a park - a complex of garden and park sculpture, which was created over the course of 40 years.	Pashkov House Louvre (Paris)

© Sukhacheva Elena Vladimirovna, 2008-2009.

TRIANGLES.

§ 17. SYMMETRY RELATIVELY TO THE RIGHT STRAIGHT.

1. Figures that are symmetrical to each other.

Let's draw some figure on a sheet of paper with ink, and with a pencil outside it - an arbitrary straight line. Then, without allowing the ink to dry, we bend the sheet of paper along this straight line so that one part of the sheet overlaps the other. This other part of the sheet will thus produce an imprint of this figure.

If you then straighten the sheet of paper again, then there will be two figures on it, which are called symmetrical relative to a given line (Fig. 128).

Two figures are called symmetrical relative to a certain straight line if, when bending the drawing plane along this straight line, they are combined.

The straight line with respect to which these figures are symmetrical is called their axis of symmetry.

From the definition of symmetrical figures it follows that all symmetrical figures are equal.

You can get symmetrical figures without using bending of the plane, but with the help geometric construction. Let it be necessary to construct a point C" symmetrical to a given point C relative to straight line AB. Let us drop a perpendicular from point C
CD to straight line AB and as its continuation we will lay down the segment DC" = DC. If we bend the drawing plane along AB, then point C will align with point C": points C and C" are symmetrical (Fig. 129).

Suppose now we need to construct a segment C "D", symmetrical to a given segment CD relative to the straight line AB. Let's construct points C" and D", symmetrical to points C and D. If we bend the drawing plane along AB, then points C and D will coincide, respectively, with points C" and D" (Drawing 130). Therefore, segments CD and C "D" will coincide , they will be symmetrical.

Let us now construct a figure symmetrical to the given polygon ABCDE relative to the given axis of symmetry MN (Fig. 131).

To solve this problem, let’s drop the perpendiculars A A, IN b, WITH With, D d and E e to the axis of symmetry MN. Then, on the extensions of these perpendiculars, we plot the segments
A A" = A A, b B" = B b, With C" = Cs; d D"" =D d And e E" = E e.

The polygon A"B"C"D"E" will be symmetrical to the polygon ABCDE. Indeed, if you bend the drawing along a straight line MN, then the corresponding vertices of both polygons will align, and therefore the polygons themselves will align; this proves that the polygons ABCDE and A" B"C"D"E" are symmetrical about the straight line MN.

2. Figures consisting of symmetrical parts.

Often there are geometric figures that are divided by some straight line into two symmetrical parts. Such figures are called symmetrical.

So, for example, an angle is a symmetrical figure, and the bisector of the angle is its axis of symmetry, since when bending along it, one part of the angle is combined with the other (Fig. 132).

In a circle, the axis of symmetry is its diameter, since when bending along it, one semicircle is combined with another (Fig. 133). The figures in drawings 134, a, b are exactly symmetrical.

Symmetrical figures are often found in nature, construction, and jewelry. The images placed on drawings 135 and 136 are symmetrical.

It should be noted that symmetrical figures can be combined simply by moving along a plane only in some cases. To combine symmetrical figures, as a rule, it is necessary to turn one of them with the opposite side,

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 a transformation of space (plane), in which each point M goes to point M" such that the segment MM" perpendicular to the plane α (straight line A) and divides it in half. Plane α (straight A) is called plane (axis) C.

Reflection is an example of an orthogonal transformation (See Orthogonal transformation) 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 a significant role in the study of geometric shapes.

2) Symmetry (in the broad sense) - a property of a geometric figure F, characterizing some regularity of form F, its invariability under the action of movements and reflections. More precisely, the figure F has S. (symmetric) if there is a non-identical orthogonal transformation that takes 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).

Thus, a flat figure that transforms into itself upon reflection is symmetrical with respect to a straight line - the C axis. ( rice. 1 ); here the symmetry group consists of two elements. If the figure F on the plane is such that rotations relative to any point O through an angle of 360°/ n, n- integer ≥ 2, convert it to itself, then F possesses S. n-th order relative to the point ABOUT- center C. An example of such figures are regular polygons ( rice. 2 ); group S. here - so-called. cyclic group n-th order. A circle has a circle of infinite order (since it can be combined with itself by rotating through any angle).

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

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

In art, composition has become widespread 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 having, respectively, one or more transfer mechanisms in combination with reflections) ( rice. 6 , 7 ).

Combinations of symbols generated by reflections and rotations (exhausting all types of symbols of geometric figures), as well as translations, are of interest and are the subject of research in various areas natural sciences. For example, helical S., carried out by rotation at a certain angle around an axis, supplemented by transfer along the same axis, is observed in the arrangement of leaves in plants ( rice. 8 ) (for more details, see the article. Symmetry in biology). C. configuration of molecules, affecting their physical and chemical characteristics, matters when theoretical analysis 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 structure of crystals and lattices, they acquire important ideas about S. in the general sense (see below). Thus, the symmetry of physical space-time, expressed in its homogeneity and isotropy (see Relativity theory), allows us to establish the so-called. Conservation laws; generalized symmetry plays a significant 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 system of the laws of relativity is determined by their invariance with respect to Lorentz transformations (See Lorentz transformations). Definition of a set of transformations that leave all structural relationships of an object unchanged, i.e., definition of a group G its automorphisms, has become the guiding principle of modern mathematics and physics, allowing deep insight into internal structure the object as a whole and its parts.

Since such an object can be represented by elements of some space R, endowed with a corresponding characteristic structure for it, insofar as transformations of an object are transformations R. That. a group representation is obtained G in transformation group R(or just in R), and the study of the S. object comes down to the study of action G on R and finding invariants of this action. In the same way, S. physical laws 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 for 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 linear transformation T g in linear space R solutions to this equation. Correspondence g → T g is a linear representation G and knowledge of all such representations of it allows one 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 need for mathematics and physics to develop 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; Coxeter G.S.M., Introduction to Geometry, trans. from English, M., 1966; Weil G., Symmetry, trans. from English, M., 1968; Wigner E., Studies on Symmetry, trans. from English, M., 1971.

M. I. Voitsekhovsky.

Rice. 3. A cube with straight line AB as the axis of symmetry of the third order, straight line CD as the axis of symmetry of the fourth order, and point O as the center of symmetry. Points M and M" of the cube are symmetrical both with respect to the axes AB and CD, and with respect to the center O.

II Symmetry

in physics. If the laws that establish relationships between quantities that characterize a physical system, or that determine the change in these quantities over time, do not change under certain operations (transformations) to which the system can be subjected, then these laws are said to have S. (or are invariant) with respect to data transformations. Mathematically, S. transformations form a group (See Group).

Experience shows that physical laws are symmetrical with respect to the following most general transformations.

Continuous transformation

1) Transfer (shift) of the system as a whole in space. This and subsequent spatiotemporal transformations can be understood in two senses: as an active transformation - a real transfer physical system relative to the selected reference system or as a passive transformation - parallel transfer of the reference system. The symbol of physical laws regarding shifts in space means the equivalence of all points in space, that is, the absence of any distinguished points in space (homogeneity of space).

2) Rotation of the system as a whole in space. S. physical laws regarding this transformation mean the equivalence of all directions in space (isotropy of space).

3) Changing the start of time (time shift). S. regarding this transformation means that physical laws do not change over time.

4) Transition to a reference system moving relative to a given system with a constant (in direction and magnitude) speed. S. relative to this transformation means, in particular, the equivalence of all inertial reference systems (See Inertial reference system) (See Relativity theory).

5) Gauge transformations. The laws that describe the interactions of particles with any charge (electric charge (See Electric charge), baryon charge (See Baryon charge), leptonic charge (See Lepton charge), Hypercharge) are symmetrical with respect to gauge transformations of the 1st kind. These transformations consist in the fact that the wave functions (See Wave function) of all particles can be simultaneously multiplied by an arbitrary phase factor:

where ψ j- particle wave function j, z j is the charge corresponding to the particle, expressed in units of elementary charge (for example, elementary electric charge e), β is an arbitrary numerical factor.

A→A + grad f, , (2)

Where f(x,at, z, t) - arbitrary function of coordinates ( X,at,z) and time ( t), With- speed of light. In order for transformations (1) and (2) to be carried out simultaneously in the case of electromagnetic fields, it is necessary to generalize gauge transformations of the 1st kind: it is necessary to require that the interaction laws be symmetrical with respect to transformations (1) with the value β, which is an arbitrary function of coordinates and time: η - Planck's constant. The connection between gauge transformations of the 1st and 2nd kind for electromagnetic interactions is due to the dual role of the electric charge: on the one hand, the electric charge is a conserved quantity, and on the other hand, it acts as an interaction constant characterizing the connection electromagnetic field with charged particles.

Transformations (1) correspond to the laws of conservation of various charges (see below), as well as to some internal interactions. If charges are not only conserved quantities, but also sources of fields (like an electric charge), then the fields corresponding to them must also be gauge fields (similar to electromagnetic fields), and transformations (1) are generalized to the case when the quantities β are arbitrary functions of coordinates and time (and even operators (See Operators) that transform the states of the internal system). This approach to the theory of interacting fields leads to various gauge theories of strong and weak interactions(the so-called Yang-Mills theory).

Discrete transformations

The types of systems listed above are characterized by parameters that can continuously change in a certain range of values ​​(for example, a shift in space is characterized by three displacement parameters along each of the coordinate axes, a rotation by three angles of rotation around these axes, etc.). Along with continuous systems, discrete systems are of great importance in physics. The main ones are the following.

Symmetry and conservation laws

According to Noether’s theorem (See Noether’s theorem), each transformation of a system, characterized by one continuously changing parameter, corresponds to a value that is conserved (does not change with time) for a system that has this transformation. From the system of physical laws regarding shift closed system in space, rotating it as a whole and changing the origin of time follow, respectively, the laws of conservation of momentum, angular momentum and energy. From the system regarding gauge transformations of the 1st kind - the laws of conservation of charges (electric, baryon, etc.), from isotopic invariance - the conservation of isotopic spin (See Isotopic spin) in strong interaction processes. As for discrete systems, in classical mechanics they do not lead to any conservation laws. However, in quantum mechanics, in which the state of the system is described by a wave function, or for wave fields (for example, the electromagnetic field), where the superposition principle is valid, the existence of discrete systems implies conservation laws for some specific quantities that have no analogues in classical mechanics. The existence of such quantities can be demonstrated by the example of spatial parity (See Parity), the conservation of which follows from the system with respect to spatial inversion. Indeed, let ψ 1 be the wave function describing some state of the system, and ψ 2 be the wave function of the system resulting from the spaces. inversion (symbolically: ψ 2 = Rψ 1, where R- operator of spaces. inversion). Then, if there is a system with respect to spatial inversion, ψ 2 is one of the possible states of the system and, according to the principle of superposition, possible conditions systems are superpositions of ψ 1 and ψ 2: a symmetric combination ψ s = ψ 1 + ψ 2 and an antisymmetric combination ψ a = ψ 1 - ψ 2. During inversion transformations, the state of ψ 2 does not change (since Pψ s = Pψ 1 + Pψ 2 = ψ 2 + ψ 1 = ψ s), and the state ψ a changes sign ( Pψ a = Pψ 1 - Pψ 2 = ψ 2 - ψ 1 = - ψ a). In the first case, they say that the spatial parity of the system is positive (+1), in the second - negative (-1). If the wave function of the system is specified using quantities that do not change during spatial inversion (such as angular momentum and energy), then the parity of the system will also have a very definite value. The system will be in a state with either positive or negative parity (and transitions from one state to another under the influence of forces symmetrical with respect to spatial inversion are absolutely prohibited).

Symmetry of quantum mechanical systems and stationary states. Degeneration

The conservation of quantities corresponding to various quantum mechanical systems is a consequence of the fact that the operators corresponding to them commute with the Hamiltonian of the system if it does not depend explicitly on time (see Quantum mechanics, Commutation relations). This means that these quantities are measurable simultaneously with the energy of the system, i.e., they can take on completely definite values ​​for a given energy value. Therefore, from them it is possible to compose the so-called. a complete set of quantities that determine the state of the system. Thus, stationary states (See Stationary State) (states with a given energy) of a system are determined by quantities corresponding to the stability of the system under consideration.

The presence of S. leads to the fact that the various states of motion of a quantum mechanical system, which are obtained from each other by transformation of S., have the same values physical quantities that do not change during these transformations. Thus, the system of systems, as a rule, leads to degeneration (See Degeneration). For example, a certain value of the energy of a system can correspond to several different states that are transformed through each other during transformations of the system. Mathematically, these states represent the basis of the irreducible representation of the group of the system (see Group). This determines the fruitfulness of the application of group theory methods in quantum mechanics.

In addition to the degeneracy of energy levels associated with the explicit control of a system (for example, with respect to rotations of the system as a whole), in a number of problems there is additional degeneracy associated with the so-called. hidden S. interaction. Such hidden oscillators exist, for example, for the Coulomb interaction and for the isotropic oscillator.

If a system that has any system is in a field of forces that violate this system (but are weak enough to be considered as a small disturbance), a splitting of the degenerate energy levels of the original system occurs: different states that, due to the system. systems had the same energy, under the influence of “asymmetrical” disturbances they acquire different energy displacements. In cases where the disturbing field has a certain value that is part of the value of the original system, the degeneracy of the energy levels is not completely removed: some of the levels remain degenerate in accordance with the value of the interaction that “includes” the disturbing field.

The presence of energy-degenerate states in a system, in turn, indicates the existence of a systemic interaction and makes it possible, in principle, to find this system when it is not known in advance. The last circumstance plays vital role, for example, in particle physics. The existence of groups of particles with similar masses and the same other characteristics, but different electric charges(the so-called isotopic multiplets) made it possible to establish the isotopic invariance of strong interactions, and the possibility of combining particles with the same properties into broader groups led to the discovery S.U.(3)-C. strong interactions and interactions that violate this system (see Strong interactions). There are indications that strong interaction has an even wider group C.

The concept of the so-called is very fruitful. dynamic system, which arises when transformations are considered that include transitions between states of the system with different energies. An irreducible representation of a dynamic system group will be the entire spectrum of stationary states of the system. The concept of a dynamic system can also be extended to cases when the Hamiltonian of a system depends explicitly on time, and in this case all states of a quantum mechanical system that are not stationary (that is, do not have a given energy) are combined into one irreducible representation of the dynamic group of the system. ).

Lit.: Wigner E., Studies on Symmetry, trans. from English, M., 1971.

S. S. Gershtein.

III Symmetry

in chemistry is manifested in the geometric configuration of molecules, which affects the specifics of physical and chemical properties molecules in an isolated state, in an external field and in interaction with other atoms and molecules.

Most simple molecules have elements of spatial symmetry of the equilibrium configuration: axes of symmetry, planes of symmetry, etc. (see Symmetry in mathematics). Thus, the ammonia molecule NH 3 has the correct symmetry triangular pyramid, methane molecule CH 4 - tetrahedral symmetry. In complex molecules, the symmetry of the equilibrium configuration as a whole is, as a rule, absent, but the symmetry of its individual fragments is approximately preserved (local symmetry). Most full description symmetry of both equilibrium and nonequilibrium configurations of molecules is achieved on the basis of ideas about the so-called. dynamic symmetry groups - groups that include not only operations of spatial symmetry of the nuclear configuration, but also operations of rearrangement of identical nuclei in different configurations. For example, the dynamic symmetry group for the NH 3 molecule also includes the inversion operation of this molecule: the transition of the N atom from one side of the plane formed by H atoms to its other side.

The symmetry of the equilibrium configuration of nuclei in a molecule entails a certain symmetry of the wave functions (See Wave function) of the various states of this molecule, which makes it possible to classify states according to types of symmetry. A transition between two states associated with the absorption or emission of light, depending on the types of symmetry of the states, can either appear in the molecular spectrum (See Molecular spectra) or be forbidden, so that the line or band corresponding to this transition will be absent in the spectrum. The types of symmetry of states between which transitions are possible affect the intensity of lines and bands, as well as their polarization. For example, in homonuclear diatomic molecules transitions between electronic states of the same parity, the electronic wave functions of which behave in the same way during the inversion operation, are prohibited and do not appear in the spectra; in benzene molecules and similar compounds transitions between non-degenerate electronic states of the same type of symmetry are prohibited, etc. The symmetry selection rules are supplemented for transitions between various conditions selection rules associated with the spin of these states.

For molecules with paramagnetic centers, the symmetry of the environment of these centers leads to a certain type of anisotropy g-factor (Lande multiplier), which affects the structure of the electron paramagnetic resonance spectra (See Electron paramagnetic resonance), while in molecules whose atomic nuclei have non-zero spin, the symmetry of individual local fragments leads to a certain type of energy splitting of states with different projections nuclear spin, which affects the structure of nuclear magnetic resonance spectra (See Nuclear magnetic resonance).

In approximate approaches of quantum chemistry, using the idea of ​​molecular orbitals, classification by symmetry is possible not only for the wave function of the molecule as a whole, but also for individual orbitals. If the equilibrium configuration of a molecule has a symmetry plane in which the nuclei lie, then all the orbitals of this molecule are divided into two classes: symmetric (σ) and antisymmetric (π) with respect to the operation of reflection in this plane. Molecules in which the highest (in energy) occupied orbitals are π-orbitals form specific classes of unsaturated and conjugated compounds with properties characteristic of them. Knowledge of the local symmetry of individual fragments of molecules and the molecular orbitals localized on these fragments makes it possible to judge which fragments are more easily excited and change more strongly during chemical transformations, for example in photochemical reactions.

Concepts of symmetry are important in the theoretical analysis of the structure of complex compounds, their properties and behavior in various reactions. Crystal field theory and ligand field theory establish relative position occupied and vacant orbitals complex compound based on data on its symmetry, the nature and degree of splitting of energy levels when the symmetry of the ligand field changes. Knowledge of the symmetry of a complex alone very often allows one to qualitatively judge its properties.

In 1965, P. Woodward and R. Hoffman put forward the principle of preserving orbital symmetry in chemical reactions, which was subsequently confirmed by extensive experimental material and had a great influence on the development of preparative science. organic chemistry. This principle (the Woodward-Hoffman rule) states that individual elementary acts chemical reactions pass while maintaining the symmetry of molecular orbitals, or orbital symmetry. The more the symmetry of orbitals is violated during an elementary event, the more difficult the reaction is.

Taking into account the symmetry of molecules is important when searching and selecting substances used in the creation of chemical lasers and molecular rectifiers, when constructing models of organic superconductors, when analyzing carcinogenic and pharmacological substances. active substances etc.

Lit.: Hochstrasser R., Molecular aspects of symmetry, trans. from English, M., 1968; Bolotin A. B., Stepanov N. f.. Group theory and its applications in quantum mechanics of molecules, M., 1973; Woodward R., Hoffman R., Conservation of Orbital Symmetry, trans. from English, M., 1971.

N. F. Stepanov.

IV Symmetry

in biology (biosymmetry). The phenomenon of S. in living nature was noticed back in Ancient Greece Pythagoreans (5th century BC) in connection with their development of the doctrine of harmony. In the 19th century There were a few works devoted to the synthesis of plants (French scientists O. P. Decandolle and O. Bravo), animals (German - E. Haeckel), and biogenic molecules (French scientists - A. Vechan, L. Pasteur, and others). In the 20th century biological objects were studied from the standpoint of the general theory of crystallization (Soviet scientists Yu. V. Wulf, V. N. Beklemishev, B. K. Vainshtein, Dutch physical chemist F. M. Yeger, English crystallographers led by J. Bernal) and the doctrine of rightness and leftism (Soviet scientists V.I. Vernadsky, V.V. Alpatov, G.F. Gause and others; German scientist W. Ludwig). These works led to the identification in 1961 of a special direction in the study of S. - biosymmetry.

The structural S. of biological objects has been studied most intensively. The study of biostructures - molecular and supramolecular - from the standpoint of structural structure makes it possible to identify in advance the possible types of structure for them, and thereby the number and type of possible modifications, and to strictly describe the external form and internal structure of any spatial biological objects. This led to the widespread use of the concepts of structural S. in zoology, botany, and molecular biology. Structural S. manifests itself primarily in the form of one or another regular repetition. IN classical theory structural structure, developed by the German scientist I. F. Hessel, E. S. Fedorov (See Fedorov) and others, the appearance of the structure of an object can be described by a set of elements of its structure, that is, such geometric elements (points, lines , planes) relative to which identical parts of an object are ordered (see Symmetry in mathematics). For example, the species S. phlox flower ( rice. 1 , c) - one 5th order axis passing through the center of the flower; produced through its operation - 5 rotations (72, 144, 216, 288 and 360°), with each of which the flower coincides with itself. View of S. butterfly figure ( rice. 2 , b) - one plane dividing it into 2 halves - left and right; the operation performed through the plane is a mirror reflection, “making” the left half right, the right half left, and the figure of the butterfly combining with itself. Species S. radiolaria Lithocubus geometricus ( rice. 3 , b), in addition to the axes of rotation and planes of reflection, it also contains center C. Any straight line drawn through such a single point inside the radiolaria meets identical (corresponding) points of the figure on both sides of it and at equal distances. The operations performed through the S. center are reflections at a point, after which the figure of the radiolaria is also combined with itself.

In living nature (as in inanimate nature), due to various restrictions, a significantly smaller number of species of S. is usually found than is theoretically possible. For example, at the lower stages of the development of living nature, representatives of all classes of point structure are found - up to organisms characterized by the structure of regular polyhedra and the ball (see. rice. 3 ). However, at higher stages of evolution, plants and animals are found mainly so-called. axial (type n) and actinomorphic (type n(m)WITH. (in both cases n can take values ​​from 1 to ∞). Biological objects with axial S. (see. rice. 1 ) are characterized only by the C axis of order n. Bioobjects of sactinomorphic S. (see. rice. 2 ) are characterized by one axis of order n and planes intersecting along this axis m. The most common species in wildlife are S. spp. n = 1 and 1․ m = m, is called, respectively, asymmetry (See Asymmetry) and bilateral, or bilateral, S. Asymmetry is characteristic of the leaves of most plant species, bilateral S. - to a certain extent, for the external shape of the body of humans, vertebrates, and many invertebrates. In mobile organisms, such movement is apparently associated with differences in their movement up and down and forward and back, while their movements to right and left are the same. Violation of their bilateral S. would inevitably lead to inhibition of the movement of one of the sides and the transformation of translational movement into a circular one. In the 50-70s. 20th century The so-called dissymmetric biological objects ( rice. 4 ). The latter can exist in at least two modifications - in the form of the original and its mirror image (antipode). Moreover, one of these forms (no matter which) is called right or D (from Latin dextro), the other is called left or L (from Latin laevo). When studying the form and structure of D- and L-bioobjects, the theory of dissymmetrizing factors was developed, proving the possibility for any D- or L-object of two or more (up to an infinite number) modifications (see also rice. 5 ); at the same time it contained formulas for determining the number and type of the latter. This theory led to the discovery of the so-called. biological isomerism (See Isomerism) (different biological objects of the same composition; on rice. 5 16 isomers of linden leaf are shown).

When studying the occurrence of biological objects, it was found that in some cases D-forms predominate, in others L-forms, in others they are represented equally often. Bechamp and Pasteur (40s of the 19th century), and in the 30s. 20th century Soviet scientist G.F. Gause and others showed that the cells of organisms are built only or predominantly from L-amino acids, L-proteins, D-deoxyribonucleic acids, D-sugars, L-alkaloids, D- and L-terpenes, etc. d. So fundamental and characteristic feature living cells, called by Pasteur the dissymmetry of protoplasm, provides the cell, as was established in the 20th century, with a more active metabolism and is maintained through complex biological and physicochemical mechanisms that arose in the process of evolution. Sov. scientist V.V. Alpatov in 1952, using 204 species of vascular plants, found that 93.2% of plant species belong to the type with L-, 1.5% - with D-course of helical thickenings of the walls of blood vessels, 5.3% of species - to racemic type (the number of D-vessels is approximately equal to the number of L-vessels).

When studying D- and L-bioobjects, it was found that equality between D- and L-shapes in some cases, it is impaired due to differences in their physiological, biochemical and other properties. Similar feature living nature was called the dissymmetry of life. Thus, the exciting effect of L-amino acids on the movement of plasma in plant cells tens and hundreds of times superior to the same effect of their D-forms. Many antibiotics (penicillin, gramicidin, etc.) containing D-amino acids are more bactericidal than their forms with L-amino acids. The more common screw-shaped L-kop sugar beets are 8-44% (depending on the variety) heavier and contain 0.5-1% more sugar than D-kop sugar beets.

Goals:

• educational:
  • give an idea of ​​symmetry;
  • introduce the main types of symmetry on the plane and in space;
  • develop strong skills in constructing symmetrical figures;
  • expand your understanding of famous figures by introducing properties associated with symmetry;
  • show the possibilities of using symmetry in solving various problems;
  • consolidate acquired knowledge;
• general education:
  • teach yourself how to prepare yourself for work;
  • teach how to control yourself and your desk neighbor;
  • teach to evaluate yourself and your desk neighbor;
• developing:
  • intensify independent activity;
  • develop cognitive activity;
  • learn to summarize and systematize the information received;
• educational:
  • develop a “shoulder sense” in students;
  • cultivate communication skills;
  • instill a culture of communication.

PROGRESS OF THE LESSON

In front of each person are scissors and a sheet of paper.

Task 1(3 min).

- Let's take a sheet of paper, fold it into pieces and cut out some figure. Now let's unfold the sheet and look at the fold line.

Question: What function does this line serve?

Suggested answer: This line divides the figure in half.

Question: How are all the points of the figure located on the two resulting halves?

Suggested answer: All points of the halves are at an equal distance from the fold line and at the same level.

– This means that the fold line divides the figure in half so that 1 half is a copy of 2 halves, i.e. this line is not simple, it has a remarkable property (all points relative to it are at the same distance), this line is an axis of symmetry.

Task 2 (2 min).

– Cut out a snowflake, find the axis of symmetry, characterize it.

Task 3 (5 min).

– Draw a circle in your notebook.

Question: Determine how the axis of symmetry goes?

Suggested answer: Differently.

Question: So how many axes of symmetry does a circle have?

Suggested answer: Many.

– That’s right, a circle has many axes of symmetry. An equally remarkable figure is a ball (spatial figure)

Question: What other figures have more than one axis of symmetry?

Suggested answer: Square, rectangle, isosceles and equilateral triangles.

– Consider three-dimensional figures: cube, pyramid, cone, cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry do the square, rectangle, equilateral triangle and the proposed three-dimensional figures have?

I distribute halves of plasticine figures to students.

Task 4 (3 min).

– Using the information received, complete the missing part of the figure.

Note: the figure can be both planar and three-dimensional. It is important that students determine how the axis of symmetry runs and complete the missing element. The correctness of the work is determined by the neighbor at the desk and evaluates how correctly the work was done.

A line (closed, open, with self-intersection, without self-intersection) is laid out from a lace of the same color on the desktop.

Task 5 (group work 5 min).

– Visually determine the axis of symmetry and, relative to it, complete the second part from a lace of a different color.

The correctness of the work performed is determined by the students themselves.

Elements of drawings are presented to students

Task 6 (2 min).

– Find the symmetrical parts of these drawings.

To consolidate the material covered, I suggest the following tasks, scheduled for 15 minutes:

Name all equal elements of the triangle KOR and KOM. What type of triangles are these?

2. Draw several isosceles triangles in your notebook with a common base of 6 cm.

3. Draw a segment AB. Construct a line segment AB perpendicular and passing through its midpoint. Mark points C and D on it so that the quadrilateral ACBD is symmetrical with respect to the straight line AB.

– Our initial ideas about form date back to the very distant era of the ancient Stone Age - the Paleolithic. For hundreds of thousands of years of this period, people lived in caves, in conditions little different from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and during the late Paleolithic era they embellished their existence by creating works of art, figurines and drawings that reveal a remarkable sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity entered a new Stone Age, in the Neolithic.
Neolithic man had a keen sense of geometric form. Firing and painting clay vessels, making reed mats, baskets, fabrics, and later metal processing developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
– Where does symmetry occur in nature?

Suggested answer: wings of butterflies, beetles, tree leaves...

– Symmetry can also be observed in architecture. When constructing buildings, builders strictly adhere to symmetry.

That's why the buildings turn out so beautiful. Also an example of symmetry is humans and animals.

Homework:

1. Come up with your own ornament, draw it on an A4 sheet (you can draw it in the form of a carpet).
2. Draw butterflies, mark where elements of symmetry are present.