In this lesson we will look at another characteristic of some figures - axial and central symmetry. We encounter axial symmetry every day when we look in the mirror. Central symmetry is very common in living nature. At the same time, figures that have symmetry have a number of properties. In addition, we will later learn that axial and central symmetries are types of movements with the help of which a whole class of problems is solved.

This lesson is devoted to axial and central symmetry.

Definition

The two points are called symmetrical relatively straight if:

In Fig. 1 shows examples of points symmetrical with respect to a straight line and , and .

Rice. 1

Let us also note the fact that any point on a line is symmetrical to itself relative to this line.

Figures can also be symmetrical relative to a straight line.

Let us formulate a strict definition.

Definition

The figure is called symmetrical relative to straight, if for each point of the figure, a point symmetrical to it relative to this straight line also belongs to the figure. In this case the line is called axis of symmetry. The figure has axial symmetry.

Let's look at a few examples of figures that have axial symmetry and their axes of symmetry.

Example 1

The angle has axial symmetry. The axis of symmetry of the angle is the bisector. Indeed: let’s lower a perpendicular to the bisector from any point of the angle and extend it until it intersects with the other side of the angle (see Fig. 2).

Rice. 2

(since - the common side, (property of a bisector), and triangles are right-angled). Means, . Therefore, the points are symmetrical with respect to the bisector of the angle.

It follows from this that an isosceles triangle also has axial symmetry with respect to the bisector (altitude, median) drawn to the base.

Example 2

An equilateral triangle has three axes of symmetry (bisectors/medians/altitudes of each of the three angles (see Fig. 3).

Rice. 3

Example 3

A rectangle has two axes of symmetry, each of which passes through the midpoints of its two opposite sides (see Fig. 4).

Rice. 4

Example 4

A rhombus also has two axes of symmetry: straight lines, which contain its diagonals (see Fig. 5).

Rice. 5

Example 5

A square, which is both a rhombus and a rectangle, has 4 axes of symmetry (see Fig. 6).

Rice. 6

Example 6

For a circle, the axis of symmetry is any straight line passing through its center (that is, containing the diameter of the circle). Therefore, a circle has infinitely many axes of symmetry (see Fig. 7).

Rice. 7

Let us now consider the concept central symmetry.

Definition

The points are called symmetrical relative to the point if: - the middle of the segment.

Let's look at a few examples: in Fig. 8 shows the points and , as well as and , which are symmetrical with respect to the point , and the points and are not symmetrical with respect to this point.

Rice. 8

Some figures are symmetrical about a certain point. Let us formulate a strict definition.

Definition

The figure is called symmetrical about the point, if for any point of the figure the point symmetrical to it also belongs to this figure. The point is called center of symmetry, and the figure has central symmetry.

Let's look at examples of figures with central symmetry.

Example 7

For a circle, the center of symmetry is the center of the circle (this is easy to prove by recalling the properties of the diameter and radius of a circle) (see Fig. 9).

Rice. 9

Example 8

For a parallelogram, the center of symmetry is the point of intersection of the diagonals (see Fig. 10).

Rice. 10

Let's solve several problems on axial and central symmetry.

Task 1.

How many axes of symmetry does the segment have?

A segment has two axes of symmetry. The first of them is a line containing a segment (since any point on a line is symmetrical to itself relative to this line). The second is the perpendicular bisector to the segment, that is, a straight line perpendicular to the segment and passing through its middle.

Answer: 2 axes of symmetry.

Task 2.

How many axes of symmetry does a straight line have?

A straight line has infinitely many axes of symmetry. One of them is the line itself (since any point on the line is symmetrical to itself relative to this line). And also the axes of symmetry are any lines perpendicular to a given line.

Answer: there are infinitely many axes of symmetry.

Task 3.

How many axes of symmetry does the beam have?

The ray has one axis of symmetry, which coincides with the line containing the ray (since any point on the line is symmetrical to itself relative to this line).

Answer: one axis of symmetry.

Task 4.

Prove that the lines containing the diagonals of a rhombus are its axes of symmetry.

Proof:

Consider a rhombus. Let us prove, for example, that the straight line is its axis of symmetry. It is obvious that the points are symmetrical to themselves, since they lie on this line. In addition, the points and are symmetrical with respect to this line, since . Let us now choose an arbitrary point and prove that the point symmetric with respect to it also belongs to the rhombus (see Fig. 11).

Rice. eleven

Draw a perpendicular to the line through the point and extend it until it intersects with . Consider triangles and . These triangles are right-angled (by construction), in addition, they have: - a common leg, and (since the diagonals of a rhombus are its bisectors). So these triangles are equal: . This means that all their corresponding elements are equal, therefore: . From the equality of these segments it follows that the points and are symmetrical with respect to the straight line. This means that it is the axis of symmetry of the rhombus. This fact can be proven similarly for the second diagonal.

Proven.

Task 5.

Prove that the point of intersection of the diagonals of a parallelogram is its center of symmetry.

Proof:

Consider a parallelogram. Let us prove that the point is its center of symmetry. It is obvious that the points and , and are pairwise symmetrical with respect to the point , since the diagonals of a parallelogram are divided in half by the point of intersection. Let us now choose an arbitrary point and prove that the point symmetric with respect to it also belongs to the parallelogram (see Fig. 12).

You will need

• - properties of symmetrical points;
• - properties of symmetrical figures;
• - ruler;
• - square;
• - compass;
• - pencil;
• - paper;
• - a computer with a graphics editor.

Instructions

Draw a straight line a, which will be the axis of symmetry. If its coordinates are not specified, draw it arbitrarily. Place an arbitrary point A on one side of this line. You need to find a symmetrical point.

Helpful advice

Symmetry properties are used constantly in AutoCAD. To do this, use the Mirror option. To construct an isosceles triangle or isosceles trapezoid, it is enough to draw the lower base and the angle between it and the side. Reflect them using the specified command and extend the sides to the required size. In the case of a triangle, this will be the point of their intersection, and for a trapezoid, this will be a given value.

You constantly come across symmetry in graphic editors when you use the “flip vertically/horizontally” option. In this case, the axis of symmetry is taken to be a straight line corresponding to one of the vertical or horizontal sides of the picture frame.

Sources:

• how to draw central symmetry

Constructing a cross section of a cone is not such a difficult task. The main thing is to follow a strict sequence of actions. Then this task will be easily accomplished and will not require much labor from you.

You will need

• - paper;
• - pen;
• - circle;
• - ruler.

Instructions

When answering this question, you must first decide what parameters define the section.
Let this be the straight line of intersection of the plane l with the plane and the point O, which is the intersection with its section.

The construction is illustrated in Fig. 1. The first step in constructing a section is through the center of the section of its diameter, extended to l perpendicular to this line. The result is point L. Next, draw a straight line LW through point O, and construct two guide cones lying in the main section O2M and O2C. At the intersection of these guides lie point Q, as well as the already shown point W. These are the first two points of the desired section.

Now draw a perpendicular MS at the base of the cone BB1 ​​and construct generatrices of the perpendicular section O2B and O2B1. In this section, through point O, draw a straight line RG parallel to BB1. Т.R and Т.G are two more points of the desired section. If the cross section of the ball were known, then it could be built already at this stage. However, this is not an ellipse at all, but something elliptical that has symmetry with respect to the segment QW. Therefore, you should build as many section points as possible in order to connect them later with a smooth curve to obtain the most reliable sketch.

Construct an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and construct the corresponding guides O2A and O2N. Through t.O, draw a straight line passing through PQ and WG until it intersects with the newly constructed guides at points P and E. These are two more points of the desired section. Continuing in the same way, you can find as many points as you want.

True, the procedure for obtaining them can be slightly simplified using symmetry with respect to QW. To do this, you can draw straight lines SS’ in the plane of the desired section, parallel to RG until they intersect with the surface of the cone. The construction is completed by rounding the constructed polyline from chords. It is enough to construct half of the desired section due to the already mentioned symmetry with respect to QW.

Video on the topic

Tip 3: How to graph a trigonometric function

You need to draw schedule trigonometric functions? Master the algorithm of actions using the example of constructing a sinusoid. To solve the problem, use the research method.

You will need

• - ruler;
• - pencil;
• - knowledge of the basics of trigonometry.

Instructions

Video on the topic

note

If the two semi-axes of a single-strip hyperboloid are equal, then the figure can be obtained by rotating a hyperbola with semi-axes, one of which is the above, and the other, different from the two equal ones, around the imaginary axis.

Helpful advice

When examining this figure relative to the Oxz and Oyz axes, it is clear that its main sections are hyperbolas. And when this spatial figure of rotation is cut by the Oxy plane, its section is an ellipse. The neck ellipse of a single-strip hyperboloid passes through the origin of coordinates, because z=0.

The throat ellipse is described by the equation x²/a² +y²/b²=1, and the other ellipses are composed by the equation x²/a² +y²/b²=1+h²/c².

Sources:

• Ellipsoids, paraboloids, hyperboloids. Rectilinear generators

The shape of a five-pointed star has been widely used by man since ancient times. We consider its shape beautiful because we unconsciously recognize in it the relationships of the golden section, i.e. the beauty of the five-pointed star is justified mathematically. Euclid was the first to describe the construction of a five-pointed star in his Elements. Let's join in with his experience.

You will need

• ruler;
• pencil;
• compass;
• protractor.

Instructions

The construction of a star comes down to the construction and subsequent connection of its vertices to each other sequentially through one. In order to build the correct one, you need to divide the circle into five.
Construct an arbitrary circle using a compass. Mark its center with point O.

Mark point A and use a ruler to draw line segment OA. Now you need to divide the segment OA in half; to do this, from point A, draw an arc of radius OA until it intersects the circle at two points M and N. Construct the segment MN. The point E where MN intersects OA will bisect segment OA.

Restore the perpendicular OD to the radius OA and connect points D and E. Make a notch B on OA from point E with radius ED.

Now, using line segment DB, mark the circle into five equal parts. Label the vertices of the regular pentagon sequentially with numbers from 1 to 5. Connect the dots in the following sequence: 1 with 3, 2 with 4, 3 with 5, 4 with 1, 5 with 2. Here is the regular five-pointed star, into a regular pentagon. This is exactly the way I built it

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 of a precious stone or a salt crystal pleases the eye; it is difficult 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 of the 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.

The axial symmetry of points relative to a straight line provides that this straight line intersects the middle of the segment connecting the points and is perpendicular to it. Examples are the bisector of an undeveloped angle of an isosceles triangle, any line drawn through the center of a 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.

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.

DURING THE CLASSES

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

Exercise 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 minutes).

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

Task 3 (5 minutes).

– 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: A lot of.

– 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 minutes).

– 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, 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, note where elements of symmetry are present.

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 the structure of geometric figures.

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 has 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) with respect 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 the symmetrical points Ф ( 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 having several transfer axes play an important role in the study of crystal lattices (See Crystal lattice).

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 that have, respectively, one or more S. transfers in combination with reflections) ( rice. 6 , 7 ).

Combinations of symmetry generated by reflections and rotations (exhausting all types of symmetry of 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 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). The symmetry of 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 structure of crystals and lattices, the concept of structure in the general sense acquires important significance (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 one to deeply penetrate into the internal structure of an 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 Tg in linear space R solutions to this equation. Correspondence 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 need for mathematics and physics to develop 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 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 space-time 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. 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, it acts as an interaction constant characterizing the connection of the 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 system. From the system of physical laws regarding the shift of a 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, the possible states of the system are the superpositions ψ 1 and ψ 2: symmetric combination ψ s = ψ 1 + ψ 2 and antisymmetric 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 quantum mechanics leads to the fact that the different states of motion of a quantum mechanical system, which are obtained from each other by transformation of quantum mechanics, have the same values ​​of 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 may correspond to several different states that are transformed through each other during transformations of the system. In mathematical terms, 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 latter circumstance plays a crucial role, for example, in elementary particle physics. The existence of groups of particles with similar masses and identical other characteristics, but different electric charges (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 the strong interaction has an even broader 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 it manifests itself in the geometric configuration of molecules, which affects the specific physical and chemical properties of molecules in an isolated state, in an external field and when interacting 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 symmetry of a regular triangular pyramid, the methane molecule CH 4 has the symmetry of a tetrahedron. 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). The most complete description of the 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, etc. are prohibited. Symmetry selection rules are supplemented for transitions between different states by 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, during 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 the relative positions of occupied and vacant orbitals of a 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 conservation of orbital symmetry in chemical reactions, which was subsequently confirmed by extensive experimental material and had a great influence on the development of preparative organic chemistry. This principle (the Woodward-Hoffman rule) states that individual elementary acts of chemical reactions take place while maintaining the symmetry of molecular orbitals, or orbital symmetry. The more the symmetry of orbitals is violated during an elementary act, 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 pharmacologically 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 harmony in living nature was noticed by the Pythagoreans in ancient Greece (5th century BC) in connection with their development of the doctrine of harmony. In the 19th century A few works appeared on 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. Weinstein, the Dutch physical chemist F. M. Yeger, English crystallographers led by J. Bernal) and the doctrine of rightism 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 the classical theory of 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 limitations, a significantly smaller number of S. species are 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. Such a fundamental and characteristic feature of 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, established 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 the equality between the D- and L-forms is violated in a number of cases due to differences in their physiological, biochemical and other properties. This feature of living nature was called the dissymmetry of life. Thus, the exciting effect of L-amino acids on the movement of plasma in plant cells is tens and hundreds of times greater than 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 beet is 8-44% (depending on the variety) heavier and contains 0.5-1% more sugar than D-kop.