Symmetry is relatively straight. Axes of 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 - tetrahedron 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 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 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.

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 general theory S. (Soviet scientists Yu. V. Wulf, V. N. Beklemishev, B. K. Weinstein, Dutch physical chemist F. M. Eger, 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 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 one) 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. e. 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, 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 beet is 8-44% (depending on the variety) heavier and contains 0.5-1% more sugar than D-kop.

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.

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 whole series properties. In addition, we subsequently learn that the axial and central symmetry 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 the 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 straight line itself (since any point on the straight line is symmetrical to itself relative to this straight 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. 11

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).