I ... Symmetry in mathematics :

Basic concepts and definitions.

Axial symmetry (definitions, construction plan, examples)

Central symmetry (definitions, construction plan, formeasures)

Summary table (all properties, features)

II ... Symmetry Applications:

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 n R goes 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 a person. And it was used by sculptors as early as the 5th century BC. NS. The word "symmetry" is Greek, it means "proportionality, proportionality, uniformity in the arrangement of parts." It is widely used by all areas of modern science without exception. Many great people thought about this pattern. For example, LN 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 indeed pleasing to the eye. Who has not admired the symmetry of nature's creations: leaves, flowers, birds, animals; or human creations: buildings, technology, - everything that surrounds us from childhood, those that strive for beauty and harmony. Hermann Weil said: "Symmetry is the idea through which man, for centuries, has tried to comprehend and create order, beauty and perfection." Hermann Weil is a German mathematician. His activity falls on the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what criteria to perceive the presence or, conversely, the absence of symmetry in one or another case. Thus, a mathematically rigorous concept was formed relatively recently - at the beginning of the twentieth century. It's quite complicated. We will 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 symmetric with respect to the straight line a if this straight line passes through the middle of the segment AA 1 and is perpendicular to it. Each point of the straight line a is considered symmetrical to itself.

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

2.2 Building plan

And so, to build a symmetrical figure relative to a straight line from each point, we draw a perpendicular to this straight line and extend it by the same distance, mark the resulting point. We do this with each point, we get the symmetrical vertices of the new shape. Then we connect them in series and get a symmetrical figure of this 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 symmetric 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 called symmetric about point O if for each point of the figure the point symmetric to it about point O also belongs to this figure.

3.2 Build plan

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

To draw a point symmetrical to a point A relative to point O, it is enough to draw a straight line OA(fig. 46 ) and on the other side of the point O postpone a segment equal to the segment OA. In other words , points A and ; In and ; With and are symmetric with respect to some point O. In Fig. 46 built a triangle symmetrical to the triangle ABC relative to point O. These triangles are equal.

Draws symmetrical points about the center.

In the figure, points M and M 1, N and N 1 are symmetric about point O, and points P and Q are not symmetric about this point.

In general, figures symmetrical about some point are equal .

3.3 Examples

Here are some examples of figures with central symmetry. The simplest figures with central symmetry are the circle and the 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.

The straight line also has central symmetry, however, unlike the circle and the parallelogram, which have only one center of symmetry (point O in the figure), the straight line has infinitely many of them - any point of the straight line is its center of symmetry.

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

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

4. Lesson summary

Let's summarize the knowledge gained. Today in the lesson we got acquainted with two main types of symmetry: central and axial. Let's look at the screen and systematize the knowledge gained.

Summarizing table

	Axial symmetry	Central symmetry
Peculiarity	All points of the figure must be symmetrical about some straight line.	All points of the shape must be symmetrical about the point selected as the center of symmetry.
Properties	1. Symmetrical points lie on perpendiculars to a straight line. 3. Straight lines turn into straight lines, angles into equal angles. 4. Sizes and shapes of figures are saved.	1. Symmetric points lie on a straight line passing through the center and this point figures. 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. Applying symmetry

Maths	In algebra lessons, we studied the graphs of the functions y = x and y = x The figures 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 Russian - palindromes that can be read the same way in two directions.	A D L M P T V W- vertical axis V E Z K S E Y - horizontal axis J N O X- both vertical and horizontal B G I Y R U Y Z- no axis Radar hut Alla Anna
Literature	Can be palindromic and sentences. Bryusov wrote a poem "The Voice of the Moon", in which each line is a palindrome. Look at the quatrains of A.S. Pushkin "The 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 go with the sword of the judge. (Derzhavin) "Search for a taxi" «Аргентина манит негра», "The Argentinean appreciates the negro", "Lesha found a bug on the shelf." The Neva was dressed in granite; Bridges hung over the waters; Dark green gardens The islands were covered with it ...

Biology

The human body is built according to the principle of bilateral symmetry. Most of us view the brain as a single structure; in fact, it is divided into two halves. These two parts - the two hemispheres - fit snugly together. In full accordance with the general symmetry of the human body, each hemisphere is an almost exact mirror image of the other. The 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 side controls the left side.

Botany

A flower is considered symmetrical when each perianth is composed of an equal number of parts. Flowers, having paired parts, are considered to be flowers with double symmetry, etc. Triple symmetry is common for monocotyledonous plants, quintuple symmetry for dicots Characteristic feature the structure of plants and their development is helicity. Pay attention to the shoots of the leaf arrangement - this is also a kind of spiral - helical. Even Goethe, who was not only a great poet, but also a natural scientist, considered helicity one of the characteristic features of all organisms, a manifestation of the most intimate essence of life. The antennae of plants are spirally twisted, tissue grows in the trunks of trees in a spiral, the seeds in the sunflower are arranged in a spiral, spiral movements are observed during the growth of roots and shoots.

A characteristic feature of the structure of plants and their development is helicity. Look at the pinecone. The scales on its surface are arranged in a strictly regular way - along two spirals, which intersect at approximately right angles. 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 shape, as well as the relative position of body parts located on opposite sides of the dividing line. With radial or radiant symmetry, the body has the form of a short or long cylinder or a vessel with a central axis, from which parts of the body radiate out in a radial order. These are coelenterates, echinoderms, starfish. With bilateral symmetry, there are three axes of symmetry, but there is only one pair of symmetrical sides. Because the other two sides - the ventral and dorsal - are not alike. This type of symmetry is typical for most animals, including insects, fish, amphibians, reptiles, birds, and mammals.

Axial symmetry

	Different kinds symmetry physical phenomena: symmetry of electric and magnetic fields (fig. 1) In mutually perpendicular planes, the distribution is symmetric electromagnetic waves(fig. 2)	fig. 1 fig. 2
Art	Mirror symmetry can often be observed in works of art. Mirror "symmetry is widespread in the works of art of primitive civilizations and in ancient painting. Medieval religious paintings are also characterized by this kind of symmetry. One of Raphael's best early works, The Betrothal of Mary, was created in 1504. A valley crowned with a white-stone temple stretches under the sunny blue sky. Foreground: the betrothal ceremony. The high priest brings the hands of Mary and Joseph closer. Behind Mary - a group of girls, behind Joseph - young men. Both parts of the symmetrical composition are held together by the oncoming movement of the characters. For modern taste, the composition of such a picture is boring, since the symmetry is too obvious.

Chemistry	The water molecule has a plane of symmetry (straight vertical line). DNA molecules (deoxyribonucleic acid) play an extremely important role in the world of wildlife. It is a double-stranded high molecular weight polymer, the monomer of which is nucleotides. DNA molecules have a double helix structure built on the principle of complementarity.
Architeculture	Since ancient times, man has used symmetry in architecture. The ancient architects used the symmetry in architectural structures especially brilliantly. Moreover, the ancient Greek architects were convinced that in their works they were guided by the laws that govern nature. 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 - park - a complex of landscape gardening sculptures, which was created over 40 years.	Pashkov House Louvre (Paris)

© Elena Vladimirovna Sukhacheva, 2008-2009.

You will need

• - properties of symmetric points;
• - properties of symmetrical figures;
• - ruler;
• - square;
• - compasses;
• - pencil;
• - paper;
• - a computer with a graphic editor.

Instructions

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

Helpful advice

Symmetry properties are constantly used in AutoCAD. For this, the Mirror option is used. To build an isosceles triangle or isosceles trapezoid, it is enough to draw the lower base and the angle between it and the side. Flip them with the command indicated and extend the sides as needed. In the case of a triangle, this will be the point of their intersection, and for a trapezoid, a given value.

You constantly encounter symmetry in graphic editors when you use the "flip vertically / horizontally" option. In this case, the line corresponding to one of the vertical or horizontal sides of the picture frame is taken as the axis of symmetry.

Sources:

• how to draw central symmetry

Cone sectioning is not so 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;
• - a pen;
• - circus;
• - ruler.

Instructions

When answering this question, you first need to decide what parameters the section is given.
Let it be the line of intersection of the plane l with the plane and the point O, which is the point of 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. As a result, point L is obtained. Then, through point O draw a straight line LW, and construct two guide cones lying in the main section O2M and O2C. At the intersection of these guides lie the point Q, as well as the already shown point W. These are the first two points of the desired section.

Now draw at the base of the cone BB1 ​​perpendicular to the MC and construct the generators of the perpendicular section О2В and О2В1. In this section, through T.O, draw a straight line RG parallel to BB1. T.R and T.G - two more points of the desired section. If the cross-section of the ball is known, then it could be built already at this stage. However, this is not an ellipse at all, but something elliptical, having symmetry about the segment QW. Therefore, you should build as many points of the section as possible in order to connect them in the future with a smooth curve to obtain the most reliable sketch.

Draw an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and draw the corresponding guides O2A and O2N. Through so, draw a straight line passing through PQ and WG, until it intersects with the just drawn guides at points P and E. These are two more points of the desired section. Continuing the same way and further, you can arbitrarily desired points.

True, the procedure for obtaining them can be slightly simplified using the 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 suffices to construct half of the sought-for section due to the already mentioned symmetry with respect to QW.

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Tip 3: How to build a graph trigonometric function

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

You will need

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

Instructions

Related Videos

note

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

Helpful advice

When considering this figure relative to the Oxz and Oyz axes, it can be seen that its main sections are hyperbolas. And when a given spatial figure of rotation is cut by the Oxy plane, its section is an ellipse. The throat ellipse of a single-strip hyperboloid passes through the origin, since z = 0.

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

Sources:

• Ellipsoids, paraboloids, hyperboloids. Straight generators

The form five pointed star has been widely used by humans since ancient times. We consider its form to be beautiful, since we unconsciously distinguish the ratio of the golden section in it, i.e. the beauty of the five-pointed star is mathematically based. Euclid was the first to describe the construction of the five-pointed star in his "Elements". Let's share his experience.

You will need

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

Instructions

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

Mark point A and use the ruler to draw line segment OA. Now you need to divide segment OA in half, for this, draw an arc from point A with radius OA until it intersects with the circle at two points M and N. Construct segment MN. Point E, at which MN intersects OA, will bisect OA.

Restore OD perpendicular to radius OA and connect point D and E. Resume B on

TRIANGLES.

§ 17. SYMMETRY REGARDING THE LINE.

1. Shapes symmetrical to each other.

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

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

Two figures are called symmetrical with respect to some straight line if they are aligned when bending the drawing plane along this straight line.

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.

It is possible to obtain symmetrical figures without using the bending of the plane, but with the help of a geometric construction. Suppose it is required to construct a point C "symmetric to a given point C with respect to line AB. Let us drop from point C the perpendicular
CD on line AB and on its continuation set aside the segment DC "= DC. If we bend the plane of the drawing along AB, then point C will be combined with point C": points C and C "are symmetric (Fig. 129).

Let it now be required to construct a segment C "D" symmetric to a given segment CD relative to the straight line AB. Let's construct points C "and D", symmetric to points C and D. If we bend the plane of the drawing along AB, then points C and D will be aligned with points C "and D", respectively (Fig. 130). Therefore, the segments CD and C "D" will match , they will be symmetrical.

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

To solve this problem, we drop the perpendiculars А a, V b, WITH with, D d and E e on the axis of symmetry МN. Then, on the extensions of these perpendiculars, we postpone the segments
a A "= A a, b B "= B b, with C "= Cc; d D "" = D d and e E "= E e.

Polygon A "B" C "D" E "will be symmetric to the polygon ABCDE. Indeed, if you bend the drawing along the straight line MN, then the corresponding vertices of both polygons will coincide, which means that the polygons themselves will combine; 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 shapes that are divided by some straight line into two symmetrical parts. Such figures are called symmetrical.

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

In a circle, the axis of symmetry is its diameter, since when bending along it, one semicircle is aligned with the other (Fig. 133). In the same way, the figures in drawings 134, a, b are symmetrical.

Symmetrical shapes are often found in nature, construction, and jewelry. The images shown in drawings 135 and 136 are symmetrical.

It should be noted that symmetrical figures can be combined by simple movement on a plane only in some cases. To combine symmetrical shapes, as a rule, you need to turn one of them with the back side,

Goals:

• educational:
  • give an idea of ​​symmetry;
  • to acquaint with the basic types of symmetry on the plane and in space;
  • develop strong skills in building symmetrical figures;
  • expand the understanding of known figures, introducing the properties associated with symmetry;
  • show the possibilities of using symmetry when solving different tasks;
  • consolidate the knowledge gained;
• general educational:
  • teach yourself to set yourself up for work;
  • teach to control yourself and your neighbor on your desk;
  • teach how to evaluate yourself and your deskmate;
• developing:
  • to intensify independent activity;
  • develop cognitive activity;
  • teach to generalize and systematize the information received;
• educational:
  • to instill in students a "sense of the shoulder";
  • educate communication;
  • instill a culture of communication.

DURING THE CLASSES

In front of each 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 figurine. Now expand the sheet and look at the fold line.

Question: What is the function of this line?

Supposed answer: This line divides the shape in half.

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

Supposed answer: All points of the halves are at the same 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 are at the same distance relative to it), this line is the axis of symmetry.

Assignment 2 (2 minutes).

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

Assignment 3 (5 minutes).

- Draw a circle in a notebook.

Question: Determine how the axis of symmetry runs?

Supposed answer: Differently.

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

Supposed answer: Many.

- That's right, a circle has many axes of symmetry. The same remarkable figure is the ball (spatial figure)

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

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

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

I distribute to students the halves of plasticine figures.

Assignment 4 (3 min).

- Using the information received, fill in the missing part of the figure.

Note: the figure can be both planar and volumetric. It is important that the students determine how the axis of symmetry goes and complete the missing piece. The correctness of the execution is determined by the neighbor on the desk, assesses how correctly the work has been done.

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

Assignment 5 (group work 5 min).

- Determine visually the axis of symmetry and build the second part from a lace of a different color relative to it.

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

The elements of the drawings are presented to the students

Assignment 6 (2 minutes).

Find the symmetrical parts of these patterns.

To consolidate the material covered, I propose the following tasks, provided for 15 minutes:

Name all equal elements of the triangle KOR and KOM. What is the appearance of these triangles?

2. In a notebook, draw several isosceles triangles with common ground equal to 6 cm.

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

- Our initial ideas about the form date back to a very distant era of the ancient Stone Age - the Paleolithic. For hundreds of millennia of this period, people lived in caves, in conditions that did not differ much from the life of animals. Humans made tools for hunting and fishing, developed languages ​​to communicate with each other, and in the late Paleolithic era adorned their existence, creating works of art, figurines and drawings that reveal a wonderful sense of form.
When there was a transition from simple gathering of food to active production, from hunting and fishing to agriculture, humanity enters a new stone Age, in the Neolithic.
Neolithic man had a keen sense of geometric shape. The burning and painting of earthen vessels, the making of reed mats, baskets, fabrics, and later - the processing of metals developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
- Where does symmetry occur in nature?

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

“Symmetry can be seen in architecture as well. When constructing buildings, builders adhere to symmetry.

That is why the buildings are so beautiful. Also, an example of symmetry is a person, animals.

Home assignment:

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

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annotation

Lessons at school are a significant part of the life of schoolchildren, requiring elementary comfort and favorable communication. The effectiveness of the educational process depends not only on the ability of diligence and hard work of students, the presence of purposeful motivation of the teacher, but also on the form of lessons.

The use of information technology allows you to save time when explaining new material, to present the material in a visual, comprehensible form, to influence different systems of students' perception, thereby ensuring a better assimilation of the material.

Much attention is paid to the application of the acquired knowledge of mathematics in everyday life. Acquaintance with beauty in life and art not only fosters the mind and feeling of a child, but also contributes to the development of imagination and fantasy. I believe that a lesson with elements creative activity helps to activate the mental activity of schoolchildren and therefore takes place at a high emotional level, which allows you to consider a large number of theoretical questions and tasks, to involve all students in the class. In order to increase the activity of students throughout the lesson, an alternation of activities is used.

At the final stage of the lesson, students perform verification work in the form of a test, they conduct a self-test, evaluating their work according to the specified criteria. The most active group of students was offered additional material on the topics studied.

Reflection at the end of the lesson helps to determine the level of assimilation of the material and set goals for further work.

Homework consists of two parts, which allows not only to continue consolidating the knowledge gained, but to develop the creative abilities of children.

In my opinion, such lessons give the teacher the opportunity to create, seek, work for high results, form universal educational actions in students - thus, prepare them for continuing education and for life in constantly changing conditions.

Lesson objectives:

• acquaintance with the concept of axial symmetry;
• the formation of skills to build figures symmetrical about a straight line and to reveal axial symmetry as a property of some geometric figures;
• disclosing the connections of mathematics with wildlife, art, technology, architecture;
• development of skills to apply knowledge of theory in practice, development of skills of self-control and mutual control, self-esteem and introspection learning activities;
• development of attention, observation, thinking, interest in the subject, mathematical speech, desire for creativity;
• the formation of aesthetic perception of the world around, the education of independence.
• preparing students for the study of geometry, deepening existing knowledge;

Lesson type: a lesson of "discovery" of new knowledge.

Equipment: computer, pin or compasses, projector, cards, geometric shapes from paper.

DURING THE CLASSES

1. Organizational moment

(Slide 1) It is easy to find examples of beauty, but how difficult it is to explain why they are beautiful. (Plato)

- Today in the lesson we will try to understand some of the features of creating beauty !!!

2. Updating

- Look at the maple leaf, snowflake, butterfly. (Slide 2) What unites them, what do they have in common? That they are symmetrical.
- Remind me, please, what the word "symmetry" means.
- "Symmetry" in Greek means "proportionality, proportionality, uniformity in the arrangement of parts." If you put a mirror along the straight line drawn in each drawing, then the half of the figure reflected on the mirror will complement it to a whole. Therefore, this symmetry is called mirror (axial).

(The teacher shows the experience on a Christmas tree cut out of colored paper)

- The straight line along which the mirror is placed is called axis of symmetry... If you bend the sheet along this straight line, then these figures fully coincide, and we can see only one figure. What do you think is the topic of today's lesson? (Axial symmetry)

(Slides 3-4)

- Guys, today we will learn how to build figures symmetrical about a straight line, and you will also learn where axial symmetry is applied.
- How do you get symmetrical shapes?
- First, let's look at the easiest way to get symmetrical shapes.
Each of you has a sheet of white paper on your table. Take a piece of paper and fold it in half. Now on one side build a triangle(1 row - acute-angled, 2 row - rectangular, 3 row - obtuse).
Further pierce the vertices of this figure so that both halves are pierced. Now unfold the sheet and connect the resulting hole points along the ruler... Thus, we have built figures that are symmetrical to the data relative to a straight line (inflection line). Make sure of this... To do this, fold the sheet along the fold line. and look through it into the light.
- What do you see? (The figures matched.)
- This is the easiest way to build symmetrical shapes.
- But is it always in practice, in this way, we will be able to build symmetrical figures?
- What have we done to build symmetrical triangles?
- We bent the sheet in half.
- Ie drew an axis of symmetry... Farther.
- Pierced the vertices of the triangle.
- Ie plotted the points that bound our triangle.
- And this means that before we build a symmetrical figure for the given one, we must learn to build in the first place what? (A point symmetrical to this one.)
- How can this be done, let's figure it out.

3. Now let's execute practical work:

- Mark a point Aa. From point A lower the perpendicular JSC on a straight line a... Now from point O set aside a perpendicular OA1 = AO... Two points A and A1 are called symmetric about a straight line a... This line is called the axis of symmetry.

(The teacher builds on the blackboard, students in notebooks).

- What two points are called symmetrical about a straight line?
- And how to construct a figure symmetrical with respect to some straight line?
- Let's try to build a triangle symmetrical about a straight line.

(The teacher calls the student who wishes to the blackboard, the rest work in notebooks).

After the work done, the students make a conclusion together with the teacher.

Output: To build a geometric figure symmetric to a given one with respect to some straight line, you need plot points, symmetrical significant points (to the heights) of this figure relative to this straight line and then connect these points with segments.

- Guys, symmetrical may be not only 2 figures, in some figures you can also draw an axis of symmetry. They say that such figures possess axial symmetry. Name the axially symmetric shapes.

(The teacher names and shows geometric shapes cut from colored paper)

- How many axes of symmetry do you think isosceles triangle, rectangle, square? (A rectangle has 2 axes of symmetry. A square has 4 axes of symmetry) – And at the circle? (A circle has infinitely many axes of symmetry).

(Slides 7-11)

- Name the shapes that do not have an axis of symmetry. (Parallelogram, versatile triangle, irregular polygon).

- The principles of symmetry play an important role in physics and mathematics, chemistry and biology, engineering and architecture, painting and sculpture, poetry and music. Almost all vehicles, household items (furniture, dishes), some musical instruments are symmetrical.
- Give examples of objects with axial symmetry.

– Nature laws, managing the phenomenon, which is inexhaustible in its diversity, in turn, also obey the principles of symmetry. Close observation shows that symmetry is the basis of the beauty of many forms created by nature.

(Slides 12-15)

Symmetry is common in human-made objects.
Symmetry is already found at the origins of human development. Since ancient times, man has used symmetry in architecture. Ancient temples, towers of medieval castles, modern buildings it gives harmony, completeness.

(Slides 18-19)

Symmetry in the visual arts gives impressive results. (Slides 20-21)
Renaissance artists often used the language of symmetry in their compositions. This followed from their logic of understanding the picture as an image of an ideal world order, where rational organization and balance reigns, which a person can cognize and comprehend.
In amazing the painting "The Betrothal of the Virgin Mary" great Raphael reproduced such an image of the world, existing according to the laws of harmony and strict logic. The principle of symmetry used creates the impression of calmness and solemnity and at the same time a certain detachment from the viewer. The entrance to the graceful rotunda and the ring that Joseph puts on Mary's hand coincide with the central axis of symmetry of the painting.
In work Leonardo "The Last Supper" dominated by strict constructions of the perspective of the interior. The compositional development here is based on the mirror repetition of the right and left parts. Of course, more often than not in the visual arts we say incomplete symmetry.
In the picture "Three heroes" by Russian artist V. Vasnetsov the heroes themselves are full of pent-up power. Due to these small deviations from strict symmetry, there is a feeling inner freedom characters, their readiness to move.
The letters of the Russian language can also be considered from the point of view of symmetry. (Slides 22-23)
The whole alphabet is divided into 4 groups, what do you think, by what criteria did I do this?
The letters A, M, T, W, P have a vertical axis of symmetry, B, Z, K, S, E, B, E - a horizontal one. And the letters Ж, Н, О, Ф, Х have two axes of symmetry.
Symmetry can also be seen in the words: Cossack, hut. There are whole phrases with this property (if you do not take into account the spaces between words): “To look for a taxi”, “Argentina beckons the Negro”, “The Argentinean appreciates the Negro”. Such words are called palindromes ... Many poets were fond of them.
Consider examples of words that have a horizontal axis of symmetry:
SNOW, CALL, SKATE, NOSE
Words with a vertical axis of symmetry:

NS	T
O	O
L	NS
O	O
D	T

Some composers, including the great Bach, wrote musical palindromes.

(Slide 24) Those who are fortunate enough to have a symmetrical face have probably already noticed that they are enjoying success with the opposite sex. It may also indicate their good health... The fact is that a face with ideal proportions is a sign that the body of its owner is well prepared to fight infections. The common cold, asthma, and flu are highly likely to recede in people whose left side is exactly like the right side.

Physical education(Slide 25)

One - rise, stretch,
Two - bend, straighten.
Three - three claps in your hands,
The Tory head nods.
Four - arms wider,
Five - wave your hands,
Six - sit at the desk again.

(Slide 26-27)

A test is carried out followed by a self-check.

- Let's not forget about the gymnastics of the mind. Our examples today are also symmetrical. Who has already completed the task, you can count orally these symmetrical examples. (Slide 30)

Option 1 Option 2

1) B 2) D 3) B 4) A 5) C 1) C 2) B 3) B 4) D 5) D

Evaluation of the work performed according to the relevant criteria:

"5" - 5 tasks;
"4" - 4 tasks;
"3" - 3 tasks;
"2" - less than three tasks.

- Try to answer the question which figure is superfluous and why? (Slide 31)

(Figure No. 3, because it does not have an axis of symmetry)

- Well done!

5. Lesson summary. Reflection

- Our lesson is coming to an end, but the acquaintance with symmetry continues. Throughout the lesson, we completed a variety of tasks.
- What concept did you meet today?
- What goals did we set for the lesson? Have we met our goals? Who worked the best? Who distinguished themselves in the lesson? What was the most difficult task for you? What theoretical material helped to cope with the task?
- What was the most interesting task for you? What new "have you discovered" for yourself in the lesson? What do you think each of you should work on?

- Guys, thank you for your work! Without the help and support of each other, we would not have been able to achieve the goal. I am very pleased with your work in the lesson. Do you think that we did not spend these minutes together in vain? Share your impressions of our lesson.

(Slides 32-33)

7. Conclusion

Really symmetrical objects literally surround us on all sides, we are dealing with symmetry wherever there is some kind of ordering. Symmetry opposes chaos, disorder. It turns out that symmetry is poise, orderliness, beauty, perfection.
The whole world can be seen as a manifestation of the unity of symmetry and asymmetry. Symmetry is manifold, omnipresent. She creates beauty and harmony.
And to the question: "Is there a future without symmetry?" we can answer with the words of the classic of modern natural science, the thinker Vladimir Ivanovich Vernadsky "The principle of symmetry covers more and more new areas ..."