Presentation on the topic "movement in space central symmetry axial symmetry mirror symmetry parallel translation." Symmetry in space

Symmetry in space is a beautiful, harmonious and balanced proportional relationship of parts or elements of various forms of objects, organisms or objects. In the space around us we can observe a lot of inanimate objects of symmetrical shape. Living organisms, both simple and highly complex, also have elements of symmetry in their structure.

Striving for Excellence

A symmetrical shape can be identified with perfection and harmony. It is not for nothing that words such as “symmetry” and “perfection” are synonymous in the languages of many peoples.

Symmetry in space is found everywhere. The variety of forms of plants and living organisms amazes with their proportionality, consistency and ergonomic form. Everything here is thought out to the smallest detail: amazing beauty, elegance of proportions and nothing superfluous. Everything is provided for the best functionality of life.

Central symmetry

In the space of the world around us, inanimate nature is clearly visible in the structure of crystals. This type of symmetry is clearly visible in the structure of snowflakes, which are ice crystals. Their forms are strikingly diverse. But they are all centrally symmetrical.

An example of central or radial symmetry is plant flowers: sunflower, chamomile, iris, aster. This type of symmetry is also called rotational. If the petals of a flower or the rays of a snowflake are rotated relative to the center, they will overlap each other.

Mirror symmetry

Mirror symmetry in the space of the natural world around us is observed in plants and animals. oak or fern, beetle or butterfly, spider or caterpillar, mouse or hare - these are just some examples where you can see bilateral or mirror symmetry in living organisms. A person is symmetrical, as are parts of the body: arms, legs. In these forms we observe a kind of mirror reflection of one half of the object from the other. If you place an object in a plane, then its image can be mentally bent in the middle, and one half will overlap the other.

Hypothesis of the emergence of symmetry

In the scientific world, there are several hypotheses that try to explain how symmetry arose in the space of our world. According to one of them, everything that grows up or down is subject to the law, and everything that is formed parallel to the earth's surface or inclined to it takes on a mirror-symmetrical shape. They try to explain these properties by gravity from the center of the planet and varying degrees of illumination of objects by sunlight, depending on their location.

Symmetry in science and art

Symmetry in space was appreciated by artists, sculptors and architects in ancient times. We see elements of symmetry in ancient rock paintings, in the ornamental decorations of ancient objects and weapons. Egyptian and Mayan pyramids, domes of Slavic cathedrals, Greek temples and palaces, ancient arches and amphitheaters, the facade of the White House and the Moscow Kremlin are just some examples of the desire for sublime beauty and true perfection.

The concepts of symmetry were seriously developed by mathematicians. The mathematical studies carried out made it possible to identify the main patterns of symmetry on the plane and in space. Physics and chemistry also did not ignore this interesting natural pattern. Academician V.I. Vernadsky believed that “symmetry... covers the properties of all fields with which a physicist and a chemist deals.” Due to the symmetrical structure of atoms, molecules enter into various reactions and determine the physical properties of crystal formation. Even if the laws of physics that establish physical quantities remain unchanged under various transformations, we can say that these laws have invariance or symmetry with respect to these transformations.

Symmetry of space

Tell me what is symmetry of space?

You need to start with definitions to get to the bottom of things. Many of your physical laws are far from reality, but simply an attempt to describe multidimensional processes with three-dimensional thinking. Symmetry is the design of a certain order of movement and focusing of energy. The universe is large and diverse, the types of forms of creation are infinitely diverse. Therefore, symmetry in your understanding and symmetry within the entire universe are different things. This is the same as comparing the decimal number system that you have adopted with, say, the binary or septal number system. Understand? These are different approaches to organizing structuring. You have countless dice. You can stack them any way you want: in many piles of two or five or seven cubes. In two big piles. In five big piles and so on. Next, in each pile you also define a certain system for distributing the cubes. This is the process of structuring space. Since the Divine light is infinite, the number of structuring cubes is also infinite, therefore the variations in the addition of these divine cubes are infinite, and therefore the variations in the symmetry of space are infinite.

Your concept of symmetry comes from its binary nature, from systems of single reflection, these are the symmetry properties of the dual world in which you reside. In your world, any form has a symmetrical mirror reflection, any concept and direction of movement has a reflected double.

A reflected double? What do you mean.

It's like the other side of the coin. The same medal, but viewed from the opposite side. A look from the outside and a look from the inside. The reflected double is a view from the inside. Any phenomenon and any action can be viewed differently from different points of perception.

Wait, let's go in order. In nature, symmetry is widespread precisely binary symmetry. Snowflakes, plant leaves, crystal lattices, flowers, fruits and much more. Even in the structure of atoms there is symmetry. Why?

Let's go back to the perception filter again. You are the source of Divine light, enclosed in a lamp form. The border shape of your lamp is subtle but strong. And it can be organized in different ways. Now there are two holes in it, relatively speaking. Therefore, if your light comes out outside of you, it always comes out in binary form. When your light comes out of your holes-sensors of space, then outside of you it also encounters binary rays emanating from other forms reflecting you, is reflected from these rays, refracted and returns to you again through your two holes. This is a very simplified model, it is a model of binary perception. Model of dual reflection. As your awareness expands, new openings-perceptions open in you and everything seems to become more complicated, multivariance increases, and the symmetry of space becomes more complex.

When you talk about the symmetry of, say, a leaf of a tree, you see this symmetry in a planar version. But imagine the symmetry of a plant leaf in a three-dimensional version, when the reflection mirrors are placed in such a way that three identical parts are created. It’s difficult for you, because in your world everything has a pair. Then try to imagine a quaternary system of symmetry, when two leaves intersect in a longitudinal trunk. Or four sheets of paper, like in a book, are united by a common binding. Now imagine that the book has an infinite number of pages and the interweaving of these pages is also infinite.

I feel like your three-dimensional thinking and imagination are confused, this is normal. It’s difficult to change your mind right away, but you must believe that your system of perception, which is actually hidden very deeply in you and others, allows you to create and perceive any multidimensionality. Therefore, I will give you examples of spatial models and complicate them, so that you gradually get used to multidimensional perception not only mentally, but also in your imagination, although in fact they are the same thing.

So we take a point in space and an infinite number of rays emanating from it. As you understand, this is a description of you in the universe. For if the number of rays emanating from a point is infinite, then it describes all possible rays of space around you. But there are also countless such points. The points from which the rays emanate are the forms of God. As you can see, the symmetry of space was inherent initially in you and in the space around you. For every ray emanating from a point of reflection will find a reflected pair. But there will be not two such rays, but many pairs. Next, these rays encounter, say, a mirror and are reflected from it. If you imagine a ray as a straight line, then its reflection gives refraction, a bend in the other direction of this straight line. And accordingly, the dual pair of this beam will also be reflected from this mirror and give a symmetrical bend, as if in the other direction. This is how fractality is born, that is, the symmetry of reflections or reflected symmetry. Now let’s imagine that there is only one point from which the rays emanate, and there are an infinite number of mirrors, then there will be an infinite number of fractal reflections. Now imagine that what they reflect is not mirrors placed by someone. But simply the rays emanating from you as points of perception are reflected from myriads of rays of countless other forms of perception, from which countless rays also emanate. This is the multidimensional symmetry of space.

But in your concept, symmetry is the identical equality of halves. But if you look at a plant leaf or a fruit, then the symmetry there still undergoes distortions. That is, the reflections do not completely coincide down to the micron and beyond. So in your perception, the symmetry of space is also partially broken. When both rays that touch and reflect from each other have the same strength and direction, then the reflection symmetry created is more accurate, when this is not the case, then the reflection of one ray is different from the reflection of the other ray. But this is if we talk about space as a whole. But your reflected ray then returns to you, and therefore for you, as for everyone, the power of direction and the power of reflection are equal, since this is your power.

Then tell me, in nature we observe certain symmetrical figures: spheres, triangles, rectangles. These figures are present in everything. Why? Moreover, there are experiments with sound. When sand poured onto the surface of a speaker takes on certain geometric shapes under the influence of sound vibrations.

There are many questions here. But again you're trying to think linearly. Let's take a snowflake whose symmetry you can see. She is beautiful and never repeats herself. Why? Because microscopic snow particles are structured in a certain order, each time representing a different reflection of energy on the parameters of the cold, on the parameters of the environment in which they are reflected. But if you imagine a snowball, then it contains a huge number of snowflakes, a huge number of non-repeating symmetries. And if you could examine this new pattern, you would find a certain symmetry in it. That is, everything is structured in interaction with each other.

Vibrations of sound are precisely reflected energy. Its fluctuations in the reflective spectrum. In principle, everything is reflected energy and its fluctuations in the reflective spectrum. It’s just that you can perceive some of these vibrations with your eyes, some with your ears, some with your sense of smell, and so on. And some of them are not yet able to perceive.

Now let's move on. You observe the world around you and see in it the symmetry of reflections in the form of certain figures and symbols. But if you look deep into you, then there is also an infinity of symmetry and reflections. You just haven’t learned to look deep into yourself yet. You have created instruments in the form of microscopes and magnifying structures, but with the power of your thoughts you yourself can penetrate into all your components down to the primordial particles, and if you do this, you will discover amazing fractality and symmetry deep inside yourself. You have been looking outside of yourself all the time. But inside you there is the same infinite world, what you call microcosm, it is not known to you at all.

So now in our example, countless rays emanate from a point not only outside the point but also inside the point, in the opposite direction. And these rays of perception are also reflected, structured, fractalized.

There are many experiments with water, when the sounds of certain vibrations, say kind words or classical music, structure snowflakes into very beautiful patterns. There are many examples of the harmonizing effect on a person of music, certain colors and smells, paintings in the form of symmetrical mandalas, and so on. What is it? What happens?

Reflection. For example, a mandala is an energetic image of certain interconnections of rays of perception, arranged symmetrically. For you it's just a picture. But imagine it as an energy picture. When you meditate on it, your directed energy is reflected from the energy of the mandala and, as it were, copies it, makes a cast of it, and is reflected symmetrically to it. Understand? And it returns to you, structures your energy in a certain way and is again reflected outside. If you sit in mandala meditation for a long time, you seem to tune in. If you turn off all other sources of perception and completely focus on the mandala, then gradually your internal structuring becomes similar to the structure of the mandala, it is symmetrically reflected from it and a mandala is also born inside you, somewhat similar to the reflected one, but still possessing your characteristics and characteristics. The same thing happens with music, and with smells, and with flowers, and so on. You simply perceive more deeply the symmetry of another form and structure your form accordingly.

Why do the sounds of nature or certain music or certain signs harmonize a person? If everything is just a type of reflection and its diversity, why do we equally not tolerate, say, a cacaphony of sounds or, for example, the smells of decomposition? If there are no bad and good perceptions, why are we fairly equally attuned to certain perceptions?

Sustainability. Why is so much symmetrical around you? Because symmetrical configurations are stable. It's like a chair with one leg, three or four. What you call harmony is the most stable viable configurations of space. Unstable configurations disintegrate. If you bend the paper sequentially and symmetrically and fold it many times, then you can roll it up to a point, to a small ball, while there will be symmetry inside it, and many edges of the sheet of paper will have a huge number of contacts and adhesion to each other. And if a sheet of paper is simply crumpled, then there will be much less contact between the points of the paper and, accordingly, less adhesion, and the volume of the crumpled sheet will be greater. This design is less stable. If you, say, sit on a folded sheet of paper, then it is almost not deformed and, more importantly, the connections are not deformed. But if you sit on a crumpled sheet of paper, then it is deformed and many connections-contacts are broken. Therefore, symmetry is a consistent compaction.

So there is some kind of primordial unmanifest chaos, which under a certain creative influence takes on symmetrical forms?

Everything is mixed up for you. Non-manifestation is the absence of movement. Movement itself is either chaos or symmetry, that is, when particles move chaotically, this is already manifestation. When the rays are reflected asymmetrically, this is also manifestation. There are simply different types of manifestation, and chaotic movement is no worse than symmetrical movement, it is just different. There are various types of space structures in the universe, including what you call chaos.

But you say that symmetrical configurations are more stable. Then why chaotic configurations?

These are various forms of creating space, its organization and structuring. Sometimes chaotic movements provide new directions for structuring. Just as you cannot reject the energy of destruction, since it is also used in creation, so you should not reject chaotic structuring, which is also used in creation. The symmetrical structuring of space is more stable, but also more rigid and less mobile. It's like a pre-created zone for choosing the movement of energy, you know? If you take your freedom of choice, this is precisely chaos. If we take any hierarchy, it is rigid symmetry and fractality.

It turns out that chaotic structuring was introduced into the symmetry of space?

Or vice versa, symmetry was introduced into the chaotic structure.

If everything I see around me is just an agreement between people on how to see it, then why do I see space symmetrically and not chaotically? If everything is energy, then why do all people see the symmetry of a flower in a certain way? Why not chaos?

Because the reflected rays of a flower as a form of God are symmetrical. And you perceive precisely the direction of these rays. Look with light vision. When you look at a luminous object, then when you close your eyes, light configurations appear on the inner screen, this is light vision. If you imagine the world around you in the form of energy, you will see vibrations and movement of light lines and points of other figures. When you look at objects that seem formless to you and give them form in your imagination, as in the case of clouds, this means that either the object does not have strict structurization connections, that is, elements of chaos predominate, or you are simply not able to perceive such structuring. It's like a snowball, inside of which there are billions of snow with amazing symmetry, but the ball of snow itself is not very symmetrical.

I'm asking about the bystander effect. If the movement of, say, elementary particles depends on the observer, does this mean that the observed symmetry of the space of nature also depends on us, on the observers of this symmetry, and not on the space itself?

Certainly. Remember the example with your reflected rays. The reflection of your beam depends on you. That is, from the properties of the beam itself. By passing Divine light through your prism of perception, you give it certain characteristics of perception, a certain degree of reflection. Therefore, the observer effect consists precisely in the fact that you and only you are reflected in your own way from other rays of perception. But at some point or in some space of a certain extent, your rays are combined, this is a reflection of the external world, this is your general picture of the world, this is the symmetry of space visible to you.

So, if we start to reflect chaotically, the picture of the world will change?

You place your accents a little wrong. You are always reflecting. It’s just that some of you and God’s forms reflect more symmetrically, and some more chaotically. Therefore, those who reflect more chaotically come into contact, intersect their perceptions with those who also reflect more chaotically. This is the law of similarity; like does not just attract like. Like only intersects with like. You cannot intersect with someone who is directed, relatively speaking, in the other direction. Like non-intersecting roads in your world, they exist and lead in certain directions. But your road is in a different area and goes in a different direction. But if your road encircles the entire globe, then sooner or later it will intersect with all other roads.

Therefore, if you see symmetry in the surrounding space, it is simply the intersection of your perception with those who are also reflected more symmetrically.

Does this mean that somewhere there are worlds and spaces where everything is asymmetrical?

Certainly. Again, in your world, the concept of chaos has a negative connotation. Imagine if you lived in a universe that is primarily built on the chaotic movement of energy. Then any symmetry would seem to you something alien and negative and dark in your assessment of duality.

That is, the fact that we are directed towards light and goodness is only a consequence of the fact that our universe is more built on the symmetry of space?

Yes. You got it right. However, your concept of light is the opposite of the concept of darkness. But everything, both light in your understanding and darkness in your understanding, is the reflected light of God, the reflected energy of God. Therefore, light in your understanding is a symmetrical reflection of the energy of God. And darkness is a chaotic reflection of the energy of God. And in fact, your universe is an attempt to balance both. Give symmetry to chaos, and add chaotic components to symmetry. To get something in between. Because the symmetrical configuration is more stable, and the chaotic configuration is more variable.

It seems to me that harmony, that is, symmetry, still wins. If you look at nature, this is clearly visible.

The development of any form and any system has stages of direction. Symmetry replaces chaos. Chaos gives way to symmetry. Now you are at the stage of a symmetrical infusion of configurations, like the process of crystallization of, say, salt, your space is crystallizing into certain harmonious structures and new forms of connection, new configurations, new crystals are created. But then, in order to test the stability of these forms, a period of chaotic movement will begin, like the effect of wind and rain on geological rocks and mountains. And then the mountains undergo changes. Is a mountain symmetry or not? It is a combination of both. When a symmetrical form, under the influence of chaotic processes, changes its configuration, and this configuration is neither bad nor good. It's just a new combination of symmetry and chaos.

How can a person use the symmetry of space other than to harmonize himself?

This is a very interesting question and you still have a lot to understand on this topic. He can use this symmetry in everything. For example, he can configure himself symmetrically to an external object and thus repeat, copy it. That is, to become similar to this object.

Did I understand correctly: if a person copies, say, the configuration of a plant, then he will become that plant?

It almost will, since it will sooner be somewhat different from the original. It will only be a copy. But you got it right. Those magicians who could transform into plants and animals did just that, copied the energy configuration of another object.

But that's not all. Knowing the configuration and symmetry of space, you can get from one point in space to any other. Now you are doing this chaotically by chance in your dreams and over very short distances. But it’s like a network of roads, a coordinate grid of the space of the universe. Knowing the coordinates, you seem to know a picture of the configuration, a picture of the symmetry of space, and by reproducing it with your consciousness, thus rearranging your configuration, you find yourself aligned with this space, as if you find yourself in a puzzle. If, by your configuration, you cannot fit into the picture like a puzzle, then you cannot perceive the boundaries of contact with other puzzles in the picture, understand? And there is much more you have to master in the symmetry of space. But it’s too early to talk about this.

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Slide captions:

SYMMETRY IN SPACE A A 1 O Points A and A1 are called symmetrical relative to point O (center of symmetry) if O is the middle of the segment AA1. Point O is considered symmetrical to itself.

SYMMETRY IN SPACE Points A and A1 are called symmetrical with respect to a straight line (axis of symmetry) if the straight line passes through the middle of the segment AA1 and is perpendicular to this segment. Each point of a line a is considered symmetrical to itself. A leaf, a snowflake, a butterfly are examples of axial symmetry. A 1 A a

SYMMETRY IN SPACE Points A and A 1 are called symmetrical relative to a plane (plane of symmetry) if this plane passes through the middle of the segment AA 1 and is perpendicular to this segment. Each point of the plane is considered symmetrical to itself. A A 1

A point (straight line, plane) is called a center (axis, plane) of symmetry of a figure if each point of the figure is symmetrical relative to it to some point of the same figure. If a figure has a center (axis, plane) of symmetry, then it is said to have central (axial, mirror) symmetry. A 1 A O A 1 A O

We often encounter symmetry in nature, architecture, technology, and everyday life. Thus, many buildings are symmetrical relative to the plane, for example, the main building of Moscow State University; some types of parts have an axis of symmetry. Almost all crystals found in nature have a center, axis, or plane of symmetry. In geometry, the center, axes, and planes of symmetry of a polyhedron are called the symmetry elements of that polyhedron.

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Methodological justification of the lesson. Using knowledge from physics, astronomy, MHC, biology in a geometry lesson when summarizing the systematization of information on the topic: “Symmetry in space. Rules...

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Lesson type: combined.

Lesson objectives:

Consider axial, central and mirror symmetries as properties of some geometric figures.
Teach to construct symmetrical points and recognize figures with axial symmetry and central symmetry.
Improve problem solving skills.

Lesson objectives:

Formation of spatial representations of students.
Developing the ability to observe and reason; developing interest in the subject through the use of information technology.
Raising a person who knows how to appreciate beauty.

Lesson equipment:

Use of information technology (presentation).
Drawings.
Homework cards.

Lesson progress

I. Organizational moment.

Inform the topic of the lesson, formulate the objectives of the lesson.

II. Introduction.

What is symmetry?

The outstanding mathematician Hermann Weyl highly appreciated the role of symmetry in modern science: “Symmetry, no matter how broadly or narrowly we understand this word, is an idea with the help of which man tried to explain and create order, beauty and perfection.”

We live in a very beautiful and harmonious world. We are surrounded by objects that please the eye. For example, a butterfly, a maple leaf, a snowflake. Look how beautiful they are. Have you paid attention to them? Today we will touch on this wonderful mathematical phenomenon - symmetry. Let's get acquainted with the concept of axial, central and mirror symmetries. We will learn to build and identify figures that are symmetrical relative to the axis, center and plane.

The word “symmetry” translated from Greek sounds like “harmony”, meaning beauty, proportionality, proportionality, uniformity in the arrangement of parts. Man has long used symmetry in architecture. It gives harmony and completeness to ancient temples, towers of medieval castles, and modern buildings.

In the most general form, “symmetry” in mathematics is understood as such a transformation of space (plane), in which each point M goes to another point M" relative to some plane (or line) a, when the segment MM" is perpendicular to the plane (or line) a and divides it in half. The plane (straight line) a is called the plane (or axis) of symmetry. The fundamental concepts of symmetry include plane of symmetry, axis of symmetry, center of symmetry. A plane of symmetry P is a plane that divides a figure into two mirror-like equal parts, located relative to each other in the same way as an object and its mirror image.

III. Main part. Types of symmetry.

Central symmetry

Symmetry about a point or central symmetry is a property of a geometric figure when any point located on one side of the center of symmetry corresponds to another point located on the other side of the center. In this case, the points are located on a line segment passing through the center, dividing the segment in half.

Practical task.

Points are given A, IN And M M relative to the middle of the segment AB.
Which of the following letters have a center of symmetry: A, O, M, X, K?
Do they have a center of symmetry: a) a segment; b) beam; c) a pair of intersecting lines; d) square?

Axial symmetry

Symmetry about a line (or axial symmetry) is a property of a geometric figure when any point located on one side of the line will always correspond to a point located on the other side of the line, and the segments connecting these points will be perpendicular to the axis of symmetry and divided by it in half.

Practical task.

Given two points A And IN, symmetrical with respect to some line, and a point M. Construct a point symmetrical to the point M relative to the same line.
Which of the following letters have an axis of symmetry: A, B, D, E, O?
How many axes of symmetry does: a) a segment have? b) straight; c) beam?
How many axes of symmetry does the drawing have? (see Fig. 1)

Mirror symmetry

Points A And IN are called symmetrical relative to the plane α (plane of symmetry) if the plane α passes through the middle of the segment AB and perpendicular to this segment. Each point of the α plane is considered symmetrical to itself.

Practical task.

Find the coordinates of the points to which points A (0; 1; 2), B (3; -1; 4), C (1; 0; -2) go with: a) central symmetry relative to the origin; b) axial symmetry relative to the coordinate axes; c) mirror symmetry relative to coordinate planes.
Does the right glove go into the right or left glove with mirror symmetry? axial symmetry? central symmetry?
The figure shows how the number 4 is reflected in two mirrors. What will be visible in place of the question mark if the same is done with the number 5? (see Fig. 2)
The picture shows how the word KANGAROO is reflected in two mirrors. What happens if you do the same with the number 2011? (see Fig. 3)

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This is interesting.

Symmetry in living nature.

Almost all living beings are built according to the laws of symmetry; it is not without reason that the word “symmetry” means “proportionality” when translated from Greek.

Among flowers, for example, there is rotational symmetry. Many flowers can be rotated so that each petal takes the position of its neighbor, the flower aligns with itself. The minimum angle of such rotation is not the same for different colors. For the iris it is 120°, for the bellflower – 72°, for the narcissus – 60°.

There is helical symmetry in the arrangement of leaves on plant stems. Positioned like a screw along the stem, the leaves seem to spread out in different directions and do not obscure each other from the light, although the leaves themselves also have an axis of symmetry. Considering the general plan of the structure of any animal, we usually notice a certain regularity in the arrangement of body parts or organs, which are repeated around a certain axis or occupy the same position in relation to a certain plane. This regularity is called body symmetry. The phenomena of symmetry are so widespread in the animal world that it is very difficult to indicate a group in which no symmetry of the body can be noticed. Both small insects and large animals have symmetry.

Symmetry in inanimate nature.

Among the infinite variety of forms of inanimate nature, such perfect images are found in abundance, whose appearance invariably attracts our attention. Observing the beauty of nature, you can notice that when objects are reflected in puddles and lakes, mirror symmetry appears (see Fig. 4).

Crystals bring the charm of symmetry to the world of inanimate nature. Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have rotational symmetry and, in addition, mirror symmetry.

One cannot help but see symmetry in faceted gemstones. Many cutters try to give diamonds the shape of a tetrahedron, cube, octahedron or icosahedron. Since the garnet has the same elements as the cube, it is highly prized by gemstone connoisseurs. Artistic items made from garnets were discovered in the graves of Ancient Egypt dating back to the pre-dynastic period (over two millennia BC) (see Fig. 5).

In the Hermitage collections, gold jewelry of the ancient Scythians receives special attention. The artistic work of gold wreaths, tiaras, wood and decorated with precious red-violet garnets is unusually fine.

One of the most obvious uses of the laws of symmetry in life is in architectural structures. This is what we see most often. In architecture, axes of symmetry are used as means of expressing architectural design (see Fig. 6). In most cases, patterns on carpets, fabrics, and indoor wallpaper are symmetrical about the axis or center.

Another example of a person using symmetry in his practice is technology. In engineering, symmetry axes are most clearly designated where it is necessary to estimate the deviation from the zero position, for example, on the steering wheel of a truck or on the steering wheel of a ship. Or one of the most important inventions of mankind that has a center of symmetry is the wheel; the propeller and other technical means also have a center of symmetry.

"Look in the mirror!"

Should we think that we only see ourselves in a “mirror image”? Or, at best, can we only find out in photographs and film what we “really” look like? Of course not: it is enough to reflect the mirror image a second time in the mirror to see your true face. Trellis come to the rescue. They have one large main mirror in the center and two smaller mirrors on the sides. If you place such a side mirror at right angles to the middle one, then you can see yourself exactly in the form in which others see you. Close your left eye, and your reflection in the second mirror will repeat your movement with your left eye. Before the trellis, you can choose whether you want to see yourself in a mirror image or in a direct image.

It is easy to imagine what kind of confusion would reign on Earth if the symmetry in nature were broken!


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IV. Physical education minute.

« Lazy Eights» – activate structures that ensure memorization, increase the stability of attention.
Draw the number eight in the air in a horizontal plane three times, first with one hand, then with both hands at once.
« Symmetrical drawings » – improve hand-eye coordination and facilitate the writing process.
Draw symmetrical patterns in the air with both hands.

V. Independent testing work.

Ι option

ΙΙ option

In the rectangle MPKH O is the point of intersection of the diagonals, RA and BH are perpendiculars drawn from the vertices P and H to the straight line MK. It is known that MA = OB. Find the angle POM.
In the rhombus MPKH the diagonals intersect at the point ABOUT. On the sides MK, KH, PH points A, B, C are taken, respectively, AK = KV = RS. Prove that OA = OB and find the sum of the angles POC and MOA.
Construct a square along the given diagonal so that the two opposite vertices of this square lie on opposite sides of the given acute angle.

VI. Summing up the lesson. Assessment.

What types of symmetry did you learn about in class?
Which two points are called symmetrical with respect to a given line?
Which figure is called symmetrical with respect to a given line?
Which two points are said to be symmetrical about a given point?
Which figure is called symmetrical about a given point?
What is mirror symmetry?
Give examples of figures that have: a) axial symmetry; b) central symmetry; c) both axial and central symmetry.
Give examples of symmetry in living and inanimate nature.