How to build axial symmetry. Construction of a segment symmetrical to a given one

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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 a subject through use information technologies.
• Raising a person who knows how to appreciate beauty.

Lesson equipment:

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

During the classes

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 the word, is an idea with the help of which man has 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 Maple Leaf, snowflake. Look how beautiful they are. Have you paid attention to them? Today we will touch upon 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 view"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 is divided in half by it . 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 such a property 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 straight line segment passing through the center, dividing the segment in half.

Practical task.

1. Points are given A, IN And M M relative to the middle of the segment AB.
2. Which of the following letters have a center of symmetry: A, O, M, X, K?
3. 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.

1. 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.
2. Which of the following letters have an axis of symmetry: A, B, D, E, O?
3. How many axes of symmetry does: a) a segment have? b) straight; c) beam?
4. 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.

1. 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 coordinate axes; c) mirror symmetry relative to coordinate planes.
2. Does the right glove go into the right or left glove in mirror symmetry? axial symmetry? central symmetry?
3. 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)
4. 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)

Rice. 2

This is interesting.

Symmetry in living nature.

Almost all living beings are built according to the laws of symmetry, not without reason translated from Greek word"symmetry" means "proportionality".

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 a rotation for various colors not the same. 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 overall plan 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. Art products of garnets were found in graves Ancient Egypt, dating back to the predynastic period (over two millennia BC) (see Fig. 5).

In the Hermitage collections special attention used gold jewelry of the ancient Scythians. Extraordinarily thin artwork golden wreaths, tiaras, wood and decorated with precious red-violet garnets.

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 expression 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 consider that we see ourselves only in “ mirror image"? Or in best case scenario Only in photographs and film can we find out 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!

Rice. 4	Rice. 5	Rice. 6

IV. Physical education minute.

• « Lazy Eights» – activate structures that ensure memorization, increase 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

1. 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.
2. 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.
3. Construct a square along the given diagonal so that two opposite vertices of this square lay on different sides of this 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.

VII. Homework.

1. Individual: complete the structure using axial symmetry (see Fig. 7).

Rice. 7

2. Construct a figure symmetrical to the given one with respect to: a) a point; b) straight (see Fig. 8, 9).

Rice. 8	Rice. 9

3. Creative task: "In the animal world". Draw a representative from the animal world and show the axis of symmetry.

VIII. Reflection.

• What did you like about the lesson?
• What material was most interesting?
• What difficulties did you encounter when completing this or that task?
• What would you change during the lesson?

I . Symmetry in mathematics :

Basic concepts and definitions.

Axial symmetry (definitions, construction plan, examples)

Central symmetry (definitions, construction plan, whenmeasures)

Summary table (all properties, features)

II . Applications of symmetry:

1) in mathematics

2) in chemistry

3) in biology, botany and zoology

4) in art, literature and architecture

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1. Basic concepts of symmetry and its types.

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

2. Axial symmetry.

2.1 Basic definitions

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

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

2.2 Construction plan

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

2.3 Examples of figures with axial symmetry.

3. Central symmetry

3.1 Basic definitions

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

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

3.2 Construction plan

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

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

Construction of symmetrical points relative to the center.

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

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

3.3 Examples

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

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

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

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

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

4. Lesson summary

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

Summary table

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

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

Biology

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

Botany

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

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

Zoology

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

Axial symmetry

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

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

© Sukhacheva Elena Vladimirovna, 2008-2009

Today we will talk about a phenomenon that each of us constantly encounters in life: symmetry. What is symmetry?

We all roughly understand the meaning of this term. The dictionary says: symmetry is proportionality and complete correspondence of the arrangement of parts of something relative to a straight line or point. There are two types of symmetry: axial and radial. Let's look at the axial one first. This is, let’s say, “mirror” symmetry, when one half of an object is completely identical to the second, but repeats it as a reflection. Look at the halves of the sheet. They are mirror symmetrical. The halves of the human body are also symmetrical (front view) - identical arms and legs, identical eyes. But let’s not be mistaken; in fact, in the organic (living) world, absolute symmetry cannot be found! The halves of the sheet copy each other far from perfectly, the same applies to human body(take a closer look for yourself); The same is true for other organisms! By the way, it is worth adding that any symmetrical body is symmetrical relative to the viewer only in one position. It’s worth, say, turning a sheet of paper, or raising one hand, and what happens? – you see for yourself.

People achieve true symmetry in the products of their labor (things) - clothes, cars... In nature, it is characteristic inorganic formations, for example, crystals.

But let's move on to practice. You shouldn’t start with complex objects like people and animals; let’s try to finish drawing the mirror half of the sheet as the first exercise in a new field.

Drawing a symmetrical object - lesson 1

We make sure that it turns out as similar as possible. To do this, we will literally build our soul mate. Don’t think that it’s so easy, especially the first time, to draw a mirror-corresponding line with one stroke!

Let's mark several reference points for the future symmetrical line. We proceed like this: with a pencil, without pressing, we draw several perpendiculars to the axis of symmetry - the midrib of the leaf. Four or five is enough for now. And on these perpendiculars we measure to the right the same distance as on the left half to the line of the edge of the leaf. I advise you to use a ruler, don’t rely too much on your eye. As a rule, we tend to reduce the drawing - this has been observed from experience. We do not recommend measuring distances with your fingers: the error is too large.

Let's connect the resulting points with a pencil line:

Now let’s look meticulously at whether the halves are really the same. If everything is correct, we will circle it with a felt-tip pen and clarify our line:

The poplar leaf has been completed, now you can take a swing at the oak leaf.

Let's draw a symmetrical figure - lesson 2

In this case, the difficulty lies in the fact that the veins are marked and they are not perpendicular to the axis of symmetry and not only the dimensions but also the angle of inclination will have to be strictly observed. Well, let’s train our eye:

So a symmetrical oak leaf has been drawn, or rather, we built it according to all the rules:

How to draw a symmetrical object - lesson 3

And let’s consolidate the theme - we’ll finish drawing a symmetrical lilac leaf.

He has too interesting shape- heart-shaped and with ears at the base, you’ll have to puff:

This is what they drew:

Take a look at the resulting work from a distance and evaluate how accurately we were able to convey the required similarity. Here's a tip: look at your image in the mirror and it will tell you if there are any mistakes. Another way: bend the image exactly along the axis (we have already learned how to bend it correctly) and cut out the leaf along the original line. Look at the figure itself and at the cut paper.

Goals:

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

DURING THE CLASSES

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

Exercise 1(3 min).

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

Question: What function does this line serve?

Suggested answer: This line divides the figure in half.

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

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

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

Task 2 (2 minutes).

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

Task 3 (5 minutes).

– Draw a circle in your notebook.

Question: Determine how the axis of symmetry goes?

Suggested answer: Differently.

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

Suggested answer: A lot of.

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

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

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

– Let’s consider volumetric 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 minutes).

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

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

Elements of drawings are presented to students

Task 6 (2 minutes).

– Find the symmetrical parts of these drawings.

To consolidate the material covered, I suggest next tasks provided for 15 minutes:

Name them all equal elements triangle KOR and COM. What type of triangles are these?

2. Draw several isosceles triangles in your notebook with common ground equal to 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, note where elements of symmetry are present.

People's lives are filled with symmetry. It’s convenient, beautiful, and there’s no need to invent new standards. But what is it really and is it as beautiful in nature as is commonly believed?

Symmetry

Since ancient times, people have sought to organize the world around them. Therefore, some things are considered beautiful, and some are not so much. From an aesthetic point of view, the golden and silver ratios are considered attractive, as well as, of course, symmetry. This term has Greek origin and literally means “proportionality”. Of course we're talking about not only about coincidence on this basis, but also on some others. IN in a general sense symmetry is a property of an object when, as a result of certain formations, the result is equal to the original data. It is found in both living and inanimate nature, as well as in objects made by man.

First of all, the term "symmetry" is used in geometry, but finds application in many scientific fields, and its meaning remains generally unchanged. This phenomenon occurs quite often and is considered interesting, since several of its types, as well as elements, differ. The use of symmetry is also interesting, because it is found not only in nature, but also in patterns on fabric, borders of buildings and many other man-made objects. It is worth considering this phenomenon in more detail, because it is extremely fascinating.

Use of the term in other scientific fields

In what follows, symmetry will be considered from a geometric point of view, but it is worth mentioning that given word used not only here. Biology, virology, chemistry, physics, crystallography - all this is an incomplete list of areas in which this phenomenon studied with various sides and in different conditions. For example, the classification depends on what science this term refers to. Thus, the division into types varies greatly, although some basic ones, perhaps, remain unchanged throughout.

Classification

There are several main types of symmetry, of which three are the most common:

In addition, in geometry there are also following types, they are much less common, but no less interesting:

• sliding;
• rotational;
• point;
• progressive;
• screw;
• fractal;
• etc.

In biology, all species are called slightly differently, although in essence they may be the same. Division into certain groups occurs on the basis of the presence or absence, as well as the quantity of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.

Basic elements

The phenomenon has certain features, one of which is necessarily present. So called basic elements include planes, centers and axes of symmetry. It is in accordance with their presence, absence and quantity that the type is determined.

The center of symmetry is the point inside a figure or crystal at which the lines connecting everything in pairs converge parallel friend to the other side. Of course, it does not always exist. If there are sides to which there is no parallel pair, then such a point cannot be found, since it does not exist. According to the definition, it is obvious that the center of symmetry is that through which a figure can be reflected onto itself. An example would be, for example, a circle and a point in its middle. This element is usually designated as C.

The plane of symmetry, of course, is imaginary, but it is precisely it that divides the figure into two parts equal to each other. It can pass through one or more sides, be parallel to it, or divide them. For the same figure, several planes can exist at once. These elements are usually designated as P.

But perhaps the most common is what is called “axis of symmetry”. This is a common phenomenon that can be seen both in geometry and in nature. And it is worthy of separate consideration.

Axles

Often the element in relation to which a figure can be called symmetrical is

a straight line or segment appears. In any case, we are not talking about a point or a plane. Then the figures are considered. There can be a lot of them, and they can be located in any way: dividing the sides or being parallel to them, as well as intersecting corners or not doing so. Axes of symmetry are usually designated as L.

Examples include isosceles and In the first case there will be a vertical axis of symmetry, on both sides of which equal faces, and in the second the lines will intersect each angle and coincide with all bisectors, medians and heights. Ordinary triangles do not have this.

By the way, the totality of all the above elements in crystallography and stereometry is called the degree of symmetry. This indicator depends on the number of axes, planes and centers.

Examples in geometry

Conventionally, we can divide the entire set of objects of study by mathematicians into figures that have an axis of symmetry and those that do not. All circles, ovals, as well as some special cases automatically fall into the first category, while the rest fall into the second group.

As in the case when it was said about the axis of symmetry of a triangle, this element for a quadrilateral does not always exist. For a square, rectangle, rhombus or parallelogram it is, and for irregular figure, accordingly, no. For a circle, the axis of symmetry is the set of straight lines that pass through its center.

In addition, it is interesting to consider three-dimensional figures from this point of view. At least one axis of symmetry in addition to all regular polygons and the ball will have some cones, as well as pyramids, parallelograms and some others. Each case must be considered separately.

Examples in nature

In life it is called bilateral, it occurs most
often. Any person and many animals are an example of this. Axial is called radial and is much less common, usually in flora. And yet they exist. For example, it is worth thinking about how many axes of symmetry a star has, and does it have any at all? Of course, we are talking about marine life, and not about the subject of study by astronomers. And the correct answer would be: it depends on the number of rays of the star, for example five, if it is five-pointed.

In addition, radial symmetry is observed in many flowers: daisies, cornflowers, sunflowers, etc. There are a huge number of examples, they are literally everywhere around.

Arrhythmia

This term, first of all, reminds most of medicine and cardiology, but it initially has a slightly different meaning. IN in this case a synonym would be “asymmetry”, that is, the absence or violation of regularity in one form or another. It can be found as an accident, and sometimes it can become a wonderful technique, for example in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous one is slightly tilted, and although it is not the only one, it is the most famous example. It is known that this happened by accident, but this has its own charm.

In addition, it is obvious that the faces and bodies of people and animals are not completely symmetrical either. There have even been studies that show that “correct” faces are judged to be lifeless or simply unattractive. Still, the perception of symmetry and this phenomenon in itself are amazing and have not yet been fully studied, and therefore are extremely interesting.