Human life is filled with symmetry. It is convenient, beautiful, no need to invent new standards. But what is she really and is she as beautiful in nature as is commonly believed?

Symmetry

Since ancient times, people have sought to streamline the world around them. Therefore, something is considered beautiful, and something not so. From an aesthetic point of view, golden and silver sections are considered attractive, as well as, of course, symmetry. This term is of Greek origin and literally means "proportion". Of course, we are talking not only about coincidence on this basis, but also on some others. In a general sense, symmetry is such 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 animate 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 is quite common 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 ornaments on fabric, building borders and many other man-made objects. It is worth considering this phenomenon in more detail, because it is extremely exciting.

Use of the term in other scientific fields

In the future, symmetry will be considered from the point of view of geometry, but it is worth mentioning that this word is used not only here. Biology, virology, chemistry, physics, crystallography - all this is an incomplete list of areas in which this phenomenon is studied from different angles and under different conditions. The classification, for example, depends on which science this term refers to. Thus, the division into types varies greatly, although some basic ones, perhaps, remain unchanged everywhere.

Classification

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

In addition, the following types are also distinguished in geometry, they are much less common, but no less curious:

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

In biology, all species are called somewhat differently, although in fact they can be the same. The division into certain groups occurs on the basis of the presence or absence, as well as the number of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.

Basic elements

Some features are distinguished in the phenomenon, one of which is necessarily present. The 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 called the point inside the figure or crystal, at which the lines converge, connecting in pairs all sides parallel to each other. Of course, it doesn't always exist. If there are sides to which there is no parallel pair, then such a point cannot be found, since there is none. According to the definition, it is obvious that the center of symmetry is that through which the figure can be reflected to itself. An example is, for example, a circle and a point in its middle. This element is usually referred to as C.

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

But perhaps the most common is what is called "axes of symmetry." This frequent phenomenon can be seen both in geometry and in nature. And it deserves separate consideration.

axes

Often the element with respect to which the figure can be called symmetrical,

is a straight line or a segment. 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: divide sides or be parallel to them, as well as cross corners or not. Axes of symmetry are usually denoted as L.

Examples are isosceles and In the first case there will be a vertical axis of symmetry, on both sides of which there are 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 it.

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

It is conditionally possible to divide the entire set of objects of study of 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 the triangle, this element for the quadrilateral does not always exist. For a square, rectangle, rhombus or parallelogram, it is, but for an irregular figure, accordingly, it is not. 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 volumetric 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 very many animals are an example of this. The axial one is called radial and is much less common, as a rule, in the plant world. And yet they are. For example, it is worth considering how many axes of symmetry a star has, and does it have them at all? Of course, we are talking about marine life, and not about the subject of study of astronomers. And the correct answer would be this: 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: chamomile, cornflowers, sunflowers, etc. Examples great amount 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 this case, the synonym will 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 be a beautiful device, for example, in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous one is slightly inclined, and although it is not the only one, this 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 also not completely symmetrical. There have even been studies, according to the results of which the "correct" faces were regarded as inanimate or simply unattractive. Still, the perception of symmetry and this phenomenon in itself are amazing and have not yet been fully studied, and therefore extremely interesting.

Mathematics teacher Kochkina L.K.

Topic AXIAL AND CENTRAL SYMMETRIES

Purpose of the lesson task:

To teach how to build symmetrical points and recognize figures with axial symmetry and central symmetry, the formation of students' spatial representations. Developing the ability to observe and reason; development of interest in the subject through the use of information technology. Development of mathematical competence of students. Raising a person who knows how to appreciate the beautiful.

Expected result Students will be able to build symmetrical figures about the center and the line.

Lesson equipment:

Use of information technologies (presentation).

During the classes

I. Organizational moment.

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

II. Presentation showing: "Symmetric World"(for students)

III. work on the topic of the lesson(group work)

Students complete assignments on their own. At the end, information is exchanged.

1 option

item 47

axial symmetry

Option 2

item 47

central symmetry

Not really

Not really

Consider the rules for constructing symmetrical figures.

1 .Central symmetry is symmetry about a point.

Points A and B are symmetrical with respect to some point O if the point O is the midpoint of the segment AB.

Algorithm for constructing a centrally symmetrical figure

We construct a triangle A 1 B 1 C 1, symmetrical to the triangle ABC, with respect to the center (point) O.

For this:

Connect the points A, B, C with the center O and continue these segments;

2. We measure the segments AO, VO, CO and set aside on the other side of the point O, segments equal to them (AO \u003d A 1 O 1, VO \u003d B 1 O 1, CO \u003d C 1 O 1);

3. Connect the resulting points with segments A 1 B 1, A 1 C 1, B 1 C 1.

4. Received ∆A 1 AT 1 FROM 1 symmetrical ∆ABC.

Point O is called the center of symmetry of the figure, and the figure is called centrally symmetrical.

Task number 1 The figure shows a part of the figure, the center of symmetry of which is the point M. Explain its construction

Task number 2 Check the correctness of the construction of the figure from No. 1 with a neighbor in the desk. Construct a quadrangle in his notebook and mark the point O, which does not belong to this quadrangle. Take your notebook back and construct a quadrilateral symmetrical to the given one with respect to point O.

Check the correctness of the completed task.

2. Axial symmetry - this is symmetry about the drawn axis (straight line).

Points A and B are symmetrical with respect to some line a if these points lie on a line perpendicular to the given one and at the same distance.

The axis of symmetry is called a straight line when bent along which the "halves" coincide, and the figure is called symmetrical about some axis.

Algorithm for constructing a figure symmetrical with respect to some straight line

We construct a triangle A 1 B 1 C 1 , symmetrical to the triangle ABC with respect to the line a.

For this:

1. We draw straight lines from the vertices of the triangle ABC perpendicular to the straight line a and continue them further.

2. We measure the distances from the vertices of the triangle to the resulting points on the straight line and plot the same distances on the other side of the straight line.

3. Connect the resulting points with segments A 1 B 1, B 1 C 1, B 1 C 1.

4. Received ∆ A 1 AT 1 FROM 1 symmetrical ∆ABC.

Tasks according to the textbook No. 248-252, No. 261

perform the construction of a figure symmetrical with respect to the straight line a (on the board and in notebooks).

VI. Summing up the lesson.

Reflection What types of symmetry did you meet in the lesson?

Homework:

Repeat definitions. Creative work: Having studied the Russian alphabet (for option 1) and the Latin alphabet (for option 2), choose those letters that have symmetry. To issue results of researches in the A4 format. Those who are interested in this topic can take part in the creative project "Symmetry in my favorite school"

Task number 4 Fill the table:

Line segment

Straight

Ray

Square

One center of symmetry

Infinitely many centers of symmetry

One axis of symmetry

Two axes of symmetry

Four axes of symmetry

Infinitely many axes of symmetry

1 option

item 47

axial symmetry

Option 2

item 47

central symmetry

Axial symmetry is symmetry about ____________

Central symmetry is symmetry about ________________

Two points A and A 1 are called symmetrical with respect to the line a if ____________

Two points A and A 1 are called symmetrical about the point O if _____________

The straight line a is called _______________

Point O is called _________________

A figure is called symmetrical with respect to a straight line, if for each point of the figure, the point symmetrical to it belongs to _________

A figure is called symmetrical with respect to the point O if for each point of the figure, the point symmetrical to it belongs to _______

Are figures that are symmetrical with respect to a straight line equal?

Not really

Are figures that are symmetrical about a point equal?

"Point of symmetry" - Such a figure has central symmetry. rotation symmetry. All solids are made up of crystals. The point O is called the center of symmetry. Symmetry in nature. Examples of symmetry of plane figures. The parallelogram has only central symmetry. A straight prism has mirror symmetry. Examples of the above types of symmetry.

"Central symmetry in geometry" - Which point passes into itself with central symmetry. Draw a triangle symmetrical to triangle OAB. Does the parallelogram have a center of symmetry? Properties. Which points are called symmetrical with respect to the point. Draw triangle A'B'C' symmetrical to triangle ABC. Lines with central symmetry transform into themselves.

"Central symmetry" - Properties of central symmetry. Symmetry in art. Examples of symmetry in architecture. Central symmetry is movement (isometry). In three-dimensional space Central symmetry in three-dimensional space is also called spherical symmetry. Types of symmetry of flowers and plants.

"Symmetry about a point and a line" - Think! The symmetry of the figure with respect to the point. Tasks. Task Construct a point C1 symmetrical to point C with respect to the line a. AO=OA1. 4. Talk about symmetry in nature. Axial and central symmetry. Symmetry on the coordinate plane. Which of these letters has a center of symmetry? Which of these figures have an axis of symmetry?

“Axial and central symmetry” - Do they have a center of symmetry: AO \u003d BO, AB a Point C is symmetrical to itself with respect to the line a. Points A and M are called symmetrical with respect to the point O if the point O is the midpoint of the segment AM. central symmetry. Axial symmetry. The line a is called the axis of symmetry of the figure. A segment, a ray, a pair of intersecting lines, a square?

"Axial and central symmetry" - 1) How many axes of symmetry does the figure have? 7) Find an object that has axial and central symmetry. plant symmetry. Geometric ornaments. Symmetry in the animal world. 4) Find figures that have a center of symmetry and axial symmetry. Symmetry in architecture. 2) Find a figure that does not have central symmetry.

There are 11 presentations in total in the topic

"Point of symmetry" - Symmetry in architecture. Examples of symmetry of plane figures. Two points A and A1 are called symmetric with respect to O if O is the midpoint of the segment AA1. Examples of figures with central symmetry are the circle and the parallelogram. Point C is called the center of symmetry. Symmetry in science and technology.

"Construction of geometric shapes" - Educational aspect. Control and correction of assimilation. The study of the theory on which the method is based. In stereometry - not strict constructions. Stereometric constructions. algebraic method. Transformation method (similarity, symmetry, parallel translation, etc.). For example: straight; angle bisector; median perpendicular.

"Human Figure" - The shape and movement of the human body is largely determined by the skeleton. Fair with a theatrical performance. Do you think there is a job for an artist in a circus? The skeleton plays the role of a frame in the structure of the figure. Main Body (belly, chest) Didn't pay attention Head, face, hands. A. Mathis. Proportions. Ancient Greece.

"Symmetry about a line" - Symmetry about a line is called axial symmetry. The straight line a is the axis of symmetry. Symmetry about a straight line. Bulavin Pavel, 9B class. How many axes of symmetry does each figure have? A figure can have one or more axes of symmetry. central symmetry. Equosceles trapezoid. Rectangle.

"Squares of figures geometry" - Pythagorean theorem. Areas of various figures. Solve the puzzle. Figures with equal areas are called equal areas. Area units. Area of ​​a triangle. Rectangle, triangle, parallelogram. square centimeter. Figures of equal area. Equal figures b). square millimeter. in). What will be the area of ​​the figure made up of figures A and D.

"Limit of a function at a point" - Then in this case. When striving. Limit of a function at a point. Continuous at a point. Equal to the value of the function in. But when calculating the limit of the function at. Equal to value. Expression. Aspiration. Or you can say this: in a sufficiently small neighborhood of the point. Compiled from. Solution. Continuous on intervals. In between.

Consider axial and central symmetries as properties of some geometric figures; Consider axial and central symmetries as properties of some geometric figures; Be able to build symmetrical points and be able to recognize figures that are symmetrical about a point or a line; Be able to build symmetrical points and be able to recognize figures that are symmetrical about a point or a line; Improving problem solving skills; Improving problem solving skills; Continue work on the accuracy of recording and performing a geometric drawing; Continue work on the accuracy of recording and performing a geometric drawing;

Oral work "Gentle poll" Oral work "Gentle poll" What point is called the midpoint of the segment? Which triangle is called an isosceles triangle? What property do the diagonals of a rhombus have? Formulate the property of the bisector of an isosceles triangle. Which lines are called perpendicular? What is an equilateral triangle? What property do the diagonals of a square have? What figures are called equal?

What new concepts did you learn in class? What new concepts did you learn in class? What have you learned about geometric shapes? What have you learned about geometric shapes? Give examples of geometric figures with axial symmetry. Give examples of geometric figures with axial symmetry. Give an example of figures with central symmetry. Give an example of figures with central symmetry. Give examples of objects from the surrounding life that have one or two types of symmetry. Give examples of objects from the surrounding life that have one or two types of symmetry.