Axial and central symmetry drawings. Central and axial symmetry

Homothety and similarity.Homothety is a transformation in which each point M (plane or space) is assigned to a point M", lying on OM (Fig. 5.16), and the ratio OM":OM= λ the same for all points other than ABOUT. Fixed point ABOUT called the center of homothety. Attitude OM": OM considered positive if M" and M lie on one side of ABOUT, negative - by different sides. Number X called the homothety coefficient. At X< 0 homothety is called inverse. Atλ = - 1 homothety turns into a symmetry transformation about a point ABOUT. With homothety, a straight line goes into a straight line, the parallelism of straight lines and planes is preserved, angles (linear and dihedral) are preserved, each figure goes into it similar (Fig. 5.17).

The converse is also true. A homothety can be defined as an affine transformation in which the lines connecting corresponding points, pass through one point - the homothety center. Homothety is used to enlarge images (projection lamp, cinema).

Central and mirror symmetry. Symmetry (in in a broad sense) - a property of a geometric figure F, characterizing a certain correctness of its shape, its invariability under the action of movements and reflections. A figure Φ has symmetry (symmetrical) if there are non-identical orthogonal transformations that take this figure into itself. The set of all orthogonal transformations that combine the figure Φ with itself is the group of this figure. So, flat figure(Fig. 5.18) with a dot M, transforming-

looking into yourself in the mirror reflection, symmetrical about the straight axis AB. Here the symmetry group consists of two elements - a point M converted to M".

If the figure Φ on the plane is such that rotations relative to any point ABOUT to an angle of 360°/n, where n > 2 is an integer, translate it into itself, then the figure Ф has nth-order symmetry with respect to the point ABOUT - center of symmetry. An example of such figures is regular polygons, for example star-shaped (Fig. 5.19), which has eighth-order symmetry relative to its center. The symmetry group here is the so-called nth order cyclic group. The circle has symmetry of infinite order (since it is compatible with itself by rotating through any angle).

The simplest types of spatial symmetry are central symmetry (inversion). In this case, relative to the point ABOUT the figure Ф is combined with itself after successive reflections from three mutually perpendicular planes, i.e. a point ABOUT - the middle of the segment connecting the symmetrical points F. So, for a cube (Fig. 5.20) the point ABOUT is the center of symmetry. Points M and M" cube

Central symmetry. Central symmetry is movement.

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Symmetry

“Symmetry in Nature” - In the 19th century, in Europe, isolated works devoted to the symmetry of plants appeared. . Axial Central. One of the main properties geometric shapes is symmetry. The work was completed by: Zhavoronkova Tanya Nikolaeva Lera Supervisor: Artemenko Svetlana Yuryevna. By symmetry in a broad sense we understand any regularity in internal structure bodies or figures.

“Symmetry in art” - II.1. Proportion in architecture. Each end of the pentagonal star represents a golden triangle. II. Central axis symmetry present in almost every architectural object. Place des Vosges in Paris. Periodicity in art. Content. Sistine Madonna. Beauty is multifaceted and many-sided.

"Point of Symmetry" - Crystals rock salt, quartz, aragonite. Symmetry in the animal world. Examples of the above types of symmetry. B A O Any point on a line is a center of symmetry. This figure has central symmetry. A circular cone has axial symmetry; the axis of symmetry is the axis of the cone. An equilateral trapezoid has only axial symmetry.

“Movement in Geometry” - Movement in Geometry. How movement is used in various fields human activity? What is movement? What sciences does movement apply to? A group of theorists. Mathematics is beautiful and harmonious! Can we see movement in nature? The concept of movement Axial symmetry Central symmetry.

"Mathematical symmetry" - Symmetry. Symmetry in mathematics. Types of symmetry. In x and m and i. Rotational. Mathematical symmetry. Central symmetry. Rotational symmetry. Physical symmetry. Secret mirror world. However, complex molecules, as a rule, there is no symmetry. HAS A LOT IN COMMON WITH PROGRESSAL SYMMETRY IN MATHEMATICS.

“Symmetry around us” - Central. One kind of symmetry. Axial. In geometry there are figures that have... Rotations. Rotation (rotary). Symmetry on a plane. Horizontal. Axial symmetry is relatively straight. Greek word symmetry means “proportionality”, “harmony”. Two types of symmetry. Central relative to a point.

There are a total of 32 presentations in the topic

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 Argentinean 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. Choosing symmetrical shapes, 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)

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, as well as, of course, symmetry, are considered attractive. This term is of 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. This occurs both in living and in 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 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

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 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 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 also not completely symmetrical. 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.

« Symmetry" - word Greek origin. It means proportionality, presence of a certain order, patterns in the arrangement of parts.

Since ancient times, people have used symmetry in drawings, ornaments, and household items.
Symmetry is widespread in nature. It can be observed in the form of leaves and flowers of plants, in the arrangement various organs animals, shaped crystalline bodies, in a fluttering butterfly, a mysterious snowflake, a mosaic in a temple, a starfish.
Symmetry is widely used in practice, in construction and technology. This is strict symmetry in the form of ancient buildings, harmonious ancient Greek vases, the Kremlin building, cars, airplanes and much more. (slide 4) Examples of using symmetry are parquet and borders. (see hyperlink about the use of symmetry in borders and parquets) Let's look at a few examples where you can see symmetry in various subjects, using a slideshow (enable icon).

Definition: – is symmetry about a point.
Definition: Points A and B are symmetrical about some point O if point O is the midpoint of segment AB.
Definition: Point O is called the center of symmetry of the figure, and the figure is called centrally symmetrical.
Property: Figures that are symmetrical about a certain point are equal.
Examples:

Algorithm for constructing a centrally symmetrical figure
1. Construct a triangle A 1B 1 C 1, symmetrical to a triangle ABC, relative to the center (point) O. To do this, we connect points A, B, C with center O and continue these segments;
2. Measure the segments AO, BO, CO and lay off on the other side of point O, segments equal to them (AO=A 1 O 1, BO=B 1 O 1, CO=C 1 O 1);

3. Connect the resulting points with segments A 1 B 1; A 1 C 1; B1 C 1.
We got ∆A 1 B 1 C 1 symmetrical ∆ABC.

– this is symmetry about the drawn axis (straight line).
Definition: Points A and B are symmetrical about a certain line a if these points lie on a line perpendicular to this one and at the same distance.
Definition: An axis of symmetry is a straight line when bent along which the “halves” coincide, and a figure is called symmetrical about a certain axis.
Property: Two symmetrical figures are equal.
Examples:

Algorithm for constructing a figure symmetrical with respect to some straight line
Let's construct a triangle A1B1C1, symmetrical to triangle ABC with respect to straight line a.
For this:
1. Draw from the vertices triangle ABC straight lines perpendicular to straight line a and continue them further.
2. 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 A1B1, B1C1, B1C1.

We obtained ∆A1B1C1 symmetrical ∆ABC.