Examples of figures with axial and central symmetry. Presentation Types of symmetry

Axial symmetry and the concept of perfection

Axial symmetry is inherent in all forms in nature and is one of the fundamental principles of beauty. Since ancient times, man has tried

comprehend the meaning of perfection. This concept was first substantiated by artists, philosophers and mathematicians Ancient Greece. And the very word "symmetry" was coined by them. It denotes the proportionality, harmony and identity of the parts of the whole. The ancient Greek thinker Plato argued that only an object that is symmetrical and proportionate can be beautiful. And indeed, those phenomena and forms that have proportionality and completeness are “pleasant to the eye”. We call them correct.

Axial symmetry as a concept

Symmetry in the world of living beings is manifested in the regular arrangement of identical parts of the body relative to the center or axis. More often in

nature is axially symmetrical. It causes not only general structure organism, but also the possibility of its subsequent development. geometric shapes and the proportions of living beings are formed by "axial symmetry". The definition of it is formulated as follows: it is the property of objects to be combined under various transformations. The ancients believed that the sphere possesses the principle of symmetry to the fullest extent. They considered this form harmonious and perfect.

Axial symmetry in wildlife

If you look at any creature, the symmetry of the structure of the body immediately catches the eye. Man: two arms, two legs, two eyes, two ears, and so on. Each type of animal has a characteristic color. If a pattern appears in the coloring, then, as a rule, it is mirrored on both sides. This means that there is a certain line along which animals and people can be visually divided into two identical halves, that is, their geometric structure is based on axial symmetry. Nature creates any living organism not chaotically and senselessly, but according to the general laws of the world order, because nothing in the Universe has a purely aesthetic, decorative purpose. Availability various forms also due to natural necessity.

Axial symmetry in inanimate nature

In the world, we are surrounded everywhere by such phenomena and objects as: a typhoon, a rainbow, a drop, leaves, flowers, etc. Their mirror, radial, central, axial symmetry are obvious. To a large extent, it is due to the phenomenon of gravity. Often, the concept of symmetry is understood as the regularity of the change of any phenomena: day and night, winter, spring, summer and autumn, and so on. In practice, this property exists wherever there is order. And the very laws of nature - biological, chemical, genetic, astronomical - are subject to the principles of symmetry common to all of us, since they have an enviable consistency. Thus, balance, identity as a principle has a universal scope. Axial symmetry in nature is one of the "cornerstone" laws on which the universe as a whole is based.

"SYMMETRY - A SYMBOL OF BEAUTY, HARMONY AND PERFECTION"

FROM symmetry(ancient Greek - "proportionality") - the regular arrangement of similar (identical) parts of the body or forms of a living organism, the totality of living organisms relative to the center or axis of symmetry. This implies that proportionality is part of harmony, the correct combination of parts of the whole.

G armony - Greek word, denoting "consistency, proportionality, unity of parts and the whole." Outwardly, harmony can manifest itself in melody, rhythm, symmetry and proportion. The law of harmony reigns in everything, And everything in the world is rhythm, chord and tone. J. Dryden

FROM perfection - highest degree, the limit of any positive quality, ability, or skill.

“Freedom is the main inner sign of every being, created in the image and likeness of God; in this sign lies the absolute perfection of the plan of creation.” N. A. Berdyaev Symmetry is the fundamental principle of the structure of the world.

Symmetry is a common phenomenon, its universality serves effective method knowledge of nature. Symmetry in nature is needed to maintain stability. Inside the external symmetry lies the internal symmetry of the construction, which guarantees balance.

Symmetry is a manifestation of the desire of matter for reliability and strength.

Symmetrical forms provide repeatability of successful forms, therefore they are more resistant to various influences. Symmetry is multifaceted.

In nature and, in particular, in living nature, symmetry is not absolute and always contains some degree of asymmetry. Asymmetry - (Greek α- - "without" and "symmetry") - lack of symmetry.

Symmetry in nature

Symmetry, like proportion, was revered necessary condition harmony and beauty.

Looking closely at nature, you can see the common even in the most insignificant things and details, find manifestations of symmetry. The shape of a tree leaf is not random: it is strictly regular. The leaf is, as it were, glued together from two more or less identical halves, one of which is mirrored relative to the other. The symmetry of the leaf is persistently repeated, whether it be a caterpillar, a butterfly, a bug, etc.

There is a very complex multilevel classification of symmetry types. Here we will not consider these difficulties of classification, we will note only the fundamental provisions and recall the simplest examples.

Actually top level There are three types of symmetry: structural, dynamic and geometric. Each of these types of symmetry at the next level is divided into classical and non-classical.

Below are the following hierarchical levels. Graphic image all levels of subordination gives a branched dendrogram.

In everyday life, we most often encounter the so-called mirror symmetry. This is the structure of objects when they can be divided into right and left or upper and lower halves by an imaginary axis, called the axis of mirror symmetry. In this case, the halves located on opposite sides of the axis are identical to each other.

Reflection in the plane of symmetry. Reflection is the most well-known and most commonly occurring type of symmetry in nature. The mirror reproduces exactly what it "sees", but the order considered is reversed: right hand your doppelgänger will actually end up on the left, since the fingers are in reverse order on it. Mirror symmetry can be found everywhere: in the leaves and flowers of plants. Moreover, mirror symmetry is inherent in the bodies of almost all living beings, and such a coincidence is by no means accidental. Mirror symmetry has everything that can be divided into two mirror equal halves. Each of the halves serves as a mirror reflection of the other, and the plane separating them is called the plane of mirror reflection, or simply the mirror plane.

rotational symmetry. The appearance of the pattern will not change if it is rotated by some angle around the axis. The symmetry that arises in this case is called rotational symmetry. The leaves and flowers of many plants exhibit radial symmetry. This is such a symmetry in which a leaf or flower, turning around the axis of symmetry, passes into itself. On cross sections of the tissues that form the root or stem of a plant, radial symmetry is clearly visible. The inflorescences of many flowers also have radial symmetry.

Flowers, mushrooms, trees have radial-beam symmetry. Here it can be noted that on unplucked flowers and mushrooms, growing trees, the symmetry planes are always oriented vertically. Determining the spatial organization of living organisms, the right angle organizes life by the forces of gravity. The biosphere (layer of being of living beings) is orthogonal to the vertical line of the earth's gravity. Vertical stems of plants, tree trunks, horizontal surfaces of water spaces and the earth's crust as a whole form a right angle. The right angle underlying the triangle governs the space of symmetry of similarities, and similarity, as already mentioned, is the goal of life. Both nature itself and the original part of man are in the power of geometry, subject to symmetry both as essences and as symbols. No matter how the objects of nature are built, each has its own main feature, which is displayed by the form, whether it is an apple, a grain of rye or a person.

Examples of radial symmetry.

The simplest type of symmetry is mirror (axial), which occurs when a figure rotates around the axis of symmetry.

In nature, mirror symmetry is characteristic of plants and animals that grow or move parallel to the surface of the Earth. For example, the wings and body of a butterfly can be called the standard of mirror symmetry.

Axial symmetry this is the result of rotating exactly the same elements around a common center. Moreover, they can be located at any angle and with different frequencies. The main thing is that the elements rotate around a single center. In nature, examples of axial symmetry are most often found among plants and animals that grow or move perpendicular to the Earth's surface.

Also exists screw symmetry.

Translation can be combined with reflection or rotation, and new symmetry operations arise. Rotation by a certain number of degrees, accompanied by translation to a distance along the axis of rotation, generates helical symmetry - the symmetry of a spiral staircase. An example of helical symmetry is the arrangement of leaves on the stem of many plants. If we consider the arrangement of leaves on a tree branch, we will notice that the leaf is separated from the other, but also rotated around the axis of the trunk.

The leaves are arranged on the trunk along a helical line so as not to obscure each other sunlight. The head of a sunflower has processes arranged in geometric spirals that unwind from the center outwards. The youngest members of the spiral are in the center. In such systems, one can notice two families of spirals that unwind in opposite directions and intersect at angles close to right. But no matter how interesting and attractive the manifestations of symmetry in the world of plants are, there are still many secrets that control the development processes. Following Goethe, who spoke of the striving of nature for a spiral, we can assume that this movement is carried out along a logarithmic spiral, starting each time from a central, fixed point and combining forward movement(stretch) with rotation rotation.

Based on this, it is possible to formulate in a somewhat simplified and schematized form (from two points) the general law of symmetry, which is clearly and everywhere manifested in nature:

1. Everything that grows or moves vertically, i.e. up or down relative to earth's surface, obeys radial-beam symmetry in the form of a fan of intersecting planes of symmetry. The leaves and flowers of many plants exhibit radial symmetry. This is such a symmetry in which a leaf or flower, turning around the axis of symmetry, passes into itself. On cross sections of the tissues that form the root or stem of a plant, radial symmetry is clearly visible. The inflorescences of many flowers also have radial symmetry.

2. Everything that grows and moves horizontally or obliquely with respect to the earth's surface is subject to bilateral symmetry, leaf symmetry.

This universal law of two postulates obeys not only flowers, animals, easily mobile liquids and gases, but also hard, unyielding stones. This law affects the changing forms of clouds. On a calm day, they have a dome shape with more or less clearly expressed radial-radial symmetry. The influence of the universal law of symmetry is, in fact, purely external, rough, imposing its stamp only on the external form of natural bodies. Their internal structure and details escape from his power.

Symmetry is based on similarity. It means such a relationship between elements, figures, when they repeat and balance each other.

Similarity symmetry. Another type of symmetry is similarity symmetry, associated with the simultaneous increase or decrease of similar parts of the figure and the distances between them. Matryoshka is an example of this kind of symmetry. Such symmetry is very widespread in wildlife. It is demonstrated by all growing organisms.

The basis of the evolution of living matter is the symmetry of similarity. Consider a rose flower or a head of cabbage. An important role in the geometry of all these natural bodies is played by the similarity of their similar parts. Such parts, of course, are interconnected by some common geometrical law, not yet known to us, which makes it possible to derive them from each other. The symmetry of similarity, realized in space and time, manifests itself everywhere in nature on everything that grows. But it is precisely the growing forms that countless figures of plants, animals and crystals belong to. The shape of the tree trunk is conical, strongly elongated. Branches are usually arranged around the trunk in a helix. This is not a simple helix: it gradually narrows towards the top. And the branches themselves decrease as they approach the top of the tree. Therefore, here we are dealing with a helical axis of symmetry of similarity.

Living nature in all its manifestations reveals the same goal, the same meaning of life: every living object repeats itself in its own kind. The main task life is life, and the accessible form of being lies in the existence of separate integral organisms. And not only primitive organizations, but also complex cosmic systems, such as man, demonstrate an amazing ability to literally repeat from generation to generation the same forms, the same sculptures, character traits, the same gestures, manners.

Nature discovers similarity as its global genetic program. The key to change also lies in similarity. Similarity governs living nature as a whole. Geometric similarity - general principle spatial organization of living structures. A maple leaf is like a maple leaf, a birch leaf is like a birch leaf. Geometric similarity permeates all branches of the tree of life. Whatever metamorphoses it undergoes in the process of growth in the future living cell, which belongs to the whole organism and performs the function of its reproduction into a new, special, single object of being, it is the point of "beginning", which, as a result of division, will be transformed into an object similar to the original one. This unites all types of living structures, for this reason there are stereotypes of life: a person, a cat, a dragonfly, an earthworm. They are endlessly interpreted and varied by division mechanisms, but remain the same stereotypes of organization, form and behavior.

For living organisms, the symmetrical arrangement of parts of the body organs helps them to maintain balance during movement and functioning, ensures their vitality and better adaptation to the surrounding world, which is also true in flora. For example, the trunk of a spruce or pine is most often straight and the branches are evenly spaced relative to the trunk. The tree, developing under the action of gravity, reaches a stable position. Towards the top of the tree, its branches become smaller in size - it takes on the shape of a cone, since light must fall on the lower branches, as well as on the upper ones. In addition, the center of gravity should be as low as possible, the stability of the tree depends on this. Laws natural selection and gravity contributed to the fact that the tree is not only aesthetically beautiful, but arranged expediently.

It turns out that the symmetry of living organisms is associated with the symmetry of the laws of nature. At the everyday level, when we see the manifestation of symmetry in animate and inanimate nature, we involuntarily feel a sense of satisfaction with the universal, as it seems to us, order that reigns in nature.

As the ordering of living organisms, their complication in the course of the development of life, asymmetry more and more prevails over symmetry, displacing it from biochemical and physiological processes. However, a dynamic process also takes place here: symmetry and asymmetry in the functioning of living organisms are closely related. Outwardly, man and animals are symmetrical, but their internal structure significantly asymmetrical. If in lower biological objects, for example, lower plants, reproduction proceeds symmetrically, then in higher ones there is a clear asymmetry, for example, the division of sexes, where each sex introduces genetic information peculiar only to it into the process of self-reproduction. Thus, the stable preservation of heredity is a manifestation of symmetry in a certain sense, while asymmetry is manifested in variability. In general, the deep internal connection of symmetry and asymmetry in living nature determines its emergence, existence and development.

The universe is an asymmetric whole, and life as it is presented must be a function of the asymmetry of the universe and its consequences. Unlike non-living molecules, molecules organic matter have a pronounced asymmetric character (chirality). Giving great importance asymmetry of living matter, Pasteur considered it to be precisely the only, clearly demarcating line that can currently be drawn between animate and inanimate nature, i.e. what distinguishes living matter from the inanimate. modern science proved that in living organisms, as in crystals, changes in structure correspond to changes in properties.

It is assumed that the resulting asymmetry occurred abruptly as a result of the Big Biological Bang (by analogy with the Big Bang, which resulted in the formation of the Universe) under the influence of radiation, temperature, electromagnetic fields, etc. and found its reflection in the genes of living organisms. This process is essentially also a process of self-organization.

Scientific and practical conference

MOU "Secondary School No. 23"

the city of Vologda

section: natural - scientific

design and research work

TYPES OF SYMMETRY

The work was done by a student of the 8th "a" class

Kreneva Margarita

Head: higher mathematics teacher

year 2014

Project structure:

1. Introduction.

2. Goals and objectives of the project.

3. Types of symmetry:

3.1. Central symmetry;

3.2. Axial symmetry;

3.3. Mirror symmetry (symmetry with respect to the plane);

3.4. Rotational symmetry;

3.5. Portable symmetry.

4. Conclusions.

Symmetry is the idea through which man has tried for centuries to comprehend and create order, beauty and perfection.

G. Weil

Introduction.

The topic of my work was chosen after studying the section "Axial and Central Symmetry" in the course "Geometry Grade 8". I was very interested in this topic. I wanted to know: what types of symmetry exist, how they differ from each other, what are the principles of construction symmetrical figures in each type.

Objective : Introduction to different types of symmetry.

Tasks:

Study the literature on this subject.

Summarize and systematize the studied material.

Prepare a presentation.

In ancient times, the word "SYMMETRY" was used in the meaning of "harmony", "beauty". Translated from Greek, this word means “proportionality, proportionality, the sameness in the arrangement of parts of something on opposite sides of a point, line or plane.

There are two groups of symmetries.

The first group includes the symmetry of positions, shapes, structures. This is the symmetry that can be directly seen. It can be called geometric symmetry.

The second group characterizes the symmetry physical phenomena and the laws of nature. This symmetry lies at the very basis of the natural-science picture of the world: it can be called physical symmetry.

I stop to studygeometric symmetry .

In turn, there are also several types of geometric symmetry: central, axial, mirror (symmetry relative to the plane), radial (or rotary), portable, and others. I will consider today 5 types of symmetry.

Central symmetry

Two points A and A 1 are called symmetric with respect to the point O if they lie on a straight line passing through m O and are on opposite sides of it at the same distance. The point O is called the center of symmetry.

The figure is called symmetrical with respect to the pointO , if for each point of the figure the point symmetrical to it with respect to the pointO also belongs to this figure. DotO called the center of symmetry of the figure, the figure is said to have central symmetry.

Examples of figures with central symmetry are the circle and the parallelogram.

The figures shown on the slide are symmetrical with respect to some point

2. Axial symmetry

Two dotsX and Y called symmetrical with respect to the linet , if this line passes through the midpoint of segment XY and is perpendicular to it. It should also be said that each point of the linet considered symmetrical to itself.

Straightt is the axis of symmetry.

The figure is said to be symmetrical with respect to a straight line.t, if for each point of the figure a point symmetrical to it with respect to a straight linet also belongs to this figure.

Straighttcalled the axis of symmetry of the figure, the figure is said to have axial symmetry.

Axial symmetry is possessed by an undeveloped angle, isosceles and equilateral triangles, a rectangle and a rhombus,letters (see presentation).

Mirror symmetry (symmetry about a plane)

Two P points 1 and P are called symmetric with respect to the plane a if they lie on a straight line perpendicular to the plane a and are at the same distance from it

Mirror symmetry well known to everyone. It connects any object and its reflection in flat mirror. One figure is said to be mirror symmetrical to the other.

On the plane, the figure with an infinite number of axes of symmetry was a circle. In space, an infinite number of planes of symmetry has a ball.

But if the circle is the only one of its kind, then in the three-dimensional world there are a number of bodies that have an infinite number of planes of symmetry: a straight cylinder with a circle at the base, a cone with a circular base, a ball.

It is easy to establish that each symmetrical plane figure can be combined with itself with the help of a mirror. It is surprising that such complex figures as a five-pointed star or an equilateral pentagon are also symmetrical. As follows from the number of axes, they are distinguished precisely by their high symmetry. And vice versa: it is not so easy to understand why such a seemingly correct figure, as an oblique parallelogram, is not symmetrical.

4. P rotational symmetry (or radial symmetry)

Rotational symmetry is symmetry that preserves the shape of an objectwhen rotating around some axis through an angle equal to 360 ° /n(or a multiple of this value), wheren= 2, 3, 4, … The indicated axis is called the rotary axisn-th order.

Atn=2 all points of the figure are rotated by an angle of 180 0 ( 360 0 /2 = 180 0 ) around the axis, while the shape of the figure is preserved, i.e. each point of the figure goes to a point of the same figure (the figure is transformed into itself). The axis is called the axis of the second order.

Figure 2 shows the axis of the third order, Figure 3 - 4th order, Figure 4 - 5th order.

An object can have more than one rotary axis: fig.1 - 3 axes of rotation, fig.2 - 4 axes, fig. 3 - 5 axes, fig. 4 - only 1 axis

The well-known letters "I" and "F" have rotational symmetry. If you rotate the letter "I" by 180 ° around an axis perpendicular to the plane of the letter and passing through its center, then the letter will be aligned with itself. In other words, the letter "I" is symmetrical with respect to rotation by 180°, 180°= 360°: 2,n=2 , so it has second-order symmetry.

Note that the letter "F" also has a rotational symmetry of the second order.

In addition, the letter and has a center of symmetry, and the letter Ф has an axis of symmetry

Let's return to examples from life: a glass, a cone-shaped pound of ice cream, a piece of wire, a pipe.

If we take a closer look at these bodies, we will notice that all of them, one way or another, consist of a circle, through an infinite number of axes of symmetry of which an infinite number of planes of symmetry pass. Most of these bodies (they are called bodies of revolution) have, of course, also a center of symmetry (the center of a circle), through which passes at least one rotary axis of symmetry.

Clearly visible, for example, is the axis of the ice cream cone. It runs from the middle of the circle (sticking out of the ice cream!) to the sharp end of the funky cone. We perceive the set of symmetry elements of a body as a kind of symmetry measure. The ball, without a doubt, in terms of symmetry is an unsurpassed embodiment of perfection, an ideal. The ancient Greeks perceived it as the most perfect body, and the circle, of course, as the most perfect flat figure.

To describe the symmetry of a particular object, it is necessary to specify all the rotation axes and their order, as well as all symmetry planes.

Consider, for example, geometric body, composed of two identical regular quadrangular pyramids.

It has one rotary axis of the 4th order (axis AB), four rotary axes of the 2nd order (axes CE,D.F., MP, NQ), five planes of symmetry (planesCDEF, AFBD, ACBE, AMBP, ANBQ).

5 . Portable symmetry

Another kind of symmetry isportable With symmetry.

They speak of such symmetry when, when a figure is moved along a straight line for some distance “a” or a distance that is a multiple of this value, it is combined with itself The straight line along which the transfer is made is called the transfer axis, and the distance "a" is called the elementary transfer, period or symmetry step.

A periodically repeating pattern on a long ribbon is called a border. In practice, borders are found in various forms (wall painting, cast iron, plaster bas-reliefs or ceramics). Borders are used by painters and artists when decorating a room. To perform these ornaments, a stencil is made. We move the stencil, turning it over or not turning it over, draw a contour, repeating the pattern, and we get an ornament (visual demonstration).

The border is easy to build using a stencil (original element), shifting or flipping it and repeating the pattern. The figure shows five types of stencils:a ) asymmetrical;b, c ) having one axis of symmetry: horizontal or vertical;G ) centrally symmetrical;d ) having two axes of symmetry: vertical and horizontal.

The following transformations are used to build borders:

a ) parallel transfer;b ) symmetry about the vertical axis;in ) central symmetry;G ) symmetry about the horizontal axis.

Similarly, you can build sockets. For this, the circle is divided inton equal sectors, in one of them a sample pattern is performed and then the latter is sequentially repeated in the remaining parts of the circle, turning the pattern each time by an angle of 360 ° /n .

A good example of the use of axial and translational symmetry is the fence shown in the photograph.

Conclusion: So there are different kinds symmetries, symmetrical points in each of these types of symmetry are built according to certain laws. In life, we everywhere meet one or another type of symmetry, and often in the objects that surround us, several types of symmetry can be noted at once. This creates order, beauty and perfection in the world around us.

LITERATURE:

Handbook of elementary mathematics. M.Ya. Vygodsky. - Publishing house "Science". - Moscow 1971. – 416pp.

Modern vocabulary foreign words. - M.: Russian language, 1993.

History of mathematics at schoolIX - Xclasses. G.I. Glaser. - Publishing house "Enlightenment". – Moscow 1983 – 351pp.

Visual geometry 5 - 6 classes. I.F. Sharygin, L.N. Erganzhiev. - Publishing house "Drofa", Moscow, 2005. - 189p.

Encyclopedia for children. Biology. S. Ismailova. – Publishing house “Avanta+”. – Moscow 1997 – 704pp.

Urmantsev Yu.A. Symmetry of nature and the nature of symmetry - M.: Thought architecture / arhkomp2. htm, , en.wikipedia.org/wiki/

The concept of movement

Let us first consider such a concept as movement.

Definition 1

A plane mapping is called a plane motion if the mapping preserves distances.

There are several theorems related to this concept.

Theorem 2

The triangle, when moving, passes into an equal triangle.

Theorem 3

Any figure, when moving, passes into a figure equal to it.

Axial and central symmetry are examples of movement. Let's consider them in more detail.

Axial symmetry

Definition 2

Points $A$ and $A_1$ are said to be symmetric with respect to the line $a$ if this line is perpendicular to the segment $(AA)_1$ and passes through its center (Fig. 1).

Picture 1.

Consider axial symmetry using the problem as an example.

Example 1

Construct a symmetrical triangle for the given triangle with respect to any of its sides.

Solution.

Let us be given a triangle $ABC$. We will construct its symmetry with respect to the side $BC$. The side $BC$ in case of axial symmetry will go into itself (follows from the definition). The point $A$ will go to the point $A_1$ as follows: $(AA)_1\bot BC$, $(AH=HA)_1$. Triangle $ABC$ will turn into triangle $A_1BC$ (Fig. 2).

Figure 2.

Definition 3

A figure is called symmetric with respect to the line $a$ if each symmetric point of this figure is contained on the same figure (Fig. 3).

Figure 3

Figure $3$ shows a rectangle. It has axial symmetry with respect to each of its diameters, as well as with respect to two straight lines that pass through the centers of opposite sides of the given rectangle.

Central symmetry

Definition 4

Points $X$ and $X_1$ are said to be symmetric with respect to the point $O$ if the point $O$ is the center of the segment $(XX)_1$ (Fig. 4).

Figure 4

Let's consider the central symmetry on the example of the problem.

Example 2

Construct a symmetrical triangle for the given triangle at any of its vertices.

Solution.

Let us be given a triangle $ABC$. We will construct its symmetry with respect to the vertex $A$. The vertex $A$ under central symmetry will go into itself (follows from the definition). The point $B$ will go to the point $B_1$ as follows $(BA=AB)_1$, and the point $C$ will go to the point $C_1$ as follows: $(CA=AC)_1$. Triangle $ABC$ goes into triangle $(AB)_1C_1$ (Fig. 5).

Figure 5

Definition 5

A figure is symmetric with respect to the point $O$ if each symmetric point of this figure is contained on the same figure (Fig. 6).

Figure 6

Figure $6$ shows a parallelogram. It has central symmetry about the point of intersection of its diagonals.

Task example.

Example 3

Let us be given a segment $AB$. Construct its symmetry with respect to the line $l$, which does not intersect the given segment, and with respect to the point $C$ lying on the line $l$.

Solution.

Let us schematically depict the condition of the problem.

Figure 7

Let us first depict the axial symmetry with respect to the straight line $l$. Since axial symmetry is a movement, then by Theorem $1$, the segment $AB$ will be mapped onto the segment $A"B"$ equal to it. To construct it, we do the following: through the points $A\ and\ B$, draw the lines $m\ and\ n$, perpendicular to the line $l$. Let $m\cap l=X,\ n\cap l=Y$. Next, draw the segments $A"X=AX$ and $B"Y=BY$.

Figure 8

Let us now depict the central symmetry with respect to the point $C$. Since the central symmetry is a motion, then by Theorem $1$, the segment $AB$ will be mapped onto the segment $A""B""$ equal to it. To construct it, we will do the following: draw the lines $AC\ and\ BC$. Next, draw the segments $A^("")C=AC$ and $B^("")C=BC$.

Figure 9

Axial symmetry and the concept of perfection

Axial symmetry is inherent in all forms in nature and is one of the fundamental principles of beauty. Since ancient times, man has tried

comprehend the meaning of perfection. This concept was first substantiated by artists, philosophers and mathematicians of Ancient Greece. And the very word "symmetry" was coined by them. It denotes the proportionality, harmony and identity of the parts of the whole. The ancient Greek thinker Plato argued that only an object that is symmetrical and proportionate can be beautiful. And indeed, those phenomena and forms that have proportionality and completeness are “pleasant to the eye”. We call them correct.

Axial symmetry as a concept

Symmetry in the world of living beings is manifested in the regular arrangement of identical parts of the body relative to the center or axis. More often in

nature is axially symmetrical. It determines not only the general structure of the organism, but also the possibilities of its subsequent development. The geometric shapes and proportions of living beings are formed by "axial symmetry". The definition of it is formulated as follows: it is the property of objects to be combined under various transformations. The ancients believed that the sphere possesses the principle of symmetry to the fullest extent. They considered this form harmonious and perfect.

Axial symmetry in wildlife

If you look at any living creature, the symmetry of the structure of the body immediately catches your eye. Man: two arms, two legs, two eyes, two ears, and so on. Each type of animal has a characteristic color. If a pattern appears in the coloring, then, as a rule, it is mirrored on both sides. This means that there is a certain line along which animals and people can be visually divided into two identical halves, that is, their geometric structure is based on axial symmetry. Nature creates any living organism not chaotically and senselessly, but according to the general laws of the world order, because nothing in the Universe has a purely aesthetic, decorative purpose. The presence of various forms is also due to a natural need.

Axial symmetry in inanimate nature