Since ancient times, man has developed ideas about beauty. All creations of nature are beautiful. People are beautiful in their own way, animals and plants are delightful. The spectacle of a precious stone or a salt crystal pleases the eye, it is difficult not to admire a snowflake or a butterfly. But why is this happening? It seems to us that the appearance of objects is correct and complete, the right and left halves of which look the same, as in a mirror image.

Apparently, people of art were the first to think about the essence of beauty. Ancient sculptors who studied the structure of the human body, back in the 5th century BC. began to use the concept of "symmetry". This word is of Greek origin and means harmony, proportionality and similarity in the arrangement of the constituent parts. Plato argued that only that which is symmetrical and proportionate can be beautiful.

In geometry and mathematics, three types of symmetry are considered: axial symmetry (with respect to a straight line), central (with respect to a point) and mirror (with respect to a plane).

If each of the points of an object has its own exact mapping relative to its center within it, then there is a central symmetry. Its examples are such geometric bodies as a cylinder, a ball, a regular prism, etc.

Axial symmetry of points relative to a straight line provides that this straight line intersects the midpoint of the segment connecting the points and is perpendicular to it. Examples of a bisector of a non-expanded angle of an isosceles triangle, any line drawn through the center of a circle, etc. If axial symmetry is characteristic, the definition of mirror points can be visualized simply by bending it along the axis and folding equal halves “face to face”. The desired points will touch each other.

With mirror symmetry, the points of the object are located equally relative to the plane that passes through its center.

Nature is wise and rational, therefore almost all her creations have a harmonious structure. This applies to both living beings and inanimate objects. The structure of most life forms is characterized by one of three types of symmetry: bilateral, radial or spherical.

Most often, axial can be observed in plants that develop perpendicular to the soil surface. In this case, symmetry is the result of rotating identical elements around a common axis located in the center. The angle and frequency of their location may be different. Trees are an example: spruce, maple and others. In some animals, axial symmetry also occurs, but this is less common. Of course, mathematical precision is rarely inherent in nature, but the similarity of the elements of an organism is still striking.

Biologists often consider not axial symmetry, but bilateral (bilateral). Its examples are the wings of a butterfly or dragonfly, plant leaves, flower petals, etc. In each case, the right and left parts of the living object are equal and are mirror images of each other.

Spherical symmetry is characteristic of the fruits of many plants, some fish, mollusks and viruses. And examples of ray symmetry are some types of worms, echinoderms.

In the eyes of a person, asymmetry is most often associated with irregularity or inferiority. Therefore, in most of the creations of human hands, symmetry and harmony can be traced.

Symmetry in nature is an objective property, one of the main ones in modern natural science. This is a universal and general characteristic of our material world.

Symmetry in nature is a concept that reflects the existing order in the world, proportionality and proportionality between the elements of various systems or objects of nature, the balance of the system, orderliness, stability, that is, a certain

Symmetry and asymmetry are opposite concepts. The latter reflects the disorder of the system, the lack of balance.

Symmetry shapes

Modern natural science defines a number of symmetries that reflect the properties of the hierarchy of individual levels of organization of the material world. Various types or forms of symmetries are known:

• space-time;
• calibration;
• isotopic;
• mirror;
• permutation.

All listed types of symmetries can be divided into external and internal.

External symmetry in nature (spatial or geometric) is represented by a huge variety. This applies to crystals, living organisms, molecules.

Internal symmetry is hidden from our eyes. It manifests itself in laws and mathematical equations. For example, Maxwell's equation, which determines the relationship between magnetic and electrical phenomena, or Einstein's property of gravity, which links space, time, and gravity.

Why is symmetry important in life?

Symmetry in living organisms was formed in the process of evolution. The very first organisms that originated in the ocean had a perfect spherical shape. In order to take root in a different environment, they had to adapt to new conditions.

One of the ways of such adaptation is the symmetry in nature at the level of physical forms. The symmetrical arrangement of body parts provides balance in movement, vitality and adaptation. The external forms of humans and large animals are quite symmetrical. In the plant world, too, there is symmetry. For example, the conical shape of the spruce crown has a symmetrical axis. This is a vertical trunk, thickened downwards for stability. Separate branches are also symmetrical with respect to it, and the shape of the cone allows rational use of solar energy by the crown. The external symmetry of animals helps them to maintain balance when moving, to enrich themselves with energy from the environment, using it rationally.

Symmetry is also present in chemical and physical systems. So, the most stable are molecules that have high symmetry. Crystals are highly symmetrical bodies; three dimensions of an elementary atom are periodically repeated in their structure.

Asymmetry

Sometimes the internal arrangement of organs in a living organism is asymmetric. For example, the heart is located in a person on the left, the liver is on the right.

Plants in the process of life from the soil absorb chemical mineral compounds from symmetrical molecules and in their body convert them into asymmetric substances: proteins, starch, glucose.

Asymmetry and symmetry in nature are two opposite characteristics. These are categories that are always in struggle and unity. Different levels of development of matter can have the properties of either symmetry or asymmetry.

If we assume that equilibrium is a state of rest and symmetry, and movement and non-equilibrium are caused by asymmetry, then we can say that the concept of equilibrium in biology is no less important than in physics. Biological is characterized by the principle of stability of thermodynamic equilibrium It is the asymmetry, which is a stable dynamic equilibrium, that can be considered a key principle in solving the problem of the origin of life.

We get used to the concept of symmetry from childhood. We know that a butterfly is symmetrical: it has the same right and left wings; a wheel is symmetrical, the sectors of which are the same; symmetrical patterns of ornaments, stars of snowflakes.

Truly boundless literature is devoted to the problem of symmetry. From textbooks and scientific monographs to works that pay attention not so much to drawings and formulas as to artistic images.

The very term "symmetry" in Greek means "proportion", which the ancient philosophers understood as a special case of harmony - the coordination of parts within the framework of the whole. Since ancient times, many peoples have owned the idea of ​​symmetry in a broad sense - as the equivalent of balance and harmony.

Symmetry is one of the most fundamental and one of the most general laws of the universe: inanimate, living nature and society. We see her everywhere. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human knowledge; it is widely used by all areas of modern science without exception. Truly symmetrical objects surround us literally from all sides, we are dealing with symmetry wherever there is any order. It turns out that symmetry is balance, orderliness, beauty, perfection. It is diverse, ubiquitous. She creates beauty and harmony. Symmetry literally permeates the whole world around us, which is why the topic I have chosen will always be relevant.

Symmetry expresses the preservation of something with some changes or the preservation of something, despite the change. Symmetry implies the immutability of not only the object itself, but also any of its properties in relation to the transformations performed on the object. The immutability of certain objects can be observed in relation to various operations - to rotations, translations, mutual replacement of parts, reflections, etc. In this regard, different types of symmetry are distinguished. Consider all types in more detail.

AXIAL SYMMETRY.

Symmetry about a straight line is called axial symmetry (mirror reflection about a straight line).

If the point A lies on the axis l, then it is symmetrical to itself, i.e. A coincides with A1.

In particular, if under the transformation of symmetry about the l-axis the figure F goes into itself, then it is called symmetric about the l-axis, and the l-axis is called its axis of symmetry.

CENTRAL SYMMETRY.

A figure is called centrally symmetric if there is a point about which each point of the figure is symmetrical to some point of the same figure. Namely: a movement that changes directions to opposite directions is a central symmetry.

Point O is called the center of symmetry and is fixed. This transformation has no other fixed points. Examples of figures that have a center of symmetry are a parallelogram, a circle, etc.

The familiar notions of rotation and translation are used to define the so-called translational symmetry. Let us consider translational symmetry in more detail.

1. TURN

A transformation in which each point A of a figure (body) rotates through the same angle α around a given center O is called rotation or rotation of the plane. The point O is called the center of rotation, and the angle α is called the angle of rotation. Point O is the fixed point of this transformation.

The rotational symmetry of a circular cylinder is interesting. It has an infinite number of 2nd order rotary axes and one infinitely high order rotary axis.

2. PARALLEL TRANSFER

A transformation in which each point of a figure (body) moves in the same direction by the same distance is called parallel translation.

To specify the parallel translation transformation, it is enough to specify the vector a.

3. SLIDING SYMMETRY

A sliding symmetry is a transformation in which axial symmetry and parallel translation are sequentially performed. Sliding symmetry is an isometry of the Euclidean plane. A sliding symmetry is a composition of a symmetry with respect to some line l and translation by a vector parallel to l (this vector may be zero).

A sliding symmetry can be represented as a composition of 3 axial symmetries (Schall's theorem).

MIRROR SYMMETRY

What could be more like my hand or my ear than their own reflection in the mirror? And yet the hand I see in the mirror cannot be put in the place of the real hand.

Immanuel Kant.

If a symmetry transformation with respect to a plane transforms a figure (body) into itself, then the figure is called symmetric with respect to the plane, and the given plane is called the plane of symmetry of this figure. This symmetry is called mirror symmetry. As the name itself shows, mirror symmetry relates an object and its reflection in a flat mirror. Two symmetrical bodies cannot be “inserted into each other”, because in comparison with the object itself, its trans-mirror twin turns out to be turned inside out along the direction perpendicular to the plane of the mirror.

Symmetrical figures, for all their similarities, differ significantly from each other. The double observed in the mirror is not an exact copy of the object itself. The mirror does not just copy the object, but swaps (represents) the parts of the object that are front and back with respect to the mirror. For example, if your mole is on your right cheek, then your mirror double is on your left. Bring a book to the mirror and you will see that the letters are as if turned inside out. In the mirror, everything is rearranged from right to left.

Mirror equal bodies are called bodies if, with their proper displacement, they can form two halves of a mirror symmetric body.

2.2 Symmetry in nature

A figure has symmetry if there is a movement (non-identical transformation) that transforms it into itself. For example, a figure has rotational symmetry if it is translated into itself by some rotation. But in nature, with the help of mathematics, beauty is not created, as in technology and art, but is only fixed, expressed. It not only pleases the eye and inspires poets of all times and peoples, but allows living organisms to better adapt to their environment and simply survive.

The basis of the structure of any living form is the principle of symmetry. From direct observation we can deduce the laws of geometry and feel their incomparable perfection. This order, which is a natural necessity, since nothing in nature serves purely decorative purposes, helps us find a common harmony on which the entire universe is based.

We see that nature designs any living organism according to a certain geometric pattern, and the laws of the universe have a clear justification.

The principles of symmetry underlie the theory of relativity, quantum mechanics, solid state physics, atomic and nuclear physics, elementary particle physics. These principles are most clearly expressed in the properties of the invariance of the laws of nature. In this case, we are talking not only about physical laws, but also about others, for example, biological ones.

Speaking about the role of symmetry in the process of scientific knowledge, we should highlight the use of the method of analogies. According to the French mathematician D. Poya, "there are probably no discoveries either in elementary or higher mathematics, or, perhaps, in any other area that could be made without analogies." Most of these analogies are based on common roots, general patterns that manifest themselves in the same way at different levels of the hierarchy.

So, in the modern sense, symmetry is a general scientific philosophical category that characterizes the structure of the organization of systems. The most important property of symmetry is the preservation (invariance) of certain attributes (geometric, physical, biological, etc.) with respect to well-defined transformations. The mathematical apparatus for studying symmetry today is the theory of groups and the theory of invariants.

Symmetry in the plant world

The specificity of the structure of plants is determined by the characteristics of the habitat to which they adapt. Any tree has a base and a top, "top" and "bottom" that perform different functions. The significance of the difference between the upper and lower parts, as well as the direction of gravity determine the vertical orientation of the "tree cone" rotary axis and symmetry planes. With the help of the root system, a tree absorbs moisture and nutrients from the soil, that is, from below, and the rest of the vital functions are performed by the crown, that is, at the top. At the same time, directions in a plane perpendicular to the vertical are practically indistinguishable for a tree; in all these directions, air, light, and moisture are equally supplied to the tree.

The tree has a vertical rotary axis (cone axis) and vertical planes of symmetry.

When we want to draw a leaf of a plant or a butterfly, we have to take into account their axial symmetry. The midrib for the leaf serves as the axis of symmetry. Leaves, branches, flowers, fruits have pronounced symmetry. Leaves are mirror symmetrical. The same symmetry is also found in flowers, however, in them, mirror symmetry often appears in combination with rotational symmetry. There are often cases of figurative symmetry (twigs of acacia, mountain ash).

In the diverse world of colors, there are turning axes of different orders. However, 5th order rotational symmetry is the most common. This symmetry is found in many wildflowers (bellflower, forget-me-not, geranium, carnation, St. , bird cherry, mountain ash, wild rose, hawthorn), etc.

Academician N. Belov explains this fact by the fact that the 5th order axis is a kind of instrument of the struggle for existence, "insurance against petrification, crystallization, the first step of which would be their capture by a lattice." Indeed, a living organism does not have a crystalline structure in the sense that even its individual organs do not have a spatial lattice. However, ordered structures are very widely represented in it.

In his book “This Right, Left World”, M. Gardner writes: “On Earth, life originated in spherically symmetrical forms, and then began to develop along two main lines: the world of plants with cone symmetry was formed, and the world of animals with bilateral symmetry.”

In nature, there are bodies that have helical symmetry, that is, alignment with their original position after turning through an angle around an axis, an additional shift along the same axis.

If is a rational number, then the rotary axis is also the translation axis.

The leaves on the stem are not arranged in a straight line, but surround the branch in a spiral. The sum of all previous steps of the spiral, starting from the top, is equal to the value of the subsequent step A + B \u003d C, B + C \u003d D, etc.

Helical symmetry is observed in the arrangement of leaves on the stems of most plants. Being located by a screw along the stem, the leaves seem to spread out in all directions and do not obscure each other from the light, which is essential for plant life. This interesting botanical phenomenon is called phyllotaxis (literally "leaf arrangement").

Another manifestation of phyllotaxis is the structure of a sunflower inflorescence or scales of a spruce cone, in which the scales are arranged in the form of spirals and helical lines. This arrangement is especially clearly seen in the pineapple, which has more or less hexagonal cells that form rows running in different directions.

Symmetry in the animal world

The significance of the form of symmetry for an animal is easy to understand if we put it in connection with the way of life, environmental conditions. Symmetry in animals is understood as correspondence in size, shape and outline, as well as the relative location of body parts located on opposite sides of the dividing line.

Rotational symmetry of the 5th order is also found in the animal kingdom. This is symmetry, in which the object is aligned with itself when rotated around the rotation axis 5 times. Examples are the starfish and the shell of the sea urchin. The entire skin of starfish is, as it were, inlaid with small plates of calcium carbonate, needles extend from some of the plates, some of which are movable. An ordinary starfish has 5 planes of symmetry and 1 axis of rotation of the 5th order (this is the highest symmetry among animals). Her ancestors appear to have had lower symmetry. This is evidenced, in particular, by the structure of star larvae: they, like most living beings, including humans, have only one plane of symmetry. Starfish do not have a horizontal plane of symmetry: they have a "top" and a "bottom". Sea urchins are like living pincushions; their spherical body carries long and mobile needles. In these animals, the calcareous plates of the skin merged and formed a spherical shell shell. There is a mouth in the center of the bottom surface. Ambulacral legs (aqueous vascular system) are collected in 5 bands on the surface of the shell.

However, unlike the plant world, rotational symmetry is rarely observed in the animal world.

Insects, fish, eggs, and animals are characterized by an incompatible rotational symmetry difference between forward and backward directions.

The direction of movement is a fundamentally distinguished direction, with respect to which there is no symmetry in any insect, any bird or fish, any animal. In this direction, the animal rushes for food, in the same direction it escapes from its pursuers.

In addition to the direction of movement, the symmetry of living beings is determined by another direction - the direction of gravity. Both directions are essential; they define the plane of symmetry of the animal being.

Bilateral (mirror) symmetry is a characteristic symmetry of all representatives of the animal world. This symmetry is clearly visible in the butterfly. The symmetry of the left and right wing appear here with almost mathematical rigor.

We can say that every animal (as well as an insect, fish, bird) consists of two enantiomorphs - the right and left halves. Enantiomorphs are also paired parts, one of which falls into the right and the other into the left half of the animal's body. So, enantiomorphs are the right and left ear, right and left eye, right and left horn, etc.

Simplification of living conditions can lead to a violation of bilateral symmetry, and animals from bilaterally symmetrical become radially symmetrical. This applies to echinoderms (starfish, sea urchins, sea lilies). All marine animals have radial symmetry, in which body parts extend radially from a central axis, like the spokes of a wheel. The degree of activity of animals correlates with their type of symmetry. Radially symmetrical echinoderms are usually poorly mobile, move slowly, or are attached to the seabed. The body of a starfish consists of a central disk and 5-20 or more rays extending radially from it. In mathematical language, this symmetry is called rotational symmetry.

Finally, we note the mirror symmetry of the human body (we are talking about the external appearance and structure of the skeleton). This symmetry has always been and is the main source of our aesthetic admiration for the well-built human body. We will not yet understand whether there really is an absolutely symmetrical person. Everyone, of course, will have a mole, a strand of hair, or some other detail that breaks the external symmetry. The left eye is never exactly the same as the right, and the corners of the mouth are at different heights, at least in most people. Still, these are just minor inconsistencies. No one will doubt that outwardly a person is built symmetrically: the left hand always corresponds to the right and both hands are exactly the same.

Everyone knows that the similarity between our hands, ears, eyes and other parts of the body is the same as between an object and its reflection in a mirror. It is the issues of symmetry and mirror reflection that are given attention here.

Many artists paid close attention to the symmetry and proportions of the human body, at least as long as they were guided by the desire to follow nature as closely as possible in their works.

In modern schools of painting, the vertical size of the head is most often taken as a single measure. With a certain assumption, we can assume that the length of the body exceeds the size of the head by eight times. The size of the head is proportional not only to the length of the body, but also to the dimensions of other parts of the body. All people are built according to this principle, which is why, in general, we are similar to each other. However, our proportions agree only approximately, and therefore people are only similar, but not the same. Anyway, we are all symmetrical! In addition, some artists in their works especially emphasize this symmetry.

Our own mirror symmetry is very convenient for us, it allows us to move in a straight line and turn right and left with equal ease. Equally convenient mirror symmetry for birds, fish and other actively moving creatures.

Bilateral symmetry means that one side of the animal's body is a mirror image of the other side. This type of organization is characteristic of most invertebrates, especially annelids and arthropods - crustaceans, arachnids, insects, butterflies; for vertebrates - fish, birds, mammals. For the first time, bilateral symmetry appears in flatworms, in which the anterior and posterior ends of the body differ from each other.

Consider another type of symmetry that is found in the animal kingdom. This is helical or helical symmetry. Screw symmetry is symmetry with respect to a combination of two transformations - rotation and translation along the axis of rotation, that is, there is a movement along the axis of the screw and around the axis of the screw.

Examples of natural screws are: the tusk of a narwhal (a small cetacean living in the northern seas) - the left screw; snail shell - right screw; the horns of the Pamir ram are enantiomorphs (one horn is twisted along the left and the other along the right spiral). Spiral symmetry is not perfect, for example, the shell of mollusks narrows or widens at the end. Although external helical symmetry is rare in multicellular animals, many important molecules from which living organisms are built - proteins, deoxyribonucleic acids - DNA, have a helical structure.

Symmetry in inanimate nature

The symmetry of crystals is the property of crystals to be combined with themselves in various positions by rotations, reflections, parallel transfers, or a part or combination of these operations. The symmetry of the external shape (faceting) of a crystal is determined by the symmetry of its atomic structure, which also determines the symmetry of the physical properties of the crystal.

Consider carefully the multifaceted forms of crystals. First of all, it is clear that the crystals of different substances differ from each other in their shapes. Rock salt is always cubes; rock crystal - always hexagonal prisms, sometimes with heads in the form of triangular or hexagonal pyramids; diamond - most often regular octahedrons (octahedrons); ice - hexagonal prisms, very similar to rock crystal, and snowflakes are always six-pointed stars. What catches your eye when you look at crystals? First of all, their symmetry.

Many people think that crystals are beautiful, rare stones. They come in a variety of colors, are usually transparent and, best of all, have a beautiful regular shape. Most often, crystals are polyhedra, their sides (faces) are perfectly flat, the edges are strictly straight. They delight the eye with a wonderful play of light in the facets, an amazing regularity of the structure.

However, crystals are not a museum rarity at all. Crystals are all around us. Solids from which we build houses and machines, substances that we use in everyday life - almost all of them belong to crystals. Why don't we see this? The fact is that in nature bodies rarely come across in the form of separate single crystals (or, as they say, single crystals). Most often, the substance occurs in the form of firmly adherent crystalline grains of a very small size - less than a thousandth of a millimeter. Such a structure can only be seen with a microscope.

Bodies consisting of crystalline grains are called fine-crystalline, or polycrystalline ("poly" - in Greek "many").

Of course, fine-crystalline bodies should also be classified as crystals. Then it turns out that almost all the solid bodies around us are crystals. Sand and granite, copper and iron, paints - all these are crystals.

There are also exceptions; glass and plastics do not consist of crystals. Such solids are called amorphous.

To study crystals means to study almost all the bodies around us. It is clear how important this is.

Single crystals are immediately recognized by the correctness of their shapes. Flat faces and straight edges are a characteristic property of a crystal; the correctness of the form is undoubtedly connected with the correctness of the internal structure of the crystal. If the crystal is especially extended in some direction, it means that the structure of the crystal in this direction is somehow special.

There is a center of symmetry in the cube of rock salt, and in the octahedron of a diamond, and in the star of a snowflake. But in a quartz crystal there is no center of symmetry.

The most exact symmetry is realized in the world of crystals, but even here it is not ideal: cracks and scratches invisible to the eye always make equal faces slightly different from each other.

All crystals are symmetrical. This means that in each crystalline polyhedron one can find symmetry planes, symmetry axes, a center of symmetry or other symmetry elements so that identical parts of the polyhedron are aligned with each other.

All elements of symmetry repeat the same parts of the figure, all give it symmetrical beauty and completeness, but the center of symmetry is the most interesting. Not only the shape, but also many physical properties of the crystal may depend on whether there is a center of symmetry in the crystal or not.

Honeycombs are a real design masterpiece. They consist of a series of hexagonal cells. This is the densest packing, which makes it possible to place the larva in the cell in the most advantageous way and, with the maximum possible volume, to use the wax building material in the most economical way.

III Conclusion

Symmetry permeates literally everything around, capturing, it would seem, completely unexpected areas and objects. It, manifesting itself in the most diverse objects of the material world, undoubtedly reflects its most general, most fundamental properties. The principles of symmetry play an important role in physics and mathematics, chemistry and biology, engineering and architecture, painting and sculpture, poetry and music.

We see that nature designs any living organism according to a certain geometric pattern, and the laws of the universe have a clear justification. Therefore, the study of the symmetry of various natural objects and the comparison of its results is a convenient and reliable tool for understanding the basic laws of the existence of matter.

The laws of nature that govern the picture of phenomena, inexhaustible in its diversity, in turn, obey the principles of symmetry. There are many types of symmetry, both in the plant and animal kingdoms, but with all the diversity of living organisms, the principle of symmetry always works, and this fact once again emphasizes the harmony of our world. Symmetry underlies things and phenomena, expressing something common, characteristic of different objects, while asymmetry is associated with the individual embodiment of this common in a particular object.

So, on the plane we have four types of motions that transform the figure F into the equal figure F1:

1) parallel transfer;

2) axial symmetry (reflection from a straight line);

3) rotation around a point (Partial case - central symmetry);

4) "sliding" reflection.

In space, a mirror symmetry is added to the above types of symmetry.

I believe that the goal set in the abstract has been achieved. When writing an abstract, the greatest difficulty for me was my own conclusions. I think that my work will help schoolchildren to expand their understanding of symmetry. I hope that my essay will be included in the methodological fund of the mathematics classroom.

"Mathematical symmetry" - Types of symmetry. Symmetry in mathematics. HAS A LOT IN COMMON WITH AXIAL SYMMETRY IN MATH. In poetry, rhyme is a translational symmetry. Symmetry in chemistry and physics. physical symmetry. In x and m and and. Bilateral symmetry. The role of symmetry in the world. Spiral symmetry. Symmetry in chemistry.

"Ornament" - Types of ornament. Geometric. a) Inside the band. 1 2 3. Creating an ornament using axial symmetry and parallel translation. 2011. Transforms used to create an ornament: Planar. c) On both sides of the strip. Turn.

"Movement in geometry" - Movement in geometry. To what sciences is movement applied? Concept of motion Axial symmetry Central symmetry. What figure does a segment, angle, etc., pass into when moving? Give examples of movement. What is called movement? How is movement used in various areas of human activity? Mathematics is beautiful and harmonious!

"Symmetry in nature" - We are engaged in the school scientific society because we love to learn something new and unknown. In the 19th century, in Europe, there were single works devoted to the symmetry of plants. Symmetry in nature and in life. One of the main properties of geometric shapes is symmetry. The work was completed by: Zhavoronkova Tanya Nikolaeva Lera Supervisor: Artyomenko Svetlana Yurievna.

"Symmetry around us" - Rotations (rotary). Center point. Rotations. Symmetry on the plane. Axial symmetry relative to a straight line. Around us. Symmetry in space. Horizontal. Symmetry rules. Mirror. Two kinds of symmetry. All kinds of axial symmetry. The Greek word symmetry means “proportionality”, “harmony”.

"Point of symmetry" - Examples of the above types of symmetry. Such figures include a parallelogram other than a rectangle, a scalene triangle. We encounter symmetry in nature, everyday life, architecture and technology. Symmetry in architecture. Symmetry in nature. Symmetry of plane figures. A rectangle and a rhombus, which are not squares, have two axes of symmetry.

Total in the topic 32 presentations

SECONDARY EDUCATIONAL SCHOOL № 55

SOVETSKY DISTRICT CITY OF VORONEZH

Research work

on the topic:

"Symmetry in human life"

Completed by student

8 "B" class:

Mitin Alexey

Supervisor:

mathematic teacher

Belyaeva M.V.

Voronezh, 2015

Table of contents:

1. 
   Relevance of the topic.
2. 
   Symmetry and its types.
3. 
   Symmetry in art.
   1. 
      Architecture;
   2. 
      Painting;
   3. 
      Literature and music.
4. 
   Symmetry and technique.
5. 
   Symmetry in different sciences.
   1. 
      Biology;
   2. 
      Physics;
   3. 
      Chemistry.
6. 
   Conclusions.
7. 
   Used Books.

Relevance of the topic.

The beauty of many forms is based on symmetry or its types. This topic is very extensive and affects, in addition to mathematics, many other areas of science, art, and technology. It is symmetry that prevails in nature over asymmetry. Not everyone can imagine or remember any asymmetric animal, because there are not many of them, and mostly these are various bacteria or simple organisms, as well as animals that have received the property of asymmetry due to necessity. Knowledge of nature and life is the first task of man. And one of the main steps towards this goal is the knowledge of symmetry.

Symmetry is the idea with which man has been trying for centuries to explain and create order, beauty and perfection.

Herman Weil

Research objectives:

• 
  to study the concepts of symmetry and its types (central, axial, rotary, mirror, etc.),
• 
  conduct research on the study of symmetry phenomena in biology, physics, architecture, painting, literature, transport and technology;
• 
  acquisition of skills of independent work with large volumes of information.

Symmetry and its types.

The concept of symmetry began to take shape a very long time ago. The study of archaeological sites shows that humanity at the dawn of its culture already had an idea of ​​\u200b\u200bsymmetry and carried it out in drawing and in household items. Now it is widely used in many areas of modern science.

Symmetry is proportionality, proportionality in the arrangement of parts of something on both sides of the center.

For centuries, symmetry has remained a subject that fascinates philosophers, astronomers, mathematicians, artists, architects and physicists. The ancient Greeks were completely obsessed with it - and even today we tend to see symmetry in everything from furniture arrangement to hair cutting.

There are three main types of symmetry: mirror, axial and central. There are also sliding, helical, point, translational, fractal and other types of symmetry.

Axial Symmetry: Two points are said to be symmetrical with respect to a line if that line passes through the midpoint of the segment connecting these points and is perpendicular to it. Each point of this line is considered symmetrical to itself. A figure is called symmetrical with respect to a line if for each point of the figure the point symmetrical to it with respect to the line also belongs to this figure. The figure is also said to have axial symmetry. Classical figures with such symmetry will be a circle, a rectangle, a rhombus, a square, and they will have several axes of symmetry. Under axial symmetry, also in the natural sciences, rotational or radial symmetry is accepted - a form of symmetry in which a figure coincides with itself when an object rotates around a certain straight line. The center of symmetry of an object is the line on which all axes of bilateral symmetry intersect. Radial symmetry is possessed by geometric objects such as a circle, a ball, a cylinder, or a cone.

Central symmetry: two points A and A 1 are said to be symmetric with respect to point O if O is the midpoint of segment AA 1 . A figure is called symmetric with respect to the point O if for each point of the figure the point symmetric to it with respect to the point O also belongs to this figure. Point O is called the center of symmetry of the figure. This means that the figure has central symmetry.

Examples of figures that have this symmetry would be a circle and a parallelogram. The center of symmetry of a circle is the center of this circle, and the center of a parallelogram is the point of intersection of its diagonals. The simplest example I can give is plants, in almost any plant you can find a part that has central or axial symmetry, but the flower itself will have central symmetry only in the case of an even number of petals.

Mirror symmetry is such a mapping of space onto itself, in which any point M goes into a point M 1 symmetric to it with respect to this plane α. When we look in a mirror, we observe our reflection in it - this is an example of “mirror” symmetry. Mirroring is an example of a so-called "orthogonal" transformation that changes orientation. I think the reflection in the river would also be a good example of mirror symmetry. This symmetry is also called in other sciences bilateral and bilateral. It is especially noticeable in architecture, as well as in the animal world. A person also has it, and if you mentally draw a line in the center, then the right side will correspond to the left.

Symmetry in art.

We admire the beauty of the world around us and do not think about what underlies this beauty. Science and art are the two main principles in human culture, two complementary forms of the highest creative activity of man. Symmetry in art plays a huge role and almost no architectural structure can do without it.

Fine examples of symmetry are demonstrated by works of architecture. Science, technology and art are closely connected and strictly balanced in it. People have always sought to achieve harmony in architecture. Thanks to this desire, new inventions, designs and styles were born. Human creativity in all its manifestations gravitates toward symmetry. The famous French architect Le Corbusier spoke well on this subject, in his book “Architecture of the 20th century” he wrote: “A person needs order: without it, all his actions lose their coherence, logical reciprocity. The more perfect the order, the calmer and more confident a person feels. Architectural structures created by man are mostly symmetrical. They are pleasing to the eye, people consider them beautiful. Symmetry is perceived by a person as a manifestation of regularity, and therefore, of internal order. Externally, this internal order is perceived as beauty. Buildings of Ancient Egypt, amphitheatres, triumphal arches of the Romans, palaces and churches of the Renaissance, as well as numerous buildings of modern architecture are subject to mirror symmetry. The symmetry of a structure is associated with the organization of its functions. The projection of the plane of symmetry - the axis of the building - usually determines the location of the main entrance and the beginning of the main traffic flows. The school where I study also has this type of symmetry.

In art there is a mathematical theory of painting. This is perspective theory. Perspective is the doctrine of how to convey on a flat sheet of paper a sense of the depth of space, that is, to convey to others the world as we see it. It is based on the observance of several laws. The laws of perspective lie in the fact that the farther an object is from us, the smaller it seems to us, completely fuzzy, it has fewer details, its base is higher. The symmetrical composition is easily perceived by the viewer, immediately drawing attention to the center of the picture, in which the main thing, relative to which the action unfolds, is located. Renaissance painters often built their compositions according to the laws of symmetry. This construction allows you to achieve the impression of peace, majesty, special solemnity and significance of events. A person distinguishes objects around him by shape. Interest in the form of an object may be dictated by vital necessity, or it may be caused by the beauty of the form. The form, which is based on a combination of symmetry and the golden section, contributes to the best visual perception and the appearance of a sense of beauty and harmony. The whole always consists of parts, parts of different sizes are in a certain relationship to each other and to the whole.

In music and literature, symmetry and certain proportions are also observed. For example, in the second half of the 19th century, analyzing the works of Bach, E.K. Rosenov came to the conclusion that they "dominate the law of the golden section and the law of symmetry." In his study, the golden ratio is considered as a condition for the proportionality of a musical work, while the golden ratio should solve three problems: 1) Establish a proportional relationship between the whole and its parts; 2) to be a special place for the satisfaction of the prepared expectation in relation to the whole and its parts; 3) to direct the listener's attention to those parts of the musical work to which the author attaches the greatest importance in connection with the main idea of ​​the work. In the work of M.A. Marutaev, the golden section, along with the so-called qualitative and broken symmetry, is regarded as a prerequisite for harmony in music. Works devoted to the study of the golden section in music play an important role in comprehending the specifics of musical art. The most common type of symmetry in music is the translational type. In this case, a musical phrase, melody, or larger passages of a piece of music are repeated, remaining unchanged. All songs that repeat the chorus multiple times will have this kind of symmetry.

The proportion and symmetry of an object is always necessary for our visual perception in order for us to consider this object beautiful. The balance and proportion of the parts, relative to the whole, are indispensable for symmetry. Looking at symmetrical images is more pleasant than asymmetric ones. It is difficult to find a person who did not admire the ornaments. You can find an intricate combination of different types of symmetry in them.

Symmetry in technology.

Technical objects - airplanes, cars, rockets, hammers, nuts - almost all of them, from the smallest technical devices to huge rockets, have one or another symmetry, and this is not accidental. In technology, beauty, the proportionality of mechanisms is often associated with their reliability, stability in operation. The symmetrical shape of an airship, aircraft, submarine, car, etc. provides good streamlining with air or water, and hence minimal resistance to movement. Any machine, machine, device, mechanism, unit must be assembled around the established symmetry. At the dawn of the development of aviation, our famous scientists N. E. Zhukovsky and S. A. Chaplygin studied the flight of birds in order to draw conclusions about the best form of the wing and the conditions for its flight. Symmetry played a big role in this, of course. Even modern combat fighters such as the Su-27, MiG-29 and T-50 are basically designed according to the laws of symmetry.

Symmetry in different sciences.

All representatives of the animal kingdom - mammals, birds, fish, insects, worms, arachnids, etc., in their external forms and the structure of their skeleton, show us mirror symmetry, i.e., the equality of right and left. Considering any of these living beings, we can mentally draw a vertical plane through it, relative to which what is located on the right will be a mirror image of what is located on the left, and vice versa. This equality is not fulfilled with an accuracy of fractions of a millimeter, maybe not even up to a millimeter, but, nevertheless, with a certain degree of approximation, mirror symmetry is evident. Visually, we perceive living organisms as symmetrical. Reflections are understood as any mirror reflections - at a point, line, plane. The imaginary plane that divides the figures into two mirror halves is called the plane of symmetry. A butterfly, a leaf of a plant are the simplest examples of figures that have only one plane of symmetry, dividing it into two mirror equal parts. Therefore, this type of symmetry in biology is called bilateral or bilateral. It is believed that such symmetry is associated with differences in the movements of organisms up - down, forward - backward, while their movements to the right - to the left are exactly the same. Violation of bilateral symmetry inevitably leads to deceleration of the movement of one of the parties and a change in the translational movement. Therefore, it is no coincidence that actively mobile animals are bilaterally symmetrical. But this kind of symmetry is also found in immobile organisms and their organs. It arises in this case due to the unequal conditions in which the attached and free sides are located. Apparently, this explains the bilaterality of some leaves, flowers and rays of coral polyps. The specificity of the structure of plants and animals is determined by the characteristics of the habitat to which they adapt, the characteristics of their lifestyle. Any tree has a base and a top, "top" and "bottom" that perform different functions. The significance of the difference between the upper and lower parts, as well as the direction of gravity determine the vertical orientation of the "tree cone" rotary axis and symmetry planes. Leaves are mirror symmetrical. The same symmetry is also found in flowers, however, in them, mirror symmetry often appears in combination with rotational symmetry. Rotational symmetry is a symmetry in which an object is aligned with itself when rotated through 360°/n. There are often cases of figurative symmetry (twigs of acacia, mountain ash). Interestingly, in the flower world, the rotational symmetry of the 5th order is most common, which is fundamentally impossible in the periodic structures of inanimate nature. Academician N. Belov explains this fact by the fact that the 5th order axis is a kind of instrument of the struggle for existence, "insurance against petrification, crystallization, the first step of which would be their capture by a lattice." Indeed, a living organism does not have a crystalline structure in the sense that even its individual organs do not have a spatial lattice. However, ordered structures are very widely represented in it. Our further searches were focused on central symmetry. It is most characteristic of flowers and fruits of plants. Central symmetry is characteristic of various fruits, but we settled on berries: blueberries, blueberries, cherries, cranberries. Consider a section of any of these berries. In section, it is a circle, and the circle, as we know, has a center of symmetry. Central symmetry can be observed in the image of the following flowers: dandelion flower, coltsfoot flower, water lily flower, chamomile core, and in some cases the image of the entire chamomile flower also has central symmetry.

Symmetry is one of the fundamental concepts in modern physics, which plays an important role in the formulation of modern physical theories. The symmetries taken into account in physics are quite diverse, some of them are considered exact in modern physics, others are only approximate. In 1918, the German mathematician Noether proved a theorem according to which each continuous symmetry of a physical system corresponds to a certain conservation law. The presence of this theorem makes it possible to analyze a physical system based on the available data on the symmetry that this system possesses. From it, for example, it follows that the symmetry of the equations of motion of a body over time leads to the law of conservation of energy; symmetry with respect to shifts in space - to the law of conservation of momentum; symmetry with respect to rotations - to the law of conservation of angular momentum. If the laws that establish relationships between the quantities that characterize a physical system, or determine the change in these quantities over time, do not change under certain operations that the system can be subjected to, then these laws are said to have symmetry with respect to these transformations.

Supersymmetry is a hypothetical symmetry relating bosons and fermions in nature. The abstract supersymmetry transformation links the bosonic and fermionic quantum fields so that they can turn into each other. Figuratively, we can say that the supersymmetry transformation can transform matter into interaction (or into radiation), and vice versa. As of 2015, supersymmetry is a physical hypothesis that has not been experimentally confirmed. It has been absolutely established that our world is not supersymmetric in the sense of exact symmetry, since in any supersymmetric model fermions and bosons bound by a supersymmetric transformation must have the same mass, charge and other quantum numbers. This requirement is not met for particles known in nature. Regardless of the existence of supersymmetry in nature, the mathematical apparatus of supersymmetric theories turns out to be useful in various fields of physics. In particular, supersymmetric quantum mechanics makes it possible to find exact solutions of highly non-trivial Schrödinger equations. Supersymmetry turns out to be useful in some problems of statistical physics.

Symmetry in chemistry is manifested in the geometric configuration of molecules. Most simple molecules have elements of spatial symmetry of the equilibrium configuration: axes of symmetry, planes of symmetry, etc. The usual way of representing molecules in organic chemistry is through structural formulas. In 1810, D. Dalton, wishing to show his listeners how atoms combine to form chemical compounds, built wooden models of balls and rods. These models have proven to be excellent visual aids. The molecule of water and hydrogen has a plane of symmetry. Nothing will change if you swap paired atoms in a molecule; such an exchange is equivalent to a mirroring operation.

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. A crystal is a solid body that has the natural shape of a polyhedron. Salt, ice, sand, etc. are made up of crystals. First of all, Romeu-Delille emphasized the correct geometric shape of crystals based on the law of constancy of the angles between their faces. He wrote: “All the bodies of the mineral kingdom began to be attributed to the category of crystals, for which the figure of a geometric polyhedron was found ...” The correct form of crystals arises for two reasons. First, crystals are composed of elementary particles - molecules that themselves have the correct shape. Secondly, "such molecules have a remarkable property to connect with each other in a symmetrical order." Why are crystals so beautiful and attractive? Their physical and chemical properties are determined by their geometric structure.

Conclusion.

There are many types of symmetry, both in the plant and animal kingdoms, but with all the diversity of living organisms, the principle of symmetry always works, and this fact once again emphasizes the harmony of our world. The human idea of ​​beauty is formed under the influence of what a person sees in wildlife. In her creations, very far from each other, she can use the same principles. And man in painting, sculpture, architecture, music applies the same principles. The fundamental principles of beauty are proportions and symmetry. Without symmetry, our world would look very different. After all, it is precisely on symmetry that many laws are based. Almost everything around us has some form of symmetry. You can talk about it endlessly. Symmetry, manifesting itself in the most diverse objects of the natural world, undoubtedly reflects its most general properties. Therefore, the study of symmetry and comparison with the results is a convenient and reliable tool for understanding the harmony of the world.

Mathematics reveals order, symmetry and certainty, and these are the most important types of beauty.

Aristotle

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