Report on examples in the life of central symmetry. Axial symmetry in life and nature is quite common. Symmetry in nature

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Project work on the topic: Symmetry in human life Performed the work: Student 11 "B" MOU secondary school No. 4 Gaydukova Zhanna

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Definition of symmetry Symmetry is proportionality, the sameness in the arrangement of parts of something on opposite sides of a point, line or plane. (Ozhegov's Explanatory Dictionary) So, a geometric object is considered symmetrical if something can be done with it, after which it will remain unchanged

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Elements of symmetry When studying the structure in comparative morphology, three main elements of symmetry are used: central symmetry, axis of symmetry and plane of symmetry. These three elements of symmetry are necessary to determine the type of symmetry.

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Central symmetry This is the point around which a body rotates. During rotation, the contours of the body continuously coincide when turning through any angle in any direction. A ball can serve as an ideal figure with a center of symmetry. Among living objects, a spherical egg with a nucleus located in the center can conditionally serve as an example. Oh Oh

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Axial symmetry This is the axis of rotation, in this case there is no center of symmetry. Then rotation can only occur around the axis. In this case, the axis most often has poles of different quality. A A1 B B1 C C1

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Relative plane symmetry This is a plane passing through the axis of symmetry, coinciding with it and cutting the body into two mirror halves. These halves, located opposite each other, are called antimers a

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Symmetrical rotation A body (or figure) has rotation symmetry if, when rotated through an angle of 360 degrees / n where n is an integer, it is completely aligned with its original position near some line AB (axis of symmetry). Radial symmetry is a form of symmetry that is preserved when an object rotates around a certain point or line. Often this point coincides with the center of gravity of the object, that is, the point at which an infinite number of axes of symmetry intersect. Such objects can be a sphere, a circle, a cylinder or a cone.

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Symmetry of similarity Symmetry of similarity is a kind of analogy of the previous symmetries with the only difference that they are associated with a simultaneous decrease or increase in similar parts of the figure and the distances between them. The simplest example of such symmetry is the matryoshka doll.

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There are many other kinds of symmetry that are abstract in nature. For example: Permutation symmetry, which consists in the fact that if identical particles are interchanged, then no changes occur; Gauge symmetry is associated with a change in scale. In inanimate nature, symmetry first of all arises in such a natural phenomenon as crystals, of which almost all solid bodies are composed. It is she who determines their properties. The most obvious example of the beauty and perfection of crystals is the well-known snowflake.

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We meet with symmetry everywhere: in nature, technology, art, science. The concept of symmetry runs through the entire centuries-old history of human creativity. The principles of symmetry play an important role in physics and mathematics, chemistry and biology, engineering and architecture, painting and sculpture, poetry and music. The laws of nature also obey the principles of symmetry

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Symmetry of plants Many flowers have an interesting property: they can be rotated so that each petal takes the position of its neighbor, while the flower is combined with itself. Such a flower has an axis of symmetry. Bilateral symmetry is also possessed by plant organs, for example, the stems of many cacti. In botany, radially symmetrically built flowers are often found. 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.

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Animal Symmetry Under the symmetry in animals understand the correspondence in size, shape and shape, as well as the relative location of body parts located on opposite sides of the dividing line. The main types of symmetry are radial (radial) - it has echinoderms, coelenterates, jellyfish, etc.; or bilateral (two-sided) - we can say that every animal (whether it be an insect, fish or bird) consists of two halves - right and left. Spherical symmetry takes place in radiolarians and sunflowers. Any plane through the center divides the animal into equal halves

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Symmetry in architecture The symmetry of structures 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. Each detail in a symmetrical system exists as a twin of its obligatory pair located on the other side of the axis, and due to this it can be considered only as part of the whole. Mirror symmetry is the most common in architecture. The buildings of Ancient Egypt and the temples of ancient Greece, amphitheatres, baths, basilicas and triumphal arches of the Romans, palaces and churches of the Renaissance, as well as numerous buildings of modern architecture are subordinate to it.

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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.

Just keep in mind that once you realize this, you are likely to have an overwhelming urge to look for symmetry in everything you see.

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1. Romanesco broccoli

Perhaps when you saw Romanesco broccoli in the store, you thought it was another example of a genetically modified product. But in fact, this is another example of the fractal symmetry of nature. Each broccoli inflorescence has a logarithmic spiral pattern. Romanesco is similar in appearance to broccoli, but in taste and texture - to cauliflower. It is rich in carotenoids, as well as vitamins C and K, which makes it not only beautiful, but also healthy food.

For thousands of years, people have marveled at the perfect hexagonal shape of the honeycomb and wondered how bees can instinctively create a shape that humans can only reproduce with a compass and straightedge. How and why do bees have an urge to create hexagons? Mathematicians believe that this is the ideal shape that allows them to store the maximum amount of honey possible using the minimum amount of wax. In any case, it's all a product of nature, and it's pretty damn impressive.

3. Sunflowers

Sunflowers boast radial symmetry and an interesting type of symmetry known as the Fibonacci sequence. Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (each number is determined by the sum of the two previous numbers). If we took our time and counted the number of seeds in a sunflower, we would find that the number of spirals grows according to the principles of the Fibonacci sequence. In nature, there are so many plants (including romanesco broccoli) whose petals, seeds and leaves follow this sequence, which is why it is so difficult to find a four-leaf clover.

But why do sunflowers and other plants follow mathematical rules? Like the hexagons in the hive, it's all a matter of efficiency.

4 Nautilus Shell

In addition to plants, some animals, such as the Nautilus, follow the Fibonacci sequence. Nautilus shell twists into a "Fibonacci spiral". The shell tries to maintain the same proportional shape, which allows it to maintain it throughout its life (unlike people who change proportions throughout their lives). Not all Nautiluses have a Fibonacci shell, but they all follow a logarithmic spiral.

Before you envy mathematician clams, remember that they do not do this on purpose, it is just that this form is the most rational for them.

5. Animals

Most animals are bilaterally symmetrical, which means they can be split into two identical halves. Even humans have bilateral symmetry, and some scientists believe that human symmetry is the most important factor that influences our perception of beauty. In other words, if you have a one-sided face, then you can only hope that this is compensated by other good qualities.

Some reach complete symmetry in an effort to attract a partner, such as a peacock. Darwin was positively annoyed by this bird, and wrote in a letter that "The sight of the peacock's tail feathers, whenever I look at it, makes me sick!" To Darwin, the tail seemed cumbersome and made no evolutionary sense, as it did not fit with his theory of "survival of the fittest". He was furious until he came up with the theory of sexual selection, which claims that animals develop certain features to increase their chances of mating. Therefore, peacocks have various adaptations to attract a partner.

There are about 5,000 types of spiders, and all of them create a near-perfect circular web, with nearly evenly spaced radial support threads and a spiral web to catch prey. Scientists aren't sure why spiders love geometry so much, as tests have shown that a round web won't lure food any better than an irregularly shaped one. The scientists suggest that the radial symmetry evenly distributes the force of impact when the victim is caught in the net, resulting in fewer breaks.

Give a couple of tricksters a board, mowers, and saving darkness, and you will see that people also create symmetrical shapes. Due to the complexity of the design and incredible symmetry of crop circles, even after the creators of the circles confessed and demonstrated their skill, many people still believe that space aliens did it.

As the circles become more complex, their artificial origin becomes more and more clear. It is illogical to assume that the aliens will make their messages more and more difficult when we have not been able to decipher even the first of them.

Regardless of how they came about, crop circles are a pleasure to look at, mainly because their geometry is impressive.

Even such tiny formations as snowflakes are governed by the laws of symmetry, since most snowflakes have hexagonal symmetry. This is partly due to the way water molecules line up when they solidify (crystallize). Water molecules solidify by forming weak hydrogen bonds as they align in an ordered arrangement that balances the forces of attraction and repulsion to form the hexagonal shape of the snowflake. But at the same time, each snowflake is symmetrical, but no snowflake is alike. This is because as each snowflake falls from the sky, it experiences unique atmospheric conditions that cause its crystals to align in a certain way.

9. Milky Way Galaxy

As we have seen, symmetry and mathematical models exist almost everywhere, but are these laws of nature limited to our planet? Obviously not. A new section has recently been discovered at the edge of the Milky Way Galaxy, and astronomers believe the galaxy is a near-perfect mirror image of itself.

10. Symmetry of the Sun-Moon

Considering that the Sun is 1.4 million km in diameter and the Moon is 3474 km, it seems almost impossible that the Moon can block sunlight and provide us with about five solar eclipses every two years. How does it work? Coincidentally, along with the fact that the Sun is about 400 times wider than the Moon, the Sun is also 400 times further away. Symmetry ensures that the Sun and Moon are the same size when viewed from Earth, and so the Moon can cover the Sun. Of course, the distance from the Earth to the Sun can increase, so sometimes we see annular and partial eclipses. But every year or two, a fine alignment occurs and we witness a spectacular event known as a total solar eclipse. Astronomers don't know how common this symmetry is among other planets, but they think it's pretty rare. However, we should not assume that we are special, as this is all a matter of chance. For example, every year the Moon moves away from the Earth by about 4 cm, which means that billions of years ago, every solar eclipse would have been a total eclipse. If things continue like this, then total eclipses will eventually disappear, and this will be accompanied by the disappearance of annular eclipses. It turns out that we are simply in the right place at the right time to see this phenomenon.

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.

The theme of this work is the concept of symmetry. There is an opinion that symmetry plays a leading, although not always conscious, role in modern science, art, technology and the life around us.

What is symmetry? Why does symmetry literally permeate the entire world around us?

There are, in principle, 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 of 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.

Target : To study the manifestations of symmetry in various areas of human life and society.

Tasks:

1. Determine the main features of the concept of symmetry.

2. Determine the presence of symmetry in living and non-living nature, in linguistics, in art.

3. To study the advantages of symmetrical objects in the figurative perception of a person.

Relevance due to the fact that symmetry surrounds a person, finding its manifestation both in living and non-living nature, as well as in most of human creations: in architecture, in art, etc. Explaining the laws of symmetry is important for understanding beauty and harmony. The results of the project will be of interest to secondary school students.

In this work, I explore geometric symmetry and show that geometric symmetry is present in everything that surrounds us, which we constantly encounter in everyday life.

2. The value of symmetry in our lives.

The concept of symmetry runs through the entire centuries-old history of human creativity. Since ancient times, many peoples have owned the idea of ​​symmetry in a broad sense - as the equivalent of balance and harmony.

Forms of perception and expression in many areas of science and art are ultimately based on symmetry, which is used and manifested in specific concepts and means inherent in certain areas of science and art.

Symmetry (from the Greek symmetria - "proportionality") is a concept that means the persistence, repeatability, "invariance" of any features of the structure of the object under study when certain transformations are carried out with it.

Truly symmetrical objects surround us literally from all sides, we are dealing with symmetry wherever there is any order. Symmetry resists chaos, disorder. It turns out that symmetry is balance, orderliness, beauty, perfection.

The whole world can be considered as a manifestation of the unity of symmetry and asymmetry. A structure that is asymmetrical as a whole can be a harmonious composition of symmetrical elements.

Symmetry is manifold, ubiquitous. She creates beauty and harmony.

Over the course of millennia, in the course of social practice and knowledge of the laws of objective reality, mankind has accumulated numerous data indicating the presence of two tendencies in the surrounding world: on the one hand, towards strict orderliness, harmony, on the other hand, towards their violation. People have long paid attention to the correctness of the shape of crystals, flowers, honeycombs and other natural objects and reproduced this proportionality in works of art, in the objects they create, through the concept of symmetry.

“Symmetry,” writes the famous scientist J. Newman, “establishes a funny and amazing relationship between objects, phenomena and theories that seem to be outwardly unrelated: terrestrial magnetism, female veil, polarized light, natural selection, group theory, work habits bees in a hive, the structure of space, drawings of vases, quantum physics, flower petals, cell division of sea urchins, equilibrium configurations of crystals, Romanesque cathedrals, snowflakes, music, the theory of relativity ... "

Consider examples of symmetry in various areas of our lives.

1. Symmetry in nature.

3.1. Symmetry in inanimate nature.

A snowflake is a crystal of frozen water.

The world of crystals is a special world of symmetry, with which great discoveries are associated both in the field of mathematics and in the field of crystallography. In crystals, symmetry axes of 1,2,3,4 and 6 orders are possible.

Snowflakes are the most striking example of the beauty of axial symmetry forms. Any snowflake has a rotational axis of symmetry, and in addition, each snowflake is mirror symmetrical. (pic 1)

Fig.1 Symmetry of snowflakes: axial symmetry.

Reflection in water is the only example of horizontal symmetry in nature. (fig.2)

Fig.2 Symmetry of the lake: horizontal symmetry.

3.2 . symmetry in plants.

The symmetry of the cone characteristic of plants is clearly visible on the example of any tree (Fig. 3).

Rice. 3 Cone symmetry: axis and plane of symmetry.

The specificity of the structure of plants is determined by the characteristics of the habitat to which they adapt, the characteristics of their lifestyle. The tree absorbs moisture and nutrients from the soil due to the root system, that is, below, and the rest of the vital functions are performed by the crown, that is, at the top. Therefore, the directions "up" and "down" for the tree are significantly different. And the directions in the plane perpendicular to the vertical are practically indistinguishable for the tree: air, light, and moisture are equally supplied to the tree in all these directions. As a result, a vertical rotary axis and a vertical plane of symmetry appear.

Most flowering plants exhibit radial and bilateral symmetry. 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 for monocots, five - for dicots (Fig. 4).

Fig.4 Flower - radial symmetry (double, triple, quintuple)

Perhaps when you saw Romanesco broccoli in the store, you thought it was another example of a genetically modified product. But in fact, this is another example of the fractal symmetry of nature. Each broccoli inflorescence has a pattern of the same logarithmic spiral as the entire head (Fig. 5).

Fig.5 Brocolli - fractal symmetry

Sunflowers (Fig. 6)boast radial symmetry and an interesting type of symmetry known as the Fibonacci sequence. Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 24, 55, 89, 144, etc. (each number is determined by the sum of the two previous numbers). If we took our time and counted the number of seeds in a sunflower, we would find that the number of spirals grows according to the principles of the Fibonacci sequence. In nature, there are so many plants (including romanesco broccoli) whose petals, seeds and leaves correspond to this sequence, which is why it is so difficult to find a four-leaf clover

Fig.6 Sunflower - radial symmetry

Conclusion: In plants, we observe the following types of symmetry:

• Tree - has an axis and a plane of symmetry
• Flower - radial symmetry (coincides with itself when rotated, has many planes of symmetry passing through the center of the flower)
• Leaves of flowers - bilateral symmetry (have only one plane of symmetry)
• Broccoli - fractal symmetry

3.3 Symmetry in animals

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.

Most animals are bilaterally symmetrical, which means they can be split into two identical halves. Some reach complete symmetry in an effort to attract a partner, such as a peacock (Fig. 7).

Rice. 7 Peacock - mirror symmetry

Darwin was positively annoyed by this bird, and wrote in a letter that "The sight of the peacock's tail feathers, whenever I look at it, makes me sick!" To Darwin, the tail seemed cumbersome and made no evolutionary sense, as it did not fit with his theory of "survival of the fittest". He was furious until he came up with the theory of sexual selection, which claims that animals develop certain features to increase their chances of mating. Therefore, peacocks have various adaptations to attract a partner.

Mirror symmetry is clearly visible in the butterfly; the symmetry of left and right appears here with almost mathematical rigor (Fig. 8).

Fig. 8 Butterfly - mirror symmetry

The symmetry of the Nautilus shell is very interesting (Fig. 9).

Rice. 9 Nautilus shell - Fibonacci spiral

Nautilus shell twists into a "Fibonacci spiral". The shell tries to maintain the same proportional shape, which allows it to maintain it throughout its life (unlike people who change proportions throughout their lives). Not all Nautiluses have a Fibonacci shell, but they all follow a logarithmic spiral.

Conclusion: We see that bilateral (mirror) symmetry is a characteristic symmetry of all representatives of the animal world.

3.4. Symmetry in humans.

The human body also has bilateral symmetry (appearance and skeletal structure) (Fig. 10).

Fig.10 Bilateral symmetry

This symmetry has always been and is the main source of our aesthetic admiration for the well-built human body. Our own mirror symmetry is very convenient, it allows a person to move in a straight line and turn right and left with equal ease.

Conclusion: Man, as well as representatives of the animal world, has mirror symmetry.

4. Symmetry in Russian.

Symmetry can also be observed in the Russian language.

For example:

The letters A, M, T, W, P have a vertical axis of symmetry

B, Z, K, C, E, B, E - horizontal.

And the letters Zh, H, O, F, X have two axes of symmetry.

Symmetry can also be seen in the words: Cossack, hut.

There are whole phrases with this property (if you do not take into account the spaces between words):

“Look for a taxi”, “Argentina beckons a black man”, “Argentine appreciates a black man”,

"Lesha walked on a valve stick." And the rose fell on Azor's paw.

Such words are called palindromes.

Many poets were fond of them.

SEARCH TAXI

ARGENTINA MANIT NEGRA

LYOSHA ON THE STICK OF THE VALVE FOUND

A ROSE FALLED ON AZOR'S PAWS

Conclusion: Thus, we see an example of axial symmetry in letters, symmetry in whole phrases.

5. Symmetry in art.

5.1. Symmetry in architecture.

How long a person lives, so much he builds.

In ancient times, residential buildings were usually built symmetrically around a certain central point. Regardless of whether their shape was round,

square or rectangular, it was quite easy to determine the location of such a point in them. Very often, the hearth was placed at such a point. He was the focal point around which the life of the whole family passed.

The role of symmetry and proportions in architecture is great. It gives harmony and completeness to ancient temples, towers of medieval castles, modern buildings. Only by relentlessly following the laws of geometry, the architects of antiquity could create their masterpiece s.

Fine examples of symmetry are demonstrated by works of architecture. The general plans of buildings, facades, ornaments, cornices, columns show proportionality and harmony.

The most famous monuments are: St. Isaac's Cathedral, the Bolshoi Theater, the Winter Palace (Russia); Arc de Triomphe, Notre Dame Cathedral (France); Gugong Museum, Temple of Heaven (China); Pantheon, Milan Cathedral (Italy) (Fig. 11).

St. Isaac's Cathedral Bolshoi Theater

Winter Palace Notre Dame Cathedral

Gougong Museum Milan Cathedral

Fig.11

These architectural structures demonstrate mirror symmetry, but if we consider the individual walls of these buildings, we will see that they all have an axis of symmetry.

Symmetrical objects and buildings are more stable. Symmetry is widely used in building construction and decorative elements. This makes architectural structures more beautiful, harmonious, solemn and reliable.

Conclusion: Thus, we found out that there is mirror and axial symmetry in the buildings that surround us.

5.2. Symmetry in poetry and music.

In poetry we are dealing with the unity of symmetry and asymmetry. “The soul of music - rhythm - consists in the correct periodic repetition of parts of a musical work,” wrote the famous Russian physicist G.V. Wulf. - The correct repetition of the same parts as a whole is the essence of symmetry. We are all the more justified in applying the concept of symmetry to a piece of music in that this piece is written with the help of notes, i.e. receives a spatial geometric image, parts of which we can survey. He also wrote: “Like musical works, verbal works, especially poems, can also be symmetrical.”

The poems imply the symmetry of the alternation of rhymes, stressed syllables, that is, again, rhythm. The composer in his symphony may return to the same theme several times, gradually developing it.

Preservation of the theme and its change (development, development) - this is the unity of symmetry and asymmetry. And the more successfully the composer or poet solves the problem of the relationship between symmetry and asymmetry, the higher the artistic value of the created work of art.

Conclusion: The rhyme of poetry and the rhythm of music is one example of symmetry.

5.3. Symmetry in painting.

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 further an object is from us, the smaller it seems to us, completely fuzzy, it has fewer details, its base is higher (Fig. 12).

Fig.12 Perspective.

If we follow all the rules, then the pictures will be harmonious, they will have a sense of stability, balance. If we break some rules, then the image will immediately become original, original and interesting.

Thus, the beauty of painting is determined, first of all, by the laws of mathematics.

To analyze the symmetry of the image, one can refer to the painting “Madonna Litta” by the brilliant Italian artist and scientist Leonardo da Vinci kept in the Hermitage (Fig. 13).

Fig. 13 Madonna Litta

You can pay attention: the figures of the Madonna and the child fit into a regular triangle, which, due to its symmetry, is especially clearly perceived by the eye of the viewer. Thanks to this, the mother and child immediately find themselves in the center of attention, as if brought to the fore. The head of the Madonna is exactly, but at the same time naturally placed between two symmetrical windows in the background.

paintings. Calm horizontal lines of gentle hills and clouds are visible in the windows. All this creates a feeling of peace and tranquility, enhanced by the harmonious combination of blue with yellowish and reddish tones.

The internal symmetry of the picture is well felt.

It turns out that whenever we admire this or that work of art, we talk about harmony, beauty, emotional impact, we thereby touch on the same inexhaustible problem - the problem of the relationship between symmetry and asymmetry. As a rule, being in a museum or in a concert hall, we do not think about this problem. After all, it is impossible to feel and analyze sensation at the same time.

Conclusion: So, we see that works of art are also subject to the laws of symmetry.

6. Symmetry in mathematics.

The idea of ​​symmetry is often the starting point in the hypotheses and theories of scientists of past centuries who believed in the mathematical harmony of the universe and saw in this harmony a manifestation of the divine principle. Since ancient times, man has been actively using the idea of ​​symmetry in his reflections on the picture of the universe.

The ancient Greeks believed that the universe is symmetrical simply because symmetry is beautiful. Based on symmetry considerations, they made a number of conjectures.

So, Pythagoras (5th century BC), considering the sphere as the most symmetrical and perfect form, concluded that the Earth is spherical and moves around the sphere. At the same time, he believed that the Earth moves along the sphere of a certain “central fire”. Around the same "fire", according to Pythagoras, the six planets known at that time, as well as the Moon, the Sun, and the stars, were supposed to circulate.

Using the idea of ​​symmetry extensively, scientists liked to refer not only to the spherical shape, but also to regular convex polyhedra. Even in the days of the ancient Greeks, an amazing fact was established - there are only five correct

convex polyhedra of various shapes. Symmetries of geometric bodies attached great importance to the Greek thinkers of the Pythagorean era. They believed that in order for a body to be "perfectly symmetrical", it must have an equal number of faces meeting at the corners, and these faces must be regular polygons, that is, figures with equal sides and angles. First explored by the Pythagoreans, these five regular polyhedra were later described in detail by Plato. The ancient Greek philosopher Plato attached particular importance to regular polyhedra, considering them to be the personification of four natural elements: fire-tetrahedron (the top is always turned upwards), earth-cube (the most stable body), air-octahedron, water-icosahedron (the most "rolling" body). The dodecahedron was presented as an image of the entire universe. That is why regular polyhedra are also called Platonic solids.

geometric symmetry- this is the most famous type of symmetry for many people. A geometric object is said to be symmetrical if, after it has been transformed geometrically, it retains some of its original properties. For example, a circle rotated around its center will have the same shape and size as the original circle. Therefore, the circle is called symmetric with respect to rotation (has axial symmetry).

The simplest types of spatial symmetry are central, axial, mirror-rotation and transfer symmetry.

central symmetry.

Two points A and A1 are called symmetric with respect to the point O, if O -middle of segmentAA 1 . Point O is considered symmetrical to itself.

Axial symmetry.

Convert F shape to F shape 1 , at which each of its points goes to a point symmetric with respect to the given line, is called a symmetry transformation with respect to the line a. The line a is called the axis of symmetry.

Mirror-rotation symmetry.

If another square is inscribed inside a square with a rotation, then this will be an example of mirror-rotation symmetry.

Portable symmetry.

If, during the transfer of a flat figure F along a given straight line AB by a distance a (or a multiple of this value), the figure is combined with itself, then they speak of translational symmetry. The straight line AB is called the transfer axis, the distance is called the elementary transfer or period.

7. Conclusion

We meet with symmetry everywhere - in nature, technology, art, science. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human development. Since ancient times, man has used symmetry in architecture. It gives harmony and completeness to ancient temples, towers of medieval castles, modern buildings. Symmetry literally permeates the whole world around us.

Knowledge of the geometric laws of nature is of great practical importance. We must not only learn to understand these laws, but also make them serve us for our benefit.

The principles of symmetry play an important role in physics and mathematics, chemistry and biology, engineering and architecture, painting and sculpture, poetry and music. 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 in both the plant and animal kingdoms, but with all its diversity of living organisms, the principle of symmetry always works, and this fact once again emphasizes the harmony of our world.

8.List of literature, Internet resources.

• L. Tarasov. This amazing symmetrical world. M., 1982
• Vulf G.V. Symmetry and its manifestations in nature. M., ed. Dep. Nar. com. Enlightenment, 1991
• Editor-in-Chief Maria Aksenova. Encyclopedia for children volume 2. M., "Avanta +" 2001.
• http://bse.scilib.com/
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"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 armonia- a Greek word meaning "consistency, proportion, unity of parts and 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- the 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 as an effective method of understanding 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 considered a necessary condition for 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.

At the highest level, three types of symmetry are distinguished: 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. A graphic representation of 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: your double's right hand will actually be left, since the fingers are placed on it in reverse order. 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 sunlight from each other. 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 towards a spiral, it can be assumed that this movement is carried out along a logarithmic spiral, starting each time from a central, fixed point and combining translational movement (stretching) with a turn of 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 the earth's surface, subject to 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 of 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 is the general principle of the 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 a living cell undergoes in the process of growth in the future, belonging to an integral organism and performing 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 outside world, which is also true in the plant world. 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. To 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. The laws of natural selection and universal gravitation have contributed to the fact that the tree is not only aesthetically beautiful, but also expediently arranged.

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 experience 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. Externally, man and animals are symmetrical, but their internal structure is 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 the molecules of inanimate nature, the molecules of organic substances have a pronounced asymmetric character (chirality). Attaching great importance to the asymmetry of living matter, Pasteur considered it to be the only, clearly demarcating line that can currently be drawn between animate and inanimate nature, i.e. what distinguishes living matter from non-living matter. Modern science has 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.