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 planning our furniture to cutting our hair.

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.

Broccoli romanesco

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.

honeycombs

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.

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, 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 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.

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.

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.

Web

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.

Crop Circles

Give a pair 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 skills, 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.

Snowflakes

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 snowflake's hexagonal shape. But at the same time, each snowflake is symmetrical, but no snowflake is alike. This is because as it falls from the sky, each snowflake experiences unique atmospheric conditions that cause its crystals to align in a certain way.

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.

Sun-Moon Symmetry

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.

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Oh symmetry! I sing a hymn to you! Oh symmetry! I sing a hymn to you! I recognize you everywhere in the world. You are in the Eiffel Tower, in a small midge, You are in a Christmas tree, by the forest path. With you in friendship and a tulip, and a rose, And a snowy swarm - a creation of frost! The concept of symmetry is well known and plays an important role in everyday life. Many creations of human hands are deliberately given a symmetrical shape for both aesthetic and practical reasons. In ancient times, the word "symmetry" was used as "harmony", "beauty". Indeed, in Greek it means "proportionality, proportionality, uniformity in the arrangement of parts"

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Central and axial symmetries Central symmetry - 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. The figure is also said to have central symmetry. Axial symmetry - A figure is called symmetrical with respect to the line a, if for each point of the figure the point symmetrical to it with respect to the line a also belongs to this figure. The line a is called the axis of symmetry of the figure. The figure is also said to have axial symmetry.

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The manifestation of symmetry in living nature Beauty in nature is not created, but only fixed, expressed. Consider the manifestation of symmetry from the “global”, namely from our planet Earth. The fact that the Earth is a sphere became known to educated people in antiquity. The earth in the view of most well-read people before the era of Copernicus was the center of the universe. Therefore, they considered the lines passing through the center of the Earth to be the center of symmetry of the Universe. Therefore, even the layout of the Earth - the globe has an axis of symmetry.

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Almost all living beings are built according to the laws of symmetry, it is not without reason that the word "symmetry" translated from Greek means "proportion". Almost all living beings are built according to the laws of symmetry, it is not without reason that the word "symmetry" translated from Greek means "proportion". Among colors, for example, rotational symmetry is observed. Many flowers can be rotated so that each petal takes the position of its neighbor, the flower is aligned with itself. The minimum angle of such a rotation for different colors is not the same. For iris, it is 120°, for bluebell - 72°, for narcissus - 60°.

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In the arrangement of leaves on the stems of plants, helical symmetry is observed. Being located as a screw along the stem, the leaves, as it were, spread out in different directions and do not obscure each other from the light), although the leaves themselves also have an axis of symmetry. Helical symmetry is observed in the arrangement of leaves on plant stems. Being located by a screw along the stem, the leaves, as it were, spread out in different directions and do not obscure each other from the light), although the leaves themselves also have an axis of symmetry

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Considering the general plan of the structure of any animal, we usually notice a well-known regularity in the arrangement of parts of the body or organs that repeat around a certain axis or occupy the same position in relation to a certain plane. This correctness is called the symmetry of the body. The phenomena of symmetry are so widespread in the animal world that it is very difficult to point out a group in which no symmetry of the body can be noticed. Both small insects and large animals have symmetry. Considering the general plan of the structure of any animal, we usually notice a well-known regularity in the arrangement of parts of the body or organs that repeat around a certain axis or occupy the same position in relation to a certain plane. This correctness is called the symmetry of the body. The phenomena of symmetry are so widespread in the animal world that it is very difficult to point out a group in which no symmetry of the body can be noticed. Both small insects and large animals have symmetry.

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The manifestation of symmetry in inanimate nature In the world of inanimate nature, the charm of symmetry is brought by crystals. 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. What is a crystal? A rigid 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. Why are crystals so beautiful and attractive? Their physical and chemical properties are determined by their geometric structure. In crystallography (the science of crystals) there is even a section called "Geometric Crystallography". In 1867, General of Artillery, Professor of the Mikhailovsky Academy in St. Petersburg A.V. Gadolin strictly mathematically deduced all combinations of symmetry elements that characterize crystalline polyhedra. In total, there are 32 types of symmetries of ideal crystal shapes.

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SYMMETRY IN LIVING NATURE. SYMMETRY AND ASYMMETRY.

Symmetry is possessed by objects and phenomena of living nature. 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.

In living nature, the vast majority of living organisms exhibit various types of symmetries (shape, similarity, relative position). Moreover, organisms of different anatomical structures can have the same type of external symmetry.

External symmetry can act as a basis for the classification of organisms (spherical, radial, axial, etc.). Microorganisms living in conditions of weak gravity have a pronounced symmetry of shape.

Asymmetry is already present at the level of elementary particles and manifests itself in the absolute predominance of particles over antiparticles in our Universe. The famous physicist F. Dyson wrote: “The discoveries of recent decades in the field of elementary particle physics force us to pay special attention to the concept of symmetry breaking. The development of the Universe since its inception looks like a continuous sequence of symmetry breaking.
At the moment of its origin in a grandiose explosion, the Universe was symmetrical and homogeneous. As it cools, one symmetry after another is broken in it, which creates opportunities for the existence of an ever greater variety of structures. The phenomenon of life naturally fits into this picture. Life is also a violation of symmetry"
Molecular asymmetry was discovered by L. Pasteur, who was the first to single out the "right" and "left" molecules of tartaric acid: the right molecules look like the right screw, and the left ones look like the left one. Chemists call such molecules stereoisomers. Molecules stereoisomers have the same atomic composition, the same size, the same structure - at the same time, they are distinguishable, because they are mirror asymmetric, i.e. the object turns out to be non-identical with its mirror double. 67 Therefore, here the concepts of "right-left" are conditional.
At present, it is well known that the molecules of organic substances, which form the basis of living matter, have an asymmetric character, i.e. they enter into the composition of living matter only either as right or left molecules. Thus, each substance can be a part of living matter only if it has a well-defined type of symmetry. For example, the molecules of all amino acids in any living organism can only be left-handed, sugars can only be right-handed.
This property of living matter and its waste products is called dissymmetry. It is completely fundamental. Although right and left molecules are indistinguishable in chemical properties, living matter not only distinguishes them, but also makes a choice. It rejects and does not use molecules that do not have the structure it needs. How this happens is not yet clear. Molecules of opposite symmetry are poison to her.
If a living being found itself in conditions where all food would be composed of molecules of opposite symmetry, not corresponding to the dissymmetry of this organism, then it would die of starvation. In inanimate matter, right and left molecules are equal. Asymmetry is the only property due to which we can distinguish a substance of biogenic origin from non-living matter. We cannot answer the question of what life is, but we have a way to distinguish the living from the non-living.
Thus, asymmetry can be seen as a dividing line between animate and inanimate nature. Inanimate matter is characterized by the predominance of symmetry; in the transition from inanimate to living matter, asymmetry predominates already at the micro level. In wildlife, asymmetry can be seen everywhere. V. Grossman noted this very well in the novel "Life and Fate": "In a large million Russian village huts there are not and cannot be two indistinguishably similar. All living things are unique.

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. The method of analogies is based on the principle of symmetry, which involves the search for common properties in various objects. On the basis of analogies, physical models of various objects and phenomena are created. Analogies between processes make it possible to describe them by general equations.

SYMMETRY IN THE PLANT WORLD:

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. 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 tool for 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 that even its individual organs do not possess a spatial lattice. However, ordered structures are very widely represented in it.

	honeycombs- a true 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.
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 next step A + B \u003d C, B + C \u003d D, etc.
The arrangement of achenes in the head of a sunflower or leaves in the shoots of climbing plants corresponds to a logarithmic spiral

SYMMETRY IN THE WORLD OF INSECTS, FISHES, BIRDS, ANIMALS

Symmetry types in animals

1-central		3-radial	4-bilateral
5-beam	6-progressive (metamerism)		7-translational-rotational

Axis of symmetry. The axis of symmetry is the axis of rotation. In this case, animals, as a rule, lack a center of symmetry. Then rotation can only occur around the axis. In this case, the axis most often has poles of different quality. For example, in coelenterates, hydra or sea anemones, the mouth is located on one pole, and the sole, with which these motionless animals are attached to the substrate, is located on the other (Fig. 1, 2,3). The axis of symmetry may coincide morphologically with the anteroposterior axis of the body.

Plane of symmetry. The plane of symmetry 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 (anti - against; mer - part). For example, in a hydra, the plane of symmetry must pass through the mouth opening and through the sole. The antimeres of the opposite halves must have an equal number of tentacles located around the hydra's mouth. Hydra can have several planes of symmetry, the number of which will be a multiple of the number of tentacles. Anemones with a very large number of tentacles can have many planes of symmetry. In a jellyfish with four tentacles on a bell, the number of planes of symmetry will be limited to a multiple of four. Ctenophores have only two planes of symmetry - pharyngeal and tentacle (Fig. 1, 5). Finally, bilaterally symmetrical organisms have only one plane and only two mirror antimers - respectively, the right and left sides of the animal (Fig. 1, 4,6,7).

Symmetry types. There are only two main types of symmetry - rotational and translational. In addition, there is a modification from the combination of these two main types of symmetry - rotational-translational symmetry.

rotational symmetry. Any organism has rotational symmetry. For rotational symmetry, an essential characteristic element is antimers . It is important to know, when turning by what degree, the contours of the body will coincide with the original position. The minimum degree of coincidence of the contour has a ball rotating around the center of symmetry. The maximum degree of rotation is 360, when the contours of the body coincide when rotated by this amount.

If the body rotates around the center of symmetry, then many axes and planes of symmetry can be drawn through the center of symmetry. If the body rotates around one heteropolar axis, then as many planes can be drawn through this axis as the number of antimers of the given body. Depending on this condition, one speaks of rotational symmetry of a certain order. For example, six-rayed corals will have sixth order rotational symmetry. Ctenophores have two planes of symmetry and are second order symmetrical. The symmetry of the ctenophores is also called biradial (Fig. 1, 5). Finally, if an organism has only one plane of symmetry and, accordingly, two antimeres, then such a symmetry is called bilateral or bilateral (Fig.1, 4). Thin needles emanate radiantly. This helps the protozoa "soar" in the water column. Other representatives of protozoa are also spherical - rays (radiolaria) and sunflowers with ray-like processes-pseudopodia.

translational symmetry. For translational symmetry, the characteristic element is metameres (meta - one after the other; mer - part). In this case, the parts of the body are not mirrored against each other, but sequentially one after the other along the main axis of the body.

Metamerism - one of the forms of translational symmetry. It is especially pronounced in annelids, whose long body consists of a large number of almost identical segments. This case of segmentation is called homogeneous (Fig.1, 6). In arthropods, the number of segments may be relatively small, but each segment differs somewhat from neighboring ones either in shape or in appendages (thoracic segments with legs or wings, abdominal segments). This segmentation is called heteronomous.

Rotational-translational symmetry. This type of symmetry has a limited distribution in the animal kingdom. This symmetry is characterized by the fact that when turning through a certain angle, a part of the body protrudes slightly forward and each next one increases its dimensions logarithmically by a certain amount. Thus, there is a combination of acts of rotation and translational motion. Spiral chambered shells of foraminifera, as well as spiral chambered shells of some cephalopods (modern nautilus or fossil ammonite shells, Fig. 1, 7) can serve as an example. With some condition, non-chambered spiral shells of gastropod mollusks can also be included in this group.

• Symmetry in nature.

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

• Hermann Weel

Symmetry in nature.

Symmetry is possessed not only by geometric shapes or things made by human hand, but also by many creations of nature (butterflies, dragonflies, leaves, starfish, snowflakes, etc.). The symmetry properties of crystals are especially diverse... Some of them are more symmetrical, others less so. For a long time, crystallographers could not describe all types of crystal symmetry. This problem was solved in 1890 by the Russian scientist E. S. Fedorov. He proved that there are exactly 230 groups that translate crystal lattices into themselves. This discovery made it much easier for crystallographers to study the kinds of crystals that might exist in nature. However, it should be noted that the variety of crystals in nature is so great that even the use of the group approach has not yet given a way to describe all possible forms of crystals.

Symmetry in nature.

The theory of symmetry groups is very widely used in quantum physics. The equations that describe the behavior of electrons in an atom (the so-called Schrödinger wave equation) are so complex even with a small number of electrons that it is practically impossible to solve them directly. However, using the symmetry properties of an atom (the invariance of the electromagnetic field of the nucleus during rotations and symmetries, the possibility of some electrons among themselves, i.e. the symmetrical arrangement of these electrons in the atom, etc.), it is possible to study their solutions without solving equations. In general, the use of group theory is a powerful mathematical method for studying and taking into account the symmetry of natural phenomena.

Symmetry in nature.

Mirror symmetry in nature.

Golden section.

GOLDEN SECTION - theoretically, the term was formed in the Renaissance and denotes a strictly defined mathematical ratio of proportions, in which one of the two components is as many times larger than the other as it is smaller than the whole. Artists and theorists of the past often considered the golden ratio to be an ideal (absolute) expression of proportionality, but in fact the aesthetic value of this “immutable law” is limited due to the well-known imbalance of the horizontal and vertical directions. In the practice of fine arts 3. p. rarely applied in its absolute, unchanging form; the character and measure of deviations from abstract mathematical proportionality are of great importance here.

The golden ratio in nature

• Everything that took on some form formed, grew, strove to take a place in space and preserve itself. This aspiration finds realization mainly in two variants - upward growth or spreading over the surface of the earth and twisting in a spiral.

• The shell is twisted in a spiral. If you unfold it, you get a length slightly inferior to the length of the snake. A small ten-centimeter shell has a spiral 35 cm long. Spirals are very common in nature. The concept of the golden ratio will be incomplete, if not to say about the spiral.

• Fig.1. Spiral of Archimedes.

Principles of shaping in nature.

In the lizard, at first glance, proportions that are pleasant to our eyes are captured - the length of its tail relates to the length of the rest of the body as 62 to 38. Both in the plant and animal world, the formative tendency of nature persistently breaks through - symmetry with respect to the direction of growth and movement. Here the golden ratio appears in the proportions of parts perpendicular to the direction of growth. Nature has carried out the division into symmetrical parts and golden proportions. In parts, a repetition of the structure of the whole is manifested.

The golden ratio in nature

Symmetry in art.

• In art, symmetry 1 plays a huge role, many masterpieces of architecture have symmetry. In this case, mirror symmetry is usually meant. The term "symmetry" in different historical eras was used to refer to different concepts.

• Symmetry - proportionality, correctness in the arrangement of parts of the whole.

For the Greeks, symmetry meant proportionality. It was believed that two values ​​are commensurate if there is a third value by which these two values ​​are divided without a remainder. A building (or statue) was considered symmetrical if it had some easily distinguishable part, such that the dimensions of all other parts were obtained by multiplying this part by integers, and thus the original part served as a visible and understandable module.

The golden ratio in art.

Art historians unanimously argue that there are four points of increased attention on the pictorial canvas. They are located at the corners of the quadrangle, and depend on the proportions of the subframe. It is believed that whatever the scale and size of the canvas, all four points are due to the golden ratio. All four points (they are called visual centers) are located at a distance of 3/8 and 5/8 from the edges. It is believed that this is the composition matrix of any work of fine art.

Here, for example, cameo "The Judgment of Paris" received in 1785 by the State Hermitage from the Academy of Sciences. (It adorns the goblet of Peter I.) Italian stone cutters repeated this story more than once on cameos, intaglios and carved shells. In the catalog you can read that the engraving by Marcantonio Raimondi based on the lost work of Raphael served as a pictorial prototype.

The golden ratio in art.

• Indeed, one of the four points of the golden ratio falls on the golden apple in the hand of Paris. And more precisely, on the point of connection of the apple with the palm.

• Suppose Raimondi consciously calculated this point. But one can hardly believe that the Scandinavian master of the middle of the VIII century first made “golden” calculations, and based on their results he set the proportions of the bronze Odin.

• Obviously, this happened unconsciously, that is, intuitively. And if so, then the golden ratio does not need the master (artist or craftsman) to consciously worship "gold". Enough for him to worship beauty.

• Fig.2.

• Singing One from Staraya Ladoga.

• Bronze. Middle of the 8th century.

• Height 5.4 cm. GE, No. 2551/2.

The golden ratio in art.

• "The Appearance of Christ to the People" by Alexander Ivanov. The clear effect of the Messiah's approach to people arises from the fact that he has already passed the golden section point (the crosshairs of the orange lines) and is now entering the point that we will call the point of the silver section (this is a segment divided by the number π, or a segment minus segment divided by the number π).

"The Appearance of Christ to the People".

Turning to examples of the "golden section" in painting, one cannot but stop one's attention on the work of Leonardo da Vinci. His identity is one of the mysteries of history. Leonardo da Vinci himself said: “Let no one who is not a mathematician dare to read my works.” He gained fame as an unsurpassed artist, a great scientist, a genius who anticipated many inventions that were not implemented until the 20th century. There is no doubt that Leonardo da Vinci was a great artist, this was already recognized by his contemporaries, but his personality and activities will remain shrouded in mystery, since he left to posterity not a coherent presentation of his ideas, but only numerous handwritten sketches, notes that say “both everyone in the world." He wrote from right to left in illegible handwriting and with his left hand. This is the most famous example of mirror writing in existence. The portrait of Monna Lisa (Gioconda) has attracted the attention of researchers for many years, who found that the composition of the drawing is based on golden triangles that are parts of a regular star pentagon. There are many versions about the history of this portrait. Here is one of them. Once Leonardo da Vinci received an order from the banker Francesco de le Giocondo to paint a portrait of a young woman, the banker's wife, Monna Lisa. The woman was not beautiful, but she was attracted by the simplicity and naturalness of her appearance. Leonardo agreed to paint a portrait. His model was sad and sad, but Leonardo told her a fairy tale, after hearing which she became alive and interesting.

The golden ratio in the works of Leonardo da Vinci.

• And when analyzing three portraits by Leonardo da Vinci, it turns out that they have an almost identical composition. And it is built not on the golden ratio, but on √2, the horizontal line of which in each of the three works passes through the tip of the nose.

The golden section in the painting by I. I. Shishkin "Pine Grove"

In this famous painting by I. I. Shishkin, the motifs of the golden section are clearly visible. The brightly lit pine tree (standing in the foreground) divides the length of the picture according to the golden ratio. To the right of the pine tree is a hillock illuminated by the sun. It divides the right side of the picture horizontally according to the golden ratio. To the left of the main pine there are many pines - if you wish, you can successfully continue dividing the picture according to the golden section and further. The presence in the picture of bright verticals and horizontals, dividing it in relation to the golden section, gives it the character of balance and tranquility, in accordance with the artist's intention. When the artist's intention is different, if, say, he creates a picture with a rapidly developing action, such a geometric scheme of composition (with a predominance of verticals and horizontals) becomes unacceptable.

Golden spiral in Raphael's "Massacre of the Innocents"

Unlike the golden section, the feeling of dynamics, excitement, is perhaps most pronounced in another simple geometric figure - the spiral. The multi-figure composition, made in 1509 - 1510 by Raphael, when the famous painter created his frescoes in the Vatican, is just distinguished by the dynamism and drama of the plot. Rafael never brought his idea to completion, however, his sketch was engraved by an unknown Italian graphic artist Marcantinio Raimondi, who, based on this sketch, created the Massacre of the Innocents engraving.

On Raphael's preparatory sketch, red lines are drawn running from the semantic center of the composition - the point where the warrior's fingers closed around the child's ankle - along the figures of the child, the woman clutching him to herself, the warrior with a raised sword, and then along the figures of the same group on the right side sketch. If you naturally connect these pieces of the curve with a dotted line, then with very high accuracy you get ... a golden spiral! This can be checked by measuring the ratio of the lengths of the segments cut by the spiral on the straight lines passing through the beginning of the curve.

Golden section in architecture.

As G.I. Sokolov, the length of the hill in front of the Parthenon, the length of the temple of Athena and the section of the Acropolis behind the Parthenon correlate as segments of the golden ratio. When looking at the Parthenon at the location of the monumental gate at the entrance to the city (Propylaea), the ratio of the rock mass at the temple also corresponds to the golden ratio. Thus, the golden ratio was already used when creating the composition of the temples on the sacred hill.

• Many researchers who sought to uncover the secret of the harmony of the Parthenon searched for and found the golden section in the ratios of its parts. If we take the end facade of the temple as a unit of width, then we get a progression consisting of eight members of the series: 1: j: j 2: j 3: j 4: j 5: j 6: j 7, where j = 1.618.

The Golden Ratio in Literature.

Symmetry in the story "Heart of a Dog"

Golden proportions in literature. Poetry and the golden ratio

Much in the structure of poetic works makes this art form related to music. A clear rhythm, a regular alternation of stressed and unstressed syllables, an ordered dimensionality of poems, their emotional richness make poetry a sister of musical works. Each verse has its own musical form - its own rhythm and melody. It can be expected that in the structure of poems some features of musical works, patterns of musical harmony, and, consequently, the golden ratio, will appear.

Let's start with the size of the poem, that is, the number of lines in it. It would seem that this parameter of the poem can change arbitrarily. However, it turned out that this was not the case. For example, the analysis of poems by A.S. Pushkin showed from this point of view that the sizes of verses are distributed very unevenly; it turned out that Pushkin clearly prefers sizes of 5, 8, 13, 21 and 34 lines (Fibonacci numbers).

The golden section in the poem by A.S. Pushkin.

• Many researchers have noticed that poems are like pieces of music; they also have climactic points that divide the poem in proportion to the golden ratio. Consider, for example, a poem by A.S. Pushkin "Shoemaker":

Golden proportions in literature.

• One of Pushkin's last poems "I don't value high-profile rights ..." consists of 21 lines and two semantic parts are distinguished in it: in 13 and 8 lines.

Look at the faces of the people around you: one eye is a little more squinted, the other less, one eyebrow is more arched, the other less; one ear is higher, the other is lower. To the above, we add that a person uses the right eye more than the left. Watch, for example, people who shoot with a gun or a bow.

From the above examples, it can be seen that in the structure of the human body, his habits, the desire to sharply single out any direction - right or left - is clearly expressed. This is not an accident. Similar phenomena can also be noted in plants, animals and microorganisms.

Scientists have long paid attention to this. Back in the 18th century the scientist and writer Bernardin de Saint Pierre pointed out that all the seas are filled with single-leaved gastropod molluscs of countless species, in which all curls are directed from left to right, like the movement of the Earth, if you put them with holes to the north and sharp ends to the Earth.

But before proceeding to consider the phenomena of such asymmetry, we first find out what symmetry is.

In order to understand at least the main results achieved in the study of the symmetry of organisms, one must begin with the basic concepts of the theory of symmetry itself. Remember which bodies in everyday life are usually considered equal. Only those that are exactly the same, or, more precisely, that, when superimposed on each other, are combined with each other in all their details, such as, for example, the two upper petals in Figure 1. However, in the theory of symmetry, in addition to compatible equality, two more types of equality are distinguished - mirror and compatible-mirror. With mirror equality, the left petal from the middle row of Figure 1 can be exactly aligned with the right petal only after preliminary reflection in the mirror. And with compatible-mirror equality of two bodies, they can be combined with each other both before and after reflection in the mirror. The petals of the lower row in figure 1 are equal to each other and compatible and mirror.

Figure 2 shows that the presence of some equal parts in the figure is still not enough to recognize the figure as symmetrical: on the left they are irregularly located and we have an asymmetrical figure, on the right - uniformly and we have a symmetrical rim. Such a regular, uniform arrangement of equal parts of the figure relative to each other is called symmetry.

The equality and uniformity of the arrangement of the parts of the figure is revealed through symmetry operations. Symmetry operations are called rotations, translations, reflections.

For us, rotations and reflections are the most important here. Rotations are understood as ordinary 360° rotations around an axis, as a result of which equal parts of a symmetrical figure exchange places, and the figure as a whole is combined with itself. In this case, the axis around which the rotation occurs is called the simple axis of symmetry. (This name is not accidental, since in the theory of symmetry there are also various kinds of complex axes.) The number of combinations of a figure with itself during one complete revolution around the axis is called the order of the axis. Thus, the image of a starfish in Figure 3 has one simple fifth-order axis passing through its center.

This means that by rotating the image of a star around its axis by 360 °, we will be able to superimpose equal parts of its figure on top of each other five times.

Reflections are understood as any mirror reflections - at a point, line, plane. The imaginary plane that divides the figures into two mirror equal halves is called the plane of symmetry. Consider in Figure 3 a flower with five petals. It has five planes of symmetry intersecting on axes of the fifth order. The symmetry of this flower can be described as follows: 5 * m. The number 5 here means one axis of symmetry of the fifth order, and m is a plane, the point is the sign of the intersection of five planes on this axis. The general formula for the symmetry of similar figures is written as n*m, where n is the axis symbol. Moreover, it can have values ​​​​from 1 to infinity (?).

When studying the symmetry of organisms, it was found that in wildlife, symmetry of the form n * m is most common. Biologists call the symmetry of this type radial (radial). In addition to the flower and starfish shown in Figure 3, radial symmetry is inherent in jellyfish and polyps, cross sections of fruits of apples, lemons, oranges, persimmons (Figure 3), etc.

With the emergence of living nature on our planet, new types of symmetry arose and developed, which before that either did not exist at all, or there were few. This is especially well seen in the example of a special case of symmetry of the form n * m, which is characterized by only one plane of symmetry, dividing the figure into two mirror equal halves. In biology, this case is called bilateral (two-sided) symmetry. In inanimate nature, this type of symmetry does not have a predominant significance, but it is extremely richly represented in living nature (Fig. 4).

It is characteristic of the external structure of the human body, mammals, birds, reptiles, amphibians, fish, many mollusks, crustaceans, insects, worms, as well as many plants, such as snapdragon flowers.

It is believed that such symmetry is associated with differences in the movement 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 into a circular one. Therefore, it is no coincidence that actively mobile animals are bilaterally symmetrical.

The bilaterality of immobile organisms and their organs arises due to the unequal conditions of the attached and free sides. This seems to be the case with certain leaves, flowers, and rays of coral polyps.

Here it is appropriate to note that among organisms there has not yet been a symmetry, which is limited to the presence of only a center of symmetry. In nature, this case of symmetry is common, perhaps, only among crystals; this includes, among other things, the blue crystals of copper sulphate that grow magnificently from the solution.

Another main type of symmetry is characterized by only one axis of symmetry of the nth order and is called axial or axial (from the Greek word "axon" - axis). Until very recently, organisms whose form is characterized by axial symmetry (with the exception of the simplest, particular case, when n = 1) were not known to biologists. However, it has recently been discovered that this symmetry is widespread in the plant kingdom. It is inherent in the corollas of all those plants (jasmine, mallow, phlox, fuchsia, cotton, yellow gentian, centaury, oleander, etc.), the edges of the petals of which lie on top of each other fan-shaped clockwise or against it (Fig. 5).

This symmetry is also inherent in some animals, for example, the jellyfish Aurelia Insulinda (Fig. 6). All these facts led to the establishment of the existence of a new class of symmetry in living nature.

Objects of axial symmetry are special cases of bodies of dissymmetric, i.e., detuned, symmetry. They differ from all other objects, in particular, in their peculiar attitude to mirror reflection. If the egg of a bird and the body of a crayfish after mirror reflection do not change their shape at all, then (Fig. 7)

an axial pansy flower (a), an asymmetric helical mollusk shell (b) and, for comparison, a clock (c), a quartz crystal (d), an asymmetric molecule (e) after mirror reflection change their shape, acquiring a number of opposite features. The hands of a real clock and a mirror clock move in opposite directions; the lines on the page of the magazine are written from left to right, and the mirror ones are written from right to left, all letters seem to be turned inside out; the stalk of a climbing plant and the helical shell of a gastropod mollusk in front of the mirror go from left to top to right, and mirror ones go from right to top to left, etc.

As for the simplest, particular case of axial symmetry (n=1), which is mentioned above, it has long been known to biologists and is called asymmetric. For an example, it suffices to refer to the picture of the internal structure of the vast majority of animal species, including humans.

Already from the above examples, it is easy to see that dissymmetric objects can exist in two varieties: in the form of an original and a mirror reflection (human hands, mollusk shells, pansies, quartz crystals). At the same time, one of the forms (it doesn’t matter which one) is called the right P, and the other the left - L. Here it is very important to understand for yourself that right and left can be called and are called not only the hands or feet of a person known in this regard, but also any dissymmetric bodies - products of human production (screws with right and left threads), organisms, inanimate bodies.

The discovery of P-L-forms in living nature also posed a number of new and very profound questions for biology at once, many of which are now being solved by complex mathematical and physico-chemical methods.

The first question is the question of the patterns of form and structure of P- and L-biological objects.

More recently, scientists have established a deep structural unity of dissymmetric objects of animate and inanimate nature. The fact is that right-leftism is a property that is equally inherent in living and inanimate bodies. Various phenomena connected with rightism-leftism turned out to be common for them. Let us point out only one such phenomenon - dissymmetric isomerism. It shows that in the world there are many objects of different structure, but with the same set of parts that make up these objects.

Figure 8 shows the predicted and then discovered 32 forms of buttercup corollas. Here in each case the number of parts (petals) is the same - five each; only their mutual arrangement is different. Therefore, here we have an example of dissymmetric isomerism of rims.

As another example, objects of a completely different nature of the glucose molecule can serve. We can consider them along with buttercup corollas just because of the similarity of the laws of their structure. The composition of glucose is as follows: 6 carbon atoms, 12 hydrogen atoms, 6 oxygen atoms. This set of atoms can be distributed in space in quite different ways. Scientists believe that glucose molecules can exist in at least 320 different forms.

The second question is: how common are the P- and L-forms of living organisms in nature?

The most important discovery in this respect was made in the study of the molecular structure of organisms. It turned out that the protoplasm of all plants, animals and micro-organisms mainly absorbs only P-sugars. Thus, every day we eat the right sugar. But amino acids are found mainly in the L-form, and the proteins built from them are mainly in the P-form.

Let's take two protein products as an example: egg white and sheep's wool. Both of them are "right-handed". Wool and egg white "left-handed" in nature have not yet been found. If we could somehow create L-wool, i.e. such wool, in which the amino acids would be located along the walls of the screw curving to the left, then the problem of fighting moths would be solved: moths can only eat P-wool, just like that the same as people absorb only the P-protein of meat, milk, eggs. And it's not hard to understand. Moth digests wool, and man digests meat through special proteins - enzymes, which are also right in their configuration. And just as L-screw cannot be screwed into U-threaded nuts, it is impossible to digest L-wool and L-meat by means of P-enzymes, if such could be found.

Perhaps this is also the mystery of the disease known as cancer: there is evidence that in some cases cancer cells build themselves not from the right, but from the left proteins that are not digested by our enzymes.

The widely known antibiotic penicillin is produced by the mold fungus only in the U-form; its artificially prepared L-form is not antibiotically active. In pharmacies, the antibiotic chloramphenicol is sold, and not its antipode, chloramphenicol, since the latter is significantly inferior to the former in its medicinal properties.

Tobacco contains L-nicotine. It is several times more toxic than P-nicotine.

If we consider the external structure of organisms, then here we will see the same thing. In the vast majority of cases, whole organisms and their organs are found in the P- or L-form. The rear part of the body of wolves and dogs is somewhat sideways when running, so they are divided into right- and left-running. Left-handed birds fold their wings so that the left wing overlaps the right, while right-handed birds do the opposite. Some pigeons prefer to circle to the right while others fly to the left. For this, pigeons have long been divided among the people into “right” and “left”. The shell of the mollusk fruticicol lantzi is found mainly in a U-twisted form. It is remarkable that when eating carrots, the predominant P-forms of this mollusk grow beautifully, and their antipodes - L-mollusks - sharply lose weight. Due to the spiral arrangement of cilia on its body, ciliates move in a drop of water, like many other protozoa, along a left-curling corkscrew. Ciliates burrowing into the medium along the right spin are rare. Narcissus, barley, cattail, and others are right-handed: their leaves are found only in the U-screw form (Fig. 9). But the beans are left-handed: the leaves of the first tier are more often L-shaped. It is remarkable that, compared with P-leaves, L-leaves weigh more, have a larger area, volume, osmotic pressure of cell sap, and growth rate.

The science of symmetry can tell a lot of interesting facts about a person as well. As you know, on average, there are approximately 3% left-handers (99 million) and 97% right-handers (3 billion 201 million) on the globe. According to some information, there are much more left-handers in the USA and on the African continent than, for example, in the USSR.

It is interesting to note that the speech centers in the brain of right-handed people are located on the left, while those of left-handers are located on the right (according to other sources, in both hemispheres). The right half of the body is controlled by the left hemisphere, and the left by the right hemisphere, and in most cases the right half of the body and the left hemisphere are better developed. In humans, as you know, the heart is on the left side, the liver is on the right. But for every 7-12 thousand people there are people in whom all or part of the internal organs are mirrored, that is, vice versa.

The third question is the question of the properties of P- and L-forms. The examples already given make it clear that in living nature a number of properties of P- and L-forms are not the same. So, on examples with shellfish, beans and antibiotics, a difference was shown in nutrition, growth rate and antibiotic activity in their P- and L-forms.

Such a feature of the P- and L-forms of living nature is of great importance: it allows us to sharply distinguish living organisms from all those P- and L-bodies of inanimate nature, which are somehow equal in their properties, for example, from elementary particles.

What is the reason for all these features of dissymmetric bodies of living nature?

It was found that by growing the microorganisms Bacillus mycoides on agar-agar with P- and L-compounds (sucrose, tartaric acid, amino acids), its L-colonies can be converted into P-, and P- into L-forms. In some cases, these changes were of a long-term, possibly hereditary nature. These experiments indicate that the external P- or L-form of organisms depends on the metabolism and the P- and L-molecules involved in this exchange.

Sometimes the transformations of P-to L-forms and vice versa occur without human intervention.

Academician V. I. Vernadsky notes that all shells of fossil mollusks Fuzus antiquus found in England are left-handed, while modern shells are right-handed. Obviously, the causes that caused such changes changed during geological epochs.

Of course, the change in the types of symmetry in the course of the evolution of life occurred not only in dissymmetric organisms. So, some echinoderms were once bilaterally asymmetrical mobile forms. Then they switched to a sedentary lifestyle and they developed radial symmetry (although their larvae still retained bilateral symmetry). In some of the echinoderms that have switched to an active way of life for the second time, radial symmetry has again been replaced by bilateral symmetry (irregular hedgehogs, holothurians).

So far, we have been talking about the causes that determine the shape of P- and L-organisms and their organs. And why are these forms not found in equal quantities? As a rule, there are more of either P- or L-forms. The reasons for this are not known. According to one very plausible hypothesis, the causes may be dissymmetric elementary particles, for example, right-handed neutrinos prevailing in our world, as well as right-handed light, which always exists in a small excess in scattered sunlight. All this initially could create unequal occurrence of right and left forms of dissymmetric organic molecules, and then lead to unequal occurrence of P- and L-organisms and their parts.

These are just some of the questions of biosymmetry - the science of the processes of symmetrization and dissymmetrization in living nature.