• 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 cup 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 - a 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.

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

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

Axial symmetry occurs in nature. It determines not only the general structure of the organism, but also the possibilities of its subsequent development. The geometric shapes and proportions of living beings are formed by "axial symmetry". Its definition is formulated as follows: it is the property of objects to be combined under various transformations. The ancients believed that the sphere possesses the principle of symmetry to the fullest extent. They considered this form harmonious and perfect. Axial Symmetry in Living Nature If you look at any living being, the symmetry of the body structure immediately catches your eye. Man: two arms, two legs, two eyes, two ears, and so on. Each type of animal has a characteristic color. If a pattern appears in the coloring, then, as a rule, it is mirrored on both sides. This means that there is a certain line along which animals and people can be visually divided into two identical halves, that is, their geometric structure is based on axial symmetry. Nature creates any living organism not chaotically and senselessly, but according to the general laws of the world order, because nothing in the Universe has a purely aesthetic, decorative purpose. The presence of various forms is also due to a natural need.

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

Why does a person have some organs - paired (for example, lungs, kidneys), while others - in one copy?

First, let's try to answer an auxiliary question: why do some parts of the human body are symmetrical, while others are not?

Symmetry is a basic property of most living beings. Being symmetrical is very convenient. Think for yourself: if you have eyes, ears, noses, mouths and limbs from all sides, then you will have time to feel something suspicious in time, no matter from which side it sneaks up, and, depending on which it, this is suspicious - to eat it or, on the contrary, to run away from it.

The most flawless, "most symmetrical" of all symmetries - spherical, when the top, bottom, right, left, front and back parts of the body do not differ, and it coincides with itself when rotated around the center of symmetry at any angle. However, this is possible only in a medium that is itself ideally symmetrical in all directions and in which the same forces act on the body from all sides. But there is no such environment on our earth. There is at least one force - gravity - which acts only along one axis (up-down) and does not affect the others (forward-backward, right-left). She pulls everything down. And living beings have to adapt to this.

So the following type of symmetry arises - radial. Radially symmetrical creatures have a top and bottom, but no right and left, front and back. They coincide with themselves when rotating around only one axis. These include, for example, starfish and hydras. These creatures are inactive and are engaged in "silent hunting" for living creatures passing by.

But if a creature is going to lead an active lifestyle, chasing prey and running away from predators, another direction becomes important for it - front-to-back. The part of the body that is in front, when the animal moves, becomes more significant. All the sense organs “crawl” here, and at the same time the nerve nodes that analyze the information received from the sense organs (for some lucky ones, these nodes will later turn into the brain). In addition, the mouth must be in front in order to have time to grab the overtaken prey. All this is usually located on a separate part of the body - the head (in principle, radially symmetrical animals do not have a head). This is how bilateral(or bilateral) symmetry. In a bilaterally symmetrical creature, the top and bottom, front and back parts are different, and only the right and left are identical and are mirror images of each other. This type of symmetry is characteristic of most animals, including humans.

In some animals, for example, annelids, in addition to bilateral, there is one more symmetry - metameric. Their body (with the exception of the very front part) consists of identical metameric segments, and if you move along the body, the worm "coincides" with itself. In more advanced animals, including humans, there is a faint “echo” of this symmetry: in a sense, our vertebrae and ribs can also be called metameres.

So why does a person have paired organs, we figured it out. Now let's discuss where the unpaired ones came from.

To begin with, let's try to understand: what is the axis of symmetry for the simplest, radially symmetrical, primitive multicellular organisms? The answer is simple: it is the digestive system. The whole organism is built around it, and it is organized in such a way that each cell of the body is close to the “feeder” and receives a sufficient amount of nutrients. Imagine a hydra: its mouth is symmetrically surrounded by tentacles that drive prey there, and the intestinal cavity is located in the very middle of the body and is the axis around which the rest of the body is formed. The digestive system of such creatures is one by definition, because “under it” the whole organism is built.

Gradually, the animals became more complex, and their digestive system also became more and more perfect. The intestines elongated in order to digest food more efficiently, and therefore had to fold several times to fit in the abdominal cavity. Additional organs appeared - the liver, gallbladder, pancreas - which were located in the body asymmetrically and "moved" some other organs (for example, due to the fact that the liver is located on the right, the right kidney and the right ovary / testicle are shifted down relative to the left) . In humans, of the entire digestive system, only the mouth, pharynx, esophagus and anus have retained their position on the plane of symmetry of the organism. But the digestive system and all its organs remained with us in a single copy.

Now let's look at the circulatory system.

If the animal is small, it has no problem to ensure that nutrients reach every cell, because all cells are close enough to the digestive system. But the larger the living being, the more acute the problem of delivering food to "remote provinces" located at a great distance from the intestines, on the periphery of the body, arises for him. There is a need for something that would “feed” these areas, and in addition, connect the whole body together and allow distant regions to “communicate” with each other (and in some animals it would also carry oxygen from the respiratory organs throughout the body). This is how the circulatory system appears.

The circulatory system lines up along the digestive system, and therefore it consists, in the most primitive cases, of only two main vessels - the abdominal and dorsal - and several additional connecting them. If the creature is small and weakly mobile (like, for example, a lancelet), then in order for the blood to move through the vessels, it is enough to contract these vessels themselves. But for relatively large creatures that lead a more active lifestyle (for example, fish), this is not enough. Therefore, in them, part of the abdominal vessel turns into a special muscular organ, pushing the blood forward with force - the heart. Since it arose on an unpaired vessel, then it itself is “lonely” and unpaired. In fish, the heart is symmetrical in itself and in the body is located on the plane of symmetry. But in terrestrial animals, due to the appearance of the second circle of blood circulation, the left side of the heart muscle becomes larger than the right one, and the heart shifts to the left, losing both the symmetry of its position and its own symmetry.

Vera Bashmakova
"Elements"

A regular honeycomb pattern can be made if the cells are triangular, square or hexagonal. The hexagonal shape more than the others allows you to save on the walls, that is, less wax will be spent on honeycombs with such cells. For the first time, such “thriftiness” of bees was noticed in the 4th century AD. e., and at the same time it was suggested that the bees in the construction of honeycombs "are guided by a mathematical plan." However, researchers from Cardiff University believe that the engineering fame of bees is greatly exaggerated: the correct geometric shape of the hexagonal cells of honeycombs arises from the physical forces acting on them, and insects are only helpers here.

A variant of a non-periodic mosaic covering a plane is proposed, in which tiles of the same shape but two different colorings are used.

Ian Stewart

For many centuries, symmetry has remained a key concept for artists, architects and musicians, but in the 20th century physicists and mathematicians also appreciated its deep meaning. It is symmetry that today underlies such fundamental physical and cosmological theories as the theory of relativity, quantum mechanics and string theory. From ancient Babylon to the most advanced frontiers of modern science, Ian Stewart, a British mathematician of world renown, traces the path of studying symmetry and discovering its fundamental laws.

Caustics are ubiquitous optical surfaces and curves that occur when light is reflected and refracted. Caustics can be described as lines or surfaces along which light rays are concentrated.

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.

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.





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