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 left - L. Here it is very important to understand that right and left can be called and are called not only human hands or legs known in this respect, 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 it were possible in some way to 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 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 has always been the mark of perfection and beauty in classical Greek illustration and aesthetics. The natural symmetry of nature in particular has been the subject of study by philosophers, astronomers, mathematicians, artists, architects and physicists such as Leonardo Da Vinci. We see this perfection every second, although we do not always notice it. Here are 10 beautiful examples of symmetry that we ourselves are a part of.

Broccoli Romanesco

This type of cabbage is known for its fractal symmetry. This is a complex pattern where the object is formed in the same geometric figure. In this case, the entire broccoli is made up of the same logarithmic spiral. Broccoli Romanesco is not only beautiful, but also very healthy, rich in carotenoids, vitamins C and K, and tastes like cauliflower.

Honeycomb

For thousands of years, bees have instinctively produced perfectly shaped hexagons. Many scientists believe that bees produce honeycombs in this form in order to retain the most honey while using the least amount of wax. Others are not so sure and believe that this is a natural formation and wax is formed when the bees make their home.

sunflowers

These children of the sun have two forms of symmetry at once - radial symmetry, and numerical symmetry of the Fibonacci sequence. The Fibonacci sequence manifests itself in the number of spirals from the seeds of a flower.

Nautilus shell

Another natural Fibonacci sequence appears in the Nautilus shell. The shell of the Nautilus grows in a “Fibonacci spiral” in a proportional shape, which allows the nautilus inside to maintain the same shape throughout its lifespan.

Animals

Animals, like people, are symmetrical on both sides. This means there is a centerline where they can be split into two identical halves.

spider web

Spiders create perfect circular webs. The web web consists of equally spaced radial levels that spiral out from the center, intertwining with each other with maximum strength.

Crop Circles.

Crop circles do not occur "naturally" at all, however it is quite amazing the symmetry that humans can achieve. Many believed that crop circles were the result of UFO visits, but in the end it turned out that this was the work of man. Crop circles show various forms of symmetry, including Fibonacci spirals and fractals.

Snowflakes

You will definitely need a microscope to witness the beautiful radial symmetry in these miniature six-sided crystals. This symmetry is formed during the crystallization process in the water molecules that form the snowflake. When water molecules freeze, they create hydrogen bonds with the hexagonal shapes.

Milky Way Galaxy

Earth is not the only place that adheres to natural symmetry and mathematics. The Milky Way Galaxy is a striking example of mirror symmetry and is made up of two main arms known as the Perseus and Scutum Centaurus. Each of these arms has a nautilus shell-like logarithmic spiral with a Fibonacci sequence that starts at the center of the galaxy and expands.

Lunar-solar symmetry

The sun is much larger than the moon, in fact four hundred times larger. However, solar eclipse events occur every five years when the lunar disk completely blocks out sunlight. The symmetry happens because the Sun is four hundred times farther from the Earth than the Moon.

In fact, symmetry is inherent in nature itself. Mathematical and logarithmic perfection creates beauty around and within us.

Symmetry (dr. gr. συμμετρία - symmetry) - the preservation of the properties of the location of the elements of the figure relative to the center or axis of symmetry in an unchanged state during any transformations.

The word "symmetry" is familiar to us from childhood. Looking in the mirror, we see symmetrical halves of the face, looking at the palms, we also see mirror-symmetrical objects. Taking a chamomile flower in our hand, we are convinced that by turning it around the stem, we can achieve the combination of different parts of the flower. This is another type of symmetry: rotary. There are a large number of types of symmetry, but all of them invariably follow one general rule: with some transformation, a symmetrical object invariably coincides with itself.

Nature does not tolerate exact symmetry . There are always at least minor deviations. So, our hands, feet, eyes and ears are not completely identical to each other, even if they are very similar. And so for each object. Nature was created not according to the principle of uniformity, but according to the principle of consistency, proportionality. Proportionality is the ancient meaning of the word "symmetry". Philosophers of antiquity considered symmetry and order to be the essence of beauty. Architects, artists and musicians have known and used the laws of symmetry since ancient times. And at the same time, a slight violation of these laws can give objects a unique charm and downright magical charm. So, it is with a slight asymmetry that some art critics explain the beauty and magnetism of the mysterious smile of the Mona Lisa by Leonardo da Vinci.

Symmetry gives rise to harmony, which is perceived by our brain as a necessary attribute of beauty. This means that even our consciousness lives according to the laws of a symmetrical world.

According to Weil, an object is called symmetric if it is possible to perform some kind of operation with which, as a result, the initial state is obtained.

Symmetry in biology is a regular arrangement of similar (identical) body parts or forms of a living organism, a set of living organisms relative to the center or axis of symmetry.

Symmetry in nature

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

The Pythagoreans paid attention to the phenomena of symmetry in living nature in Ancient Greece in connection with the development of the doctrine of harmony (V century BC). In the 19th century, single works appeared devoted to symmetry in the plant and animal world.

In the 20th century, the efforts of Russian scientists - V. Beklemishev, V. Vernadsky, V. Alpatov, G. Gause - created a new direction in the theory of symmetry - biosymmetry, which, by studying the symmetries of biostructures at the molecular and supramolecular levels, makes it possible to determine in advance possible symmetry options in biological objects, to strictly describe the external shape and internal structure of any organisms.

Symmetry in plants

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.

Plants are characterized by the symmetry of the cone, which is clearly visible in the example of any tree. 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. The tree absorbs moisture and nutrients from the soil through the root system, that is, below, and the rest of the vital functions are performed by the crown, that is, at the top. Therefore, the directions "up" and "down" for the tree are significantly different. And the directions in the plane perpendicular to the vertical are practically indistinguishable for the tree: air, light, and moisture are equally supplied to the tree in all these directions. As a result, a vertical rotary axis and a vertical plane of symmetry appear.

Most flowering plants exhibit radial and bilateral symmetry. A flower is considered symmetrical when each perianth consists of an equal number of parts. Flowers, having paired parts, are considered flowers with double symmetry, etc. Triple symmetry is common for monocotyledonous plants, five - for dicotyledons.

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

Symmetry in animals

Symmetry in animals is understood as correspondence in size, shape and outline, as well as the relative location of body parts located on opposite sides of the dividing line.

Spherical symmetry occurs in radiolarians and sunfish, whose bodies are spherical, and parts are distributed around the center of the sphere and move away from it. Such organisms have neither anterior, nor posterior, nor lateral parts of the body; any plane drawn through the center divides the animal into identical halves.

With radial or radiative symmetry, the body has the form of a short or long cylinder or vessel with a central axis, from which parts of the body extend in a radial order. These are coelenterates, echinoderms, starfish.

With mirror symmetry, there are three axes of symmetry, but only one pair of symmetrical sides. Because the other two sides - the abdominal and dorsal - are not similar to each other. This kind of symmetry is characteristic of most animals, including insects, fish, amphibians, reptiles, birds, and mammals.

Insects, fish, birds, and animals are characterized by an incompatible rotational symmetry difference between forward and backward directions. The fantastic Tyanitolkai, invented in the famous fairy tale about Dr. Aibolit, seems to be an absolutely incredible creature, since its front and back halves are symmetrical. The direction of movement is a fundamentally distinguished direction, with respect to which there is no symmetry in any insect, any fish or bird, any animal. In this direction, the animal rushes for food, in the same direction it escapes from its pursuers.

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

Bilateral (mirror) symmetry is a characteristic symmetry of all representatives of the animal world. This symmetry is clearly visible in the butterfly; the symmetry of left and right appears here with almost mathematical rigor. We can say that every animal (as well as an insect, fish, bird) consists of two enantiomorphs - the right and left halves. Enantiomorphs are also paired parts, one of which falls into the right and the other into the left half of the body of the animal. So, right and left ear, right and left eye, right and left horn, etc. are enantiomorphs.

Symmetry in humans

The human body has bilateral symmetry (appearance and skeletal structure). This symmetry has always been and is the main source of our aesthetic admiration for the well-built human body. The human body is built on the principle of bilateral symmetry.

Most of us think of the brain as a single structure, in fact it is divided into two halves. These two parts - the two hemispheres - fit snugly together. In full accordance with the general symmetry of the human body, each hemisphere is an almost exact mirror image of the other.

The control of the basic movements of the human body and its sensory functions is evenly distributed between the two hemispheres of the brain. The left hemisphere controls the right side of the brain, while the right hemisphere controls the left side.

The physical symmetry of the body and brain does not mean that the right side and the left side are equal in all respects. It is enough to pay attention to the actions of our hands to see the initial signs of functional symmetry. Only a few people are equally proficient with both hands; most have the dominant hand.

Symmetry types in animals

1. central

2. axial (mirror)

3. radial

4. bilateral

5. double beam

6. translational (metamerism)

7. translational-rotational

Symmetry types

Only two main types of symmetry are known - 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. Antimers are an essential characteristic element for rotational symmetry. It is important to know that when turning by any 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 0 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. Finally, if an organism has only one plane of symmetry and, accordingly, two antimeres, then such symmetry is called bilateral or bilateral. Thin needles emanate radiantly. This helps the protozoa "soar" in the water column. Other representatives of the protozoa are also spherical - rays (radiolaria) and sunflowers with ray-like processes-pseudopodia.

translational symmetry. For translational symmetry, metameres are a characteristic element (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 is a form 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 homonomous. 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. An example is the spiral chambered shells of foraminifera, as well as the spiral chambered shells of some cephalopods. With some condition, non-chambered spiral shells of gastropod mollusks can also be attributed to this group.

Mirror symmetry

If you stand in the center of the building and you have the same number of floors, columns, windows to your left as to the right, then the building is symmetrical. If it were possible to bend it along the central axis, then both halves of the house would coincide when superimposed. This symmetry is called mirror symmetry. This type of symmetry is very popular in the animal kingdom, the man himself is tailored according to its canons.

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 intestinal cavities, 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. The axis of symmetry may coincide morphologically with the anteroposterior axis of the body.

With mirror symmetry, the right and left parts of the object change.

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. Finally, bilaterally symmetrical organisms have only one plane and only two mirror antimeres, respectively, the right and left sides of the animal.

The transition from radial or radial to bilateral or bilateral symmetry is associated with the transition from a sedentary lifestyle to active movement in the environment. For sedentary forms, relations with the environment are equivalent in all directions: radial symmetry exactly corresponds to such a way of life. In actively moving animals, the anterior end of the body becomes biologically not equivalent to the rest of the body, the head is formed, and the right and left sides of the body become distinguishable. Due to this, radial symmetry is lost, and only one plane of symmetry can be drawn through the body of the animal, dividing the body into right and left sides. Bilateral symmetry means that one side of the animal's body is a mirror image of the other side. This type of organization is characteristic of most invertebrates, especially annelids and arthropods - crustaceans, arachnids, insects, butterflies; for vertebrates - fish, birds, mammals. For the first time, bilateral symmetry appears in flatworms, in which the anterior and posterior ends of the body differ from each other.

In annelids and arthropods, metamerism is also observed - one of the forms of translational symmetry, when parts of the body are arranged sequentially one after another along the main axis of the body. It is especially pronounced in annelids (earthworm). Annelids owe their name to the fact that their body consists of a series of rings or segments (segments). Both internal organs and body walls are segmented. So an animal consists of about a hundred more or less similar units - metameres, each of which contains one or a pair of organs of each system. The segments are separated from each other by transverse septa. In an earthworm, almost all segments are similar to each other. Annelids include polychaetes - marine forms that swim freely in the water, dig in the sand. Each segment of their body has a pair of lateral projections bearing a dense tuft of setae. Arthropods got their name for their characteristic jointed paired appendages (as swimming organs, walking limbs, mouthparts). All of them are characterized by a segmented body. Each arthropod has a strictly defined number of segments, which remains unchanged throughout life. Mirror symmetry is clearly visible in the butterfly; the symmetry of left and right appears here with almost mathematical rigor. We can say that every animal, insect, fish, bird consists of two enantiomorphs - the right and left halves. So, right and left ear, right and left eye, right and left horn, etc. are enantiomorphs.

Radial symmetry

Radial symmetry is a form of symmetry in which a body (or figure) coincides with itself when an object rotates around a certain point or line. Often this point coincides with the center of symmetry of the object, that is, the point at which an infinite number of axes of bilateral symmetry intersect.

In biology, one speaks of radial symmetry when one or more axes of symmetry pass through a three-dimensional being. Moreover, radially symmetrical animals may not have planes of symmetry. Thus, the Velella siphonophore has a second-order symmetry axis and no symmetry planes.

Usually two or more planes of symmetry pass through the axis of symmetry. These planes intersect in a straight line - the axis of symmetry. If the animal will rotate around this axis by a certain degree, then it will be displayed on itself (coincide with itself).
There can be several such axes of symmetry (polyaxon symmetry) or one (monaxon symmetry). Polyaxon symmetry is common among protists (such as radiolarians).

As a rule, in multicellular animals, the two ends (poles) of a single axis of symmetry are unequal (for example, in jellyfish, one pole (oral) has a mouth, and the opposite (aboral) has a bell top. Such symmetry (a variant of radial symmetry) in comparative anatomy is called uniaxial-heteropole. In a two-dimensional projection, radial symmetry can be preserved if the axis of symmetry is directed perpendicular to the projection plane. Others In other words, the preservation of radial symmetry depends on the viewing angle.
Radial symmetry is characteristic of many cnidarians, as well as most echinoderms. Among them there is the so-called pentasymmetry, based on five planes of symmetry. In echinoderms, radial symmetry is secondary: their larvae are bilaterally symmetrical, while in adult animals, external radial symmetry is violated by the presence of a madrepore plate.

In addition to typical radial symmetry, there is two-beam radial symmetry (two planes of symmetry, for example, in ctenophores). If there is only one plane of symmetry, then the symmetry is bilateral (this symmetry is bilaterally symmetrical).

In flowering plants, radially symmetrical flowers are often found: 3 planes of symmetry (frog watercress), 4 planes of symmetry (Potentilla straight), 5 planes of symmetry (bellflower), 6 planes of symmetry (colchicum). Flowers with radial symmetry are called actinomorphic, flowers with bilateral symmetry are called zygomorphic.

If the environment surrounding the animal is more or less homogeneous on all sides and the animal evenly contacts it with all parts of its surface, then the shape of the body is usually spherical, and the repeating parts are located in radial directions. Many radiolarians, which are part of the so-called plankton, are spherical; aggregates of organisms suspended in the water column and incapable of active swimming; spherical chambers have a few planktonic representatives of foraminifera (protozoa, inhabitants of the seas, marine shell amoeba). Foraminifera are enclosed in shells of various, bizarre shapes. The spherical body of sunflowers sends in all directions numerous thin, filamentous, radially located pseudopodia, the body is devoid of a mineral skeleton. This type of symmetry is called equiaxed, since it is characterized by the presence of many identical axes of symmetry.

The equiaxed and polysymmetric types are found mainly among low-organized and poorly differentiated animals. If 4 identical organs are located around the longitudinal axis, then the radial symmetry in this case is called four-beam. If there are six such organs, then the order of symmetry will be six-ray, and so on. Since the number of such organs is limited (often 2,4,8 or a multiple of 6), then several planes of symmetry can always be drawn, corresponding to the number of these organs. The planes divide the body of the animal into identical sections with repeating organs. This is the difference between radial symmetry and polysymmetric type. Radial symmetry is characteristic of sedentary and attached forms. The ecological significance of ray symmetry is clear: a sedentary animal is surrounded on all sides by the same environment and must enter into relationships with this environment with the help of identical organs repeating in the radial directions. It is a sedentary lifestyle that contributes to the development of radiant symmetry.

Rotational symmetry

Rotational symmetry is "popular" in the plant world. Take a chamomile flower in your hand. The combination of different parts of the flower occurs if they are rotated around the stem.

Very often flora and fauna borrow external forms from each other. Sea stars, leading a plant lifestyle, have rotational symmetry, and the leaves are mirror-like.

Plants chained to a permanent place clearly distinguish only up and down, and all other directions are more or less the same for them. Naturally, their appearance is subject to rotational symmetry. For animals, it is very important what is in front and what is behind, only “left” and “right” remain equal for them. In this case, mirror symmetry prevails. It is curious that animals that change from a mobile life to a stationary one and then return to a mobile life again pass from one type of symmetry to another a corresponding number of times, as happened, for example, with echinoderms (starfish, etc.).

Helical or spiral symmetry

Screw symmetry is symmetry with respect to a combination of two transformations - rotation and translation along the rotation axis, i.e. there is movement along the axis of the screw and around the axis of the screw. There are left and right screws.

Examples of natural screws are: the tusk of a narwhal (a small cetacean living in the northern seas) - the left screw; snail shell - right screw; the horns of the Pamir ram are enantiomorphs (one horn is twisted along the left and the other along the right spiral). Spiral symmetry is not perfect, for example, the shell of mollusks narrows or widens at the end.

Although external helical symmetry is rare in multicellular animals, many important molecules from which living organisms are built - proteins, deoxyribonucleic acids - DNA, have a helical structure. The real realm of natural screws is the world of "living molecules" - molecules that play a fundamentally important role in life processes. These molecules include, first of all, protein molecules. There are up to 10 types of proteins in the human body. All parts of the body, including bones, blood, muscles, tendons, hair, contain proteins. A protein molecule is a chain made up of separate blocks and twisted in a right-handed helix. It's called the alpha helix. The tendon fiber molecules are triple alpha helices. Twisted repeatedly with each other, alpha helices form molecular screws, which are found in hair, horns, and hooves. The DNA molecule has the structure of a double right helix, discovered by American scientists Watson and Crick. The double helix of the DNA molecule is the main natural screw.

Conclusion

All forms in the world obey the laws of symmetry. Even "eternally free" clouds have symmetry, albeit distorted. Freezing in the blue sky, they resemble jellyfish slowly moving in sea water, obviously gravitating towards rotational symmetry, and then, driven by the rising breeze, they change symmetry to a mirror one.

Symmetry, manifesting itself in the most diverse objects of the material world, undoubtedly reflects its most general, most fundamental properties. Therefore, the study of the symmetry of various natural objects and the comparison of its results is a convenient and reliable tool for understanding the basic laws of the existence of matter.

Symmetry - this is equality in the broadest sense of the word. This means that if there is symmetry, then something will not happen and, therefore, something will necessarily remain unchanged, will be preserved.

Sources

1. Urmantsev Yu. A. “Symmetry of nature and the nature of symmetry”. Moscow, Thought, 1974.

2. V.I. Vernadsky. Chemical structure of the Earth's biosphere and its environment. M., 1965.

3. http://www.worldnature.ru

4.http://otherreferats

The topic of the abstract was chosen after studying the section "Axial and Central Symmetry". I stopped on this topic not by chance, I wanted to know the principles of symmetry, its types, its diversity in animate and inanimate nature.

Introduction……………………………………………………………………………3

Section I. Symmetry in mathematics…………………………………………………5

Chapter 1. Central symmetry…………………………………………………..5

Chapter 2. Axial symmetry………………………………………………………….6

Chapter 4. Mirror symmetry……………………………………………………7

Section II. Symmetry in wildlife…………………………………………….8

Chapter 1. Symmetry in living nature. Asymmetry and symmetry…………8

Chapter 2. Symmetry of plants……………………………………………………10

Chapter 3. Symmetry of animals…………………………………………………….12

Chapter 4

Conclusion………………………………………………………………………….16

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Municipal budgetary educational institution

Secondary school №3

Essay on mathematics on the topic:

"Symmetry in Nature"

Prepared by: 6th grade student "B" Zvyagintsev Denis

Teacher: Kurbatova I.G.

With. Safe, 2012

Introduction……………………………………………………………………………3

Section I. Symmetry in mathematics…………………………………………………5

Chapter 1. Central symmetry…………………………………………………..5

Chapter 2. Axial symmetry………………………………………………………….6

Chapter 4. Mirror symmetry……………………………………………………7

Section II. Symmetry in wildlife…………………………………………….8

Chapter 1. Symmetry in nature. Asymmetry and symmetry…………8

Chapter 2 Symmetry of plants………………………………………………………………………………………………………………………………10

Chapter 3. Symmetry of animals…………………………………………………….12

Chapter 4

Conclusion………………………………………………………………………….16

Introduction

Symmetry (from the Greek symmetria - proportionality) in a broad sense is understood as the correctness in the structure of the body and figure. The doctrine of symmetry is a large and important branch closely related to the sciences of various branches. We often meet with symmetry in art, architecture, technology, everyday life. Thus, the facades of many buildings have axial symmetry. In most cases, patterns on carpets, fabrics, and room wallpapers are symmetrical about the axis or center. Many details of mechanisms are symmetrical, for example, gear wheels.

It was interesting, because this topic affects not only mathematics, although it underlies it, but also other areas of science, technology, and nature. Symmetry, it seems to me, is the foundation of nature, the concept of which has been formed over tens, hundreds, thousands of generations of people.

I noticed that in many things, the basis of the beauty of many forms created by nature is symmetry, or rather, all its types - from the simplest to the most complex. One can speak of symmetry as the harmony of proportions, as "proportionality", regularity and orderliness.

This is important for us, because for many people mathematics is a boring and complex science, but mathematics is not only numbers, equations and solutions, but also beauty in the structure of geometric bodies, living organisms, and even is the foundation for many sciences from simple to the most complex.

The objectives of the abstract were:

reveal the features of the types of symmetry;
to show all the attractiveness of mathematics as a science and its relationship with nature in general.

Tasks:

collection of material on the topic of the abstract and its processing;
generalization of the processed material;
conclusions about the work done;
summary of the material.

Section I. Symmetry in mathematics

Chapter 1

The concept of central symmetry is as follows: “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. Therefore, the figure is said to have central symmetry.

There is no concept of a center of symmetry in Euclid's Elements, however, in the 38th sentence of the XI book, the concept of a spatial axis of symmetry is contained. The concept of a center of symmetry was first encountered in the 16th century. In one of the Clavius theorems, which says: "if a box is cut by a plane passing through the center, then it is split in half and, conversely, if the box is cut in half, then the plane passes through the center." Legendre, who first introduced elements of the doctrine of symmetry into elementary geometry, shows that a right parallelepiped has 3 planes of symmetry perpendicular to the edges, and a cube has 9 planes of symmetry, of which 3 are perpendicular to the edges, and the other 6 pass through the diagonals of the faces.

Examples of figures with central symmetry are the circle and the parallelogram. The center of symmetry of a circle is the center of the circle, and the center of symmetry of a parallelogram is the point of intersection of its diagonals. Any straight line also has central symmetry. However, unlike a circle and a parallelogram, which have only one center of symmetry, a straight line has an infinite number of them - any point on a straight line is its center of symmetry. An example of a figure that does not have a center of symmetry is an arbitrary triangle.

In algebra, when studying even and odd functions, their graphs are considered. The graph of an even function when plotted is symmetrical about the y-axis, and the graph of an odd function is about the origin, i.e. points O. Hence, the odd function has central symmetry, and the even function has axial symmetry.

Thus, two centrally symmetrical plane figures can always be superimposed on each other without taking them out of the common plane. To do this, it is enough to turn one of them through an angle of 180 ° near the center of symmetry.

Both in the case of mirror and in the case of central symmetry, a plane figure certainly has a second-order symmetry axis, but in the first case this axis lies in the plane of the figure, and in the second it is perpendicular to this plane.

Chapter 2

The concept of axial symmetry is represented as follows: “A figure is called symmetric with respect to the line a, if for each point of the figure the point symmetric to it with respect to the line a also belongs to this figure. The straight line a is called the axis of symmetry of the figure. Then we say that the figure has axial symmetry.

In a narrower sense, the axis of symmetry is called the axis of symmetry of the second order and they speak of “axial symmetry”, which can be defined as follows: a figure (or body) has axial symmetry about some axis, if each of its points E corresponds to a point F belonging to the same figure that the segment EF is perpendicular to the axis, intersects it and at the point of intersection is divided in half. The pair of triangles considered above (Chapter 1) has (in addition to the central one) axial symmetry. Its axis of symmetry passes through the point C perpendicular to the plane of the drawing.

Let us give examples of figures with axial symmetry. An undeveloped angle has one axis of symmetry - a straight line on which the bisector of the angle is located. An isosceles (but not equilateral) triangle also has one axis of symmetry, and an equilateral triangle has three axes of symmetry. A rectangle and a rhombus, which are not squares, each have two axes of symmetry, and a square has four axes of symmetry. A circle has an infinite number of them - any straight line passing through its center is an axis of symmetry.

There are figures that do not have any axis of symmetry. Such figures include a parallelogram other than a rectangle, a scalene triangle.

Chapter 3

Mirror symmetry is well known to every person from everyday observation. As the name itself shows, mirror symmetry connects any object and its reflection in a flat mirror. One figure (or body) is said to be mirror symmetrical to another if together they form a mirror symmetrical figure (or body).

Billiards players have long been familiar with the action of reflection. Their "mirrors" are the sides of the playing field, and the trajectories of the balls play the role of a beam of light. Having hit the board near the corner, the ball rolls to the side located at a right angle, and, reflected from it, moves back parallel to the direction of the first impact.

It is important to note that two bodies that are symmetrical to each other cannot be nested or superimposed on each other. So the glove of the right hand cannot be put on the left hand. Symmetrically mirrored figures, for all their similarities, differ significantly from each other. To verify this, it is enough to bring a piece of paper to a mirror and try to read a few words printed on it, the letters and words will simply be turned right to left. For this reason, symmetrical objects cannot be called equal, so they are called mirror equal.

Consider an example. If the plane figure ABCDE is symmetrical with respect to the plane P (which is possible only if the planes ABCDE and P are mutually perpendicular), then the line KL, along which the mentioned planes intersect, serves as an axis of symmetry (of the second order) of the figure ABCDE. Conversely, if a plane figure ABCDE has an axis of symmetry KL lying in its plane, then this figure is symmetrical with respect to the plane P, drawn through KL perpendicular to the plane of the figure. Therefore, the KE axis can also be called the mirror L of the straight plane figure ABCDE.

Two mirror-symmetric plane figures can always be superimposed
Each other. However, for this it is necessary to remove one of them (or both) from their common plane.

In general, bodies (or figures) are called mirror equal bodies (or figures) in the event that, with their proper displacement, they can form two halves of a mirror symmetrical body (or figure).

Section II. Symmetry in nature

Chapter 1. Symmetry in nature. Asymmetry and symmetry

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 wildlife, the vast majority of living organisms exhibit various types of symmetry (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 the last decades in the field of physics of elementary particles make us pay special attention to the concept of symmetry disorders. The development of the Universe from the moment of its origin looks like a continuous sequence of symmetry disorders. At the time of its occurrence with a grand explosion, the universe was symmetrical and uniformly unleashed. As it cools in it, one symmetry is disturbed by one symmetry, which creates one symmetry for another, which creates one symmetry for another, which creates one symmetry for another, which creates one symmetry for the other. The possibilities for the existence of an increasing and 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.

Stereoisomer molecules 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. 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, sugar ~ only 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. Everything living is 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.

Chapter 2

Images on the plane of many objects of the world around us have an axis of symmetry or a center of symmetry. Many tree leaves and flower petals are symmetrical about the middle stem.

Rotational symmetries of different orders are observed among the colors. Many flowers have the characteristic property that a flower can be rotated so that each petal takes the position of its neighbor, while the flower is aligned with itself. Such a flower has an axis of symmetry. The minimum angle by which the flower must be rotated around the axis of symmetry so that it is aligned with itself is called the elementary angle of rotation of the axis. This angle is not the same for different colors. For iris, it is 120º, for bell - 72º, for narcissus - 60º. A rotary axis can also be characterized by another quantity, called the order of the axis, which indicates how many times the alignment will occur during a 360º rotation. The same flowers of iris, bluebell and narcissus have axes of the third, fifth and sixth orders, respectively. Especially often among the flowers there is fifth-order symmetry. These are such wild flowers as a bell, forget-me-not, St. John's wort, goose cinquefoil, etc .; flowers of fruit trees - cherry, apple, pear, tangerine, etc., flowers of fruit and berry plants - strawberries, blackberries, raspberries, wild roses; garden flowers - nasturtium, phlox, etc.

There are bodies in space that have helical symmetry, i.e., those that coincide with their original position after rotation through an angle around an axis, supplemented by a shift along the same axis.

Helical symmetry is observed in the arrangement of leaves on the stems of most plants. Being located by a screw along the stem, the leaves seem to spread out in all directions and do not obscure each other from the light, which is essential for plant life. This interesting botanical phenomenon is called phyllotaxis, which literally means leaf structure. Another manifestation of phyllotaxis is the structure of a sunflower inflorescence or scales of a spruce cone, in which the scales are arranged in the form of spirals and helical lines. This arrangement is especially clearly seen in the pineapple, which has more or less hexagonal cells that form rows running in different directions.

Chapter 3

Careful observation reveals that the basis of the beauty of many forms created by nature is symmetry, or rather, all of its types - from the simplest to the most complex. Symmetry in the structure of animals is almost a general phenomenon, although there are almost always exceptions to the general rule.

Symmetry in animals is understood as correspondence in size, shape and outline, as well as the relative location of body parts located on opposite sides of the dividing line. The body structure of many multicellular organisms reflects certain forms of symmetry, such as radial (radial) or bilateral (bilateral), which are the main types of symmetry. By the way, the tendency to regenerate (recovery) depends on the type of symmetry of the animal.

In biology, we talk about radial symmetry when two or more planes of symmetry pass through a three-dimensional being. These planes intersect in a straight line. If the animal will rotate around this axis by a certain degree, then it will be displayed on itself. In a 2D projection, radial symmetry can be maintained if the axis of symmetry is directed perpendicular to the projection plane. In other words, the preservation of radial symmetry depends on the viewing angle.

With radial or radiative symmetry, the body has the form of a short or long cylinder or vessel with a central axis, from which parts of the body extend in a radial order. Among them there is the so-called pentasymmetry, based on five planes of symmetry.

Radial symmetry is characteristic of many cnidarians, as well as most echinoderms and coelenterates. Adult forms of echinoderms approach radial symmetry, while their larvae are bilaterally symmetrical.

We also see ray symmetry in jellyfish, corals, sea anemones, starfish. If you rotate them around their own axis, they will “align with themselves” several times. If you cut off any of the five tentacles from a starfish, it will be able to restore the entire star. Two-beam radial symmetry (two planes of symmetry, for example, ctenophores), as well as bilateral symmetry (one plane of symmetry, for example, bilaterally symmetrical) are distinguished from radial symmetry.

With bilateral symmetry, there are three axes of symmetry, but only one pair of symmetrical sides. Because the other two sides - the abdominal and dorsal - are not similar to each other. This kind of symmetry is characteristic of most animals, including insects, fish, amphibians, reptiles, birds, and mammals. For example, worms, arthropods, vertebrates. Most multicellular organisms (including humans) have a different type of symmetry - bilateral. The left half of their body is, as it were, "the right half reflected in the mirror." This principle, however, does not apply to individual internal organs, which is demonstrated, for example, by the location of the liver or heart in humans. The planarian flatworm is bilaterally symmetrical. If you cut it along the axis of the body or across, new worms will grow from both halves. If you grind the planaria in some other way, most likely nothing will come of it.

We can also say that every animal (be it an insect, a fish or a bird) consists of two enantiomorphs - the right and left halves. Enantiomorphs are a pair of mirror-asymmetrical objects (figures) that are mirror images of one another (for example, a pair of gloves). In other words, this is an object and its mirror-like counterpart, provided that the object itself is mirror-asymmetric.

Spherical symmetry takes place in radiolarians and sunfish, whose body is spherical, and its parts are distributed around the center of the sphere and move away from it. Such organisms have neither anterior, nor posterior, nor lateral parts of the body; any plane drawn through the center divides the animal into identical halves.

Sponges and lamellar do not show symmetry.

Chapter 4

We will not yet understand whether there really is an absolutely symmetrical person. Everyone, of course, will have a mole, a strand of hair, or some other detail that breaks the external symmetry. The left eye is never exactly the same as the right, and the corners of the mouth are at different heights, at least in most people. Still, these are just minor inconsistencies. No one will doubt that outwardly a person is built symmetrically: the left hand always corresponds to the right hand and both hands are exactly the same! BUT! It's worth stopping here. If our hands really were exactly the same, we could change them at any time. It would be possible, say, by transplantation, to transplant the left hand to the right hand, or, more simply, the left glove would then fit the right hand, but in fact this is not the case. Everyone knows that the similarity between our hands, ears, eyes and other parts of the body is the same as between an object and its reflection in a mirror. Many artists paid close attention to the symmetry and proportions of the human body, at least as long as they were guided by the desire to follow nature as closely as possible in their works.

The canons of proportions compiled by Albrecht Dürer and Leonardo da Vinci are known. According to these canons, the human body is not only symmetrical, but also proportional. Leonardo discovered that the body fits into a circle and a square. Dürer was looking for a single measure that would be in a certain ratio with the length of the torso or leg (he considered the length of the arm to the elbow as such a measure). In modern schools of painting, the vertical size of the head is most often taken as a single measure. With a certain assumption, we can assume that the length of the body exceeds the size of the head by eight times. At first glance, this seems strange. But we must not forget that most tall people are distinguished by an elongated skull and, conversely, it is rare to find a short fat man with an elongated head. The size of the head is proportional not only to the length of the body, but also to the dimensions of other parts of the body. All people are built according to this principle, which is why, in general, we are similar to each other. However, our proportions agree only approximately, and therefore people are only similar, but not the same. Anyway, we are all symmetrical! In addition, some artists in their works especially emphasize this symmetry. And in clothes, a person also, as a rule, tries to maintain the impression of symmetry: the right sleeve corresponds to the left, the right leg corresponds to the left. The buttons on the jacket and on the shirt sit exactly in the middle, and if they recede from it, then at symmetrical distances. But against the background of this general symmetry in small details, we deliberately allow asymmetry, for example, combing our hair in a side part - on the left or right, or making an asymmetrical haircut. Or, say, placing an asymmetrical pocket on the chest on the suit. Or by wearing a ring on the ring finger of only one hand. Orders and badges are worn only on one side of the chest (more often on the left). Complete perfect symmetry would look unbearably boring. It is small deviations from it that give characteristic, individual features. And at the same time, sometimes a person tries to emphasize, strengthen the difference between left and right. In the Middle Ages, men at one time flaunted pantaloons with legs of different colors (for example, one red and the other black or white). In the not-so-distant days, jeans with bright patches or color streaks were popular. But such fashion is always short-lived. Only tactful, modest deviations from symmetry remain for a long time.

Conclusion

We meet with symmetry everywhere ~ in nature, technology, art, science. The concept of symmetry runs through the entire centuries-old history of human creativity. The principles of symmetry play an important role in physics and mathematics, chemistry and biology, engineering and architecture, painting and sculpture, poetry and music. The laws of nature that govern the picture of phenomena, inexhaustible in its diversity, in turn, obey the principles of symmetry. There are many types of symmetry in both the plant and animal kingdoms, but with all the diversity of living organisms, the principle of symmetry always works, and this fact once again emphasizes the harmony of our world.

Another interesting manifestation of the symmetry of life npoifeccoe are biological rhythms (biorhythms), cyclic fluctuations of biological processes and their characteristics (heart contractions, respiration, fluctuations in the intensity of cell division, metabolism, motor activity, the number of plants and animals), often associated with the adaptation of organisms to geophysical cycles. The study of biorhythms is a special science - chronobiology. In addition to symmetry, there is also the concept of asymmetry; 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.

Axial symmetry and the concept of perfection

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

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

Axial symmetry as a concept

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

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

Axial symmetry in wildlife

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

Axial symmetry in inanimate nature

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.