What are the types of symmetry in nature? Wonder Wild World: Symmetry in nature. Milky Way Galaxy

30.03.2022

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 don't happen "naturally" at all, but it's 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.

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 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 the natural need

Central symmetry in nature

Symmetry can be found everywhere if you look closely at the reality around us. It is present in snowflakes, leaves of trees and grasses, insects, flowers, animals. The central symmetry of plants and living organisms is completely determined by the influence of the external environment, which still forms the appearance of the inhabitants of planet Earth.

Introduction 2

Symmetry in nature 3

Symmetry in plants 3

Symmetry in animals 4

Human symmetry 5

Symmetry types in animals 5

Symmetry types 6

Mirror symmetry 7

Radial symmetry 8

Rotational symmetry 10

Helical or spiral symmetry 10

Conclusion 12

Sources 13

"...to be beautiful means to be symmetrical and proportionate"

Plato

Introduction

If you look closely at everything that surrounds us, you will notice that we live in a rather symmetrical world. All living organisms, to one degree or another, comply with the laws of symmetry: people, animals, fish, birds, insects - everything is built according to its laws. Snowflakes, crystals, leaves, fruits are symmetrical, even our spherical planet has almost perfect symmetry.

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.

Word "symmetry" known to us since 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, through the efforts of Russian scientists - V. Beklemishev, V. Vernadsky, V. Alpatov, G. Gause - a new direction in the study of symmetry was created - biosymmetry, which, by studying the symmetries of biostructures at the molecular and supramolecular levels, makes it possible to determine in advance the possible variants of symmetry in biological objects, strictly describe the external form 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 due to the root system, that is, below, and the rest of the vital functions are performed by the crown, that is, at the top. Therefore, the directions "up" and "down" for the tree are significantly different. And the directions in the plane perpendicular to the vertical are practically indistinguishable for the tree: air, light, and moisture are equally supplied to the tree in all these directions. As a result, a vertical rotary axis and a vertical plane of symmetry appear.

Most flowering plants exhibit radial and bilateral symmetry. A flower is considered symmetrical when each perianth consists of an equal number of parts. Flowers, having paired parts, are considered flowers with double symmetry, etc. Triple symmetry is common for 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. This fact is explained by academician N. Belov by the fact that the 5th order axis is a kind of instrument of the struggle for existence, "insurance against petrification, crystallization, the first step of which would be their capture by a lattice." Indeed, a living organism does not have a crystalline structure in the sense that even its individual organs do not have a spatial lattice. However, ordered structures are very widely represented in it.

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 depart 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 being.

Bilateral (mirror) symmetry is a characteristic symmetry of all representatives of the animal world. This symmetry is clearly visible in the butterfly; the symmetry of 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 animal's body. 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

central

axial (mirror)

radial

bilateral

two-beam

translational (metamerism)

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.

M.: Thought, 1974. Khoroshavina S.G. concept of modern...

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All-Russiantocompetition of student essays "Krugozor"

MOU "Secondary School with. Petropavlovka, Dergachevsky district

Saratov region»

ESSAY

mathematics, biology, ecologyon the topic:

"Symmetry in Nature"

6th grade studentMOU

Leaders:Kutishcheva Nina Semyonovna,

Rudenko Ludmila Viktorovna,

Introduction

1. Theoretical part

1.1.1 Developing the doctrine of symmetry

1.1.2 Axial symmetry of figures

1.1.3 Central symmetry

1.1.4 Symmetry about the plane

2. Practical part

2.2 Rationale for the cause of symmetry in plants

Conclusion

Literature

symmetry plant geometry point

Introduction

"Symmetry is that idea, with the help of

which man has been trying to explain for centuries

and create order, beauty and perfection" Hermann Weil.

In the summer, I rested on the banks of the Volga in a wonderful place in the Saratov region "Chardym". I, a resident of the steppe Trans-Volga region, was struck by the surrounding riot of greenery, the diversity of plants, and I examined the nature around me with interest. I involuntarily wondered: is there something in common in the forms of plants and animals? Perhaps there is some pattern, some reasons that give such an unexpected similarity to the most diverse leaves, flowers, and the animal world? Carefully looking at the surrounding nature, I noticed that the shape of the leaves of all plants obeys a strict pattern: the leaf, as it were, is glued together from two more or less identical halves. Butterflies have the same property. We can mentally divide them lengthwise into two mirror equal parts.

In mathematics lessons, we considered symmetry on a plane with respect to a point and a line, figures in space that are symmetrical with respect to a plane. So that's what it's all about! Here is the regularity that I felt in my observations, but could not explain! The laws of symmetry - this is how such similarity in leaves, flowers, and the animal world can be explained.

And I set out to find out whether there is symmetry in the plant kingdom and what causes it. For its implementation, I formulated the following tasks:

1. Get to know more about the geometric laws of symmetry.

2. Reveal the reasons for the symmetry in nature.

1. Theoretical part

1.1 Basic concepts of symmetry and geometry of plants

1.1.1 The developing doctrine of symmetry

The word "symmetry" comes from the Greek symmetria, meaning proportionality. It is she who will allow to cover a wide variety of bodies from a single geometric position.

Symmetry is one of the most fundamental and one of the most general laws of the universe: living, inanimate nature and society. The concept of symmetry runs through the entire centuries-old history of human creativity. The famous academician V.I. Vernadsky believed that “... the concept of symmetry was formed over tens, hundreds, thousands of generations. Its correctness has been verified by real experience and observation, by the life of mankind in the most diverse natural conditions.

The concept of "symmetry" has grown on the study of living organisms and living matter, primarily man. The very concept associated with the concept of beauty or harmony was given by the great Greek sculptors, and the word “symmetry” corresponding to this phenomenon is attributed to the sculpture of Pythagoras from Regnum (Southern Italy, then Great Greece), who lived in the 5th century BC.

And another well-known academician A.V. Shubnikov (1887-1970), in the preface to his book "Symmetry" wrote: "The study of archaeological sites shows that humanity at the dawn of its culture already had an idea of \u200b\u200bsymmetry and carried it out in drawing and in household items. It must be assumed that the use of symmetry in primitive production was determined not only by aesthetic motives, but also to a certain extent by the person's confidence in the greater suitability for the practice of regular forms.

This confidence continues to exist to this day, being reflected in many areas of human activity: art, science, technology, etc.”

But what is the significance of this undeniably classical concept? There are many definitions of symmetry:

1. "Dictionary of foreign words": "Symmetry - [Greek. symmetria] - full mirror correspondence in the arrangement of parts of the whole relative to the midline, center; proportionality".

2. "Concise Oxford Dictionary": "Symmetry - beauty, due to the proportionality of parts of the body or any whole, balance, similarity, harmony, consistency."

3. Dictionary of S.I. Ozhegov": "Symmetry is proportionality, proportionality of parts of something located on both sides of the middle, center."

4. V.I. Vernadsky. “The chemical structure of the Earth's biosphere and its environment”: “In the sciences of nature, symmetry is an expression of geometrically spatial regularities empirically observed in natural bodies and phenomena. It, consequently, manifests itself, obviously, not only in space, but also on a plane and on a line.

But the opinion of Yu.A. Urmantseva: “Symmetry is any figure that can be combined with itself as a result of one or several successively produced reflections in planes. In other words, one can say about a symmetrical figure: “Eadem mutate resurgo” - “Changed, I resurrect the same” - the inscription under the logarithmic spiral that fascinated Jacob Bernoulli (1654-1705).

1.1.2 Axial symmetry of figures

Two points A and A1 are called symmetrical with respect to the line a if this line passes through the midpoint of the segment AA 1 and is perpendicular to it.

A figure is called symmetric with respect to a 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.

Looking at various figures, we notice that some of them are symmetrical about the axis, i.e. are mapped onto themselves if they are symmetrical about this axis.

The axis of symmetry divides such a figure into two symmetrical figures located in different half-planes determined by the axis of symmetry. (Fig. 1.)

Some figures have multiple axes of symmetry. For example, a circle (Fig. 2) is symmetrical with respect to any straight line passing through its center. By bending the drawing along the diameter of the drawn circle, you can make sure that the two parts of the circle coincide. Therefore, any diameter lies on the axis of symmetry of the circle.

The segment has two axes of symmetry: it is symmetric with respect to a straight line perpendicular to it, passing through its middle, and relative to the straight line on which this segment lies (Fig. 3).

1.1.3 Central symmetry

Two points A and A 1 are called symmetrical with respect to the point O if O is the midpoint of the segment AA 1.

A figure is called symmetric with respect to the point O if for each point of the figure the point symmetric to it with respect to the point O also belongs to this figure.

Central symmetry, as a particular kind of rotation around a given point, has all the properties of rotation. In particular, distances are preserved under central symmetry, so central symmetry is displacement. It follows that if one of the two figures is mapped to the other by central symmetry, then these figures are equal.

The straight line passing through the center of symmetry is displayed by the central symmetry on itself.

For each point of the plane there is a unique point symmetrical to it relative to the given center; if point A coincides with the center of symmetry, then the point B symmetric to it coincides with the center of symmetry.

Just as axial symmetry is uniquely defined by its axis, so central symmetry is uniquely defined by its center.

Some figures have a center of symmetry - this means that for each point of this figure, the point that is centrally symmetrical to it also belongs to this figure. Such figures are called centrally symmetrical. For example, a segment is a centrally symmetrical figure, the center of symmetry of which is its middle; straight line - a centrally symmetrical figure with respect to any of its points; circle - a centrally symmetrical figure about its center; a pair of vertical angles is a centrally symmetrical figure with the center of symmetry at the common vertex of the angles.

1.1.4 Symmetry about the plane (mirror symmetry)

Two points A and A1 are called symmetric about the plane b if this plane passes through the middle of the segment AA1 and is perpendicular to it (Fig. 4).

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A figure is called symmetric with respect to the plane b, if for each point of the figure the point symmetric to it with respect to the plane also belongs to this figure (Fig. 5).

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In the following, we will most often deal with three types of symmetry elements: plane, axes, and center.

So, we got acquainted with an exhaustive list of symmetry elements. We have at our disposal a complete set of different symmetry elements for finite figures. For a complete characterization of such figures, it is necessary to take into account the totality of all symmetry elements present on a given object.

1.2 Form and symmetry of plants

We encounter axial symmetry not only in geometry, but also in nature. In biology, it is customary and correct to speak not about axial, but about bilateral, bilateral symmetry or mirror symmetry of a spatial object. Bilateral symmetry is characteristic of most multicellular animals and arose in connection with active movement. Insects and some plants also have bilateral symmetry. For example, the shape of a leaf is not random, it is strictly natural. It is, as it were, glued together from two more or less identical halves. One of these halves is mirrored relative to the other, just like the reflection of an object in the mirror and the object itself are located relative to each other. In order to make sure of what has been said, let's put a mirror with a straight edge on the line running along the handle and dividing the leaf blade in half. Looking in the mirror, we will see that the reflection of the right half of the sheet more or less exactly replaces its left half and, conversely, the left half of the sheet in the mirror, as it were, moves to the place of the right half. The plane dividing the sheet into two mirror equal parts is called the plane of symmetry. Botanists call this symmetry bilateral or twice lateral. But not only a tree leaf has such symmetry. Mentally, you can cut an ordinary caterpillar into two mirror equal parts. Yes, and we ourselves can be divided into two equal halves. Everything that grows and moves horizontally or obliquely with respect to the earth's surface is subject to bilateral symmetry. The same symmetry is preserved in organisms that have gained the ability to move. Albeit without a specific direction. These creatures include starfish and urchins.

Radiation symmetry is typical, as a rule, for animals leading an attached lifestyle. Hydra is one of these animals. If an axis is drawn along the body of the hydra, then its tentacles will diverge from this axis in all directions, like rays. If we consider the petals of chamomile, we can see that they also have a plane of symmetry. This is not all. After all, there are many petals and a plane of symmetry can be drawn along each. This means that this flower has many planes of symmetry, and they all intersect at its center. This whole fan or bundle of intersecting planes of symmetry. The geometry of the sunflower, cornflower, bluebell can be characterized in a similar way. Such symmetry, as in daisies, mushrooms, spruce, is called radial-radial. In the marine environment, such symmetry does not prevent animals from directional swimming. This symmetry has a jellyfish. Pushing water out from under itself with the lower edges of the body, similar in shape to a bell (sea urchins, stars). Thus, we can conclude that everything that grows or moves vertically down or up relative to the earth's surface is subject to radial-beam symmetry.

The symmetry of the cone, characteristic of plants, is clearly visible on the example of any tree.

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

Very rarely, the body of a plant is built the same in all directions. For the most part, you can distinguish between the upper (front) and lower (back) end. The line connecting these two ends is called the longitudinal axis. With respect to this longitudinal axis, plant organs and tissues can be distributed differently.

1) If at least two planes can be drawn through the longitudinal axis, dividing the considered part of the plant into identical symmetrical halves, then the arrangement is called radial (multisymmetric arrangement). Most of the roots, stems and flowers are built according to the ray type.

2) If only one plane can be drawn through the longitudinal axis, dividing the plant into symmetrical halves, then they speak of a dorsiventral (monosymmetric) arrangement. In the absence of symmetry planes, the organ is called asymmetric. Finally, bisymmetric or bilateral organs are those in which the right and left, anterior and posterior sides can be distinguished, and the right is symmetrical to the left, the anterior to the posterior, but the right and anterior, left and posterior are completely different. Thus, there are two unequal planes of symmetry here. Such an arrangement is obtained, for example, if a cylindrical organ is flattened in one direction. Thus, the flattened stems of Opuntia cacti are bisymmetrical, and the thallus of many seaweeds, such as Fucus, Laminaria, and so on, is bisymmetrical. Bisymmetrical organs are usually formed from ray organs, which is especially well seen on cacti or fucus. With regard to flowers in particular, the rays are often called stellate (actinomorphic), and dorsiventral - zygomorphic.

2. Practical part

2.1 Features of each type of symmetry

Two kinds of symmetry recur with unusual persistence around us. I was convinced of this by looking at photographs taken during the rest.

I was surrounded by various flowers, trees. A breeze blew, and a leaf from a tree fell right on my sleeve. Its form is not random, it is strictly natural. The leaf is, as it were, glued together from two more or less identical halves. One of these halves is mirrored relative to the other, just as the reflection of an object in a mirror and the object itself are located relative to each other. To verify this, I put a pocket mirror with a straight edge on the line that runs along the handle and divides the leaf blade in half. Looking in the mirror, I saw that the reflection of the right half of the sheet more or less exactly replaces its left half and, conversely, the left half of the sheet in the mirror, as it were, moves to the place of the right half.

The plane dividing the sheet into two mirror-equal parts (which now coincides with the plane of the mirror) is called the “plane of symmetry”. Botanists and zoologists call this symmetry bilateral (translated from Latin twice lateral).

Is it only a tree leaf that has this symmetry?

If you look at a beautiful butterfly with bright colors, it also consists of two identical halves. Even the spotted pattern on her wings obeys this geometry.

And a bug that looked out of the grass, and a midge that flashed by, and a plucked branch - everything obeys "bilateral symmetry". So, everywhere in the forest we come across bilateral symmetry. It may be that any creature has a plane of symmetry and therefore fits under bilateral symmetry.

At first glance, it may seem that it fits, but not everything is as simple as it seems. Near the bush, an ordinary popovnik (chamomile) modestly peeps out of the grass. I tore it off and examined it. Around the yellow center, like the rays around the sun in a child's drawing, there are white petals.

Does such a “flower sun” have a plane of symmetry? Certainly! Without any difficulty, you can cut it into two mirror-equal halves along a line passing through the center of the flower and continuing through the middle of any of the petals or between them. This, however, is not all. After all, there are a lot of petals, and along each petal you can find a plane of symmetry. This means that this flower has many planes of symmetry, and they all intersect at its center. Similarly, the geometry of sunflower, cornflower, bluebell can be covered.

Everything that grows and moves vertically, that is, up or down relative to the earth's surface, is subject to radial-beam symmetry in the form of a fan of intersecting symmetry planes. Everything that grows and moves horizontally or obliquely with respect to the earth's surface is subject to bilateral symmetry.

Not only plants, but also animals are obedient to this universal law.

2.2 Justification of the causes of symmetry in plants

I have carried out research work, the purpose of which is to find out the reasons for the symmetry in the plant kingdom. I placed bean sprouts in two transparent tubes. One tube was placed in a horizontal position, and the other in a vertical position. A week later, I found that as soon as the root and stem grew beyond the horizontal tube, the root began to grow straight down, and the stem up. I believe that the downward growth of the root is due to gravity; stem growth upward - by the influence of light. Experiments carried out by cosmonauts aboard the orbital station under weightless conditions showed that in the absence of gravity, the habitual spatial orientation of seedlings is disturbed. Therefore, under the conditions of gravity, the presence of symmetry allows plants to take a stable position.

Conclusion: Most often, central symmetry is found in flowering plants and in gymnosperms in leaves. In axial symmetry, the largest number of plants are algae (root and leaves), green mosses (root, stem, leaves), horsetails (root, stem, leaves), club mosses (root, stem, leaves), ferns (root, leaves), gymnosperms and flowers. In mirror symmetry, plant species such as ferns (leaves), gymnosperms (stem, fruits) and flowering plants are found.

What is the main reason for the emergence of different symmetry in plants? This is the force of gravity, or gravity.

Studying geometry, biology and physics in high school will help me to find out more deeply the causes of symmetry in nature, to determine the type of symmetry in any plant.

Conclusion

It is difficult to find a person who would not have any idea about the symmetry that explains the presence of a certain order, patterns in the arrangement of parts of the world around. In each flower there is a similarity with others, but there is also a difference.

Having considered and studied the above on the pages of the abstract, I can now assert: everything that grows vertically, that is, up or down relative to the earth's surface, is subject to radial-ray symmetry in the form of a fan of intersecting symmetry planes; everything that grows horizontally or obliquely with respect to the earth's surface is subject to bilateral symmetry. I also proved in practice that the orderliness and proportionality of plants is due to two factors:

Gravity;

The influence of light.

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

In my abstract, I paid more attention to the symmetry of living nature, but this is only a small part that is accessible to my understanding. In the future, I would like to explore the world of symmetry more deeply.

Sources

1. Atanasyan L.S. Geometry 7-9. M.: Enlightenment, 2004. p. 110.

2. Atanasyan L.S. Geometry 10-11. M.: Enlightenment, 2007. p. 68.

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

4. Vulf G.V. Symmetry and its manifestations in nature. M., ed. Dep. Nar. com. Enlightenment, 1991. p. 135.

5. A. V. Shubnikov, Symmetry. M., 1940.

6. Urmantsev Yu.A. Symmetry in nature and the nature of symmetry. M., Thought, 1974. p. 230.

7. Shafranovsky I.I. Symmetry in nature. 2nd ed., revised. L.

8. http://kl10sch55.narod.ru/kl/sim.htm#_Toc157753210.

9. http://www.wikiznanie.ru/ru-wz/index.php/.

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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 flat 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 such a point F belonging to the same figure, that the segment EF is perpendicular to the axis, intersects it and is divided in half at the point of intersection. 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 recent decades in the field of elementary particle physics force us to pay special attention to the concept of symmetry breaking. The development of the Universe since its inception looks like a continuous sequence of symmetry breakings. "homogeneous. As it cools, one symmetry after another is broken in it, which creates opportunities for the existence of an ever greater variety of structures. The phenomenon of life naturally fits into this picture. Life is also a violation of symmetry"

Molecular asymmetry was discovered by L. Pasteur, who was the first to single out the "right" and "left" molecules of tartaric acid: the right molecules look like the right screw, and the left ones look like the left one. Chemists call such molecules stereoisomers.

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.

In space, there are bodies that have helical symmetry, i.e., they are combined 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 depart 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, as 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 in 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.