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Introduction.

I sometimes 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, animals? Also, when my dad was telling me something about animals, he mentioned that being symmetrical is very convenient. So, if you have eyes, ears, noses, mouths and limbs on all sides, then you will have time to feel something suspicious in time, no matter from which side it sneaks up, and, depending on what it is, it is suspicious, - eat it or, conversely, run away from it.

In biology classes, I found out that the basic property of most living beings is symmetry. Perhaps it is the laws of symmetry that can explain such similarity in leaves, flowers, and the animal world.

The purpose of my work will be to determine the role of symmetry in animate and inanimate nature.

To achieve the goal of the study, it is necessary to implement the following tasks:

learn more about the concept of symmetry;

find evidence of the existence of symmetry in nature;

prepare a presentation;

present a presentation.

Theoretical part.

1. Basic concepts of symmetry

We get used to the word "symmetry" from childhood, and it seems that there can be nothing mysterious in this clear concept. 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.

A truly boundless amount of literature has been devoted to the problem of symmetry. From textbooks and scientific monographs to works that appeal not so much to a drawing and formula as to an artistic image, and combine scientific authenticity with literary sophistication.

The concept of symmetry historically grows out of aesthetic ideas. It is widely manifested in rock paintings, primitive products of labor and everyday life, which testifies to its antiquity.

The concept of symmetry originates from Ancient Greece. It was first introduced in the 5th century. BC e. sculptor Pythagoras from Rhegium, who understood the beauty of the human body and beauty in general by symmetry, and defined the deviation from symmetry by the term "asymmetry". In the works of ancient Greek philosophers (Pythagoreans, Plato, Aristotle), the concepts of “harmony”, “proportion” are more common than “symmetry”.

There are many definitions of symmetry:

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

      brief Oxford Dictionary: “Symmetry is beauty due to the proportionality of parts of the body or any whole, balance, likeness, harmony, consistency”;

      dictionary of S. I. Ozhegov: “Symmetry is proportionality, proportionality of parts of something located on both sides of the middle, center”;

      “The chemical structure of the Earth's biosphere and its environment” by V. I. Vernadsky: “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 most complete and generalizing all of the above definitions seems to me the opinion of Yu.

The word "symmetry" has a dual meaning.

In one sense, symmetrical means something very proportional, balanced; symmetry shows that way of coordinating many parts, with the help of which they are combined into a whole.

The second meaning of this word is balance. Even Aristotle spoke of symmetry as a state that is characterized by a ratio of extremes. From this statement it follows that Aristotle, perhaps, was closest to the discovery of one of the most fundamental laws of Nature - the laws of its duality. The initial concept of geometric symmetry as a harmony of proportions, as a “proportionality”, which is what the word “symmetry” means in Greek, acquired a universal character over time and was recognized as a general idea of ​​invariance (i.e., immutability) with respect to some transformations. Thus, a geometric object or physical phenomenon is considered symmetrical if something can be done to it, after which it will remain unchanged. 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.

1. Symmetry in geometry

2.1 Symmetry of geometric shapes (bodies).

Mirror symmetry. A geometric figure (Fig. 1) is called symmetric with respect to the plane S if for each point E of this figure, a point E’ of the same figure can be found, so that the segment EE’ is perpendicular to the plane S and is divided by this plane in half (EA = AE). The plane S is called the plane of symmetry. Symmetrical figures, objects and bodies are not equal to each other in the narrow sense of the word (for example, the left glove does not fit the right hand and vice versa). They are called mirror equals.

central symmetry. A geometric figure (Fig. 2) is called symmetric about the center C if for each point A of this figure a point E of the same figure can be found, so that the segment AE passes through the center C and is divided in half at this point (AC = CE). Point C is called the center of symmetry.

rotation symmetry. The body (Fig. 3) has rotational symmetry if, when rotated through an angle of 360 ° / n (here n is an integer) around some straight line AB (axis of symmetry), it completely coincides with its initial position. For n = 2 we have axial symmetry. Triangles also have axial symmetry.

Examples of the aforementioned types of symmetry (Fig. 4).

A ball (sphere) has both central, mirror and rotational symmetry. The center of symmetry is the center of the ball; the plane of symmetry is the plane of any great circle; the axis of symmetry is the diameter of the ball.

The round cone is axially symmetrical; the axis of symmetry is the axis of the cone.

A straight prism has mirror symmetry. The plane of symmetry is parallel to its bases and is located at the same distance between them.

2.2 Symmetry of plane figures.

Mirror-axial symmetry. If the plane figure ABCDE (Fig. 5 on the right) is symmetrical with respect to the plane S (which is possible only if the plane figure is perpendicular to the plane S), then the line KL, along which these planes intersect, is the axis of symmetry of the second order of the figure ABCDE. In this case, the figure ABCDE is called mirror-symmetrical.

central symmetry. If the flat figure ABCDEF has a second-order axis of symmetry perpendicular to the plane of the figure - the straight line MN (Fig. 5 on the left), then the point O, at which the straight line MN and the plane of the figure ABCDEF intersect, is the center of symmetry.

Examples of symmetry of flat figures (Fig. 6).

The parallelogram has only central symmetry. Its center of symmetry is the intersection point of the diagonals.

An isosceles trapezoid has only axial symmetry. Its axis of symmetry is a perpendicular drawn through the midpoints of the bases of the trapezoid.

The rhombus has both central and axial symmetry. Its axis of symmetry is any of its diagonals; the center of symmetry is the point of their intersection.

1. Types of symmetry in nature

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

This is how the next type of symmetry arises - radial. Radially symmetrical creatures have a top and bottom, but no right and left, front and back. They coincide with themselves when rotating around only one axis. These include, for example, starfish and hydras. These creatures are inactive and are engaged in "silent hunting" for living creatures passing by. Radial symmetry is inherent in jellyfish and polyps, cross sections of fruits of apples, lemons, oranges, persimmons (Fig. 7), etc.

But if a creature is going to lead an active lifestyle, chasing prey and running away from predators, another direction becomes important for it - anterior-posterior. The part of the body that is in front, when the animal moves, becomes more significant. All the sense organs “crawl” here, and at the same time the nerve nodes that analyze the information received from the sensory organs (for some lucky ones, these nodes will later turn into the brain). In addition, the mouth must be in front in order to have time to grab the overtaken prey. All this is usually located on a separate part of the body - the head (in principle, radially symmetrical animals do not have a head). This is how bilateral (or bilateral) symmetry arises. In a bilaterally symmetrical creature, the top and bottom, front and back parts are different, and only the right and left are identical and are mirror images of each other. In inanimate nature, this type of symmetry does not have a predominant significance, but it is extremely richly represented in living nature (Fig. 8).

In some animals, for example, annelids, in addition to bilateral there is another symmetry - metameric. Their body (with the exception of the very front part) consists of identical metameric segments, and if you move along the body, the worm "coincides" with itself. In more advanced animals, including humans, a faint “echo” of this symmetry persists: in a sense, our vertebrae and ribs can also be called metameres (Fig. 9).

So, according to numerous literary data, the laws of symmetry operate in nature, which ensure its beauty and harmony, and are explained by the action of natural selection.

I went to the mirror and saw that I have two arms, two legs, two ears, two eyes, which are mirror-symmetrical. But when I looked at myself, I noticed that one eye was a little more narrowed, the other less, one eyebrow was more curved, the other less; one ear is higher, the other is lower, the thumb of the left hand is slightly smaller than the thumb of the right. So is there symmetry in nature and can it be measured, and not just visually assessed “by eye”? Or maybe there are units of measure for symmetry?

Practical part.

Description of the methodology for collecting and processing data

To conduct a study to prove the presence and measure the symmetry of living organisms (on the advice of the pope), the method "Assessment of the ecological state of the forest by leaf asymmetry" was used, developed by a group of scientists from the Kaluga State Pedagogical University named after K. E. Tsiolkovsky. The authors of the technique use birch leaves as an object of study.

The studies were conducted on September 19, 2016. Birches grow in the yard of my house: five mature tall trees. I collected ten leaves from each tree (Fig. 10). The material was processed immediately after collection.

To measure, I folded the sheet across, in half, applying the top of the sheet to the base, then unfolded it and took measurements along the formed fold (Fig. 12).

1 - the width of the half of the sheet (counting from the top of the sheet to the base);

2 - length of the second vein of the second order from the base of the leaf;

3 - distance between the bases of the first and second veins of the second order;

4 - the distance between the ends of these veins.

I entered the measurement data into a table in the excel program, so that it would be easier to process the data later.

Calculation of the average relative difference of a feature

I estimated the magnitude of symmetry using an integral indicator - the value of the average relative difference of a trait (arithmetic mean ratio of the difference to the sum of leaf measurements on the left and right, referred to the number of traits).

Using the excel program, in the first step, I found the relative difference between the values ​​of each attribute on the left and on the right - Yi: I found the difference in the measurement values ​​for one attribute for each sheet, then the sum of the same values ​​\u200b\u200band divided the difference by the sum.

Yi = (Xl - Xp): (Xl + Xp);

The found values ​​for each attribute Y1-Y4 were entered into the table.

In the second step, I found the value of the average relative difference between the sides per feature for each leaf (Z). For this, the sum of relative differences was divided by the number of features.

Y1 + Y2 + Y3 + Y4

Z1 = ________________________________,

where N is the number of features. In my case N = 4.

Similar calculations were made for each sheet, and the values ​​\u200b\u200bwere entered in the table.

In the third step, I calculated the average relative difference per feature for the entire sample (X). To do this, all Z values ​​were added and divided by the number of these values:

Z1 + Z2 + Z3 + Z4 + Z5 + Z6 + Z7 + Z8 + Z9 + Z10

X = ____________________________________________ ,

where n is the number of Z values, i.e. the number of leaves (in our example - 10).

The obtained indicator X characterizes the degree of symmetry of the organism.

To determine the presence of symmetry, I used the scale recommended in the methodology, in which 1 point is the conditional norm and the presence of symmetry, and 5 points is the critical deviation from the symmetry hole.

Pivot data table.

Research results

From the presented data table and diagram (Fig. 13), it can be seen that all values ​​were in the range up to 0.055, which corresponds to the norm on the symmetry scale. Thus, all five birches in my yard had symmetrical leaves.

Conclusion.

As a result of my research, I became convinced that symmetry exists in nature and can be measured.

BIBLIOGRAPHY

Demyanenko T.V. "Symmetry in nature", Ukraine.

Zakharov V. M., Baranov A. S., Borisov V. I., Valetsky A. V., Kryazheva N. G., Chistyakova E. K., Chubinishvili A. T. Health of the environment: assessment methodology. - M., Center for Environmental Policy of Russia, 2000.

Roslova L.O., Sharygin I.F. Symmetry: Tutorial, M .: Publishing house of the gymnasium "Open World", 1995.

Children's encyclopedia for middle and older age vol. 3. - M .: Publishing house of the Academy of Pedagogical Sciences of the RSFSR, 1959.

I know the world: Children's Encyclopedia: Mathematics / Comp. A.P. Savin, V.V. Stanzo, A.Yu. Kotova: Ed. O.G. Hinn. - M .: LLC "Publishing House AST - LTD", 1998.

I.F. Sharygin, L.N. Erganzhieva Visual geometry grades 5-6. - M.: Bustard, 2005.

Great Computer Encyclopedia of Cyril and Methodius.

Andrushchenko A.V. The development of spatial imagination in mathematics lessons. M.: Vlados, 2003.

Ivanova O. Integrated lesson "This symmetrical world" // newspaper Mathematics. 2006. No. 6 p.32-36.

Ozhegov S.I. Explanatory dictionary of the Russian language. M. 1997.

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

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

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

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

We get used to the concept of symmetry from childhood. We know that a butterfly is symmetrical: it has the same right and left wings; a wheel is symmetrical, the sectors of which are the same; symmetrical patterns of ornaments, stars of snowflakes.

Truly boundless literature is devoted to the problem of symmetry. From textbooks and scientific monographs to works that pay attention not so much to drawings and formulas as to artistic images.

The very term "symmetry" in Greek means "proportion", which the ancient philosophers understood as a special case of harmony - the coordination of parts within the framework of the whole. Since ancient times, many peoples have owned the idea of ​​symmetry in a broad sense - as the equivalent of balance and harmony.

Symmetry is one of the most fundamental and one of the most general laws of the universe: inanimate, living nature and society. We see her everywhere. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found at the origin human knowledge; it is widely used by all areas of modern science without exception. Truly symmetrical objects surround us literally from all sides, we are dealing with symmetry wherever there is any order. It turns out that symmetry is balance, orderliness, beauty, perfection. It is diverse, ubiquitous. She creates beauty and harmony. Symmetry literally permeates the whole world around us, which is why the topic I have chosen will always be relevant.

Symmetry expresses the preservation of something with some changes or the preservation of something, despite the change. Symmetry implies the immutability of not only the object itself, but also any of its properties in relation to the transformations performed on the object. The immutability of certain objects can be observed in relation to various operations - to rotations, transfers, mutual replacement of parts, reflections, etc. In this regard, they distinguish different types symmetry. Consider all types in more detail.

AXIAL SYMMETRY.

Symmetry about a straight line is called axial symmetry (mirror reflection about a straight line).

If the point A lies on the axis l, then it is symmetrical to itself, i.e. A coincides with A1.

In particular, if under the transformation of symmetry about the l-axis the figure F goes into itself, then it is called symmetric about the l-axis, and the l-axis is called its axis of symmetry.

CENTRAL SYMMETRY.

A figure is called centrally symmetric if there is a point about which each point of the figure is symmetrical to some point of the same figure. Namely: a movement that changes directions to opposite directions is a central symmetry.

Point O is called the center of symmetry and is fixed. This transformation has no other fixed points. Examples of figures that have a center of symmetry are a parallelogram, a circle, etc.

The familiar notions of rotation and translation are used to define the so-called translational symmetry. Let us consider translational symmetry in more detail.

1. TURN

A transformation in which each point A of a figure (body) rotates through the same angle α around a given center O is called rotation or rotation of the plane. The point O is called the center of rotation, and the angle α is called the angle of rotation. Point O is the fixed point of this transformation.

The rotational symmetry of a circular cylinder is interesting. It has an infinite number of 2nd order rotary axes and one infinitely high order rotary axis.

2. PARALLEL TRANSFER

A transformation in which each point of a figure (body) moves in the same direction by the same distance is called parallel translation.

To specify the parallel translation transformation, it is enough to specify the vector a.

3. SLIDING SYMMETRY

A sliding symmetry is a transformation in which axial symmetry and parallel translation are sequentially performed. Sliding symmetry is an isometry of the Euclidean plane. A sliding symmetry is a composition of a symmetry with respect to some line l and translation by a vector parallel to l (this vector may be zero).

A sliding symmetry can be represented as a composition of 3 axial symmetries (Schall's theorem).

MIRROR SYMMETRY

What could be more like my hand or my ear than their own reflection in the mirror? And yet the hand I see in the mirror cannot be put in the place of the real hand.

Immanuel Kant.

If a symmetry transformation with respect to a plane transforms a figure (body) into itself, then the figure is called symmetric with respect to the plane, and the given plane is called the plane of symmetry of this figure. This symmetry is called mirror symmetry. As the name itself shows, mirror symmetry relates an object and its reflection in a flat mirror. Two symmetrical bodies cannot be “inserted into each other”, since, in comparison with the object itself, its trans-mirror counterpart turns out to be turned inside out along the direction perpendicular to the plane of the mirror.

Symmetrical figures, for all their similarities, differ significantly from each other. The double observed in the mirror is not an exact copy of the object itself. The mirror does not just copy the object, but swaps (represents) the parts of the object that are front and back with respect to the mirror. For example, if your mole is on your right cheek, then your mirror double is on your left. Bring a book to the mirror and you will see that the letters are as if turned inside out. In the mirror, everything is rearranged from right to left.

Mirror equal bodies are called bodies if, with their proper displacement, they can form two halves of a mirror symmetric body.

2.2 Symmetry in nature

A figure has symmetry if there is a movement (non-identical transformation) that transforms it into itself. For example, a figure has rotational symmetry if it is translated into itself by some rotation. But in nature, with the help of mathematics, beauty is not created, as in technology and art, but is only fixed, expressed. 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.

The basis of the structure of any living form is the principle of symmetry. From direct observation we can deduce the laws of geometry and feel their incomparable perfection. This order, which is a natural necessity, since nothing in nature serves purely decorative purposes, helps us find a common harmony on which the entire universe is based.

We see that nature designs any living organism according to a certain geometric scheme, and the laws of the universe have a clear justification.

The principles of symmetry underlie the theory of relativity, quantum mechanics, solid state physics, atomic and nuclear physics, elementary particle physics. These principles are most clearly expressed in the properties of the invariance of the laws of nature. In this case, we are talking not only about physical laws, but also about others, for example, biological ones.

Speaking about the role of symmetry in the process of scientific knowledge, we should highlight the use of the method of analogies. According to the French mathematician D. Poya, "perhaps there are no discoveries either in elementary or higher mathematics, or, perhaps, in any other area that could be made without analogies." Most of these analogies are based on common roots, general patterns that manifest themselves in the same way at different levels of the hierarchy.

So, in the modern sense, symmetry is a general scientific philosophical category that characterizes the structure of the organization of systems. The most important property of symmetry is the preservation (invariance) of certain attributes (geometric, physical, biological, etc.) with respect to well-defined transformations. The mathematical apparatus for studying symmetry today is the theory of groups and the theory of invariants.

Symmetry in the plant world

The specificity of the structure of plants is determined by the characteristics of the habitat to which they adapt. 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. With the help of the root system, a tree absorbs moisture and nutrients from the soil, that is, from below, and the rest of the vital functions are performed by the crown, that is, at the top. At the same time, directions in a plane perpendicular to the vertical are practically indistinguishable for a tree; in all these directions, air, light, and moisture are equally supplied to the tree.

The tree has a vertical rotary axis (cone axis) and vertical planes of symmetry.

When we want to draw a leaf of a plant or a butterfly, we have to take into account their axial symmetry. The midrib for the leaf serves as the axis of symmetry. Leaves, branches, flowers, fruits have pronounced symmetry. 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).

In the diverse world of colors, there are turning axes of different orders. However, 5th order rotational symmetry is the most common. This symmetry is found in many wild flowers (bellflower, forget-me-not, geranium, carnation, St. , bird cherry, mountain ash, wild rose, hawthorn), etc.

Academician N. Belov explains this fact 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.

In his book “This Right, Left World”, M. Gardner writes: “On Earth, life originated in spherically symmetrical forms, and then began to develop along two main lines: the world of plants with cone symmetry was formed, and the world of animals with bilateral symmetry.”

In nature, there are bodies that have helical symmetry, that is, alignment with their original position after turning through an angle around an axis, an additional shift along the same axis.

If is a rational number, then the rotary axis is also the translation axis.

The leaves on the stem are not arranged in a straight line, but surround the branch in a spiral. The sum of all previous steps of the spiral, starting from the top, is equal to the value of the subsequent step A + B \u003d C, B + C \u003d D, etc.

Helical symmetry is observed in the arrangement of leaves on the stems of most plants. Located like 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 (literally "leaf arrangement").

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.

Symmetry in the animal world

The significance of the form of symmetry for an animal is easy to understand if we put it in connection with the way of life, environmental conditions. 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.

Rotational symmetry of the 5th order is also found in the animal kingdom. This is symmetry, in which the object is aligned with itself when rotated around the rotation axis 5 times. Examples are the starfish and the shell of the sea urchin. The entire skin of starfish is, as it were, inlaid with small plates of calcium carbonate, needles extend from some of the plates, some of which are movable. An ordinary starfish has 5 planes of symmetry and 1 axis of rotation of the 5th order (this is the highest symmetry among animals). Her ancestors appear to have had lower symmetry. This is evidenced, in particular, by the structure of star larvae: they, like most living beings, including humans, have only one plane of symmetry. Starfish do not have a horizontal plane of symmetry: they have a "top" and a "bottom". Sea urchins are like living pincushions; their spherical body carries long and mobile needles. In these animals, the calcareous plates of the skin merged and formed a spherical shell shell. There is a mouth in the center of the bottom surface. Ambulacral legs (aqueous vascular system) are collected in 5 bands on the surface of the shell.

However, unlike the plant world, rotational symmetry is rarely observed in the animal world.

Insects, fish, eggs, and animals are characterized by an incompatible rotational symmetry difference between forward and backward directions.

The direction of movement is a fundamentally distinguished direction, with respect to which there is no symmetry in any insect, any bird or fish, 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 define the plane of symmetry of the animal 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 the left and right wing appear 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, enantiomorphs are the right and left ear, right and left eye, right and left horn, etc.

Simplification of living conditions can lead to a violation of bilateral symmetry, and animals from bilaterally symmetrical become radially symmetrical. This applies to echinoderms (starfish, sea ​​urchins, sea lilies). All marine animals have radial symmetry, in which body parts extend radially from a central axis, like the spokes of a wheel. The degree of activity of animals correlates with their type of symmetry. Radially symmetrical echinoderms are usually poorly mobile, move slowly, or are attached to the seabed. The body of a starfish consists of a central disk and 5-20 or more rays extending radially from it. In mathematical language, this symmetry is called rotational symmetry.

Finally, we note the mirror symmetry of the human body (we are talking about the external appearance and structure of the skeleton). This symmetry has always been and is the main source of our aesthetic admiration for the well-built human body. 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 and both hands are exactly the same.

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. It is the issues of symmetry and mirror reflection that are given attention here.

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.

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. 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.

Our own mirror symmetry is very convenient for us, it allows us to move in a straight line and turn right and left with equal ease. Equally convenient mirror symmetry for birds, fish and other actively moving creatures.

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.

Consider another type of symmetry that is found in the animal kingdom. This is helical or helical symmetry. Screw symmetry is symmetry with respect to a combination of two transformations - rotation and translation along the axis of rotation, i.e., there is a movement along the axis of the screw and around the axis of the screw.

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.

Symmetry in inanimate nature

The symmetry of crystals is the property of crystals to be combined with themselves in various positions by rotations, reflections, parallel transfers, or a part or combination of these operations. The symmetry of the external form (faceting) of a crystal is determined by the symmetry of its atomic structure, which also determines the symmetry of the physical properties of the crystal.

Consider carefully the multifaceted forms of crystals. First of all, it is clear that the crystals of different substances differ from each other in their shapes. Rock salt is always cubes; rock crystal - always hexagonal prisms, sometimes with heads in the form of triangular or hexagonal pyramids; diamond - most often regular octahedrons (octahedrons); ice - hexagonal prisms, very similar to rock crystal, and snowflakes are always six-pointed stars. What catches your eye when you look at crystals? First of all, their symmetry.

Many people think that crystals are beautiful, rare stones. They come in a variety of colors, are usually transparent and, best of all, have a beautiful regular shape. Most often, crystals are polyhedra, their sides (faces) are perfectly flat, the edges are strictly straight. They delight the eye with a wonderful play of light in the facets, an amazing regularity of the structure.

However, crystals are not a museum rarity at all. Crystals are all around us. Solids from which we build houses and machines, substances that we use in everyday life - almost all of them belong to crystals. Why don't we see this? The fact is that in nature bodies rarely come across in the form of separate single crystals (or, as they say, single crystals). Most often, the substance occurs in the form of firmly adherent crystalline grains of a very small size - less than a thousandth of a millimeter. Such a structure can only be seen with a microscope.

Bodies consisting of crystalline grains are called fine-crystalline, or polycrystalline ("poly" - in Greek "many").

Of course, fine-crystalline bodies should also be classified as crystals. Then it turns out that almost all the solid bodies around us are crystals. Sand and granite, copper and iron, paints - all these are crystals.

There are also exceptions; glass and plastics do not consist of crystals. Such solids are called amorphous.

To study crystals means to study almost all the bodies around us. It is clear how important this is.

Single crystals are immediately recognized by the correctness of their shapes. Flat faces and straight edges are a characteristic property of a crystal; the correctness of the form is undoubtedly connected with the correctness of the internal structure of the crystal. If the crystal is especially extended in some direction, it means that the structure of the crystal in this direction is somehow special.

There is a center of symmetry in the cube of rock salt, and in the octahedron of a diamond, and in the star of a snowflake. But in a quartz crystal there is no center of symmetry.

The most exact symmetry is realized in the world of crystals, but even here it is imperfect: cracks and scratches invisible to the eye always make equal faces slightly different from each other.

All crystals are symmetrical. This means that in each crystalline polyhedron one can find symmetry planes, symmetry axes, a center of symmetry or other symmetry elements so that identical parts of the polyhedron are aligned with each other.

All elements of symmetry repeat the same parts of the figure, all give it symmetrical beauty and completeness, but the center of symmetry is the most interesting. Whether a crystal has a center of symmetry or not can depend not only on its shape, but also on many physical properties crystal.

Honeycombs are a real design masterpiece. They consist of a series of hexagonal cells. This is the densest packing, which makes it possible to place the larva in the cell in the most advantageous way and, with the maximum possible volume, to use the wax building material in the most economical way.

III Conclusion

Symmetry permeates literally everything around, capturing, it would seem, completely unexpected areas and objects. It, manifesting itself in the most diverse objects of the material world, undoubtedly reflects its most general, most fundamental properties. The principles of symmetry play an important role in physics and mathematics, chemistry and biology, engineering and architecture, painting and sculpture, poetry and music.

We see that nature designs any living organism according to a certain geometric pattern, and the laws of the universe have a clear justification. 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.

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, both in 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. 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.

So, on the plane we have four types of motions that transform the figure F into the equal figure F1:

1) parallel transfer;

2) axial symmetry (reflection from a straight line);

3) rotation around a point (Partial case - central symmetry);

4) "sliding" reflection.

In space, a mirror symmetry is added to the above types of symmetry.

I believe that the goal set in the abstract has been achieved. When writing an abstract, the greatest difficulty for me was my own conclusions. I think that my work will help schoolchildren to expand their understanding of symmetry. I hope that my essay will be included in the methodological fund of the mathematics classroom.

If there were no symmetry, what would our world look like? What would be considered the standard of beauty and perfection? What does central symmetry mean for us and what role does it play? By the way, one of the most significant. To understand this, let's get acquainted with the natural law of nature closer.

Central symmetry

First, let's define the concept. What do we mean by the phrase "central symmetry"? This is proportionality, ratio, proportionality, exact similarity of the sides or parts of something relative to a conditional or well-defined rod axis.

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.

Flora

Do you like picking mushrooms? Then you know that a mushroom cut vertically has an axis of symmetry along which it forms. You can observe the same phenomenon in round, centrally symmetrical berries. And what a beautiful cut apple! Moreover, absolutely in every plant there is some part that has developed according to the laws of symmetry.

Fauna

To notice the symmetry of insects, fortunately, they do not need to be dissected. Butterflies, dragonflies - like revived and fluttering flowers. Graceful predators and domestic cats... You can endlessly admire the creations of nature.

water world

How limitless is the species diversity of the inhabitants aquatic environment, so often there is a central symmetry. Surely everyone can give a few simple examples.

Central symmetry in life

Throughout its centuries-old history from ancient temples, medieval castles and up to the present, man has known beauty, harmony and learned to create by observing nature. The urban world, in which the majority of the world's population lives, is full of symmetry. These are houses, appliances, household items, science and art. Analogy is the key to the success of any engineering structure.

Symmetry in art

Central symmetry is not only a mathematical concept. It is present in all spheres of human life. The harmony of the rhythmic composition has never left a person indifferent. The reflection of these principles can be found in the arts and crafts: the embroidery of authentic craftswomen is completely different peoples, patterned woodcarving, self-woven carpets. There is a uniform construction of repetitions even in oral songwriting and the art of versification! And, of course, craftsmen made jewelry according to the same laws. central symmetry. It is then that the decoration takes on individuality, unique beauty and becomes a real work of art. This is how symmetry educates humanity, revealing the magical principle of order, harmony and perfection.

Symmetry (dr. gr. συμμετρία - symmetry) - 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 hand, we are convinced that by turning it around the stem, we can achieve alignment different parts 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 different kinds symmetries (shape, similarity, relative position). Moreover, organisms of different anatomical structures can have the same type of external symmetry.

External symmetry can act as the 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 the possible variants of symmetry in biological objects, strictly describe the external form and internal structure 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 monocot plants, five - for dicots.

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 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 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 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 - 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. two-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 - one of the forms of translational symmetry. It is especially pronounced in annelids, whose long body consists of a large number of almost identical segments. This case of segmentation is called 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 located 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). Segmented as internal organs and body walls. 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 not equivalent (for example, in jellyfish, the mouth is on one pole (oral), and the top of the bell is on the opposite (aboral). Such symmetry (a variant of radial symmetry) in comparative anatomy is called In a 2D projection, radial symmetry can be preserved 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.
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 an immobile 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 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. IN AND. Vernadsky. Chemical structure of the Earth's biosphere and its environment. M., 1965.

For centuries, symmetry has remained the property that has occupied the minds of philosophers, astronomers, mathematicians, artists, architects and physicists. The ancient Greeks were simply obsessed with it, and even today we tend to try to apply symmetry in everything from how we arrange furniture to how we style our hair.

No one knows why this phenomenon occupies our minds so much, or why mathematicians try to see order and symmetry in the things around us - anyway, below are ten examples that symmetry really exists, as well as that we have it. surrounded. Take into account: as soon as you think about it, you will constantly involuntarily look for symmetry in the objects around you.

Broccoli romanesco

Most likely, you have repeatedly walked past a shelf with Romanesco broccoli in a store and, because of its unusual appearance, assumed that it was a genetically modified product. But in fact, this is just one more example of fractal symmetry in nature - albeit certainly striking.

In geometry, a fractal is a complex pattern, each part of which has the same geometric pattern as the entire pattern as a whole.

Therefore, in the case of Romanesco broccoli, each flower of a compact inflorescence has the same logarithmic spiral as the entire head (just in miniature form). In fact, the entire head of this cabbage is one large spiral, which consists of small cone-like buds that also grow in mini-spirals. By the way, Romanesco broccoli is a relative of both broccoli and cauliflower, although its taste and texture are more like cauliflower.

It is also rich in carotenoids and vitamins C and K, which means that it is a healthy and mathematically beautiful addition to our food.

Honeycomb

Bees are not only the leading producers of honey - they also know a lot about geometry.

For thousands of years, people have marveled at the perfection of the hexagonal shapes in honeycombs and wondered how bees can instinctively create shapes that humans can only create with a ruler and a compass.

Honeycombs are objects of wallpaper symmetry, where a repeating pattern covers a plane (for example, a tiled floor or a mosaic). So how and why do bees love to build hexagons so much?

To begin with, mathematicians believe that this perfect shape allows the bees to store the largest amount of honey using the least amount of wax. When building other shapes, the bees would have large spaces, since shapes such as a circle, for example, do not fit completely.

Other observers, who are less inclined to believe in the cleverness of bees, believe that they form a hexagonal shape quite "accidentally". In other words, the bees actually make circles, and the wax itself takes on a hexagonal shape.

In any case, it is a work of nature and quite amazing.

sunflowers

Sunflowers boast radial symmetry and an interesting type of number symmetry known as the Fibonacci sequence. The Fibonacci sequence is: 1, 2, 3, 5, 8, 13, 21, 24, 55, 89, 144, etc. (each number is determined by the sum of the two previous numbers). If we spared no time to count the number of seed spirals in a sunflower, we would find that the number of spirals coincides with the Fibonacci numbers.

Moreover, a huge number of plants (including romanesco broccoli) release petals, leaves and seeds according to the Fibonacci sequence, which is why it is so difficult to find a four-leaf clover.

Counting spirals on sunflowers can be quite difficult, so if you want to test this principle yourself, try counting spirals on larger items such as cones, pineapples, and artichokes.

But why do sunflower flowers and other plants obey mathematical rules? As with the hexagons in the hive, it's all about efficiency. Without getting too technical, we can simply say that a sunflower flower can hold the most seeds if each seed is at an angle that is an irrational number.

It turns out that the most irrational number is the golden ratio, or Phi, and it just so happens that if we divide any Fibonacci or Lucas number by the previous number in the sequence, we get a number close to Phi (+1.618033988749895...). Thus, in any plant growing according to the Fibonacci sequence, there must be an angle that corresponds to Phi (an angle equal to the golden ratio) between each of the seeds, leaves, petals, or branches.

Nautilus shell

In addition to plants, there are also some animals that demonstrate Fibonacci numbers. For example, the Nautilus shell has grown into a "Fibonacci Spiral". The spiral is formed as a result of the shell's attempt to maintain the same proportional shape as it grows outward. In the case of the nautilus, this growth trend allows it to maintain the same body shape throughout its life (unlike humans, whose bodies change proportions as they grow older). As one would expect, there are exceptions to this rule: not every nautilus shell grows into a Fibonacci spiral.

But they all grow in the form of peculiar logarithmic spirals. And, before you start thinking that these cephalopods probably know math better than you, remember that their shells grow in this form unconsciously to them, and that they are simply using an evolutionary design that allows the mollusk to grow without changing form.

Animals

Most animals are bilaterally symmetrical, which means that they can be divided into two identical halves if a dividing line is drawn across their center of the body. Even humans are bilaterally symmetrical, and some scientists believe that a person's symmetry is the most important factor in whether we consider them physically attractive or not.

In other words, if you have a lopsided face, hope that you have a whole host of compensatory, positive qualities.

One animal most likely takes the importance of symmetry in mating rituals too seriously, and that animal is the peacock. Darwin was very annoyed by this species of bird, and in his letter in 1860 he wrote that "every time I look at a peacock tail feather - I feel sick!". For Darwin, the peacock's tail seemed somewhat burdensome, since, in his opinion, such a tail did not make evolutionary sense, since it did not fit his theory of "natural selection".

He was angry until he developed the theory of sexual selection, which is that an animal develops certain qualities in itself that will give it the best chance to mate. Obviously, for peacocks, sexual selection is considered incredibly important, since they have grown themselves various options patterns to attract their ladies, starting with the bright colors, large size, symmetry of their bodies, and the repeating pattern of their tails.

spider webs

There are approximately 5,000 species of orbweb spiders, and all of them create almost perfectly round webs with almost equidistant radial supports radiating from the center and connected in a spiral for more efficient catching of prey.

Why Orb Weaving Spiders place so much emphasis on geometry has yet to be answered by scientists, as studies have shown that rounded webs do not retain prey any better than irregularly shaped webs. Some scientists speculate that spiders build circular webs because they are more durable, and the radial symmetry helps to evenly distribute the force of impact when the prey is caught in the web, resulting in fewer breaks in the web.

But the question remains: if this is indeed the best way to create a web, then why don't all spiders use it?

Some non-orbweb spiders have the ability to create the same web, but they do not. For example, a spider recently discovered in Peru builds separate parts of a web of the same size and length (which proves its ability to "measure"), but then it simply connects all these parts of the same size in a random order into a large web that does not have any particular shape. . Maybe these spiders from Peru know something that the orbweb spiders don't, or maybe they just haven't appreciated the beauty of symmetry yet?

Crop circles with crops

Give a couple of pranksters a board, a piece of string, and a cover of darkness, and it turns out that humans are good at creating symmetrical shapes, too.

In fact, it is precisely because of the incredible symmetry and complexity of crop circle design that people continue to believe that only aliens from outer space are capable of doing this, even though the people who created the crop circles have confessed. There may have once been a mixture of human-made circles with those made by aliens, but the progressive complexity of the circles is the clearest evidence that they were made by humans.

It would be illogical to assume that the aliens will make their messages even more complicated, given that people have not really figured out the meaning of simple messages yet. Most likely, people learn from each other by examples of what they have created and more and more complicate their creations. If we put aside the talk about their origin, we can definitely say that circles are pleasant to look at, in large part because they are so geometrically impressive.

Physicist Richard Taylor has done research on crop circles and found that in addition to the fact that at least one circle is created on the earth every night, most of their designs display a wide range of symmetries and mathematical patterns, including fractals and Fibonacci spirals. .

Snowflakes

Even tiny things like snowflakes also form according to the laws of order, since most snowflakes form in six-fold radial symmetry, with complex, identical patterns on each of its branches.

Understanding why plants and animals choose symmetry is difficult in itself, but inanimate objects - how do they do it? Apparently, it all comes down to chemistry, and specifically how water molecules line up as they freeze (crystallize).

Water molecules come to a solid state by forming weak hydrogen bonds with each other. These bonds align in an ordered arrangement that maximizes attractive forces and reduces repulsive forces, which is precisely what causes the snowflake to form a hexagonal shape. However, we all know that no two snowflakes are the same, so how does a snowflake form in absolute symmetry with itself, but not like other snowflakes? As each snowflake falls from the sky, it goes through unique atmospheric conditions, such as temperature and humidity, that affect how crystals "grow" on it. All branches of a snowflake go through the same conditions and therefore crystallize in the same way - each branch is an exact copy of the other. No other snowflake goes through the same conditions as it descends, so they all look a little different.

Milky Way Galaxy

As we have seen, symmetry and mathematical patterns exist everywhere we look – but are these laws of nature limited to our planet? Apparently - no.

Having recently discovered a new part of the Milky Way, astronomers believe that our galaxy is a near-perfect reflection of itself. Based on new information, scientists have confirmed their theory that there are only two huge arms in our galaxy: Perseus and the Centauri Arm. In addition to mirror symmetry, the Milky Way has another amazing design, similar to the shells of a nautilus and a sunflower, where each arm of the galaxy is a logarithmic spiral, originating at the center of the galaxy and expanding towards the outer edge.

Symmetry of the Sun and Moon

Considering that the sun is 1.4 million kilometers in diameter and the moon is only 3.474 kilometers in diameter, it is very difficult to imagine that the Moon could block out sunlight and give us about five solar eclipses every two years.

So how does this even happen?

Coincidentally, even though the sun is about four hundred times as wide as the moon, it is four hundred times further away from us than the moon. The symmetry of this ratio causes the sun and moon to appear to be the same size when viewed from Earth, so the moon can easily block the sun when they are in line with the Earth.

The distance from the Earth to the sun, of course, can increase during its orbital entry, and when an eclipse occurs at this time, we can admire an annual or partial eclipse, since the sun is not completely covered. But every year or two, everything becomes absolutely symmetrical, and we can look at the magnificent event that we call a total solar eclipse.