Symmetry has always been a 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 don’t always notice it. Here are 10 beautiful examples symmetry, of which we ourselves are a part.

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, all the 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 similar to cauliflower.

Honeycomb

For thousands of years, bees have instinctively produced perfectly shaped hexagons. Many scientists believe that bees produce honeycombs in this form to retain the most honey while using the least amount of wax. Others are not so sure and believe that it is a natural formation, and the wax is formed when bees create 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 appears in the number of spirals from the seeds of a flower.

Nautilus shell

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

Animals

Animals, like people, are symmetrical on both sides. This means that there is a center line where they can be divided into two identical halves.

Spider web

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

Crop Circles.

Crop circles don't occur "naturally" at all, but they are a pretty amazing symmetry that humans can achieve. Many believed that crop circles were the result of a UFO visit, but in the end it turned out that they were the work of man. Crop circles exhibit various forms of symmetry, including Fibonacci spirals and fractals.

Snowflakes

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

Milky Way Galaxy

The 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 composed of two main arms known as the Perseus and Centauri Shield. Each of these arms has a logarithmic spiral, similar to the shell of a nautilus, with a Fibonacci sequence that begins at the center of the galaxy and expands.

Lunar-solar symmetry

The sun is much larger than the moon, four hundred times larger in fact. However, the phenomena solar eclipse occur every five years when the lunar disk completely blocks sunlight. The symmetry occurs 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.

1 of 14

Presentation on the topic:

Slide no. 1

Slide description:

Slide no. 2

Slide description:

Slide no. 3

Slide description:

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

Slide no. 4

Slide description:

Slide no. 5

Slide description:

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

Slide no. 6

Slide description:

Slide no. 7

Slide description:

The manifestation of symmetry in living nature Beauty in nature is not created, but only recorded and expressed. Let us consider the manifestation of symmetry from the “global”, namely from our planet Earth. The fact that the Earth is a ball became known to educated people in ancient times. The earth, in the minds of most well-read people before the era of Copernicus, was the center of the universe. Therefore, they considered the straight lines passing through the center of the Earth to be the center of symmetry of the Universe. Therefore, even the model of the Earth - the globe has an axis of symmetry.

Slide no. 8

Slide description:

Almost all living beings are built according to the laws of symmetry; it is not for nothing that the word “symmetry” means “proportionality” when translated from Greek. Almost all living beings are built according to the laws of symmetry; it is not for nothing that the word “symmetry” means “proportionality” when translated from Greek. Among flowers, for example, there is rotational symmetry. Many flowers can be rotated so that each petal takes the position of its neighbor, the flower aligns with itself. The minimum angle of such a rotation for various colors not the same. For the iris it is 120°, for the bellflower – 72°, for the narcissus – 60°.

Slide no. 9

Slide description:

There is helical symmetry in the arrangement of leaves on plant stems. Arranging like a screw along the stem, the leaves seem to spread out into different sides and do not shield each other from light), although the leaves themselves also have an axis of symmetry. Helical symmetry is observed in the arrangement of leaves on plant stems. Positioned like a screw along the stem, the leaves seem to spread out in different directions and do not obscure each other from the light), although the leaves themselves also have an axis of symmetry

Slide no. 10

Slide description:

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

Slide no. 11

Slide description:

Manifestation of symmetry in inanimate nature Crystals bring the charm of symmetry to the world of inanimate nature. Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have rotational symmetry and, in addition, mirror symmetry. What is a crystal? A solid body that has the natural shape of a polyhedron. Salt, ice, sand, etc. consist of crystals. First of all, Romeu-Delisle emphasized the correct geometric shape of crystals based on the law of constancy of angles between their faces. Why are crystals so beautiful and attractive? Their physical and Chemical properties determined by their geometric structure. In crystallography (the science of crystals) there is even a section called “Geometric Crystallography”. In 1867, artillery general, professor at the Mikhailovsky Academy in St. Petersburg A.V. Gadolin strictly mathematically derived all combinations of symmetry elements that characterize crystalline polyhedra. In total, there are 32 types of symmetries of ideal crystal shapes.

Slide no. 14

Slide description:

Symmetry in nature is an objective property, one of the main ones in modern natural science. This is a universal and general characteristics our material world.

Symmetry in nature is a concept that reflects the order, proportionality and proportionality existing in the world between elements various systems or objects of nature, balance of the system, orderliness, stability, that is, a certain

Symmetry and asymmetry are opposite concepts. The latter reflects the disorder of the system, the lack of equilibrium.

Shapes of symmetries

Modern natural science defines a number of symmetries that reflect the properties of the hierarchy of individual levels of organization of the material world. Known different kinds or symmetry forms:

• spatiotemporal;
• calibration;
• isotopic;
• mirrored;
• permutable.

All listed types of symmetries can be divided into external and internal.

External symmetry in nature (spatial or geometric) is represented by a huge variety. This applies to crystals, living organisms, molecules.

Internal symmetry is hidden from our eyes. It manifests itself in laws and mathematical equations. For example, Maxwell's equation, which determines the relationship between magnetic and electrical phenomena, or Einstein's property of gravity, which connects space, time and gravity.

Why is symmetry needed in life?

Symmetry in living organisms was formed during the process of evolution. The very first organisms that arose in the ocean had a perfect spherical shape. In order to penetrate into a different environment, they had to adapt to new conditions.

One of the ways of such adaptation is symmetry in nature at the level physical forms. The symmetrical arrangement of body parts ensures balance during movement, vitality and adaptation. External forms humans and large animals have a fairly symmetrical appearance. IN flora there is also symmetry. For example, the cone-shaped crown of a spruce tree has a symmetrical axis. This vertical trunk, thickened at the bottom for stability. The individual branches are also located symmetrically in relation to it, and the shape of the cone allows the crown to rationally use solar energy. The external symmetry of animals helps them maintain balance when moving, gain energy from environment using it rationally.

In chemical and physical systems, symmetry is also present. Thus, the most stable molecules are those that have high symmetry. Crystals are highly symmetrical bodies; three dimensions of an elementary atom are periodically repeated in their structure.

Asymmetry

Sometimes the internal arrangement of organs in a living organism is asymmetrical. For example, a person’s heart is located on the left, the liver on the right.

In the process of life, plants absorb chemical mineral compounds from molecules of symmetrical shape from the soil and convert them in their bodies into asymmetric substances: proteins, starch, glucose.

Asymmetry and symmetry in nature are two opposing characteristics. These are categories that are always in struggle and unity. Different levels The development of matter can have properties of either symmetry or asymmetry.

If we assume that equilibrium is a state of rest and symmetry, and non-equilibrium movement is caused by asymmetry, then we can say that the concept of equilibrium in biology is no less important than in physics. Biological is characterized by the principle of stability of thermodynamic equilibrium. It is asymmetry, which is a stable dynamic equilibrium, that can be considered a key principle in solving the problem of the origin of life.

The text of the work is posted without images and formulas.
Full version work is available in the "Work Files" tab in PDF format

Introduction.

Sometimes I involuntarily wondered: is there something in common in the forms of plants and animals? Perhaps there is some pattern, some reason that gives 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 sense something suspicious, no matter from which side it creeps up, and, depending on what it is, it is suspicious, - eat it or, conversely, run away from it.

In biology lessons, I learned 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 living and inanimate nature.

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

learn more about the concept of symmetry;

find confirmation of the existence of symmetry in nature;

prepare a presentation;

give a presentation.

Theoretical part.

1. Basic Concepts of Symmetry

We have gotten used to the word “symmetry” since childhood, and it seems that there can be nothing mysterious in this clear concept. All forms in the world are subject to the laws of symmetry. Even “eternally free” clouds have symmetry, albeit distorted. Freezing in the blue sky, they resemble slowly moving sea ​​water jellyfish, clearly gravitating towards rotational symmetry, and then, driven by the rising wind, change their symmetry to mirror one.

A truly immeasurable amount of literature is devoted to the problem of symmetry. From textbooks and scientific monographs to works that appeal not so much to a drawing and formula, but to an artistic image, and combine scientific reliability with literary precision.

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 indicates its antiquity.

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

There are many definitions of symmetry:

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

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

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

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

But it seems to me that the most complete and generalizing of all the above definitions is the opinion of Yu. A. Urmantsev: “Symmetry is any figure that can be combined with itself as a result of one or more successively produced reflections in planes.”

The word “symmetry” has a dual interpretation.

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

The second meaning of this word is balance. Aristotle also spoke about symmetry as a state that is characterized by the relationship 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 law of its duality. The initial concept of geometric symmetry as a harmony of proportions, as “proportionality”, which is what the word “symmetry” means in translation from Greek, over time acquired a universal character and was recognized as a universal 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. Equality and sameness of arrangement of parts of a figure are revealed through symmetry operations. Symmetry operations are rotations, translations, and reflections.

1. Symmetry in geometry

2.1 Symmetry of geometric figures (solids).

Mirror symmetry. Geometric figure(Fig. 1) is called symmetrical 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 bisected by this plane (EA = AE). The S plane is called the plane of symmetry. Symmetrical figures, objects and bodies are not equal to each other in in the narrow sense words (for example, the left glove is not suitable for the right hand and vice versa). They are called mirror equals.

Central symmetry. A geometric figure (Fig. 2) is called symmetrical 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. A 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 (symmetry axis), it completely coincides with its initial position. When n = 2 we have axial symmetry. Triangles also have axial symmetry.

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

The ball (sphere) has both central, mirror, and rotation 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.

A circular cone has axial symmetry; 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 located at the same distance between them.

2.2 Symmetry flat figures .

Mirror-axis 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 straight line KL along which these planes intersect is the second-order symmetry axis of the figure ABCDE. In this case, the figure ABCDE is called mirror-symmetrical.

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

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

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

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

A 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 body does not differ in its upper, lower, right, left, front and back parts, and it coincides with itself when rotated around the center of symmetry at any angle. However, this is only possible 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 on our land there is no such environment. There is at least one force - gravity - that acts only along one axis (top-bottom) and does not affect the others (forward-backward, left-right). She's pulling everything down. And living beings have to adapt to this.

This is how the next type of symmetry arises - radial. Radially symmetrical creatures have an upper and lower part, but no right and left, front and back. They coincide with themselves when rotating around only one axis. These include, for example, sea ​​stars and hydra. These creatures are sedentary and engage in a “quiet hunt” for passing living creatures. Radial symmetry is inherent in jellyfish and polyps, cross-sections of fruits of apples, lemons, oranges, persimmons (Fig. 7), etc.

But if some creature is going to lead an active lifestyle, chasing prey and escaping from predators, another direction becomes important for it - the 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 sense organs (for some lucky people, 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 (radially symmetrical animals have no head in principle). This is how bilateral (or bilateral) symmetry arises. A bilaterally symmetrical creature has different upper and lower, front and back parts, 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 the bilateral one, there is another symmetry - metameric. Their body (with the exception of the very anterior part) consists of identical metameric segments, and if you move along the body, the worm “coincides” with itself. More developed animals, including humans, retain a weak “echo” of this symmetry: 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 had two arms, two legs, two ears, two eyes, which were located mirror-symmetrically. But when I took a closer look at myself, I noticed that one eye was squinted a little more, the other less, one eyebrow was more arched, the other less; one ear is higher, the other is lower, the thumb of the left hand is slightly smaller than the finger of the right. So is there symmetry in nature and is it possible to measure it, and not just evaluate it visually “by eye”? Or maybe there are units for measuring symmetry?

Practical part.

Description of the methodology for collecting and processing data

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

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

To measure, I folded the sheet crosswise, in half, placing the top of the sheet against the base, then unbent it and took measurements along the resulting fold (Fig. 12).

1 - the width of half a 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 excel program to make it easier to process the data later.

Calculation of the average relative difference of a characteristic

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

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

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

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

In the second step, I found the value of the average relative difference between the sides per attribute for each sheet (Z). To do this, the sum of relative differences was divided by the number of characteristics.

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 ​​were entered into a table.

In the third step, I calculated the average relative difference per trait for the entire sample (X). To do this, I added all the Z values ​​and divided them 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. number of leaves (in our example - 10).

The resulting X index 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 a critical deviation from the hole of symmetry.

Summary table of data.

Tree No.

1. Width of sheet halves, mm

2. Length of the 2nd vein, mm

3. Distance between the bases of the 1st and 2nd veins, mm

4. Distance between the ends of the 1st and 2nd veins, mm

Research results

Tree number	Indicator value (X)	Symmetry

From the presented data table and diagram (Fig. 13) it can be seen that all values ​​were in the range of up to 0.055, which corresponds to the norm on the symmetry scale. Thus, all five birch trees 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. Environmental health: assessment methodology. - M., Center for Environmental Policy of Russia, 2000.

Roslova L.O., Sharygin I.F. Symmetry: Tutorial, M.: Gymnasium Publishing House " 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 explore the world: Children's encyclopedia: Mathematics / Comp. A.P. Savin, V.V. Stanzo, A.Yu. Kotova: Under the general editorship. O.G. Hinn. - M.: LLC Publishing House AST - LTD, 1998.

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

Large computer encyclopedia of Cyril and Methodius.

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

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

Ozhegov S.I. Dictionary Russian language. M. 1997.

Wolf G.V. Symmetry and its manifestations in nature. M., Ed. Dept. 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/

WITH symmetry(ancient Greek - “proportionality”) - the regular arrangement of similar (identical) parts of the body or forms of a living organism, a collection of living organisms relative to the center or axis of symmetry. This implies that proportionality is part of harmony, the correct combination of parts of the whole.

G armonia- a Greek word meaning “coherence, proportionality, unity of parts and whole.” Externally, harmony can manifest itself in melody, rhythm, symmetry and proportionality.

The law of harmony reigns in everything, And in the world everything is rhythm, chord and tone.J. Dryden

WITH perfection- the highest degree, the limit of any positive quality, ability, or skill.

"Freedom is the fundamental internal sign every creature created in the image and likeness of God; in this attribute lies the absolute perfection of the plan of creation.”N. A. Berdyaev

Symmetry – fundamental principle devices of the world.

Symmetry is a common phenomenon, its universality serves as an effective method of understanding nature. Symmetry in nature is needed to maintain stability. Within the external symmetry lies the internal symmetry of the structure, which guarantees balance.

Symmetry is a manifestation of matter’s desire for reliability and strength.

Symmetrical shapes ensure repeatability of successful shapes and are therefore more resistant to various influences. Symmetry is diverse.

In nature and, in particular, in living nature, symmetry is not absolute and always contains some degree of asymmetry. Asymmetry - (Greek α- - “without” and “symmetry”) - lack of symmetry.

Symmetry in nature

Symmetry, like proportion, was considered a necessary condition for harmony and beauty.

By looking closely at nature, you can see the commonality even in the most insignificant things and details, and find manifestations of symmetry. The shape of a tree leaf is not random: it is strictly natural. The sheet seems to be glued together from two more or less identical halves, one of which is located mirror-image relative to the other. The symmetry of a leaf stubbornly repeats itself, be it a caterpillar, a butterfly, a bug, etc.

At the highest level, there are three types of symmetry: structural, dynamic and geometric. Each of these types of symmetry at the next level is divided into classical and non-classical.

Below are the following hierarchical levels. A graphical representation of all levels of subordination gives a branched dendrogram.

In everyday life, we most often encounter the so-called mirror symmetry. This is the structure of objects when they can be divided into right and left or upper and lower halves by an imaginary axis called the axis of mirror symmetry. Moreover, the halves located on opposite sides of the axis are identical to each other.

Reflection in the plane of symmetry. Reflection is the most famous and most often found type of symmetry in nature. The mirror reproduces exactly what it "sees", but the order considered is reversed: right hand your double will actually have a left one, since the fingers are located on it in reverse order. Mirror symmetry can be found everywhere: in the leaves and flowers of plants. Moreover, mirror symmetry is inherent in the bodies of almost all living creatures, and such a coincidence is by no means accidental. Anything that can be divided into two mirror-like halves has mirror symmetry. Each of the halves serves as a mirror image of the other, and the plane separating them is called the plane of mirror reflection, or simply the mirror plane.

Rotational symmetry. The appearance of the pattern will not change if it is rotated at a certain angle around its axis. The symmetry that arises is called rotational symmetry. The leaves and flowers of many plants exhibit radial symmetry. This is a symmetry in which a leaf or flower, turning around the axis of symmetry, turns into itself. In cross sections of tissues forming the root or stem of a plant, radial symmetry is clearly visible. The inflorescences of many flowers also have radial symmetry.

Flowers, mushrooms, and trees have radial symmetry. Here it can be noted that on unpicked flowers and mushrooms and growing trees, the planes of symmetry are always oriented vertically. Determining the spatial organization of living organisms, the right angle organizes life through the forces of gravity. The biosphere (the layer of existence of living beings) is orthogonal to the vertical line of gravity. Vertical plant stems, tree trunks, horizontal surfaces of water bodies and the earth's crust in general form a right angle. The right angle underlying the triangle rules the space of symmetry of similarities, and similarity, as already mentioned, is the goal of life. Both nature itself and the original part of man are at the mercy of geometry, subject to symmetry both as essence and as symbols. No matter how the objects of nature are built, each has its own main feature, which is reflected in the form, be it an apple, a grain of rye or a person.

Examples of radial symmetry.

The simplest type of symmetry is mirror (axial), which occurs when a figure rotates around an axis of symmetry.

In nature, mirror symmetry is characteristic of plants and animals that grow or move parallel to the surface of the Earth. For example, the wings and body of a butterfly can be called the standard of mirror symmetry.

Axial symmetry this is the result of rotation of absolutely identical elements around a common center. Moreover, they can be located at any angle and with different frequencies. The main thing is that the elements rotate around a single center. In nature, examples axial symmetry most often found among plants and animals that grow or move perpendicular to the Earth's surface.

There is also helical symmetry.

Translation can be combined with reflection or rotation, which creates new symmetry operations.

A rotation by a certain number of degrees, accompanied by a translation over a distance along the axis of rotation, generates helical symmetry - the symmetry of a spiral staircase.

An example of helical symmetry is the arrangement of leaves on the stem of many plants.

If we consider the arrangement of leaves on a tree branch, we will notice that the leaf is spaced apart from the other, but also rotated around the axis of the trunk.

The leaves are located on the trunk along a helical line so as not to block sunlight from each other. The sunflower head has shoots arranged in geometric spirals, unwinding from the center outward. The youngest members of the spiral are in the center. In such systems, one can notice two families of spirals, unwinding in opposite directions and intersecting at angles close to straight lines.

But no matter how interesting and attractive the manifestations of symmetry in the plant world are, there are still many secrets that control development processes. Following Goethe, who spoke about the tendency of nature towards a spiral, we can assume that this movement is carried out along a logarithmic spiral, each time starting from a central, fixed point and combining translational movement (stretching) with a rotation.

Based on this, we can formulate in a somewhat simplified and schematized form (from two points) the general law of symmetry, which clearly and everywhere manifests itself in nature:

1. Anything that grows or moves vertically, i.e. up or down relative to earth's surface, is subject to radial symmetry in the form of a fan of intersecting symmetry planes. The leaves and flowers of many plants exhibit radial symmetry. This is a symmetry in which a leaf or flower, turning around the axis of symmetry, turns into itself. In cross sections of tissues forming the root or stem of a plant, radial symmetry is clearly visible. The inflorescences of many flowers also have radial symmetry.

2. Everything that grows and moves horizontally or obliquely in relation to the earth’s surface is subject to bilateral symmetry, leaf symmetry.

Not only flowers, animals, easily moving liquids and gases, but also hard, inflexible stones are subject to this universal law of two postulates. This law affects the changing shapes of clouds. On a windless day, they have a dome-shaped shape with more or less clearly defined radial symmetry. The influence of the universal law of symmetry is essentially purely external, crude, leaving its mark only on the external form of natural bodies. Their internal structure and details escape his control.

Symmetry is based on similarity. It means such a relationship between elements and figures when they repeat and balance each other.

Symmetry of similarity. Another type of symmetry is similarity symmetry, associated with the simultaneous increase or decrease of similar parts of the figure and the distances between them. An example of this kind of symmetry is the matryoshka doll. Such symmetry is very widespread in living nature. It is demonstrated by all growing organisms.

The basis of the evolution of living matter is the symmetry of similarity. Consider a rose flower or a head of cabbage. An important role in the geometry of all these natural bodies is played by the similarity of their similar parts. Such parts, of course, are interconnected by some general geometric law, not yet known to us, which allows us to derive them from each other. The symmetry of similarity, realized in space and time, is everywhere manifested in nature on everything that grows. But it is precisely the growing forms that include the countless figures of plants, animals and crystals. The shape of the tree trunk is conical, highly elongated. The branches are usually located around the trunk in a helical line. This is not a simple helix: it gradually tapers towards the top. And the branches themselves become smaller as they approach the top of the tree. Consequently, here we are dealing with a helical axis of similarity symmetry.

Living nature in all its manifestations reveals the same goal, the same meaning of life: every living object repeats itself in its own kind. The main task of life is life, and the accessible form of existence lies in the existence of individual integral organisms. And not only primitive organizations, but also complex cosmic systems, such as man, demonstrate an amazing ability to literally repeat from generation to generation the same forms, the same sculptures, character traits, the same gestures, manners.

Nature discovers similarity as its global genetic program. The key to change also lies in similarity. Similarity rules living nature as a whole. Geometric similarity - general principle spatial organization of living structures. A maple leaf is similar to a maple leaf, a birch leaf is similar to a birch leaf. Geometric similarity permeates all branches of the tree of life. No matter what metamorphoses it undergoes in the process of growth in the future living cell, belonging to the entire organism and performing the function of its reproduction into a new, special, individual object of existence, it is the point of “beginning”, which as a result of division will be transformed into an object similar to the original one. This unites all types of living structures, for this reason there are stereotypes of life: man, cat, dragonfly, earthworm. They are endlessly interpreted and varied by division mechanisms, but remain the same stereotypes of organization, form and behavior.

For living organisms, the symmetrical arrangement of parts of the body organs helps them maintain balance during movement and functioning, ensures their vitality and better adaptation to the surrounding world, which is also true in the plant world. For example, the trunk of a spruce or pine tree is most often straight and the branches are evenly spaced relative to the trunk. The tree, developing under the influence of gravity, reaches a stable position. Towards the top of the tree, its branches become smaller in size - it takes on the shape of a cone, since light must fall on the lower branches, as well as on the upper ones. In addition, the center of gravity should be as low as possible; the stability of the tree depends on this. The laws of natural selection and universal gravitation contributed to the fact that the tree is not only aesthetically beautiful, but also designed expediently.

It turns out that the symmetry of living organisms is associated with the symmetry of the laws of nature. On an everyday level, when we see the manifestation of symmetry in living and inanimate nature, we involuntarily experience a feeling of satisfaction with the universal, as it seems to us, order that reigns in nature.

As living organisms become more orderly and become more complex during the development of life, asymmetry increasingly prevails over symmetry, displacing it from biochemical and physiological processes. However, a dynamic process also takes place here: symmetry and asymmetry in the functioning of living organisms are closely related. Externally, humans and animals are symmetrical, but their internal structure is significantly asymmetrical. If lower biological objects, for example lower plants, reproduction proceeds symmetrically, then in the higher there is a clear asymmetry, for example, the division of the sexes, where each sex brings into the process of self-reproduction genetic information peculiar only to it. Thus, the stable preservation of heredity is a manifestation of symmetry in a certain sense, and asymmetry is manifested in variability. In general, the deep internal connection between symmetry and asymmetry in living nature determines its emergence, existence and development.

The Universe is an asymmetrical whole, and life as it appears must be a function of the asymmetry of the Universe and the consequences that flow from it. Unlike molecules of inanimate nature, molecules of organic substances have a pronounced asymmetric character (chirality). Giving great importance asymmetry of living matter, Pasteur considered it precisely the only, clearly demarcating line that can currently be drawn between living and inanimate nature, i.e. what distinguishes living matter from nonliving matter. Modern science proved that in living organisms, as in crystals, changes in structure correspond to changes in properties.

It is assumed that the resulting asymmetry occurred abruptly as a result of the Big Biological Bang (by analogy with the Big Bang, as a result of which the Universe was formed) under the influence of radiation, temperature, electromagnetic fields, etc. and is reflected in the genes of living organisms. This process is essentially also a process of self-organization.