Monolith / 13.07.2020

Symmetry and movement in nature. Symmetry in nature. Indeed, one of the four points of the golden ratio falls on the golden apple in the hand of Paris. More precisely, on the point of connection of the apple with the palm

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 interesting type symmetry of numbers, 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. In order not to delve into technical features, you can just say that a sunflower flower can accommodate the largest number seeds if each seed is at an angle that is an irrational number.

It turns out that the most irrational number is 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 shape.

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 your ladies, starting with 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 such a heavy 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 spread the force of impact evenly when the prey is caught in the web, resulting in fewer breaks in the web.

But the question remains: if it is true The best way creating a web, 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 it was humans who made them.

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 ground every night, most of their designs display a wide range of symmetry and mathematical models, 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 hexagonal shape of the snowflake. 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 in 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 relationship 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.

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

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

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

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

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 species diversity inhabitants of the 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: embroidery by authentic craftswomen of completely different nations, 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 of 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.

Introduction 2

Symmetry in nature 3

Symmetry in plants 3

Symmetry in animals 4

Human symmetry 5

Symmetry types in animals 5

Symmetry types 6

Mirror symmetry 7

Radial symmetry 8

Rotational symmetry 10

Helical or spiral symmetry 10

Conclusion 12

Sources 13

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

Plato

Introduction

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

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

Word "symmetry" known to us since childhood. Looking in the mirror, we see symmetrical halves of the face, looking at the palms, we also see mirror-symmetrical objects. Taking a chamomile flower in our hand, we are convinced that by turning it around the stem, we can achieve the combination of different parts of the flower. This is another type of symmetry: rotary. There are a large number of types of symmetry, but they all invariably correspond to the same general rule: under some transformation, a symmetrical object is invariably aligned with itself.

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

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

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

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

Symmetry in nature

Symmetry is possessed by objects and phenomena of living nature. It allows living organisms to better adapt to their environment and simply survive.

In living nature, the vast majority of living organisms exhibit various types of symmetries (shape, similarity, relative position). Moreover, organisms of different anatomical structures can have the same type of external symmetry.

External symmetry can act as a basis for the classification of organisms (spherical, radial, axial, etc.). Microorganisms living in conditions of weak gravity have a pronounced symmetry of shape.

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

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

Symmetry in plants

The specificity of the structure of plants and animals is determined by the characteristics of the habitat to which they adapt, the characteristics of their lifestyle.

Plants are characterized by the symmetry of the cone, which is clearly visible in the example of any tree. Any tree has a base and a top, "top" and "bottom" that perform different functions. The significance of the difference between the upper and lower parts, as well as the direction of gravity determine the vertical orientation of the "tree cone" rotary axis and symmetry planes. The tree absorbs moisture from the soil and nutrients due to the root system, that is, below, and the rest of the vital functions are performed by the crown, that is, at the top. Therefore, the directions "up" and "down" for the tree are significantly different. And the directions in the plane perpendicular to the vertical are practically indistinguishable for the tree: air, light, and moisture are equally supplied to the tree in all these directions. As a result, a vertical rotary axis and a vertical plane of symmetry appear.

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

Leaves are mirror symmetrical. The same symmetry is also found in flowers, however, in them, mirror symmetry often appears in combination with rotational symmetry. There are often cases of figurative symmetry (twigs of acacia, mountain ash). Interestingly, in the flower world, the rotational symmetry of the 5th order is most common, which is fundamentally impossible in the periodic structures of inanimate nature. Academician N. Belov explains this fact by the fact that the 5th order axis is a kind of tool for the struggle for existence, "insurance against petrification, crystallization, the first step of which would be their capture by a lattice." Indeed, a living organism does not have a crystalline structure in the sense that even its individual organs do not have a spatial lattice. However, ordered structures are very widely represented in it.

Symmetry in animals

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

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

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

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

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

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

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

Symmetry in humans

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

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

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

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

Symmetry types in animals

central

axial (mirror)

radial

bilateral

two-beam

translational (metamerism)

translational-rotational

Symmetry types

Only two main types of symmetry are known - rotational and translational. In addition, there is a modification from the combination of these two main types of symmetry - rotational-translational symmetry.

rotational symmetry. Any organism has rotational symmetry. Antimers are an essential characteristic element for rotational symmetry. It is important to know that when turning by any degree, the contours of the body will coincide with the original position. The minimum degree of coincidence of the contour has a ball rotating around the center of symmetry. The maximum degree of rotation is 360 0 when the contours of the body coincide when rotated by this amount. If the body rotates around the center of symmetry, then many axes and planes of symmetry can be drawn through the center of symmetry. If the body rotates around one heteropolar axis, then as many planes can be drawn through this axis as the antimer has given body. Depending on this condition, one speaks of rotational symmetry of a certain order. For example, six-rayed corals will have sixth order rotational symmetry. Ctenophores have two planes of symmetry and are second order symmetrical. The symmetry of the ctenophores is also called biradial. Finally, if an organism has only one plane of symmetry and, accordingly, two antimeres, then such symmetry is called bilateral or bilateral. Thin needles emanate radiantly. This helps the protozoa "soar" in the water column. Other representatives of the protozoa are also spherical - rays (radiolaria) and sunflowers with ray-like processes-pseudopodia.

translational symmetry. For translational symmetry, metameres are a characteristic element (meta - one after the other; mer - part). In this case, the parts of the body are not mirrored against each other, but sequentially one after the other along the main axis of the body.

Metamerism is a form of translational symmetry. It is especially pronounced in annelids, whose long body consists of a large number almost identical segments. This case of segmentation is called homonomous. In arthropods, the number of segments may be relatively small, but each segment differs somewhat from neighboring ones either in shape or in appendages (thoracic segments with legs or wings, abdominal segments). This segmentation is called heteronomous.

Rotational-translational symmetry . This type of symmetry has a limited distribution in the animal kingdom. This symmetry is characterized by the fact that when turning through a certain angle, a part of the body protrudes slightly forward and each next one increases its dimensions logarithmically by a certain amount. Thus, there is a combination of acts of rotation and translational motion. An example is the spiral chambered shells of foraminifera, as well as the spiral chambered shells of some cephalopods. With some condition, non-chambered spiral shells of gastropod mollusks can also be attributed to this group.

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

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All-Russiantocompetition of student essays "Krugozor"

MOU "Secondary School with. Petropavlovka, Dergachevsky district

Saratov region»

ESSAY

mathematics, biology, ecologyon the topic:

"Symmetry in Nature"

6th grade studentMOU

Leaders:Kutishcheva Nina Semyonovna,

Rudenko Ludmila Viktorovna,

Introduction

1. Theoretical part

1.1.1 Developing the doctrine of symmetry

1.1.2 Axial symmetry of figures

1.1.3 Central symmetry

1.1.4 Symmetry about the plane

2. Practical part

2.2 Rationale for the cause of symmetry in plants

Conclusion

Literature

symmetry plant geometry point

Introduction

"Symmetry is that idea, with the help of

which man has been trying to explain for centuries

and create order, beauty and perfection" Hermann Weil.

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

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

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

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

2. Reveal the reasons for the symmetry in nature.

1. Theoretical part

1.1 Basic concepts of symmetry and geometry of plants

1.1.1 The developing doctrine of symmetry

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

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

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

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

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

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

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

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

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

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

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

1.1.2 Axial symmetry of figures

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

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

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

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

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

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

1.1.3 Central symmetry

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

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

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

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

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

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

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

1.1.4 Symmetry about the plane (mirror symmetry)

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

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

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

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

1.2 Form and symmetry of plants

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

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

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

The tree absorbs moisture and nutrients from the soil 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.

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

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

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

2. Practical part

2.1 Features of each type of symmetry

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

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

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

Is it only a tree leaf that has this symmetry?

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

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

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

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

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

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

2.2 Justification of the causes of symmetry in plants

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

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

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

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

Conclusion

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

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

Gravity;

The influence of light.

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

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

Sources

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

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

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

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

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

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

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

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

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

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Symmetry and movement in nature. Symmetry in nature. Indeed, one of the four points of the golden ratio falls on the golden apple in the hand of Paris. More precisely, on the point of connection of the apple with the palm