Municipal state educational institution

"Secondary school No. 1"

Research

"Symmetry and Snowflakes"

Completed by: Davtyan Anna

student of 8th grade "A"

Head: Volkova S.V.

Mathematic teacher

Shchuchye, 2016

Content

Introduction ……………………………………………………………………..……3

1. Theoretical part ……………………………………………….…….....4-5

1.1. Symmetry in nature................................................................... .......................................4

1.2. How is a snowflake born?……………………………………………..…..4

1.3. Shapes of snowflakes................................................... ...........................................4-5

1.4 Snowflake researchers...................................................…………… ……...…5

2. Practical part …………………………………………………...……6-7

2.1. Experiment 1. Are all snowflakes the same?.................…………………...…….6

2.2. Experiment 2. Let’s take a photo of a snowflake and make sure that it has six points…………………………………………………………………………………...…..6

2.3. Questioning classmates and analyzing questionnaires…………………………6-7

Conclusion ……………………………………………………………………….8

Literature ………………………………………………………………………..9

Applications .........................................................................................................10

Introduction

“...to be beautiful means to be symmetrical and proportionate”

Symmetry (ancient Greek συμμετρία - “proportionality”), in a broad sense - immutability under any transformations. The principles of symmetry play an important role in physics and mathematics, chemistry and biology, technology and architecture, painting and sculpture. “Is it possible to create order, beauty and perfection with the help of symmetry?”, “Should there be symmetry in everything in life?” - I asked myself these questions a long time ago, and I will try to answer them in this work.The subject of this study is symmetry as one of the mathematical foundations behindbeauty laws using snowflakes as an example. Relevance The problem lies in showing that beauty is an external sign of symmetry and, above all, has a mathematical basis.Goal of the work - use examples to consider and study the formation and shape of snowflakes.Job objectives: 1. collect information on the topic under consideration; 2.highlight symmetry as the mathematical basis of the laws of beauty of snowflakes.3.conduct a survey among classmates “What do you know about snowflakes?”4.competition for the most beautiful hand-made snowflake.To solve the problems, the following were usedmethods: searching for the necessary information on the Internet, scientific literature, questioning classmates and analyzing questionnaires, observation, comparison,. generalization. Practical significance research consists

in drawing up a presentation that can be used in mathematics lessons, the natural world, fine arts and technology, and extracurricular activities;

in enriching vocabulary.

1. Theoretical part. 1.1. Symmetry of snowflakes. Unlike art or technology, beauty in nature is not created, but only recorded and expressed. Among the infinite variety of forms of living and inanimate nature, such perfect images are found in abundance, whose appearance invariably attracts our attention. Such images include some crystals and many plants.Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have symmetry - rotational symmetry of the 6th order and, in addition, mirror symmetry. 1.2. How a snowflake is born. People living in northern latitudes have long been interested in why in winter when snow falls it is not round, like rain. Where do they come from?
Snowflakes also fall from clouds, just like rain, but they are not formed quite like rain. Previously, they thought that snow was frozen droplets of water and that it came from the same clouds as rain. And not so long ago, the mystery of the birth of snowflakes was solved. And then they learned that snow will never be born from droplets of water. Snow crystals form in cold clouds high above the ground when an ice crystal forms around a small speck of dust or bacteria. Ice crystals are hexagon shaped. It is because of this that most snowflakes are shaped like a six-pointed star. Then this crystal begins to grow. Its rays may begin to grow, these rays may have shoots, or, conversely, the snowflake begins to grow in thickness. Regular snowflakes have a diameter of about 5 mm and a weight of 0.004 grams. The world's largest snowflake was discovered in the USA in January 1887. The diameter of the snow beauty was as much as 38 cm! And in Moscow on April 30, 1944, the strangest snow in the history of mankind fell. Snowflakes the size of a palm circled over the capital, and their shape resembled ostrich feathers.

1.3. Snowflake shapes.

The shape and growth of snowflakes depend on air temperature and humidity.As the snowflake grows, it becomes heavier and falls to the ground, changing its shape. If a snowflake spins like a top when it falls, then its shape is perfectly symmetrical. If it falls sideways or otherwise, then its shape will be asymmetrical. The greater the distance a snowflake flies from the cloud to the ground, the larger it will be. Falling crystals stick together to form snow flakes. Most often, their size does not exceed 1-2 cm. Sometimes these flakes are of record sizes. In Serbia in the winter of 1971, snow fell with flakes up to 30 cm in diameter! Snowflakes are 95% air. This is why snowflakes fall to the ground so slowly.

Scientists studying snowflakes have identified nine main forms of snow crystals. They were given interesting names: plate, star, column, needle, fluff, hedgehog, cufflink, icy snowflake, croup-shaped snowflake. (Appendix 1)

1.4. Snezhinka researchers.

Hexagonal openwork snowflakes became the subject of study back in 1550. Archbishop Olaf Magnus of Sweden was the first to observe snowflakes with the naked eye and sketch them.His drawings suggest that he did not notice their six-pointed symmetry.

AstronomerJohannes Keplerpublished a scientific treatise “On Hexagonal Snowflakes.” He “disassembled the snowflake” from the point of view of strict geometry.
In 1635, a French philosopher, mathematician and natural scientist became interested in the shape of snowflakes.Rene Descartes. He classified the geometric shape of snowflakes.

The first photograph of a snowflake under a microscope was taken by an American farmer in 1885.Wilson Bentley. Wilson has been photographing all types of snow for nearly fifty years and has taken over 5,000 unique photographs over the years. Based on his work, it was proven that there is not a single pair of absolutely identical snowflakes.

In 1939Ukihiro Nakaya, a professor at Hokkaido University, also began to seriously study and classify snowflakes. And over time, he even created the “Ice Crystal Museum” in the city of Kaga (500 km west of Tokyo).

Since 2001, snowflakes have been grown artificially in the laboratory of Professor Kenneth Libbrecht.

Thanks to the photographerDonKomarechkafrom Canadawe havethere was an opportunity to admire the beauty and diversitysnowflakes. He takes macro photographs of snowflakes. (Appendix 2).

2. Practical part.

1.1. Experiment 1. Are all snowflakes the same?

When snowflakes began to fall from the sky to the ground, I took a magnifying glass, a notebook with a pencil and sketched the snowflakes. I managed to make drawings of several snowflakes. This means that snowflakes have different shapes.

1.2. Experiment 2. Let's take a photo of a snowflake and make sure that it has six points.

For this experiment I needed a digital camera and black velvet paper.

When the snowflakes began to fall to the ground, I took the black paper and waited for the snowflakes to fall on it. I photographed several snowflakes with a digital camera. Output the images via computer. When the pictures were enlarged, it was clearly visible that the snowflakes had 6 rays. It is impossible to get beautiful snowflakes at home. But you can “grow” your own snowflakes by cutting them out of paper. Or bake from dough. You can also draw entire snow dances. After all, everyone can do this! (Appendix 3.4).

1.3. Questioning classmates and analyzing questionnaires.

At the first stage of the study, a survey was conducted among children in grade 8A: “What do you know about snowflakes?” 24 people took part in the survey. Here's what I found out.

What is a snowflake made of?

a) I know - 17 people.

b) I don’t know - 7 people.

Are all snowflakes the same?

a) yes – 0 people.

b) no – 20 people.

c) I don’t know – 4 people.

Why is a snowflake hexagonal?

a) I know – 6 people.

b) don’t know – 18 people

Is it possible to photograph a snowflake?

a) yes – 24 people.

b) no – 0 people.

c) I don’t know – 0 people.

5. Is it possible to get a snowflake at home:

a) possible – 3 people.

b) impossible – 21 people.

Conclusion: knowledge about snowflakes is not 100%.

At the second stage, a competition was held for the most beautiful snowflake cut out of paper.

Based on the results of the survey, diagrams were constructed (Appendix 5).

Conclusion

Symmetry, manifesting itself in a wide variety of objects of the material world, undoubtedly reflects its most general, most fundamental properties.
Therefore, the study of the symmetry of various natural objects and the comparison of its results is a convenient and reliable tool for understanding the basic laws of the existence of matter. You can see that this apparent simplicity will take us far into the world of science and technology and will allow us to test the abilities of our brain from time to time (since it is the brain that is programmed for symmetry). “The principle of symmetry covers all new areas. From the field of crystallography, solid state physics, he entered the field of chemistry, the field of molecular processes and atomic physics. There is no doubt that we will find its manifestations in the world of the electron, even more distant from the complexes surrounding us, and the phenomena of quanta will be subordinate to it,” these are the words of Academician V.I. Vernadsky, who studied the principles of symmetry in inanimate nature.

Literature:

Great schoolchild encyclopedia. " Planet Earth". – Publishing house “Rosman-Press”, 2001 - 660 p. / A.Yu.Biryukova.

Everything about everything. Popular encyclopedia for children. – Publishing House

“Klyuch-S, Philological Society “Slovo”, 1994 - 488 pp. / Slavkin V.

Colors of nature: A book for elementary school students - M: Prosveshchenie, 1989 - 160 pp. / Korabelnikov V.A.

Internet resources:

http://vorotila.ru/Otdyh-turizm-oteli-kurorty/Snezhnye-tayny-i174550

Electronic children's encyclopedia "Pochemuchki".

“Mandelbrot Fractals” - There are several methods for obtaining algebraic fractals. The concept of "fractal". Lots of Julia. The role of fractals in computer graphics today is quite large. Fractals. Let's turn to the classics - the Mandelbrot set. Sierpinski triangle. Gallery of fractals. Journey into the world of fractals. The second large group of fractals are algebraic.

“Sheet of Paper” - A triangle is cut out of paper. In geometry, paper is used to: write, draw; cut; bend. The practical properties of paper give rise to a peculiar geometry. Geometry and sheet of paper. What paper actions can be used in geometry? Among the many possible actions with paper, an important place is occupied by the fact that it can be cut.

“Sine function” - Average sunset time – 18 hours. Date of. Different faces of trigonometry. Time. Using a tear-off calendar, it is easy to mark the moment of sunset. Target. Sunset schedule. Conclusions. The process of sunset is described by the trigonometric sine function. Sunset.

“Lobachevsky geometry” - Euclidean axiom about parallels. It cannot be said that non-Euclidean geometry is the only correct one. “How does Lobachevsky’s geometry differ from Euclid’s geometry?” Is non-Euclidean geometry the only correct one? Riemannian geometry got its name from B. Riemann, who laid its foundations in 1854.

“Proof of Pythagorean Theorem” - Pythagorean Theorem. The simplest proof. Geometric proof. The meaning of the Pythagorean theorem. Euclid's proof. “In a right triangle, the square of the hypotenuse is equal to the sum of the squares of the legs.” The Pythagorean theorem is one of the most important theorems in geometry. Proof of the theorem. Statement of the theorem.

"Pythagorean Theorem" - Creates the "Pythagorean" school around 510. BC. Aphorisms. Proof of the theorem. Divisibility of numbers. Here is the problem of an Indian mathematician of the 12th century. Bhaskars. The Pythagoreans had an oath with the number 36. Friendly numbers. Pythagoras began to represent numbers with dots. The number 3 is a triangle, the triangle defines a plane.

There are a total of 13 presentations in the topic

Snow is a letter from heaven, written in secret hieroglyphs.
Ukichiro Nakaya

In Japanese gardens you can find an unusual stone lantern topped with a wide roof with edges curved upward. This is the Yukimi-Toro, a lantern for admiring the snow. The Yukimi holiday is designed to give people enjoyment of the beauty of everyday life. We also decided to look at the beauty in the everyday and came a little closer to “Yukimi-Toro” than usual. On the stone roof of the lantern there are millions of tiny snowflakes, each of which is unique and worthy of close attention. Amazed by the extremely complex shape, perfect symmetry and endless variety of snowflakes, people from ancient times associated their outlines with the action of supernatural forces or divine providence.

Many great scientists dreamed of solving the mystery of snow crystals. Back in 1611, a treatise on the six-ray symmetry of snowflakes was published by the famous German mathematician and astronomer Johannes Kepler. The first systematic classification of the geometric shapes of snowflakes was created in 1635 by none other than the famous mathematician, physicist, physiologist and philosopher Rene Descartes. He was able to detect even such rare snow crystals as tipped columns and twelve-rayed snowflakes with the naked eye. The most complete study of the structure of snowflakes and their varieties was published by Japanese nuclear physicist Ukichiro Nakaya only in the middle of the last century. To unravel the mysteries of the formation of snow crystals, modern understanding of the molecular structure of ice and sophisticated research technologies, such as X-ray crystallography, were needed.

Despite the achievements of modern science, people still continue to ask questions that interested them thousands of years ago: why are snowflakes symmetrical, why is snow white, is it true that among all the snowflakes in the world, no two are alike? Caltech physics professor Kenneth Libbrecht answered our questions. He devoted a significant part of his life to the study of snow crystals, while learning how to grow snowflakes in laboratory conditions and even control their shape. In addition, Professor Libbrecht is known for having the largest and most diverse collection of snowflake photographs.

Trinity of water

Many people mistakenly believe that snowflakes are raindrops frozen on their way to the ground. Of course, such an atmospheric phenomenon also happens and is called “snow and rain,” but there are no beautiful geometrically correct snowflakes in this cocktail. Real snowflakes grow when water vapor condenses on the surface of an ice crystal, bypassing the liquid phase. Water is the only substance that can be observed in everyday life at the triple point of the phase diagram: its solid, gaseous and liquid stages can coexist at a temperature of approximately 0.01 degrees Celsius. The very first ice crystal, which serves as the foundation of a future snowflake, can be formed from a microscopic droplet of liquid water, but all further construction occurs due to the addition of water vapor molecules.

The answer to the mysterious symmetry of snowflakes lies in the crystal lattice of ice. Ice is a unique substance that can form more than ten different crystal structures. Cube Ice IX became the centerpiece of Kurt Vonnegut's novel Cat's Cradle, where it was credited with the fantastic ability to freeze all the water on Earth with just one small pellet. In fact, almost all the ice on the planet crystallizes in a hexagonal system - its molecules form regular prisms with a hexagonal base. It is the hexagonal shape of the lattice that ultimately determines the six-ray symmetry of snowflakes.

However, the connection between the structure of the crystal lattice and the shape of a snowflake, which is ten million times larger than a water molecule, is not obvious: if water molecules were attached to the crystal in a random order, the shape of the snowflake would be irregular. It's all about the orientation of the molecules in the lattice and the arrangement of free hydrogen bonds, which contributes to the formation of smooth edges. Imagine a game of Tetris: placing a smooth cube on a smooth surface is somewhat more difficult than filling a gap in a smooth line. In the first case, you have to make a choice and think through a strategy for the future. And in the second - everything is clear. Likewise, water vapor molecules are more likely to fill voids rather than adhere to smooth edges because the voids contain more free hydrogen bonds. As a result, snowflakes take the shape of regular hexagonal prisms with smooth edges. Such prisms fall from the sky at relatively low air humidity under a wide variety of temperature conditions.

Sooner or later, irregularities appear on the edges. Each bump attracts additional molecules and begins to grow. A snowflake travels through the air for a long time, and the chances of meeting new water molecules near the protruding tubercle are slightly higher than at the faces. This is how rays grow on a snowflake very quickly. One thick ray grows from each face, since molecules do not tolerate emptiness. Branches grow from the tubercles formed on this ray. During the journey of a tiny snowflake, all its faces are in the same conditions, which serves as a prerequisite for the growth of identical rays on all six faces.

Star family

It is interesting to observe a phenomenon only when you feel its diversity.

It is very difficult to classify a phenomenon that has no repetitions in nature. “All snowflakes are different, and their grouping is largely a matter of personal preference,” says Kenneth Libbrecht. The International Classification of Solid Precipitation identifies seven main types of snowflakes. The table created by Ukichiro Nakaya contains 41 morphological types. Meteorologists Magono and Lee expanded Nakai's table to 81 types. We invite you to familiarize yourself with several characteristic types of snow crystals.

Path of light

The route along which a snowflake travels from heaven to earth directly determines its appearance. In areas with different humidity, temperature and pressure, the edges and rays grow differently. A snowflake that the wind has carried over a wide area has every chance of acquiring the most bizarre shape. The longer a snowflake takes to fall to the ground, the larger it can become. The largest snowflake was recorded in 1887 in Montana, America. Its diameter was 38 cm and its thickness was 20 cm. In Moscow, the largest snowflakes, the size of a palm, fell on April 30, 1944.

Chasing snow

To get a good look at real snowflakes, you need to at least leave the house. And especially large and beautiful specimens will have to be hunted throughout the country. First, you should look at the precipitation map and select those places where it often snows. In the same way, skiers chase snow, but we are not on the same path with them: in equipped mountain resorts, as a rule, it is relatively warm, from 0 to -5 degrees. In such weather, snowflakes, approaching the ground, melt, become covered with frost, their shape is smoothed out or completely lost. For good snow you need good frost - about a couple of tens of degrees below zero. It allows snowflakes to grow confidently, maintaining the sharpness of their rays and edges all the way to the ground. However, here too it is important to know when to stop: as a rule, all the snow falls at the same -20°C, and with a further drop in temperature the air remains dry and precipitation does not form. Of course, in the polar regions, where temperatures rarely rise above -40°C and the air is very dry, it still snows. At the same time, snowflakes are tiny hexagonal prisms with perfectly smooth edges, without the slightest smoothing of the corners. But in central Russia, especially in Central Siberia, sometimes huge stars with a diameter of up to 30 cm fall out. The likelihood of seeing large snowflakes increases significantly near bodies of water: evaporation from lakes and reservoirs is an excellent building material. And of course, the absence of strong wind is highly desirable, otherwise large snowflakes will collide with each other and break. Therefore, a forest landscape is preferable to steppes and tundras.

Even Kenneth Libbrecht, traveling around the world in search of rare snow crystals, has still not been able to find an accurate way to predict where and when the snow will be best - there are too many random variables in this formula, and the result can be the most unexpected. For example, Ukichiro Nakaya discovered and photographed almost all the crystals that formed the basis of his classification in his homeland, on the island of Hokkaido in Japan.

Usually snowflakes are small, a couple of millimeters in diameter and a couple of milligrams in weight. Nevertheless, by the end of winter, the mass of snow cover in the northern hemisphere of the planet reaches 13,500 billion tons. The snow-white blanket reflects up to 90% of sunlight into space. And why, in fact, snow-white? Why does snow look white while snowflakes are made of transparent ice? Everything is explained by the complex shape of snowflakes, their large number and the ability of ice to refract and reflect light. Passing through the numerous faces of snowflakes, rays of light are refracted and reflected, changing direction unpredictably. The snow is illuminated by the sun and partly by rays of different colors reflected from surrounding objects. As a result of numerous refractions, the reflections of objects are scattered and the snow returns mostly white sunlight. A mountain of crushed ice or broken glass has exactly the same property. Of course, during numerous re-reflections, snow absorbs some of the light, and light from the red spectrum is absorbed more actively than light from the blue spectrum. On the surface, the bluish tint of snow is barely noticeable, since with a direct hit almost all the light is reflected. Try to make a deep narrow hole in the snow, to the bottom of which no light would penetrate. In the depths of the hole, you will be able to see the light passing through the thickness of the snow - and it will be blue.

Snow mythology

The symmetry and identity of all rays of snowflakes is due to the presence of an information channel between them.
Wrong. Many people find it difficult to believe in a simple explanation of the symmetry of snowflakes, which is as follows: during growth, all the faces and rays of snowflakes are in exactly the same conditions, so they may well grow the same. Trying to explain symmetry, people introduce surface energy, quantum quasiparticles phonons, excitations of the crystal lattice, and even supernatural forces into theories. Professor Kenneth suggests taking into account the fact that the vast majority of snowflakes are completely unsymmetrical, and his collection of photographs of regularly shaped snowflakes is the result of careful selection. So the only factors of symmetry are stable growth conditions and luck.

Snow made using snow cannons at ski resorts is absolutely identical to natural snow.
Wrong. Real snowflakes form when water vapor condenses on an ice crystal without passing through the liquid phase. Snow cannons spray liquid water into small droplets that freeze in the cold air and fall to the ground. Frozen drops have no edges or rays, they are just small shapeless pieces of ice. Skiing on them is no worse than on natural snow crystals, except that they crunch less loudly.

There are no two identical snowflakes in nature.
Right. Here you need to decide what is considered a snowflake and what is meant by the word “identical”. Microscopic ice crystals, consisting of several water molecules, can be absolutely identical. Although here it should be taken into account that for every 5000 water molecules there is one, which contains deuterium instead of ordinary hydrogen. Simple snowflakes, such as prisms that form in low humidity, may look the same. Although at the molecular level they will, of course, be different. But complex star-shaped snowflakes really do have a unique geometric shape that can be distinguished by the eye. And there are more variants of such forms, according to physicist John Nelson of Ritsumeikan University in Kyoto, than there are atoms in the observable Universe.

When the snowflake melts, the resulting water can be frozen, and it will take the original shape of the snowflake.
Wrong. It's the 21st century, but this fairy tale continues to be passed down from generation to generation. This is impossible both from the point of view of physics and from the point of view of common sense. Yes, water molecules can unite into clusters due to hydrogen bonds, but these bonds in the liquid phase last no more than a picosecond (10 -12 s), so water has a maiden memory. There can be no talk of any long-term memory of water at the macro level. In addition, as we have already found out, snowflakes are formed not from water, but from water vapor.

On Soviet posters you can see snowflakes with five rays. They exist?
Wrong. The artists painted snowflakes with five rays not from life, but guided by their own ideological zeal and the orders of the party.

In some cases, snow can take on completely unexpected shades. In the Arctic regions you can see red snow: it does not melt for a long time, so algae live between its crystals. In the middle of the last century, black snow fell in industrial European cities, heated mainly by coal. Residents of modern Chelyabinsk told us about black snow.

Fresh snow on a frosty day is always accompanied by a cheerful crunch underfoot. This is nothing more than the sound of crystals breaking. No one can hear one snowflake breaking, but thousands of small crystals are a solid orchestra. The lower the thermometer drops, the harder and more fragile the snowflakes become and the higher the pitch of the crunch underfoot becomes. Once you gain experience, you can use this property of snow to determine the temperature by ear.

Snow pattern

The art of growing ice crystals is not accessible to everyone: you need a diffusion chamber, a lot of measuring equipment, special knowledge and a lot of patience. Cutting snowflakes out of paper is much easier, although this art is fraught with no less creative possibilities.

You can choose patterns suggested on the pages of the magazine, or come up with your own. The most exciting moment comes when the patterned blank unfolds and turns into a large lace snowflake.

See also about snowflakes:
Photos don't melt. How to Capture the Unique Shape of Snowflakes for Story
Design in cool colors. Advice for beginning elemental masters (“Popular Mechanics” No. 1, 2008).

Introduction.
Looking at various snowflakes, we see that they are all different in shape, but each of them represents a symmetrical body.
We call bodies symmetrical if they consist of equal, identical parts. The elements of symmetry for us are the plane of symmetry (mirror image), the axis of symmetry (rotation around an axis perpendicular to the plane). There is one more element of symmetry - the center of symmetry.
Imagine a mirror, but not a big one, but a point mirror: a point at which everything is displayed as in a mirror. This point is the center

Symmetry. With this display, the reflection rotates not only from right to left, but also from the face to the wrong side.
Snowflakes are crystals, and all crystals are symmetrical. This means that in each crystalline polyhedron one can find planes of symmetry, axes of symmetry, centers of symmetry and other symmetry elements so that identical parts of the polyhedron fit together.
And indeed symmetry is one of the main properties of crystals. For many years, the geometry of crystals seemed a mysterious and insoluble riddle. The symmetry of crystals has always attracted the attention of scientists. Already in the year 79 of our chronology, Pliny the Elder mentions the flat-sided and straight-sided nature of crystals. This conclusion can be considered the first generalization of geometric crystallography.
FORMATION OF SNOWFLAKES
In 1619, the great German mathematician and astronomer Johann Kepler drew attention to the sixfold symmetry of snowflakes. He tried to explain it by saying that the crystals are built from the smallest identical balls, closely attached to each other (only six of the same balls can be tightly arranged around the central ball). Robert Hooke and M.V. Lomonosov subsequently followed the path outlined by Kepler. They also believed that the elementary particles of crystals could be likened to tightly packed balls. Nowadays, the principle of dense spherical packings underlies structural crystallography; only the solid spherical particles of ancient authors have now been replaced by atoms and ions. 50 years after Kepler, the Danish geologist, crystallographer and anatomist Nicholas Stenon first formulated the basic concepts of crystal formation: “The growth of a crystal does not occur from within, as in plants, but by superimposing on the outer planes of the crystal the smallest particles brought from outside by some liquid.” This idea about the growth of crystals as a result of the deposition of more and more layers of matter on the faces has retained its significance to this day. For each given substance there is its own, unique ideal form of its crystal. This form has the property of symmetry, that is, the property of crystals to align with themselves in different positions through rotations, reflections, and parallel transfers. Among the elements of symmetry, there are axes of symmetry, planes of symmetry, center of symmetry, and mirror axes.
The internal structure of a crystal is represented in the form of a spatial lattice, in the identical cells of which, having the shape of parallelepipeds, identical smallest particles - molecules, atoms, ions and their groups - are placed according to the laws of symmetry.
The symmetry of the external shape of a crystal is a consequence of its internal symmetry - the ordered relative arrangement in space of atoms (molecules).
Law of constancy of dihedral angles.
Over the course of many centuries, material accumulated very slowly and gradually, which made it possible at the end of the 18th century. discover the most important law of geometric crystallography - the law of constancy of dihedral angles. This law is usually associated with the name of the French scientist Romé de Lisle, who in 1783. published a monograph containing abundant material on measuring the angles of natural crystals. For each substance (mineral) he studied, it turned out to be true that the angles between the corresponding faces in all crystals of the same substance are constant.
One should not think that before Romé de Lisle, none of the scientists dealt with this problem. The history of the discovery of the law of constancy of angles has covered a long, almost two-century path before this law was clearly formulated and generalized for all crystalline substances. So, for example, I. Kepler already in 1615. pointed to the preservation of angles of 60° between individual rays of snowflakes.
All crystals have the property that the angles between the corresponding faces are constant. The edges of individual crystals may be developed differently: edges observed on some specimens may be absent on others - but if we measure the angles between the corresponding faces, then the values ​​​​of these angles will remain constant regardless of the shape of the crystal.
However, as the technique improved and the accuracy of measuring crystals increased, it became clear that the law of constant angles was only approximately justified. In the same crystal, the angles between faces of the same type are slightly different from each other. For many substances, the deviation of dihedral angles between the corresponding faces reaches 10 -20′, and in some cases even a degree.
DEVIATIONS FROM THE LAW
The faces of a real crystal are never perfect flat surfaces. They are often covered with pits or growth tubercles; in some cases, the edges are curved surfaces, such as diamond crystals. Sometimes flat areas are noticed on the faces, the position of which is slightly deviated from the plane of the face itself on which they develop. In crystallography, these regions are called vicinal faces, or simply vicinals. Vicinals can occupy most of the plane of a normal face, and sometimes even completely replace the latter.
Many, if not all, crystals split more or less easily along certain strictly defined planes. This phenomenon is called cleavage and indicates that the mechanical properties of crystals are anisotropic, i.e., not the same in different directions.
CONCLUSION
Symmetry is manifested in the diverse structures and phenomena of the inorganic world and living 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 symmetry - rotational symmetry of the 6th order and, in addition, mirror symmetry. . A characteristic feature of a particular substance is the constancy of the angles between the corresponding faces and edges for all images of crystals of the same substance.
As for the shape of the faces, the number of faces and edges and the size of the snowflakes, they can differ significantly from each other, depending on the height from which they fall.
Bibliography.
1. “Crystals”, M. P. Shaskolskaya, Moscow “science”, 1978.
2. “Essays on the properties of crystals”, M. P. Shaskolskaya, Moscow “science”, 1978.
3. “Symmetry in nature”, I. I. Shafranovsky, Leningrad “Nedra”, 1985.
4. “Crystal chemistry”, G. B. Bokiy, Moscow “science”, 1971.
5. “Living Crystal”, Ya. E. Geguzin, Moscow “science”, 1981.
6. “Essays on diffusion in crystals”, Ya. E. Geguzin, Moscow “science”, 1974.

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5. People have long noticed that a person’s home is not only his fortress, but also his mirror. Any house bears the imprint of the personality of its owner. N.V. Gogol took this trait to the limit in “Dead Souls”, and the similarity became almost grotesque. Read More...... N.A. Zabolotsky was a supporter of natural philosophy. According to this direction of philosophical thought, nature is not divided into living and nonliving. In this regard, plants, animals, and stones are equally significant. When a person dies, he also becomes part of the natural world. Poem Read More......

Snowflake symmetry

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 of 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 phenomenon of a solar eclipse occurs every five years when the lunar disk completely blocks the 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.