Examples of symmetrical figures in everyday life. Symmetry is all around us. Symmetry in nature

Dubova Olga 5L class

The work introduces symmetry in mathematics. Talks about symmetry in nature, technology, everyday life, art, and the Russian language.

Hello, my name is Olga Dubova, I am a 5L grade student. My supervisor is Dubva Polina Sergeevna. I have prepared a work on the topic “Symmetry around us”
The purpose of my work:

Familiarize yourself with symmetry in mathematics,
nature,
technology,
everyday life
art,
Russian language
Find out why symmetry is needed

We encounter symmetry everywhere - in nature, technology, art, science. Man has long used symmetry in architecture. It gives harmony and completeness to ancient temples, towers, medieval castles, and modern buildings. Symmetry literally permeates the entire world around us
In ancient times, the word “symmetry” was used as “harmony” and “beauty”. In Greek it means “proportionality, proportionality, uniformity in the arrangement of parts.”
Contents of my work

Symmetry in mathematics
Symmetry in nature
Symmetry in technology
Symmetry in everyday life
Symmetry in art
Symmetry in Russian

Symmetry in mathematics
1 type of symmetry - Central symmetry

Two points A and A1 are called symmetrical with respect to point O if O is the midpoint of segment AA1. Point O is considered symmetrical to itself.

2nd view - Axial symmetry

Two points A and A1 lying on the same perpendicular to a given line on opposite sides and at the same distance from it are called symmetrical with respect to the given line.

On the slide, a snowflake has 1 center of symmetry and 6 axes of symmetry
An angle, an isosceles triangle, an isosceles trapezoid have one axis of symmetry
Rectangle, rhombus have 2 axes of symmetry
An equilateral triangle has 3 axes, a square has 4 axes, and a circle has infinitely many axes of symmetry that pass through the center of the circle
There are figures that are not symmetrical: arbitrary triangles, parallelograms, polygons
Symmetry in nature

According to the encyclopedist scientist Academician V.I. Vernadsky, symmetry surrounds us everywhere.

In the 19th century, research in this area led to the conclusion that the symmetry of natural forms depends on the influence of the forces of gravity, which at each point has the symmetry of a cone.

On this slide we see the forest and its reflection in the water, and the shore is the axis of symmetry.
Even a leaf has an axis of symmetry
On January 27, 2013, in the city of Alatyr, at sunrise, near the Sura River, an interesting phenomenon was observed: solar light pillars. This natural phenomenon occurs during severe frost, when sunlight is reflected in crystallized precipitation in the atmosphere.

On the left photo in the center there is the sun, and on the left and right there are light pillars, on the right photo there is a close-up of the right pillar

Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have symmetry. All solids are made of crystals
In nature, the two most common types of symmetry are mirror and radial symmetries.

A butterfly, leaf or beetle has mirror symmetry and this type of symmetry is often called “leaf symmetry” or “bilateral symmetry”. We can say that every animal (as well as insects, fish, birds) consists of two: right and left halves.

Symmetry in plants

Forms with radial symmetry include mushroom, chamomile, pine tree, and often this type of symmetry is called “chamomile-mushroom” symmetry

The human body also has bilateral symmetry. But if you divide the human body in half, you will notice that not each part is equal. Someone's right leg is longer than their left, or their arm, fingers, etc. A person with a perfectly symmetrical body is considered beautiful and healthy. And not symmetry is, according to doctors, a sign of some disease. For example, the symmetry of a newborn’s face is used to judge the health of his brain and nervous system.

The human brain consists of two parts - hemispheres, tightly adjacent to each other. Each hemisphere is almost an exact mirror image of the other.

In Japan, both hemispheres are developed from childhood. This is how the Japanese can write equally with their left and right hands. And if one hemisphere is damaged, its work is performed by the other hemisphere, and human functions are not impaired.

Helical Symmetry is also found in nature. For example, in plants, crystals, shells, etc.
Symmetry in technology is observed very often. I think that the symmetrical technique is more convenient to use.
Symmetry in everyday life can be observed in Ornaments and borders
Here we see famous architectural creations: the pyramids, the Taj Mahal, the Cathedral of Christ the Savior and Moscow University.
Symmetry in poetry and music

“The soul of music - rhythm - consists in the correct periodic repetition of parts of a musical work,” wrote the famous Russian physicist G.V. in 1908. Wulf. – The correct repetition of identical parts as a whole is the essence of symmetry.
The poems imply symmetry in the alternation of rhymes and stressed syllables.
A composer may return to the same theme several times in his symphony, gradually developing it.

Everything is bright, everything is white all around.

There are light patterns on the glass,

Forty merry ones in the yard

Trees in winter silver

And softly carpeted mountains

Winter with a brilliant carpet Pushkin A.S. "Eugene Onegin"

Symmetry in Russian

The letters A, M, T, Ш, П have a vertical axis of symmetry
B, W, K, S, E, E – horizontal.
And the letters Zh, N, O, F, X each have two axes of symmetry.
Symmetry can also be seen in the words: Cossack, hut.
There are entire phrases with this property (if you do not take into account the spaces between words). Such phrases are called palindromes.
“Look for a taxi”
“Argentina beckons the Negro”
“The Argentinean appreciates the black man”
“And the rose fell on Azor’s paw”

Palindrome of V. Nabokov:

I ate moose meat, melting...
Aeolus tore aloe, laurel.

Conclusion

Symmetry can be found almost everywhere.

For a person, this is balance and harmony.

In the words of the German mathematician Hermann Weyl: “ Through symmetry man has always tried to comprehend and create order, beauty and perfection.”

O 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.
Both a tulip and a rose are in friendship with you,
And the snow swarm is a creation of frost!

Thank you for your attention

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Symmetry around us Completed by: 5th grade student of MBOU “Secondary School No. 39” Olga Dubova Supervisor: P. S. Dubova

Goal: To become familiar with symmetry in mathematics, nature, technology, everyday life, art, and the Russian language. To find out why symmetry is needed.

Introduction We encounter symmetry everywhere - in nature, technology, art, science. Man has long used symmetry in architecture. It gives harmony and completeness to ancient temples, towers, medieval castles, and modern buildings. Symmetry literally permeates the entire world around us

Definition In ancient times, the word “symmetry” was used to mean “harmony” and “beauty”. In Greek it means “proportionality, proportionality, uniformity in the arrangement of parts.”

Symmetry in mathematics 1 type of symmetry Central symmetry Two points A and A1 are called symmetrical with respect to point O if O is the middle of the segment AA1. Point O is considered symmetrical to itself.

Axial symmetry 2nd view Two points A and A1, lying on the same perpendicular to a given line on opposite sides and at the same distance from it, are called symmetrical with respect to a given line.

Symmetrical figures In the figure, a snowflake has 1 center of symmetry and 6 axes of symmetry

Figures with one axis of symmetry Angle Isosceles triangle Isosceles trapezoid

Figures with two axes of symmetry Rectangle Rhombus

Shapes with more than two axes of symmetry Equilateral triangle Square Circle

Figures that do not have axial symmetry Arbitrary triangle Parallelogram Irregular polygon

Symmetry in nature According to the encyclopedist scientist Academician V.I. Vernadsky, symmetry surrounds us everywhere. In the 19th century, research in this area led to the conclusion that the symmetry of natural forms depends on the influence of the forces of gravity, which at each point has the symmetry of a cone.

Symmetry in physics and non-living nature

Symmetry in the atmosphere On January 27, 2013 in the city of Alatyr at sunrise, near the Sura River, an interesting phenomenon was observed, solar light pillars. This natural phenomenon occurs during severe frost, when sunlight is reflected in crystallized precipitation in the atmosphere.

Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have symmetry. All solids are made of crystals.

Symmetry in the animal world In nature, the two most common types of symmetry are mirror and radial symmetry. A butterfly, leaf or beetle has mirror symmetry and this type of symmetry is often called “leaf symmetry” or “bilateral symmetry”. We can say that every animal (as well as insects, fish, birds) consists of two: right and left halves.

Symmetry in plants Forms with radial symmetry include mushroom, chamomile, pine tree, and often this type of symmetry is called “chamomile-mushroom” symmetry

Symmetry in humans The human body also has bilateral symmetry. But if you divide the human body in half, you will notice that not each part is equal. Someone's right leg is longer than their left, or their arm, fingers, etc. A person with a perfectly symmetrical body is considered beautiful and healthy. And not symmetry is, according to doctors, a sign of some disease. For example, the symmetry of a newborn’s face is used to judge the health of his brain and nervous system. The human brain consists of two parts - hemispheres, tightly adjacent to each other. Each hemisphere is almost an exact mirror image of the other. In Japan, both hemispheres are developed from childhood. This is how the Japanese can write equally with their left and right hands. And if one hemisphere is damaged, its work is performed by the other hemisphere, and human functions are not impaired.

Helical Symmetry This type of symmetry is often found in plants, crystals, shells, etc.

Symmetry in technology Symmetry in technology is observed very often. I think people do this because this technology is more convenient to use.

Symmetry in everyday life Ornament and borders

Symmetry in architecture

Symmetry in poetry and music “The soul of music - rhythm - consists in the correct periodic repetition of parts of a musical work,” wrote the famous Russian physicist G.V. in 1908. Wulf. – The correct repetition of identical parts as a whole is the essence of symmetry. The poems imply symmetry in the alternation of rhymes and stressed syllables. A composer may return to the same theme several times in his symphony, gradually developing it. Everything is bright, everything is white all around. There are light patterns on the glass, Forty merry ones in the yard, Trees in winter silver, And the softly covered mountains of Winter with a brilliant carpet Pushkin A.S. "Eugene Onegin"

Symmetry in the Russian language The letters A, M, T, Ш, P have a vertical axis of symmetry; B, Z, K, S, E, E - horizontal. And the letters Zh, N, O, F, X each have two axes of symmetry. Symmetry can also be seen in the words: Cossack, hut. There are entire phrases with this property (if you do not take into account the spaces between words). Such phrases are called palindromes. “Look for a taxi” “Argentina beckons the Negro” “The Argentinean appreciates the Negro” “And the rose fell on Azor’s paw” V. Nabokov’s palindrome: I ate moose meat, liking... I tore up Aeolus aloe, laurel.

Conclusion Symmetry can be found almost everywhere. For a person, this is balance and harmony. In the words of the German mathematician Hermann Weyl: “Through symmetry, man has always tried to comprehend and create order, beauty and perfection.” O 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!

Thank you for your attention

Municipal educational institution

"Voskhod Basic Secondary School"

Alatyrsky district of the Chuvash Republic

Scientific and practical conference

"The first step into science"

Symmetry is all around us

Work completed:

Supervisor:

mathematics teacher at Voskhodskaya secondary school

Voskhod village

Plan

1. Introduction ------- 3

2. Main part

2.1. What is symmetry? -------

2.2. Symmetry in nature 5

2.3. Why do you need to know about symmetry when studying physics - 6

2.4. Symmetry in technology 7

2.5. Symmetry in architecture, fine arts and

literature----

2.6. The use of symmetry elements in Chuvash

embroidery 8

3. Conclusion ----- 9

4. List of used literature0

Application

1. Introduction.

This research work is devoted to the search for patterns of symmetry in nature. The research topic helps to understand the connection of mathematics with other sciences and with the world around us. Why does nature create symmetry, what does it strive for when creating symmetry? It is difficult to find a person who does not have some idea of symmetry. “Symmetry” is a word of Greek origin. It, like the word “harmony,” means proportionality, the presence of a certain order, patterns in the arrangement of parts. The famous German mathematician Hermann Weyl gave a definition of symmetry: “Symmetry is the idea with the help of which man has been trying for centuries to explain and create order, beauty and perfection.”

The purpose of my work is to study symmetry and its application.

1. Through the concept of “symmetry”, reveal the most important connections between the phenomena of symmetry and living nature, art, and technology.

2. Show the direct relationship between symmetry and the surrounding world.

3. Disclosure of the basic laws of natural symmetry.

4. Determine whether there should be symmetry in everything in life.

2. Main part

2.1. What is symmetry?

“...To be beautiful means to be

symmetrical and proportionate."

In his reflections on the picture of the universe, man has actively used the idea of symmetry since ancient times. Pythagoras, considering the sphere to be the most symmetrical and perfect form, concluded that the Earth is spherical. The ancient Greeks believed that the universe was symmetrical simply because symmetry is beautiful.

In “Boyhood” there is a confession: “...Standing in front of a black board and drawing different figures on it with chalk, I was suddenly struck by the thought: why is symmetry pleasing to the eye? This is an innate feeling, I answered to myself. What is it based on? “Is there symmetry in everything in life?”

The prominent mathematician G. Weil (G.G.) noted that symmetry “is the idea through which man for centuries has tried to comprehend and create order, beauty and perfection.”

Translated from Greek, the term “symmetry” means proportionality (uniformity, proportionality, harmony). Parallels are often drawn: symmetry and balance, symmetry and perfection. The development of the doctrine of symmetry owes primarily to natural scientists who studied crystal formations in depth. These are I. Kepler, N. Stenon, P. Curie, Lodeave, Fedorov and others.

In mathematics, various types of symmetry are considered. Each of them has its own name: axial symmetry(symmetry about a straight line), central symmetry(symmetry about a point) and mirror symmetry(symmetry relative to the plane). The transformation of figures (symmetry) entered mathematics as a result of human observation of the world around us. It occurs frequently and everywhere. Therefore, even an inexperienced person usually easily sees symmetry in its relatively simple manifestations.

The following lines are dedicated to symmetry:

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.
Both a tulip and a rose are in friendship with you,
And the snow swarm is a creation of frost!

2.2. Symmetry in living nature.

Nature is an amazing creator and master. All living things in nature have the property of symmetry. If you look at any insect from above and mentally draw a straight line (plane) in the middle, then the left and right halves of the insects will be the same in location, size, and color. After all, we have never seen that a beetle or a dragonfly, or any other insect, had paws on the left that were closer to the head than on the right, or that the right wing of a butterfly or ladybug would be larger than the left. This does not happen in nature, otherwise insects would not be able to fly. Man used the property of symmetry inherent in living nature in his achievements: he invented the airplane, created unique architectural buildings. And the man himself is a symmetrical figure. Symmetry can be seen among the colors. Flowers of the Rosaceae family have axial symmetry, while flowers of the Cruciferae family have central symmetry. Symmetry can also be seen in tree leaves.

2.3. Why do you need to know about symmetry when studying physics?

Why do you need to know about symmetry when studying physics? But thanks to crystals, symmetry penetrated into the world of physical laws and became a sovereign mistress there. However, symmetry also exists where it is not visible at first glance. A physicist will say that every solid body is a crystal. The famous crystallographer Evgraf Stepanovich Fedorov said: “Crystals shine with symmetry.” A chemist will say that all bodies are made of molecules, and molecules are made of atoms. And many atoms are located in space according to the principle of symmetry. The striking, regular outlines of the crystals aroused superstitious beliefs among our ancestors. “Only the gods could do this,” they said. But we know that this is a creation of nature, that the formation of crystals occurs spontaneously, that the vast majority of solids have a crystalline structure.

Even in prehistoric times, people found natural crystals and collected them. Their imagination was struck by the constancy of the angles between the faces of a crystal of the same type. For the first time, the law of constancy of angles between crystal faces for the special case of ice crystals - snowflakes - was established by I. Kepler. (g. g.).

In his work “New Year's Gift”, or about hexagonal snowflakes, he reflected on the New Year's gift to the emperor's advisor, patron of science and philosopher. This gentleman greatly loved... Nothing because of its insignificant value, but rather as the charming amusement of a playfully twittering nightingale. Painfully sorting out what kind of object could be Nothing, Kepler suddenly noticed snowflakes quietly falling on his clothes, all of them hexagonal, with fluffy rays. Kepler will give the adviser snowflakes on New Year's Day.

Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have a hexagon shape.

2.4. Symmetry in technology.

Symmetry can also be observed in technology. Why is symmetry used in technology?

Such technical objects as airplanes, bridges, cars, rockets, hammers, nuts - almost all of them, from small to large, have some form of symmetry. Is this a coincidence? In technology, the beauty and proportionality of mechanisms is often associated with their reliability and stability in operation. The symmetrical shape of an airship, airplane, submarine, car, etc. ensures good flow around air or water, and therefore minimal resistance to movement. There is a kind of postulate in technology: the most expedient and functionally perfect products are the most beautiful. To confirm this postulate, we cite the words of the general aircraft designer: “We know very well that a beautiful plane flies well, but an ugly one flies poorly, or even won’t fly at all. This is not a superstition, but a completely materialistic position... a designer can often go from beauty to technology, from aesthetic solutions to technical solutions."

2.5. Symmetry in architecture, fine arts and literature

Works of architecture demonstrate excellent examples of symmetry. Most buildings are mirror symmetrical. General plans of buildings, facades, ornaments, cornices, columns reveal proportionality and harmony. Old Russian architecture provides many examples of the use of symmetry: bell towers, watchtowers, internal support pillars. Since ancient times, people have sought to decorate with ornaments everything that surrounded them in everyday life. The principles of symmetry and rhythmic repetition techniques are often used in the construction of ornaments.

In literary works, there are a number of funny verbal constructions based on the properties of mirror symmetry. For example, the words “topot”, “Cossack”, “hut”, type of words are called palindromes. Phrases, poems, and stories can be palindromic. For example. “I am walking with the sword of the judge” (T. Derzhavin), “And the rose fell on Azor’s paw” (A. Fet); “Argentina beckons the Negro” (Bulgakov).

Poetry is distinguished from prose by the symmetry of syllables, lines, stress and

unstressed sounds. Excerpt from a poem by A. Fet:

What sadness! End of Alley A

Again in the morning he disappeared into the dust,

Silver snakes again

They crawled through the snowdrifts. IN

There is an element of repetition here - this is symmetry. This poetic element is called iambic.

The composition of A. Rublev's painting "Trinity" is symmetrical. The symmetrical arrangement of the three angels enhances the expressiveness of the work of art. The artist in the painting "Trinity" wanted to show the poise and peace that these three angels bring

2.6. The use of symmetry elements in Chuvash embroidery.

Since ancient times, wood carving and embroidery have been common among the Chuvash. Both are distinguished by the richness of patterns that are created using symmetry. Embroidery was done in four directions: horizontally, vertically and two diagonals. Color played a big role in embroidery. Five colors were used in the patterns: black, red, yellow, blue and green. To complete the outline, black was usually used - the color of earth and fertility. This was the most important part of the work, requiring great precision from the craftswoman, because if she was wrong by just one thread, the symmetry of the design was broken. The most common color was red - the color of blood, the color of life. And the Chuvash considered yellow, the color of the sun, to be the most beautiful

3. Conclusion.

After researching various sources of information about symmetry, I came to the conclusion that nature is arranged in accordance with the laws of symmetry. All living things in nature have the property of symmetry. Symmetry can be seen among flowers and on the leaves of trees. Man used the property of symmetry inherent in living nature in his achievements: he invented the airplane, created unique architectural buildings. And the man himself is a symmetrical figure. Therefore, symmetry did not arise by chance - perhaps symmetrical objects are easier to perceive for living beings.

NOh the world can't be absolutely symmetrical. Builders of modern bridges, high-rise buildings, and towers know that the structure should not be perfectly symmetrical due to the risk of resonant vibrations that can lead to its destruction. Therefore, the symmetry of structures is deliberately broken by introducing individual asymmetrical elements into it. Some deviations from symmetry also exist in living nature. The famous artist O. Renoir spoke about this: “Two eyes, even on the most beautiful face, are always slightly different, the nose is never exactly above the middle of the mouth; an orange slice, leaves on a tree, flower petals are never exactly the same ".

Why does a person need to know about symmetry? Knowledge about symmetry can be applied in your activities: in construction, in creating household items, in decorating clothes, in decorating the interior of a home.

Slide 2

Symmetry in everyday life

Slide 3

Symmetry in science and technology.

Slide 4

Symmetry in architecture

Slide 5

Central symmetry

A geometric figure (or body) is called symmetrical with respect to center C (Fig. 105) 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 of C and is bisected at this point (AC = CE). Point C is called the center of symmetry.

Slide 6

Slide 7

Mirror symmetry.

A geometric figure is called symmetrical with respect to the plane S (Fig. 104) 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 plane S is called the plane of symmetry. Symmetrical figures, objects and bodies are not equal to each other in the narrow sense of the word (for example, the left glove does not fit the right hand and vice versa. They are called mirror equal).

Slide 8

Rotation symmetry

A body (figure) has rotational symmetry (Fig. 106) if, when rotated through an angle of 360°/n (here n is an integer) around some straight line AB (axis of symmetry), it completely coincides with its initial position. When n = 2 we have axial symmetry.

Slide 9

Examples of the above types of symmetry

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.

Slide 10

Symmetry of plane figures

Mirror-axis symmetry. If the plane figure ABCDE (Fig. 107) 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

Slide 11

Central symmetry. If a flat figure (ABCDEF, Fig. 108) has a second-order symmetry axis perpendicular to the plane of the figure (straight line MN, Fig. 108), then point O, at which straight line MN and the plane of the figure ABCDEF intersect, is the center of symmetry.

Slide 12

Examples of symmetry of plane figures

A parallelogram has only 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.

Slide 13

Symmetry in nature

Symmetry in our minds is closely related to the concept of beauty
Ideas about beauty and perfection were born and strengthened under the influence of the surrounding nature even among our distant ancestors. Crystals were especially striking with the correctness of their proportions and the impeccable repetition of shape.

Slide 14

Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have symmetry.

All solids are made of crystals
Diamond crystals
Crystals of rock salt, quartz, aragonite

Slide 15

Not just crystals, most of nature's creations usually exhibit some form of symmetry.
The earth could well be called the kingdom of symmetry.
Nature has used all its main types that can be represented by geometric considerations.
The overwhelming number of living organisms have one of its three types: spherical, radial, and bilateral symmetry.

Zaitseva Ksenia, Kirichenko Arthur, Mamadaminov Bakhrom

Project Manager:

Pavlova Olga Viktorovna

Institution:

MBOU secondary school in the village of De-Kastri, Ulchsky district, Khabarovsk Territory

In this research project in mathematics on the topic "Symmetry in life" the student makes observations, searches the literature, systematizes and analyzes the material, and as a result finds out how symmetry manifests itself in life.

In the presented research paper on mathematics on the topic “Symmetry in Life,” the author gives a general concept of symmetry, examines the types and applications of symmetry in the Russian language, in clothing, everyday life, wildlife, architecture and in objects of decorative and applied art.

During the design and research work in mathematics "Symmetry in Life", photographs of things and objects are created, they are analyzed for symmetry, axes and centers of symmetry are found.

The proposed math project, "Symmetry in Life," demonstrates what clothing would look like if it was not symmetrical on the left and right sides.

"Mathematics reveals order, symmetry and certainty, and these are the most important types of beauty."

Aristotle

Introduction
1. Definition of symmetry.
2. Types of symmetry.
3. Applications of symmetry.
4. Russian language and symmetry.

6. Symmetry in everyday life.
7. Symmetry in living nature.

9. Symmetry in objects of decorative and applied art.
Conclusion
List of sources used.

Introduction

« Standing in front of a black board and drawing different shapes on it with chalk, I was suddenly struck by the thought: why is symmetry pleasing to the eye? What is symmetry? This is an innate feeling, I answered myself.»

L.N. Tolstoy

Object of study – symmetry.

Subject of study - symmetry in life.

Goal of the work : find out how symmetry manifests itself in life.

To achieve this goal it is necessary to fulfill next tasks :

Give a general concept of symmetry, types of symmetry, symmetry in life.
Take photographs of everything we can and analyze whether they are symmetrical, find the axes and centers of symmetry.
Demonstrate how clothes will look if their clothes are not symmetrical relative to the left and right sides.
Present the observation results in a presentation.

Research hypothesis: symmetry is harmony and beauty, balance, stability.

Research methods:

Analysis of articles about symmetry in life.
Observation.
Computer modeling (photo processing using a graphic editor).
Generalization and systematization of the obtained data.

Stages of work:

Preparatory. Studying literature, drawing up a plan.
Basic. Collection of information, photography, photo processing.
Final. Systematization of the information received, making a presentation.

Relevance of the topic .
Mathematics project topic " Symmetry in life" Very relevant and interesting. Nowadays, it is probably difficult to find a person who would not have some idea of symmetry. The world in which we live is filled with the symmetry of houses and streets, mountains and fields, creations of nature and man.

We encounter symmetry literally at every step: in nature, technology, art, science. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human development. Man has long used symmetry in architecture. It gives harmony and completeness to ancient temples, towers of medieval castles, and modern buildings.

1. Definition of symmetry

Symmetry- correspondence, immutability, one of the most clearly manifested (and therefore most familiar to us) properties of composition. This is both a property - the state of the form, and a means by which the form is organized.

Symmetry is understood as any regularity in the internal structure of the body or figure.

One of the famous mathematicians Hermann Weil wrote that " symmetry is the idea through which man over the centuries has tried to comprehend and create order, beauty and perfection".

2. Types of symmetry

Type of symmetry	Definition	Example
Radial	An arrangement of body parts that allows it to be divided into 2 equal halves that mirror each other in several planes.
Bilateral (axial)	An arrangement of body parts that allows it to be divided into two equal halves that mirror each other with only one plane. This plane is called the axis of symmetry.
Central	Symmetry about a point. It assumes that there is an object on both sides of a point at equal distances.
Mirror	Mirror symmetry in architecture and nature. Reflection of coastal buildings. Optical reflection in the river of coastal trees. Reflection of a candle in the mirror.

3. Applications of symmetry

Having studied theoretical material and observed the world around us, we came to the conclusion that symmetry literally permeates everything that surrounds us.

But, at the same time, we noticed that there are constantly deviations in the forms of nature: one claw of a crab or crayfish is noticeably larger than the other.

The pattern of a zebra's stripes is not repeated on two halves of its body, etc. Asymmetry and symmetry constantly interact.

4. Russian language and symmetry

The letters of the Russian language can also be considered from the point of view of symmetry.

Vertical axis of symmetry: A; D; L; M; P; T; F; Sh.
Horizontal axis of symmetry: B; E; Z; TO; WITH; E; YU.
Both vertical and horizontal axes of symmetry: F; N; ABOUT; X.
Neither vertical nor horizontal axes: B; G; AND; Y; R; U; C; H; SCH; I.

In the Russian language there are symmetrical words - palindromes, which can be read equally in two directions:
Shalash, Cossack, radar, Alla, Anna, cook, pop.

Sentences can also be palindromic. Thousands of such sentences have been written.
« And the rose fell on Azor's paw».
« And the moon has sank».

6. Symmetry in everyday life

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

1. Symmetry………………………………………………………………………………………….. .....4

1.1. What is symmetry?........................................................ .....................................4

1.2. Types of symmetry…………………………………………………….…..…5

1.3. Symmetry in mathematics…..…………………………….….………….7

1.4. Symmetry in the Russian language..……………………………..…………………8

1.5. Symmetry in the surrounding world………………………..…….………….9

2. Symmetry around us………………………………………………………….….13

3. The role of symmetry………………………………………………………….…….…...15

Conclusion………………………………………………………………………………….…….…..16

List of sources used………………………………………………………………..17

Introduction

In mathematics lessons we studied symmetry, but it turned out that little time was devoted to this topic. And I wanted to know more about symmetry.

In this work we will consider the concept of “symmetry” more broadly, not limiting ourselves to mathematics. The world around us is largely symmetrical - insects and animals, flowers and trees, household objects and architectural structures have symmetry.

Research objectives:

Studying the concept of “symmetry”;

What role does symmetry play;

Symmetry is all around us.

Research objectives;

Prove why symmetry is important;

Consider the types of symmetry and where it occurs;

Conduct an experiment and find out whether a person’s face is symmetrical;

The object of study is symmetry, and the subject is symmetry in nature and the surrounding world.

When carrying out the work, observation methods, questionnaires, experiments and theoretical analysis were used.

Symmetry

1.1.What is symmetry?

To find out what elementary school children know, we conducted a survey about what symmetry is and where it occurs. 90 people took part in it.

From the survey we learned that students know little where symmetry occurs and what it is.

We got the following results:

Only 9 people know the correct answer to the first question. On the second

question - 16 people. Most correct answers to the third question -

57 people.

After reading encyclopedias and textbooks, I learned that the most perfect forms are created by nature, and it is nature that gives these forms unusually harmonious color combinations (butterfly, wasp, dragonfly). Since ancient times, people have used symmetry in drawings, ornaments, and household items. I noticed how strictly symmetrical the forms of ancient buildings are, how harmonious the ancient Greek vases are, and how proportionate their ornaments are. We encounter one or another manifestation of symmetry literally at every step.

So what is symmetry? We looked at several sources. In the explanatory dictionary S.I. Ozhegova:

Symmetry is proportionality, the sameness in the arrangement of parts of something on opposite sides of a point, straight line or plane.

In the explanatory dictionary V.I. Dalia:

Symmetry (Greek) - proportionality, correspondence, similarity;

In the Great Soviet Encyclopedia:

Symmetry is a property of a geometric figure that characterizes a certain regularity of shape, its invariability under the action of movements and reflections.

Of the definitions found, the most understandable for me was the definition given by S.I. Ozhigov. The definitions are different, but the word proportionality appears in all of them.

Mathematics is the queen of all sciences, a symbol of wisdom. The beauty of mathematics is unattainable among the sciences, and beauty is one of the connecting links between science and art. This is not only a harmonious system of laws, but also a unique means of experiencing beauty. In mathematics, various types of symmetry are considered. Each of them has its own name.

In nature, the most common types of symmetry are “mirror”, axial, and central symmetry.

A butterfly, leaf or beetle has “mirror” symmetry and this type of symmetry is often called “leaf symmetry”. Forms with radial symmetry include mushroom, chamomile, and pine tree. And the mirror not only copies the object, but also swaps the parts of the object that are front and rear in relation to the mirror.

I looked in the mirror and thought about the fact that my left hand in the mirror is my right and vice versa.

I learned that in the school geometry course three types of symmetry are considered: symmetry about a point (central symmetry); symmetry relative to a straight line (axial or mirror symmetry); symmetry relative to the plane. Central symmetry Two points A and A1 are called symmetrical with respect to point O if O is the midpoint of segment AA1. Point O is considered symmetrical to itself.

Axial symmetry. The transformation of figure F into figure F1, in which each of its points goes to a point symmetrical with respect to a given line, is called a symmetry transformation with respect to a line A. Straight A called the axis of symmetry.

To see this, fold a sheet of paper in half and pierce it with a needle. Unbend the sheet. We find two points A and B on it. Draw a segment AB and denote its intersection with the line L with the letter O. The segments AO and BO are equal.

Mirror symmetry . Mirror symmetry is a mapping of space onto itself, in which any point transforms into a point symmetric to it, relative to the plane.

In space, the analogue of the axis of symmetry is the plane of symmetry. The mapping of space onto itself relative to a plane is called mirror symmetry. This name is justified by the fact that both parts of the figure, located on opposite sides of the plane of symmetry, are similar to some object and its reflection in the mirror.

In our village there is a pond where the residents of our village like to go to relax. Its shore is very beautiful. Quiet. Nothing moves. Birches, bushes, and reeds are reflected in the water. This is some kind of mirror symmetry!

Rotational symmetry . Rotational symmetry is a symmetry in which an object aligns with itself when rotated around a certain axis through certain angles.

This symmetry is found in flowers. I tried turning the daisy and it worked. I examine the arrangement of leaves on a tree branch, I see that one leaf is not only at a distance from the other, but is also rotated around the axis of the trunk. For what? The encyclopedia says that the leaves are arranged on the trunk along a helical line (the principle of helical symmetry) so as not to block sunlight from each other.

Portable symmetry. If, when transferring a flat figure F along a given straight line AB to a distance A(or a multiple of this value) the figure is combined with itself, then we speak of portable symmetry. The straight line AB is called the translation axis, the distance A elementary transfer.

Symmetry is also found in our regular mathematics lessons, for example:

In geometric shapes: square, rectangle, triangle, circle.

Mirror symmetry in numbers.

Numbers consisting of the digits 8 and 0 are symmetrical.

The signs of arithmetic operations, double and curly brackets are also symmetrical:

+ = : () ( ) X

When studying the topic “Units of Mass”, we get acquainted with scales. Libra in balance is symmetrical!

When studying the multiplication and division table, we saw that the numbers and answers in it are located symmetrically relative to the diagonal axis of symmetry.

In the Russian language lesson, we noticed that symmetry also occurs, for example:

In letters:

In words:

A mirror anagram is a type of anagram, a phrase (or one word) obtained by reading another phrase in reverse order, for example, “thief” - “ditch”.

Examples of mirror anagrams

azu - bond;

beech - cube;

march - scar;

disco - oxide;

Milan - burbot;

Mirror anagrams are similar to palindromes, but for palindromes the meaning does not change when read back (Appendix 1).

Shalash, Cossack, radar, cook, Anna, priest, Alla.

And the rose fell on Azor's paw.

The shortest palindrome in the Russian language consists of just one letter - ABOUT!.

When underlining sentence members:

Predicate object definition circumstance

Our Russian language textbook uses the following conventions, they are symmetrical:

In the lessons “The World Around us” we study living and inanimate nature.

The butterfly is a prime example of mirror symmetry. You can swap the right and left halves without changing the object.

Examples of symmetry can also be found when considering plants.

Central symmetry Axial symmetry

We noticed symmetry when looking at the flags of different countries.

Canada Azerbaijan UK

Vietnam Bahamas

Man is also an object of living nature. And I wondered, is a person’s face symmetrical? In order to find the answer to this question, we will conduct an experiment.

We draw the vertical axis of symmetry:

Copy the left half. They did the same with the right one.

Combined the two left halves:

Combined the two right halves:

After conducting an experiment, we came to the conclusion that a person’s face is not symmetrical, as it seems at first glance.

Symmetry is all around us

We encounter symmetry everywhere - in nature, technology, art, science. Man has long used symmetry in architecture. It gives harmony and completeness to ancient temples, towers, medieval castles, and modern buildings. Symmetry literally permeates the entire world around us.

Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have symmetry.

Symmetry in technology is observed very often. I think people do this because it's easier to use.

Symmetry is also used in everyday life, for example, in ornaments and borders, dishes, interior items, and clothing.

Symmetry is found even in poetry and music.

“The soul of music - rhythm - consists in the correct periodic repetition of parts of a musical work,” wrote the famous Russian physicist G.V. in 1908. Wulf. The correct repetition of identical parts as a whole is the essence of symmetry.

A composer can return to the same theme several times in his symphony, gradually revealing it.

The poems imply symmetry in the alternation of rhymes and stressed syllables.

Everything is bright, everything is white circle ohm

There are light bonds on the glass ory,

Forty merry for two re,

Trees in winter silver re,

And softly covered ors

Winter brilliant carpet ohm

Pushkin A.S. "Eugene Onegin"

Thus, I realized that symmetry is everywhere in my life, you just have to be careful and observant.

The role of symmetry

We got acquainted with the concept of symmetry and its types.

Now I'm wondering, what role does symmetry play?

I asked the guys to help complete the task.

Task: It is necessary to complete the drawing of a symmetrical half and an asymmetrical one. Draw a conclusion (Appendix 2).

Conclusion: In these drawings, symmetrical objects look more harmonious than asymmetrical ones.

Symmetry is order, predictability, stability. A person loves order, predictability, stability, so symmetrical objects seem more beautiful to him.

At the same time, minor deviations from symmetry give the object individuality, and this is also good. For example, if all the trees were completely symmetrical, then we would hardly like the spruce forest. And small deviations from symmetry made it possible to turn the vase into a jug...

Conclusion

For centuries, symmetry has remained a property that has occupied the minds of philosophers, astronomers, mathematicians, artists, architects, and we began to study symmetry with great pleasure.

In the course of this work, we became acquainted with several types of symmetry: “mirror”, axial and central. We found where it was hiding and realized that symmetry is found everywhere: in living and inanimate nature, in technology, science, art, architecture, and in everyday life. We encounter symmetry in all lessons at school.

We consider everything symmetrical to be beautiful, because symmetry is order and stability, and man always strives for order and harmony. But there is no absolute symmetry in the world around us, and we found this out as a result of an experiment with photography.

Researchers have proven that small deviations from symmetry give an object personality and make it more interesting. Small deviations from symmetry are allowed in architecture, clothing, hairstyles, decorations, etc. Significant deviations from symmetry are considered unsightly and are often not accepted by people.

Symmetry plays a huge role in architecture, music, painting, technology and nature. This is stated in one poem:

Oh, symmetry! I sing a hymn to you! I recognize you everywhere in the world. You are in the Eiffel Tower, in a small midge, You are in a Christmas tree along a forest path. A tulip and a rose are in friendship with you, And a snow swarm - the creation of frost!

As a result of the study, all goals and objectives were achieved. The work was interesting and useful. I will share my knowledge with classmates and other elementary school children.

List of used sources

1.Wulf G.V. Symmetry and its manifestations in nature. M., Ed. Dept. Nar.com. Enlightenment, 1991

2. Gasparov M.L. Essay on the history of Russian verse: metrics, rhythm, rhyme, stanza. M., 1984

4. Smolina N.I. Traditions of symmetry in architecture. - M., 1990.

5. Tarasov L. This amazingly symmetrical world. - M.: Education, 1982.

6. Shubnikov A.V., Koptsik V.A. Symmetry in science and art. M., 1972.

Annex 1

Palindromes

Argentina beckons the Negro.

The leader was delirious.

City of roads.

Leps sang.