Class: 8
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The purpose of the lesson: teach how to construct axial symmetry of geometric figures.
Tasks:
- Educational:
- consider symmetrical points and figures relative to a straight line;
- teach how to construct symmetrical points and recognize figures with axial symmetry;
- consider axial symmetry as a property of some geometric figures.
- get an idea of symmetry in mathematics and the world around us.
- develop logical thinking;
- activate mental activity through the use of information technology
Forms of organizing educational activities: whole class, individual, pair.
Lesson type: Studying and primary consolidation of new knowledge.
Lesson plan:
- symmetry of a point relative to a straight line;
- construction of axial symmetry of a point on a plane;
- symmetry of the figure relative to a straight line;
- construction of axial symmetry of geometric figures;
- application of acquired knowledge when solving problems.
Equipment: projector; screen; double-sided board (chalk, marker); square; Handout; teacher's pointer; colour pencils; rulers.
During the classes
I. Organization of the beginning of the lesson
Slide.
Hello guys, sit down.
Today in class we will do a lot of creative and entertaining tasks. So, pay attention to the screen!
II. Communicating the topic, purpose and objectives of the lesson
The topic of our lesson is “Symmetry in mathematics and the world around us.”
Today in the lesson we will get acquainted with the concept of symmetry, learn how to construct points that are symmetrical relative to a straight line; We will solve problems on constructing the symmetry of geometric figures.
When completing assignments, we will evaluate the work. According to my instructions, for each correctly completed task you will fill in one of the circles located at the top Sheet 1 (appendix).
III. Assimilation of new knowledge
Slide.
Let's start by making sure we define the term "symmetry".
What do you think the word “symmetry” means?
Where can we find symmetry in life?
I will summarize your answers. Symmetry (from the Greek Symmetria - proportionality), in a broad sense, is the immutability of the structure of a material object relative to its transformations.
Symmetry plays a huge role in art and architecture. But it can be seen both in music and poetry.
Symmetry is found widely in nature, especially in crystals, plants and animals. Symmetry can be found not only in geometry, but also in other branches of mathematics, for example in algebra - when constructing graphs of functions.
There are two types of symmetry: axial and central. Let's fill out the diagram in the handout Sheet 1.
Today we will consider only axial symmetry.
Find a sentence that says which two points are called symmetrical.
OPR: Two points A and A1 are called symmetrical with respect to line a if this line passes through the middle of segment AA1 and perpendicular to it.
Let's analyze the definition. What conditions must be met in order to be able to say unambiguously that point A is symmetrical to point A1 relative to straight line a? ( AA 1⊥ a and AO=OA 1)
Let us write in more geometric language in brackets the condition for the symmetry of points A and A 1 .
Let's learn how to construct together a point symmetrical to a given line relative to a straight line. For this we will find in the handout Exercise 1. Let's take a square and a pencil in our hands. (teacher builds on the board)
Stages of solving the problem: (on screen)
- Construct a perpendicular from point A to line a;
- O – the point of intersection of the perpendicular and straight line a;
- Extend the perpendicular beyond line a;
- Place a segment equal to segment OA on the continuation of the perpendicular;
- AO=OA 1
- Points A and A 1 are symmetrical relative to straight line a.
Let's do it oral task: Which points in the pictures are symmetrical?
Answer: Figure 2 only.
Who's ready to explain?
Who agrees with the answer, raise your hands? Fill in one circle at the top Sheet 1.
Many figures also have axial symmetry.
ODA: The figure is said to be symmetrical about a straight linea, if for each point of the figure there is a point symmetrical to it relative to the straight line A also belongs to this figure.
VII. Consolidation of knowledge
Let's consider geometric shapes and determine whether they have or do not have axial symmetry.
We work with task 2 Sheet2.
- Task 2: On the geometric figures shown, draw all the axes of symmetry and write down how many there are in the “Number of axes” column.
You can consult your desk neighbor.
Figure |
Number of axes of symmetry |
Educational activities |
Unturned corner |
1 axis of symmetry - |
Student at the blackboard
|
Isosceles triangle |
1 axis of symmetry - bisector, median, height |
Teacher:
|
4 axes of symmetry |
On one's own
|
|
Circle |
there are an infinite number of axes of symmetry |
On one's own
|
So, let’s check the solution to the problem on the screen and correct any inaccuracies in solving the problem.
Raise your hands, who drew all the axes of the square? One circle has been painted.
Raise your hands, who correctly identified the axis of the circle? One circle has been painted.
Do you think all geometric figures have axes of symmetry? True, not all. Let's look at the screen.
Put your pens aside, let’s decide verbally task : How many axes does: segment; straight; Ray?
Let's reason. We analyze each case sequentially.
Who's ready to answer?
Those who agree raised their hands. Fill in one of the circles.
Gymnastics for the eyes 1 min.
- Our eyes are tired from hard work. Let's give them the opportunity to relax a little by doing some eye exercises.
VIII. Generalization and systematization
Now let’s solve two practical problems using the “lesson materials” sheet.
Task 3: Construct a segment symmetrical to the given one.
Let's analyze the condition of the problem: How to construct a segment symmetrical to a given one relative to a straight line?
What is a segment? ( Part of a straight line, limited on both sides.)
What is enough to build to solve the problem? ( The symmetry of the points that are the ends of the segment.)
Conclusion: Since the segment is limited by two points, it is enough to construct points symmetrical to points A and B relative to straight line c and connect them.
We work independently, one person at the board.
X. Lesson summary
What concept did we learn about in class today? ( Symmetry.)
What type of symmetry have we considered? ( Axial.)
What did you learn in the lesson? ( Construct a point symmetrical relative to a given line; build an axis of symmetry of geometric figures; construct a figure symmetrical to a given one relative to a given line.)
Now everyone count the filled circles.
Raise your hands, who has exactly 4 or 5 filled circles? Place a “5” next to the circles.
Raise your hands, who has exactly 3 filled circles? Place a “4” mark next to the circles.
For those who received fewer circles, don’t be upset - you just couldn’t immediately find the answer to the question posed.
In conclusion, symmetry can be found almost anywhere if you know how to look for it. Since ancient times, many peoples have had an idea of symmetry in the broad sense - as balance and harmony. Human creativity in all its manifestations tends towards symmetry. Through symmetry, man has always tried, in the words of the German mathematician Hermann Weyl, “to comprehend and create order, beauty and perfection.”
Thank you for your active work.
- Study the topic “Symmetry”
- Explore the question “Symmetry in the world around us”
- Consider different types of symmetry in natural objects
- Why does a person need to know about symmetry?
- 1. Reveal the meaning of the basic concepts of symmetry.
- 2. Show that nature is a world of symmetry.
- study of literature;
- comparison of essential features;
- analysis, comparison, generalization.
- O symmetry!
- I sing your anthem!
- I recognize you everywhere in the world.
- You are in the Eiffel Tower, in a small midge,
- Are you in Christmas tree near the forest path.
- With you in friendship and tulip and rose,
- And the snow swarm is a creation of frost!
- The topic of my scientific research work is “Many-faced symmetry”.
- I chose this topic because we encounter symmetry everywhere - in nature, architecture, art, science. I would like to become more deeply acquainted with symmetry in mathematics and biology, technology and architecture since the concept of symmetry is widely used by all areas of modern science.
- What is it symmetry ?
- What deep meaning lies in this concept?
- Why does symmetry literally permeate the entire world around us?
- Symmetry (from the Greek symmetria - “proportionality”) - a concept meaning persistence, repeatability, “invariance” of any structural features of the object being studied when certain transformations are carried out with it .
- Symmetry - this is balance,
orderliness,
beauty,
perfection.
- a) symmetry about a point (central symmetry); b) symmetry relative to a straight line (axial symmetry);
- c) symmetry relative to the plane (mirror symmetry);
- G) Rotation symmetry (turn)
- d) Sliding symmetry
OA 1 = OA
Definition
Points A and A 1 are called symmetrical about the point ABOUT, if O is the middle of the segment AA 1.
Definition
The figure is called symmetrical about the center
Symmetry of points relative to a straight line
Definition
Two points A and A 1 are called symmetrical relative to straight line a , if this line passes through the middle of segment AA 1 and is perpendicular to it.
Symmetrical figure relative to a straight line
Definition
The figure is called symmetrical relative to straight , if for each point of a figure the point symmetrical to it also belongs to this figure. Straight l called the axis of symmetry of the figure.
- Transformation in which each point A of the figure (body) is rotated by the same angle α around a given center O is called rotation or rotation of the plane. Point O is called the center of rotation, and angle α is called the angle of rotation. Point O is a fixed point of this transformation.
Central symmetry is a rotation of a figure by 180°.
- Sliding symmetry is a transformation in which axial symmetry and parallel translation are performed sequentially.
- a segment goes into an equal segment;
- the angle goes into an angle equal to it;
- the circle turns into an equal circle;
- any polygon goes into an equal polygon, etc.
- parallel lines turn into parallel, perpendicular into perpendicular.
So, on the plane we have four types of movements that translate the figure F into an equal figure F 1 :
- parallel transfer;
- axial symmetry (reflection from a straight line);
- rotation around a point (partial case - central symmetry);
- "sliding" reflection.
- RADIAL SYMMETRY
(radial symmetry) - symmetry with respect to any planes passing through the longitudinal axis of the animal’s body.
Bilateral symmetry (bilateral symmetry) - mirror reflection symmetry, in which an object has one plane of symmetry, relative to which its two halves are mirror symmetrical.
Symmetry has many faces.
It is associated with orderliness, proportionality and proportionality of parts, beauty and harmony, with expediency and usefulness.
While working on the project, I touched the mysterious mathematical beauty. Mathematics is a language, the language of nature. Without knowing the language, you cannot understand the beauty of the world around you.
But one thing is certain: The world is symmetrical!
- 1. This amazingly symmetrical world” - L. Tarasov
- 2. “Explanatory Dictionary” - V. Dalya
- “Geometry 7-9 grades” - L. Atanasyan
- Malakhov V.V. // Journal. total biology. 1977. T.38.
- I.G. Zenkevich “Aesthetics of a mathematics lesson.”
- http://900igr.net/fotografii/geometrija/Simmetrija/O-simmetrii.html
Ignatovskaya Elena, Dorokhov Anatoly
Look around! We admire a bright flower, a beautiful butterfly, a mysterious snowflake, tall trees, church domes, beautiful sculptures and slender athletes. What is the basis of this beauty? Symmetry is pleasing to the eye and is often associated with beauty. “Symmetry is the idea through which man tried to comprehend and create order, beauty and perfection,” wrote the famous scientist G. Weil. Many processes occurring in the world can be considered using a mathematical model. Having studied the mathematical foundations of the concept of symmetry, we will learn to see the beauty of the world and create it with our own hands!
The project method allows schoolchildren to move from mastering ready-made knowledge to their conscious acquisition.
This project was prepared by 8th grade students while studying the topic “Axial and central symmetry.” Its goal is to develop the concept of symmetry, the ability to see symmetry phenomena in the surrounding world, expand the understanding of the areas of application of mathematics and its connection with other subjects. In addition to the main goals, we pursued another one: touching the beautiful, various types of art.
The project was defended at the school scientific and practical conference “Mathematics in the Modern World”, it is used by the teacher in mathematics lessons when studying the topic “Axial and Central Symmetry”.
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The world around us is a world of symmetry
Ignatovskaya Elena, Dorokhov Anatoly, students of 8 “B” grade, Sigodina Larisa Vladimirovna,
mathematic teacher
MBOU "Blagoveshchensk Secondary Educational School No. 1"
Slide 1
The concept of symmetry runs through the entire centuries-old history of human creativity. Since ancient times, many peoples have had the idea of symmetry in a broad sense - as the equivalent of balance and harmony. Forms of perception and expression in many fields of science and art are ultimately based on symmetry, used and manifested in specific concepts and means inherent in individual fields of science and types of art. Today we invite you to consider the manifestation of this idea in various areas.
Slide 2
Symmetry (from the Greek “proportionality”) is the property of a geometric object to combine with itself under certain transformations that form a group.
The idea of symmetry is often the starting point in the hypotheses and theories of scientists of past centuries, who believed in the mathematical harmony of the universe and saw in this harmony a manifestation of the divine principle. The ancient Greeks believed that the universe was symmetrical simply because symmetry is beautiful.
Slide 3
The main types of symmetry are axial, central and mirror.
Slide 4
Two points A and A 1 are called symmetrical with respect to line a if this line passes through the middle of segment AA 1 and perpendicular to it.
Slide 5
Central symmetry.
Two points A and A 1 are called symmetrical with respect to point O if O is the midpoint of segment AA 1.
Slide 6
If a transformation of symmetry relative to a plane transforms a figure (body) into itself, then the figure is called symmetrical relative to the plane, and this plane is called the plane of symmetry of this figure. In some sources, this symmetry is called mirror symmetry.
Slide 7
Look at a maple leaf, a snowflake, a butterfly. What they have in common is that they are symmetrical. An ordinary leaf fell from a tree onto your sleeve. Its form is not random, it is strictly natural. The sheet is, as it were, glued together from two identical halves, one of these halves is located mirror-image relative to the other. The leaf has mirror symmetry, but it also has axial symmetry.
Slide 8
Oh, symmetry! I sing your anthem!
I recognize you everywhere in the world.
You are in the Eiffel Tower, in a small midge.
You are in a Christmas tree near a forest path.
With you in friendship is a tulip and a rose, and a snow swarm - the creation of frost!
Slide 9
Looking around, we can notice symmetry.
Slide 10
Let's look at examples of geometric shapes that have symmetry.
An isosceles triangle, a rectangle, a square, a circle, and an equilateral triangle have axial symmetry.
Slide 11
Central symmetry can be seen in a parallelogram, circle, square, rectangle.
Slide 12 Symmetry in algebra.
A parabola has axial symmetry, while a cubic parabola has central symmetry.
Slide 13
The Pythagoreans drew attention to the phenomena of symmetry in living nature back in Ancient Greece in connection with the development of the doctrine of harmony (5th century BC). In the 19th century, isolated works appeared on symmetry in the plant and animal world.
The human body is built on the principle of bilateral symmetry. Most of us
views the brain as a single structure, in reality it is divided into two halves. These two parts - the two hemispheres - fit tightly to each other. In full accordance with the general symmetry of the human body, each hemisphere is an almost exact mirror image of the other.
Slide 14
The vertical orientation of the body axis characterizes the symmetry of the tree. Leaves, flowers, branches, and fruits have pronounced symmetry.
Slide 15
Symmetry occurs widely in nature, especially in plants, such as the symmetry of a flower. 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 monocotyledons, while quintuple symmetry is common for dicotyledons.
Slide 16
Symmetry in animals means correspondence in size, shape and outline, as well as the relative arrangement of body parts located on opposite sides of the dividing line.
Spherical symmetry occurs in radiolarians and sunfishes, whose bodies are spherical in shape, and the parts are distributed around the center of the sphere and extend from it. Such organisms have neither front, nor back, nor lateral parts of the body; any plane drawn through the center divides the animal into equal halves.
With radial or radial symmetry, the body has the shape of a short or long cylinder or vessel with a central axis, from which parts of the body extend radially. These are coelenterates, echinoderms, and starfish.
With bilateral symmetry, there are three axes of symmetry, but only one pair of symmetrical sides. Because the other two sides - abdominal and dorsal - are not similar to each other. This type of symmetry is characteristic of most animals, including insects, fish, amphibians, reptiles, birds, and mammals.
Slide 17
The principles of symmetry are a tool in physics for finding new laws of nature. Among the symmetrical principles is the principle of relativity of Galileo and Einstein.
Slide 18- 19 Symmetry in chemistry.
Symmetry is discovered at the atomic level in the study of matter. It manifests itself in geometrically ordered atomic structures of molecules that are inaccessible to direct observation.
In 1810, D. Dalton, wanting to show his listeners how atoms combine to form chemical compounds, built wooden models of balls and rods. These models turned out to be excellent visual aids.
A water molecule has a plane of symmetry. Nothing changes if you swap paired atoms in a molecule; such an exchange is equivalent to a mirror operation. All solids are crystals, and crystals have symmetry.
In the picture you see crystals of topaz, beryl, and smoky quartz.
The symmetry of the external shape is clearly visible in the figure. Crystals of rock salt, quartz, aragonite.
Slide 20-23
Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have symmetry. Real natural snowflakes always have six axes of symmetry.
Slide 24-26
Symmetry plays a huge role in art, especially clear in ornaments and architecture.
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. For example, the building of the Bolshoi Theater in Moscow. The beauty of this building is associated with symmetry. Another example is St. Basil's Cathedral on Red Square in Moscow. This is a composition of ten different temples, each temple is geometrically symmetrical. However, the cathedral as a whole has neither mirror nor axial symmetry.
Slide 27
Lace provides striking examples of symmetry.
Slide 28
Symmetry was used by different peoples to color household and cultural objects.
Slide 29
A periodically repeating pattern on a long ribbon is called an ornament. In practice, ornaments are found in various forms: wall paintings, cast iron, plaster bas-reliefs or ceramics. Ornaments are used by painters and artists when decorating a room. For many centuries, people believed in the protective power of ornament, believing that it protects from troubles and brings happiness and prosperity. Gradually, the function of the amulet was lost, but its main task remained - to make the object more elegant and attractive, artistically expressive.
Slide 30
Ornaments covered the walls in ancient times, you see ancient Egyptian ornaments. The ornaments created by the modern famous Dutch artist Escher are beautiful. Dutch artist Maurice Escher uses symmetry effects with extraordinary ingenuity in his original, unique puzzle paintings. Isn’t it true that images of white, red and black lizards tightly intertwined with each other, which completely fill the entire plane of the picture, are perceived as a kind of hymn to all-pervasive symmetry.
Slide 31
Mirror symmetry is also called heraldic symmetry, as it can be seen in the coats of arms of different countries. The double-headed eagle served the Russian state well, as a symbol of the united Russian lands around a rich city and an intelligent, strong-willed leader. In 1997, the half-millennial anniversary of the Russian coat of arms was celebrated. Over 5 centuries, the historical fate of Russia has changed many times, but the state emblem of our country - its figurative name has invariably served the Motherland, and remains its main symbol today.
Slide 32
Some letters have symmetry. For example, the letter A. M, T, Sh, P have a vertical axis of symmetry. The letters B, Z, K, S, E, E have horizontal symmetry.
And the letters ZH, N, O, F, X have symmetry on both axes. Symmetry can also be seen in the words: radar, order, Cossack, hut. Such words that read the same in both directions are called palindromes. There are also entire phrases with this property (if you do not take into account the spaces between words): “Look for a taxi”, “Argentina attracts a black man”,
“The Argentinean appreciates the black man,” “Lesha found a bug on the shelf.” Many poets were fond of them.
Slide 33 Symmetry in music.
The soul of music, rhythm, consists of the correct periodic repetition of parts of a musical work. The correct repetition of identical parts as a whole is the essence of music. We can rightfully apply the concept of symmetry to a musical work, that this work is written using notes. Composition has the most direct relation to symmetry. The great German poet J. W. Goethe argued that every composition is based on hidden symmetry. To master the laws of composition means to master the laws of symmetry.
Slide 34
Truly symmetrical objects surround us literally on all sides. We are dealing with symmetry wherever there is any order. Symmetry is opposed to chaos, disorder. It turns out that symmetry is balance, orderliness, beauty, perfection.
Symmetry is diverse and omnipresent. She creates beauty and harmony.
Literature:
1. Vilenkin N.Ya. Behind the pages of a mathematics textbook. Arithmetic. Algebra. Geometry. A book for students in grades 10–11 of general education institutions: - M: Prosveshchenie, 1996.
2. Polya D. Mathematical discovery. - M.: Nauka, 1970
3. Batkin L. M. Leonardo da Vinci and the features of Renaissance creative thinking. – M.: Art, 1990
4. Gutkov A. The world of architecture: The language of architecture. –M.: Mol. Guard, 1985
5. N.V. Korneva, Yu.E. Novoselova, E.S. Timakina 9th grade integrated lesson
Regional research conference "Junior"
Research
Symmetry in the world around us
(section of exact sciences)
Performed: Merizanova Anna,
8th grade student,
Eliseenko Vera,
8th grade student
Supervisor: Kolesnikova
Lyudmila Alexandrovna,
mathematic teacher
Introduction
This school year we discussed this topic in mathematics lessons. We were interested in the topic “Symmetry”. And we decided to create a project on this topic, because... In the geometry textbook, little attention is paid to studying the topic “Symmetry”, while students often ask the question: why is it needed, where is it found, why is it studied at all.
But symmetry is found in nature, and in science, and in art - the unity and opposition of symmetry is found in everything.
Symmetry is characteristic of various phenomena that underlie all things; it describes many phenomena of life and many sciences
As a result of our work, we asked ourselves the following questions:
Why do you need to know symmetry, where in the world around you does it occur?
We have set ourselves a goal:
form ideas about symmetry , through the systematization of knowledge about symmetry, as well as through the analysis of natural phenomena and human activity.
To reveal the topic of our research work, the following tasks were set:
Learn to recognize symmetrical figures among others.
Get acquainted with the use of symmetry in nature, everyday life, art, and technology.
Demonstrate the varied applications of mathematics in real life.
Realize the degree of your interest in the subject and evaluate the possibilities of mastering it from the point of view of a future perspective (show the possibilities of applying the acquired knowledge in your future profession as an artist, architect, biologist, civil engineer).
To write the work, I used various methods:
study and analysis of scientific literature;
method of inductive generalization, specification;
use of computer equipment.
Chapter 1. First ideas about symmetry
In this chapter we describe the first ideas about symmetry, historical information on this topic; some examples of symmetrical figures are given; examples of a research nature on the topic: “Symmetry” are considered.
Historical development and understanding of the concept of symmetry
In the process of historical development and understanding of symmetry, a special stage of symmetry as a measure of beauty and harmony is associated with the work of the outstanding mathematician Hermann Weyl “Symmetry” (1952). G. Weil understood symmetry as the immeasurability (invariance) of any object during transformations: an object is symmetrical in the case when it is subjected to some operation, after which it will look the same as before the transformation.
The Greek word "symmetry" means "proportionality", "proportionality", "sameness in the arrangement of parts." However, the word “symmetry” is often understood as a broader concept: the regularity of changes in certain phenomena (seasons, day and night, etc.), the balance of left and right, the equality of natural phenomena. In fact, we are dealing with symmetry wherever any order is observed. The concept of symmetry was widely used in psychology and morality. Thus, the great Aristotle believed that symmetry has the meaning of a certain average measure to which a virtuous person should strive in his actions. The Roman physician Galen (2nd century AD) understood symmetry as a state of mind equally distant from both extremes, for example, from grief and joy, apathy and excitement. Symmetry, understood as peace and balance, is opposed to chaos and disorder. This is evidenced by the engraving of Marius Escher “Order and Chaos” (Fig. 196), where, as the artist himself wrote, “a stellated dodecahedron, a symbol of beauty and order, is surrounded by a transparent sphere. It reflects a meaningless collection of useless things."
Mathematical idea of symmetry
The ideas about symmetry outlined above are of a general nature and are not accurate and strict for mathematics.
Definition 1. Symmetry – this is proportionality, the sameness in the arrangement of parts of something on opposite sides of a point, straight line or plane.
A mathematical strict definition of symmetry was formed relatively recently - in the 19th century, when the concepts of mirror and rotational symmetry were introduced.
Rosettes and snowflakes are symmetrical and very beautiful figures.
In planimetry, there are axial (symmetry relative to a straight line), central symmetry (symmetry relative to a point), as well as rotational, mirror, and portable.
Definition 2. Two points A and A 1 are called symmetrical relative to straight line a, if this line passes through the middle of segment AA 1 and is perpendicular to it.
Every point is straight A
Definition 2 . The figure is said to be symmetrical about a straight line A, if for each point of the figure there is a point symmetrical to it relative to the straight line A also belongs to this figure. Straight A called axis of symmetry figures. They say the figure has axial symmetry. Shapes that have an axis of symmetry: rectangle, rhombus, square, equilateral triangle, isosceles triangle, circle, etc.
ABOUT 
definition 3. Two points A and A 1 are called symmetrical about point O, if O is the middle of the segment AA 1. Dot ABOUT is considered symmetrical to itself.
Definition 4. The figure is called symmetrical about point O, if for each point of the figure there is a point symmetrical to it relative to the point ABOUT also belongs to this figure. Dot ABOUT, called center of symmetry of the figure. They say the figure has central symmetry. Examples of figures that have central symmetry: circle, parallelogram, triangle, etc.
Mathematics studies many figures that have both axial and central symmetry (circle, square, etc.), only axial symmetry (for example, an isosceles triangle), and only central symmetry (for example, a general parallelogram).
To understand this topic, we carried out a number of research tasks.
Research assignments.
Exercise 1. On a straight line AB find a point whose distance is the sum of two given points M And N would be the smallest.
Discussion.1 case.
Let M And N
lie on opposite sides of , the shortest distance between them is 
, therefore, the required point X lies at the intersection and .
IN 
any other point 
straight AB does not have this property, since 
.
Case 2 . M And N lie on one side Build M 1 , symmetrical M relative to , after which the problem is reduced to case 1. if
Then the required point X is the point of intersection of the lines MN And AB.
Task 2. Given straight lines AB and dots M And N. Find on such a point such that the difference (modulo its distance from the points M And N was the greatest.
ABOUT 
discussion.1 case.
Points M And N
lie on one side of the line AB (and, moreover, at different distances from it. Then point X of the line AB, for which the difference in distances from the points M And N
the largest, is the point of intersection of the line AB with the continuation of the segment MN. Then 
any other point X 1 of the line AB does not have this property, since 
(a corollary of the triangle axiom). If M And N
is at the same distance from , the problem has no solution.
2
happening.
Points M And N
lie on opposite sides of . Then the required point 
, Where 
.
If points M And N are on opposite sides of and at the same distance from it, then the problem has no solutions.
Task 3. Investigate whether the following have a center of symmetry: 1) a segment; 2) beam; 3) square.
Discussion. 1) yes; 2) no; 3 yes
Task 4. Investigate which of the following points of the Latin alphabet have a center of symmetry: A, O, M, X.
Discussion. O and X
Discussion. 1) two; 2) “infinite set”: any line perpendicular to a given one, as well as the line itself; 3) one.
Task 6. Explore which of the following letters have an axis of symmetry: A, B, d, E, O in the alphabet.
Discussion. A, E, O
Conclusion: These examples show us that even points in the alphabet have a symmetrical position. Various geometric shapes have an axis of symmetry.
Symmetry of Old Russian ornament
D 
Russian ornament is characterized by both floral and geometric forms, as well as images of birds, animals and fantastic animals. Russian ornament is especially clearly expressed in wood carving and embroidery. The most commonly used were so-called braids - interlacing ribbons, belts, and flower stems. In the 17th century The architect Stepan Ivanov created his famous “Peacock Eye” ornament.
According to Academician B. A. Rybakov, a famous archaeologist and historian with a worldwide reputation, the basis of the Old Russian ornament included universal, different ideas about the world. The consciousness of the ancient Slav was conditioned by mythological perceptions of reality. All this was reflected in the motifs characteristic of Russian ornament.
Motive braids, characteristic of Rusal bracelets, which B. A. Rybakov interpreted as a sign of water and the kingdom of the underground ruler Pereplut.
Ancient motif goddess Mokoshi as a specific embodiment of the idea of the Great Mother, common to all peoples at a certain stage of historical existence. Mokosha (Makosh) is the only female image in ancient Russian mythology. Her name brings to mind phlegm, moisture, water.
Motive Mokosh patronized all women's activities, especially spinning, and was revered mainly by women.
tree of life.
Among the traditional patterns used for centuries in Russian decorative and applied arts is a pattern depicting the tree of life with birds symmetrically located on or near it.
The water element was represented by rows of dots and lines reproducing raindrops, as well as zigzag lines, which serves as an example of figurative symmetry.
The earth was represented by a rectangle, divided by diagonals into four parts with a repeating pattern in them. This configuration is characterized by axial symmetry in combination with central symmetry. These types of symmetry predominate in the depiction of the plant world. 
Since ancient times, Russian ornament has developed a special system of arrangement of symbols representing the movement of the Sun around the Earth. There are several types of sun signs; they are characterized by rotational symmetry. The most common is a circle divided by radii into different sectors (“Wheel of Jupiter”), as well as a circle with a cross inside.
Conclusion: Having analyzed the literature on this issue, we came to the conclusion that symmetrical symbols are often found in ancient Russian ornaments. In traditional national jewelry and household items you can find all types of symmetry on a plane: central, axial, rotary, portable.
1.4. Symmetry through the centuries
In his reflections on the picture of the world, people have been actively using the idea of symmetry for a long time. According to legend, the term “symmetry” was coined by the sculptor Pythagoras of Rhegium, who lived in the city of Regulus. He defined deviation from symmetry by the term “asymmetry”. The ancient Greeks believed that the universe was symmetrical simply because it was beautiful. Considering the sphere to be the most symmetrical and perfect form, they concluded that the Earth was spherical and that it moved on a sphere around a certain “central fire”, where the 6 then known planets also moved along with the moon, the Sun, and the stars.
Representatives of the first scientific school in human history, followers of Pythagoras of Samoa, tried to connect symmetry with number.
Widely using the idea of harmony and symmetry, ancient scientists loved to turn not only to spherical forms, but also to regular polyhedra, for the construction of which they used the “golden ratio”. Regular polyhedra have faces that are regular polygons of the same type, and the angles between the faces are equal. The ancient Greeks established an astonishing fact: there are only five regular convex polyhedra, the names of which are associated with the number of faces - tetrahedron, octahedron, icosahedron, cube, dodecahedron.
Chapter 2. Symmetry around us
This chapter describes a theory that indicates various representations of symmetry in nature; in this chapter we prove that structures created by man also have symmetrical figures.
2.1. The role of symmetry in knowledge of nature
The symmetry of crystals is a consequence of their internal structure: their atoms and molecules have an ordered mutual arrangement, forming a symmetrical lattice of atoms - the so-called crystal lattice.
The missing elements of symmetry were identified by academician Axel Vilhelmovich Gadolin (1828-1892). The famous professor of mineralogy from the German city of Marburg Johann Hessel in 1830. Published his work on the symmetry of crystals. For some reason, his work went unnoticed. But in 1897 Hessel's work was republished, and since then his name has gone down in the history of science.
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Thus, they learned to study and compare the symmetry of crystals. There are 9 symmetry elements and only 32 different sets of symmetry elements - symmetry groups, which determine the external shape of crystals. But since the number of symmetry elements of crystals is finite, then the number of their sets - combinations that describe the symmetry of the external form - is finite. It follows that symmetry is a strict and comprehensive law governing the kingdom of crystals. It determines the shape of the crystal, the number of its faces and edges, and it also dictates its internal structure.
Symmetry can be found in sea creatures such as starfish, sea urchins and some jellyfish.
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Leaves, branches, flowers and fruits of plants have strong symmetry. Some of them are characterized by only mirror symmetry, or only rotational symmetry, sliding.
It is interesting that among plants of the same species there are those that have both left and right leaf structures.
Living nature is characterized not only by well-known types of symmetry. Thus, the curved stem of a plant and the twisted shape of a mollusk are no less symmetrical than a crystal. But this is a different symmetry - curvilinear, which was discovered in 1926.
And in 1960 Academician A.V. Shubnikov introduced the symmetry of similarity into consideration. Similar figures are considered to be of the same shape. Symmetry of similarity consists of transferring (rotating) a figure while simultaneously decreasing or increasing its size.
Symmetry in architectural structures
Symmetry dominates not only in nature, but also in human creativity. Works of architecture demonstrate excellent examples of symmetry. Old Russian buildings are interesting, in particular wooden churches. Slender and expressive, cut into figures of eight, i.e. with symmetrical octagonal tents, they perfectly corresponded to the concept of beauty in medieval Rus'.
An example is St. Basil's Cathedral on Red Square in Moscow. The temple consists of ten different temples, each of which is strictly symmetrical, but as a whole it has neither mirror nor rotational symmetry.
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It is possible to give many examples of the use of symmetry and asymmetry in sculpture. For example, the sculpture of the Peloponnesian master from the school of Pythagoras “The Delphic Charioteer”, which depicts the winner in horse-drawn chariot competitions. The figure of a young man in a long chiton is generally symmetrical, but a slight rotation of the torso and head breaks the mirror symmetry, which creates the illusion of movement, and the statue seems alive.
Louis Pasteur believed that it was asymmetry that distinguishes living from non-living, believing that symmetry is the guardian of peace, and asymmetry is the engine of life. An example of how the paradox of symmetry serves not only to convey movement, but also to enhance impression is the image of a Greek vase from the Kamares Cave on the island of Crete.
Conclusion
Symmetry is something common, characteristic of different phenomena, underlying all things, and asymmetry expresses certain individual characteristics of things and phenomena. In nature, in science, and in art, the unity and opposition of symmetry and asymmetry is revealed in everything. The world exists thanks to the unity of these two opposites.
After analyzing the work, we came to the conclusion that symmetry is often found in art, architecture, technology, and everyday life. Thus, the facades of many buildings have axial symmetry. In most cases, patterns on carpets, fabrics, and indoor wallpaper are symmetrical about the axis or center. Many parts of mechanisms, such as gears, are symmetrical.
As a result of the project:
expanded knowledge about symmetry;
learned what phenomena from life and
some sciences are described by symmetry;
new practical techniques: work with educational, scientific and educational literature;
generalized the concepts, ideas, knowledge that the project result is aimed at obtaining: we looked at where symmetry occurs in life.
Bibliography
Afanasyev A.N., Mythology of Ancient Rus'. – M.: Eksmo, 2006.
Weil G. Symmetry. – Ed. 2nd, erased – M.: Unified URSS, 2003.
Gnedengo B.V. Essays on the history of mathematics in Russia. – 2nd ed., rev. and additional – M.: KomKniga, 2005.
Fine motifs in Russian folk embroidery. Museum of Folk Art. – M.: Soviet Russia, 1990.
Klimova N. T. Folk ornament in the composition of artistic products.
We get used to the concept of symmetry from childhood. We know that a butterfly is symmetrical: its right and left wings are the same; a symmetrical wheel whose sectors are identical; symmetrical patterns of ornaments, stars of snowflakes.
A truly vast literature is devoted to the problem of symmetry. From textbooks and scientific monographs to works that pay attention not so much to drawings and formulas, but to artistic images.
The very term “symmetry” in Greek means “proportionality,” which ancient philosophers understood as a special case of harmony - the coordination of parts within the whole. Since ancient times, many peoples have had the idea of symmetry in the broad sense - as the equivalent of balance and harmony.
Symmetry is one of the most fundamental and one of the most general patterns of the universe: inanimate, living nature and society. We meet her everywhere. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human knowledge; it is widely used by all areas of modern science without exception. Truly symmetrical objects surround us literally on all sides; we are dealing with symmetry wherever any order is observed. It turns out that symmetry is balance, orderliness, beauty, perfection. It is diverse, omnipresent. She creates beauty and harmony. Symmetry literally permeates the entire world around us, which is why the topic I have chosen will always be relevant.
Symmetry expresses the preservation of something despite some changes or the preservation of something despite a change. Symmetry presupposes the invariability not only of the object itself, but also of any of its properties in relation to transformations performed on the object. The immutability of certain objects can be observed in relation to various operations - rotations, translations, mutual replacement of parts, reflections, etc. In this regard, different types of symmetry are distinguished. Let's look at all types in more detail.
AXIAL SYMMETRY.
Symmetry about a straight line is called axial symmetry (mirror reflection about a straight line).
If point A lies on the l axis, then it is symmetrical to itself, i.e. A coincides with A1.
In particular, if, when transforming symmetry with respect to the l axis, the figure F transforms into itself, then it is called symmetric with respect to the l axis, and the l axis is called its symmetry axis.
CENTRAL SYMMETRY.
A figure is called centrally symmetric if there is a point relative to which each point of the figure is symmetrical to some point of the same figure. Namely: movement that changes directions to opposite ones is central symmetry.
Point O is called the center of symmetry and is motionless. This transformation has no other fixed points. Examples of figures that have a center of symmetry are a parallelogram, a circle, etc.
The familiar concepts of rotation and parallel translation are used in the definition of so-called translational symmetry. Let's look at translation symmetry in more detail.
1. TURN
A transformation in which each point A of a figure (body) is rotated by the same angle α around a given center O is called rotation or rotation of the plane. Point O is called the center of rotation, and angle α is called the angle of rotation. Point O is a fixed point of this transformation.
The rotational symmetry of the circular cylinder is interesting. It has an infinite number of 2nd order rotary axes and one infinitely high order rotary axis.
2. PARALLEL TRANSFER
A transformation in which each point of a figure (body) moves in the same direction by the same distance is called parallel translation.
To specify a parallel translation transformation, it is enough to specify the vector a.
3. SLIDING SYMMETRY
Sliding symmetry is a transformation in which axial symmetry and parallel translation are performed sequentially. Sliding symmetry is an isometry of the Euclidean plane. Gliding symmetry is a composition of symmetry with respect to some line l and translation to a vector parallel to l (this vector may be zero).
Gliding symmetry can be represented as a composition of 3 axial symmetries (Chales' theorem).
MIRROR SYMMETRY
What could be more like my hand or my ear than their own reflection in the mirror? And yet the hand that I see in the mirror cannot be put in the place of the real hand.
Immanuel Kant.
If a transformation of symmetry relative to a plane transforms a figure (body) into itself, then the figure is called symmetrical relative to the plane, and this plane is called the plane of symmetry of this figure. This symmetry is called mirror symmetry. As the name itself suggests, mirror symmetry connects an object and its reflection in a plane mirror. Two symmetrical bodies cannot be “nested into each other”, since in comparison with the object itself, its mirror-mirror double turns out to be turned out along the direction perpendicular to the plane of the mirror.
Symmetrical figures, for all their similarities, differ significantly from each other. The double observed in the mirror is not an exact copy of the object itself. The mirror does not simply copy the object, but swaps (represents) the front and rear parts of the object in relation to the mirror. For example, if your mole is on your right cheek, then your looking-glass double’s is on your left. Hold a book up to the mirror and you will see that the letters seem to be turned inside out. Everything in the mirror is rearranged from right to left.
Bodies are called mirror-equal bodies if, with proper displacement, they can form two halves of a mirror-symmetrical body.
2. 2 Symmetry in nature
A figure has symmetry if there is a movement (non-identical transformation) that transforms it into itself. For example, a figure has rotational symmetry if it is translated into itself by some rotation. But in nature, with the help of mathematics, beauty is not created, as in technology and art, but is only recorded and expressed. It not only pleases the eye and inspires poets of all times and peoples, but allows living organisms to better adapt to their environment and simply survive.
The structure of any living form is based on the principle of symmetry. From direct observation we can deduce the laws of geometry and feel their incomparable perfection. This order, which is a natural necessity, since nothing in nature serves purely decorative purposes, helps us find the general harmony on which the entire universe is based.
We see that nature designs any living organism according to a certain geometric pattern, and the laws of the universe have a clear justification.
The principles of symmetry underlie the theory of relativity, quantum mechanics, solid state physics, atomic and nuclear physics, and particle physics. These principles are most clearly expressed in the invariance properties of the laws of nature. We are talking not only about physical laws, but also others, for example, biological ones.
Speaking about the role of symmetry in the process of scientific knowledge, we should especially highlight the use of the method of analogies. According to the French mathematician D. Polya, “there are, perhaps, no discoveries either in elementary or higher mathematics, or, perhaps, in any other field that could be made without analogies.” Most of these analogies are based on common roots, general patterns that manifest themselves in the same way at different levels of the hierarchy.
So, in the modern understanding, symmetry is a general scientific philosophical category that characterizes the structure of the organization of systems. The most important property of symmetry is the preservation (invariance) of certain features (geometric, physical, biological, etc.) in relation to well-defined transformations. The mathematical apparatus for studying symmetry today is the theory of groups and the theory of invariants.
Symmetry in the plant world
The specific structure of plants is determined by the characteristics of the habitat to which they adapt. Any tree has a base and a top, a “top” and a “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 rotary axis of the “wood cone” and the planes of symmetry. A tree, with the help of its root system, absorbs moisture and nutrients from the soil, that is, from below, and the remaining vital functions are performed by the crown, that is, above. At the same time, directions in a plane perpendicular to the vertical are virtually indistinguishable for a tree; in all these directions, air, light, and moisture enter the tree equally.
The tree has a vertical rotary axis (cone axis) and vertical planes of symmetry.
When we want to draw a leaf of a plant or a butterfly, we have to take into account their axial symmetry. The midrib for the leaf serves as an axis of symmetry. Leaves, branches, flowers, and fruits have pronounced symmetry. The leaves are characterized by mirror symmetry. The same symmetry is also found in flowers, but in them mirror symmetry often appears in combination with rotational symmetry. There are also frequent cases of figurative symmetry (acacia branches, rowan trees).
In the diverse world of colors, there are rotary axes of different orders. However, the most common is 5th order rotational symmetry. This symmetry is found in many wildflowers (bell, forget-me-not, geranium, carnation, St. John's wort, cinquefoil), in the flowers of fruit trees (cherry, apple, pear, tangerine, etc.), in the flowers of fruit and berry plants (strawberries, raspberries, viburnum , bird cherry, rowan, rose hip, hawthorn), etc.
Academician N. Belov explains this fact by the fact that the 5th order axis is a kind of instrument of the struggle for existence, “insurance against petrification, crystallization, the first step of which would be their capture by the grid.” 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 represented very widely in it.
In his book “This Right, Left World,” M. Gardner writes: “On Earth, life originated in spherically symmetrical forms, and then began to develop along two main lines: the world of plants with cone symmetry was formed, and the world of animals with bilateral symmetry.”
In nature, there are bodies that have helical symmetry, that is, alignment with their original position after rotation by an angle around an axis, with an additional shift along the same axis.
If is a rational number, then the rotary axis also turns out to be the translation axis.
The leaves on the stem are not arranged in a straight line, but surround the branch in a spiral. The sum of all previous steps of the spiral, starting from the top, is equal to the value of the subsequent step A+B=C, B+C=D, etc.
Helical symmetry is observed in the arrangement of leaves on the stems of most plants. Arranging in a spiral along the stem, the leaves seem to spread out in all directions and do not block each other from the light, which is extremely necessary for plant life. This interesting botanical phenomenon is called phyllotaxis (literally “leaf arrangement”).
Another manifestation of phyllotaxis is the structure of the inflorescence of a sunflower or the scales of a fir cone, in which the scales are arranged in the form of spirals and helical lines. This arrangement is especially clear in the pineapple, which has more or less hexagonal cells that form rows running in different directions.
Symmetry in the animal world
The significance of the form of symmetry for an animal is easy to understand if it is connected with the way of life and environmental conditions. Symmetry in animals means correspondence in size, shape and outline, as well as the relative arrangement of body parts located on opposite sides of the dividing line.
Rotational symmetry of the 5th order is also found in the animal world. This is a symmetry in which an object aligns with itself when rotated around a rotary axis 5 times. Examples include the starfish and sea urchin shell. The entire skin of starfish is as if encrusted with small plates of calcium carbonate; needles extend from some of the plates, some of which are movable. An ordinary starfish has 5 planes of symmetry and 1 axis of rotation of the 5th order (this is the highest symmetry among animals). Her ancestors appear to have had lower symmetry. This is evidenced, in particular, by the structure of the star larvae: they, like most living beings, including humans, have only one plane of symmetry. Starfish do not have a horizontal plane of symmetry: they have a “top” and a “bottom.” Sea urchins are like living pincushions; their spherical body bears long and movable needles. In these animals, the calcareous plates of the skin merged and formed a spherical carapace. There is a mouth in the center of the lower surface. The ambulacral legs (water-vascular system) are collected in 5 stripes on the surface of the shell.
However, unlike the plant world, rotational symmetry is rarely observed in the animal world.
Insects, fish, eggs, and animals are characterized by a difference between the directions “forward” and “backward” that is incompatible with rotational symmetry.
The direction of movement is a fundamentally selected direction, with respect to which there is no symmetry in any insect, any bird or fish, any animal. In this direction the animal rushes for food, in the same direction it escapes from its pursuers.
In addition to the direction of movement, the symmetry of living beings is determined by another direction - the direction of gravity. Both directions are significant; they define the plane of symmetry of an animal being.
Bilateral (mirror) symmetry is the characteristic symmetry of all representatives of the animal world. This symmetry is clearly visible in the butterfly. The symmetry of the left and right wings appears here with almost mathematical rigor.
We can say that every animal (as well as insects, fish, birds) 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. Thus, enantiomorphs are the right and left ear, right and left eye, right and left horn, etc.
Simplification of living conditions can lead to a violation of bilateral symmetry, and animals from being bilaterally symmetrical become radially symmetrical. This applies to echinoderms (starfish, sea urchins, crinoids). All marine animals have radial symmetry, in which parts of the body radiate away from a central axis, like the spokes of a wheel. The degree of activity of animals correlates with their type of symmetry. Radially symmetrical echinoderms are usually poorly mobile, move slowly, or are attached to the seabed. The body of a starfish consists of a central disk and 5-20 or more rays radiating from it. In mathematical language, this symmetry is called rotational symmetry.
Let us finally note the mirror symmetry of the human body (we are talking about the appearance and structure of the skeleton). This symmetry has always been and is the main source of our aesthetic admiration for the well-proportioned human body. Let’s not figure out for now whether an absolutely symmetrical person actually exists. Everyone, of course, will have a mole, a strand of hair or some other detail that breaks the external symmetry. The left eye is never exactly the same as the right, and the corners of the mouth are at different heights, at least for most people. Yet these are only minor inconsistencies. No one will doubt that outwardly a person is built symmetrically: the left hand always corresponds to the right and both hands are exactly the same.
Everyone knows that the similarity between our hands, ears, eyes and other parts of the body is the same as between an object and its reflection in a mirror. It is the issues of symmetry and mirror reflection that are given attention here.
Many artists paid close attention to the symmetry and proportions of the human body, at least as long as they were guided by the desire to follow nature as closely as possible in their works.
In modern schools of painting, the vertical size of the head is most often taken as a single measure. With a certain assumption, we can assume that the length of the body is eight times the size of the head. The size of the head is proportional not only to the length of the body, but also to the size of other parts of the body. All people are built on this principle, which is why we are, in general, similar to each other. However, our proportions are only approximately consistent, and therefore people are only similar, but not the same. In any case, we are all symmetrical! In addition, some artists especially emphasize this symmetry in their works.
Our own mirror symmetry is very convenient for us, it allows us to move straight and turn right and left with equal ease. Mirror symmetry is equally convenient for birds, fish and other actively moving creatures.
Bilateral symmetry means that one side of an animal's body is a mirror image of the other side. This type of organization is characteristic of most invertebrates, especially annelids and arthropods - crustaceans, arachnids, insects, butterflies; for vertebrates - fish, birds, mammals. Bilateral symmetry first appears in flatworms, in which the anterior and posterior ends of the body differ from each other.
Let's consider another type of symmetry that is found in the animal world. This is helical or spiral symmetry. Helical symmetry is symmetry with respect to the combination of two transformations - rotation and translation along the axis of rotation, i.e. there is movement along the axis of the screw and around the axis of the screw.
Examples of natural propellers are: tusk of a narwhal (a small cetacean that lives in the northern seas) - left propeller; snail shell – right screw; The horns of the Pamir ram are enantiomorphs (one horn is twisted in a left-handed spiral, and the other in a right-handed spiral). Spiral symmetry is not ideal, for example, the shell of mollusks narrows or widens at the end. Although external helical symmetry is rare in multicellular animals, many important molecules from which living organisms are built - proteins, deoxyribonucleic acids - DNA have a helical structure.
Symmetry in inanimate nature
Crystal symmetry is the property of crystals to align with themselves in various positions by rotation, reflection, parallel translation, or part or combination of these operations. The symmetry of the external shape (cut) of a crystal is determined by the symmetry of its atomic structure, which also determines the symmetry of the physical properties of the crystal.
Let's take a closer look at the multifaceted shapes of crystals. First of all, it is clear that crystals of different substances differ from each other in their shapes. Rock salt is always cubes; rock crystal - always hexagonal prisms, sometimes with heads in the form of trihedral or hexagonal pyramids; diamond - most often regular octahedrons (octahedrons); ice is hexagonal prisms, very similar to rock crystal, and snowflakes are always six-pointed stars. What catches your eye when you look at crystals? First of all, their symmetry.
Many people think that crystals are beautiful, rare stones. They come in different colors, are usually transparent and, best of all, have a beautiful, regular shape. Most often, crystals are polyhedra, their sides (faces) are perfectly flat, and their edges are strictly straight. They delight the eye with the wonderful play of light in their edges and the amazing correctness of their structure.
However, crystals are not museum rarities at all. Crystals surround us everywhere. The solids from which we build houses and machines, the substances that we use in everyday life - almost all of them belong to crystals. Why don't we see this? The fact is that in nature one rarely comes across bodies in the form of separate single crystals (or, as they say, single crystals). Most often, the substance is found in the form of firmly adhered crystalline grains of a very small size - less than a thousandth of a millimeter. This structure can only be seen through a microscope.
Bodies consisting of crystalline grains are called finely crystalline, or polycrystalline (“poly” - in Greek “many”).
Of course, finely crystalline bodies should also be classified as crystals. Then it will turn out that almost all the solid bodies around us are crystals. Sand and granite, copper and iron, paints - all these are crystals.
There are exceptions; glass and plastics do not consist of crystals. Such solids are called amorphous.
Studying crystals means studying almost all the bodies around us. It's clear how important this is.
Single crystals are immediately recognizable by their regular shape. Flat faces and straight edges are a characteristic property of the crystal; the correctness of the form is undoubtedly related to the correctness of the internal structure of the crystal. If a crystal is especially elongated in a certain direction, it means that the structure of the crystal in that direction is somehow special.
There is a center of symmetry in a cube of rock salt, in the octahedron of a diamond, and in the star of a snowflake. But in a quartz crystal there is no center of symmetry.
The most accurate symmetry is achieved in the world of crystals, but even here it is not ideal: cracks and scratches invisible to the eye always make equal faces slightly different from each other.
All crystals are symmetrical. This means that in each crystalline polyhedron one can find planes of symmetry, axes of symmetry, a center of symmetry or other symmetry elements so that identical parts of the polyhedron are aligned with each other.
All elements of symmetry repeat the same parts of the figure, all give it symmetrical beauty and completeness, but the center of symmetry is the most interesting. Not only the shape, but also many physical properties of the crystal can depend on whether a crystal has a center of symmetry or not.
Honeycombs are a real design masterpiece. They consist of a number of hexagonal cells. This is the densest packaging, allowing the most advantageous placement of the larva in the cell and, with the maximum possible volume, the most economical use of the building material - wax.
III Conclusion
Symmetry permeates literally everything around, capturing seemingly completely unexpected areas and objects. It, manifesting itself in the most diverse objects of the material world, undoubtedly reflects its most general, most fundamental properties. The principles of symmetry play an important role in physics and mathematics, chemistry and biology, technology and architecture, painting and sculpture, poetry and music.
We see that nature designs any living organism according to a certain geometric pattern, and the laws of the universe have a clear justification. Therefore, the study of the symmetry of various natural objects and the comparison of its results is a convenient and reliable tool for understanding the basic laws of the existence of matter.
The laws of nature that govern the inexhaustible picture of phenomena in their diversity, in turn, are subject to the principles of symmetry. There are many types of symmetry, both in the plant and animal world, but with all the diversity of living organisms, the principle of symmetry always operates, and this fact once again emphasizes the harmony of our world. Symmetry underlies things and phenomena, expressing something common, characteristic of different objects, while asymmetry is associated with the individual embodiment of this common thing in a specific object.
So, on the plane we have four types of movements that transform figure F into an equal figure F1:
1) parallel transfer;
2) axial symmetry (reflection from a straight line);
3) rotation around a point (Partial case - central symmetry);
4) “sliding” reflection.
In space, mirror symmetry is added to the above types of symmetry.
I believe that the goal set in the abstract has been achieved. When writing my essay, the greatest difficulty for me was drawing my own conclusions. I think that my work will help schoolchildren expand their understanding of symmetry. I hope that my essay will be included in the methodological fund of the mathematics classroom.
