Drawings with an axis of symmetry. Axial symmetry

I . Symmetry in mathematics :

Basic concepts and definitions.

Axial symmetry (definitions, construction plan, examples)

Central symmetry (definitions, construction plan, whenmeasures)

Summary table (all properties, features)

II . Applications of symmetry:

1) in mathematics

2) in chemistry

3) in biology, botany and zoology

4) in art, literature and architecture

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1. Basic concepts of symmetry and its types.

The concept of symmetry R goes back through the entire history of mankind. It is found already at the origins of human knowledge. It arose in connection with the study of a living organism, namely man. And it was used by sculptors back in the 5th century BC. e. The word “symmetry” is Greek and means “proportionality, proportionality, sameness in the arrangement of parts.” It is widely used by all areas of modern science without exception. Many great people have thought about this pattern. For example, L.N. Tolstoy said: “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 clear to the eye? What is symmetry? This is an innate feeling, I answered myself. What is it based on?” The symmetry is truly pleasing to the eye. Who hasn’t admired the symmetry of nature’s creations: leaves, flowers, birds, animals; or human creations: buildings, technology, everything that surrounds us since childhood, everything that strives for beauty and harmony. Hermann Weyl said: “Symmetry is the idea through which man throughout the ages has tried to comprehend and create order, beauty and perfection.” Hermann Weyl is a German mathematician. His activities span the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what criteria one can determine the presence or, conversely, absence of symmetry in a given case. Thus, a mathematically rigorous concept was formed relatively recently - at the beginning of the twentieth century. It's quite complicated. Let us turn and once again remember the definitions that were given to us in the textbook.

2. Axial symmetry.

2.1 Basic definitions

Definition. 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 is perpendicular to it. Each point of a line a is considered symmetrical to itself.

Definition. 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 the axis of symmetry of the figure. The figure is also said to have axial symmetry.

2.2 Construction plan

And so, to construct a symmetrical figure relative to a straight line, from each point we draw a perpendicular to this straight line and extend it to the same distance, mark the resulting point. We do this with each point and get symmetrical vertices of a new figure. Then we connect them in series and get a symmetrical figure of a given relative axis.

2.3 Examples of figures with axial symmetry.

3. Central symmetry

3.1 Basic definitions

Definition. Two points A and A 1 are called symmetrical with respect to point O if O is the middle of the segment AA 1. Point O is considered symmetrical to itself.

Definition. A figure is said to be symmetrical with respect to point O if, for each point of the figure, a point symmetrical with respect to point O also belongs to this figure.

3.2 Construction plan

Construction of a triangle symmetrical to the given one relative to the center O.

To construct a point symmetrical to a point A relative to the point ABOUT, it is enough to draw a straight line OA(Fig. 46 ) and on the other side of the point ABOUT set aside a segment equal to the segment OA. In other words , points A and ; In and ; C and symmetrical about some point O. In Fig. 46 a triangle is constructed that is symmetrical to a triangle ABC relative to the point ABOUT. These triangles are equal.

Construction of symmetrical points relative to the center.

In the figure, points M and M 1, N and N 1 are symmetrical relative to point O, but points P and Q are not symmetrical relative to this point.

In general, figures that are symmetrical about a certain point are equal .

3.3 Examples

Let us give examples of figures that have central symmetry. The simplest figures with central symmetry are the circle and parallelogram.

Point O is called the center of symmetry of the figure. In such cases, the figure has central symmetry. The center of symmetry of a circle is the center of the circle, and the center of symmetry of a parallelogram is the point of intersection of its diagonals.

A straight line also has central symmetry, but unlike a circle and a parallelogram, which have only one center of symmetry (point O in the figure), a straight line has an infinite number of them - any point on the straight line is its center of symmetry.

The pictures show an angle symmetrical relative to the vertex, a segment symmetrical to another segment relative to the center A and a quadrilateral symmetrical about its vertex M.

An example of a figure that does not have a center of symmetry is a triangle.

4. Lesson summary

Let us summarize the knowledge gained. Today in class we learned about two main types of symmetry: central and axial. Let's look at the screen and systematize the knowledge gained.

Summary table

	Axial symmetry	Central symmetry
Peculiarity	All points of the figure must be symmetrical relative to some straight line.	All points of the figure must be symmetrical relative to the point chosen as the center of symmetry.
Properties	1. Symmetrical points lie on perpendiculars to a line. 3. Straight lines turn into straight lines, angles into equal angles. 4. The sizes and shapes of the figures are preserved.	1. Symmetrical points lie on a line passing through the center and a given point of the figure. 2. The distance from a point to a straight line is equal to the distance from a straight line to a symmetrical point. 3. The sizes and shapes of the figures are preserved.

II. Application of symmetry

Man has long used symmetry in architecture.

The ancient architects made especially brilliant use of symmetry in architectural structures. Moreover, the ancient Greek architects were convinced that in their works they were guided by the laws that govern nature. By choosing symmetrical forms, the artist thereby expressed his understanding of natural harmony as stability and balance.

The city of Oslo, the capital of Norway, has an expressive ensemble of nature and art. This is Frogner - a park - a complex of garden and park sculpture, which was created over the course of 40 years.

Pashkov House Louvre (Paris)

© Elena Vladimirovna Sukhacheva, 2008-2009.

Movement concept

Let us first examine the concept of movement.

Definition 1

A mapping of a plane is called a motion of the plane if the mapping preserves distances.

There are several theorems related to this concept.

Axial and central symmetry are examples of motion. Let's look at them in more detail.

Axial symmetry

Definition 2

Points $A$ and $A_1$ are called symmetrical with respect to the line $a$ if this line is perpendicular to the segment $(AA)_1$ and passes through its center (Fig. 1).

Picture 1.

Let's consider axial symmetry using an example problem.

Example 1

Construct a symmetrical triangle for a given triangle relative to any of its sides.

Solution.

Let us be given a triangle $ABC$. We will construct its symmetry with respect to the side $BC$. The side $BC$ with axial symmetry will transform into itself (follows from the definition). Point $A$ will go to point $A_1$ as follows: $(AA)_1\bot BC$, $(AH=HA)_1$. Triangle $ABC$ will transform into triangle $A_1BC$ (Fig. 2).

Figure 2.

Definition 3

A figure is called symmetrical with respect to straight line $a$ if every symmetrical point of this figure is contained in the same figure (Fig. 3).

Figure 3.

Figure $3$ shows a rectangle. It has axial symmetry with respect to each of its diameters, as well as with respect to two straight lines that pass through the centers of opposite sides of a given rectangle.

Central symmetry

Definition 4

Points $X$ and $X_1$ are called symmetrical with respect to point $O$ if point $O$ is the center of the segment $(XX)_1$ (Fig. 4).

Figure 4.

Let's consider central symmetry using an example problem.

Example 2

Construct a symmetrical triangle for a given triangle at any of its vertices.

Solution.

Let us be given a triangle $ABC$. We will construct its symmetry relative to the vertex $A$. The vertex $A$ with central symmetry will transform into itself (follows from the definition). Point $B$ will go to point $B_1$ as follows: $(BA=AB)_1$, and point $C$ will go to point $C_1$ as follows: $(CA=AC)_1$. Triangle $ABC$ will transform into triangle $(AB)_1C_1$ (Fig. 5).

Figure 5.

Definition 5

A figure is symmetrical with respect to point $O$ if every symmetrical point of this figure is contained in the same figure (Fig. 6).

Figure 6.

Figure $6$ shows a parallelogram. It has central symmetry about the point of intersection of its diagonals.

Example task.

Example 3

Let us be given a segment $AB$. Construct its symmetry with respect to the line $l$, which does not intersect the given segment, and with respect to the point $C$ lying on the line $l$.

Solution.

Let us schematically depict the condition of the problem.

Figure 7.

Let us first depict axial symmetry with respect to straight line $l$. Since axial symmetry is a movement, then by Theorem $1$, the segment $AB$ will be mapped onto the segment $A"B"$ equal to it. To construct it, we will do the following: draw straight lines $m\ and\n$ through points $A\ and\B$, perpendicular to straight line $l$. Let $m\cap l=X,\ n\cap l=Y$. Next we draw the segments $A"X=AX$ and $B"Y=BY$.

Figure 8.

Let us now depict the central symmetry with respect to the point $C$. Since central symmetry is a motion, then by Theorem $1$, the segment $AB$ will be mapped onto the segment $A""B""$ equal to it. To construct it, we will do the following: draw the lines $AC\ and\ BC$. Next we draw the segments $A^("")C=AC$ and $B^("")C=BC$.

Figure 9.

Goals:

• educational:
  • give an idea of ​​symmetry;
  • introduce the main types of symmetry on the plane and in space;
  • develop strong skills in constructing symmetrical figures;
  • expand your understanding of famous figures by introducing properties associated with symmetry;
  • show the possibilities of using symmetry in solving various problems;
  • consolidate acquired knowledge;
• general education:
  • teach yourself how to prepare yourself for work;
  • teach how to control yourself and your desk neighbor;
  • teach to evaluate yourself and your desk neighbor;
• developing:
  • intensify independent activity;
  • develop cognitive activity;
  • learn to summarize and systematize the information received;
• educational:
  • develop a “shoulder sense” in students;
  • cultivate communication skills;
  • instill a culture of communication.

DURING THE CLASSES

In front of each person are scissors and a sheet of paper.

Exercise 1(3 min).

- Let's take a sheet of paper, fold it into pieces and cut out some figure. Now let's unfold the sheet and look at the fold line.

Question: What function does this line serve?

Suggested answer: This line divides the figure in half.

Question: How are all the points of the figure located on the two resulting halves?

Suggested answer: All points of the halves are at an equal distance from the fold line and at the same level.

– This means that the fold line divides the figure in half so that 1 half is a copy of 2 halves, i.e. this line is not simple, it has a remarkable property (all points relative to it are at the same distance), this line is an axis of symmetry.

Task 2 (2 minutes).

– Cut out a snowflake, find the axis of symmetry, characterize it.

Task 3 (5 minutes).

– Draw a circle in your notebook.

Question: Determine how the axis of symmetry goes?

Suggested answer: Differently.

Question: So how many axes of symmetry does a circle have?

Suggested answer: A lot of.

– That’s right, a circle has many axes of symmetry. An equally remarkable figure is a ball (spatial figure)

Question: What other figures have more than one axis of symmetry?

Suggested answer: Square, rectangle, isosceles and equilateral triangles.

– Consider three-dimensional figures: cube, pyramid, cone, cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry do the square, rectangle, equilateral triangle and the proposed three-dimensional figures have?

I distribute halves of plasticine figures to students.

Task 4 (3 min).

– Using the information received, complete the missing part of the figure.

Note: the figure can be both planar and three-dimensional. It is important that students determine how the axis of symmetry runs and complete the missing element. The correctness of the work is determined by the neighbor at the desk and evaluates how correctly the work was done.

A line (closed, open, with self-intersection, without self-intersection) is laid out from a lace of the same color on the desktop.

Task 5 (group work 5 min).

– Visually determine the axis of symmetry and, relative to it, complete the second part from a lace of a different color.

The correctness of the work performed is determined by the students themselves.

Elements of drawings are presented to students

Task 6 (2 minutes).

– Find the symmetrical parts of these drawings.

To consolidate the material covered, I suggest the following tasks, scheduled for 15 minutes:

Name all equal elements of the triangle KOR and KOM. What type of triangles are these?

2. Draw several isosceles triangles in your notebook with a common base of 6 cm.

3. Draw a segment AB. Construct a line segment AB perpendicular and passing through its midpoint. Mark points C and D on it so that the quadrilateral ACBD is symmetrical with respect to the straight line AB.

– Our initial ideas about form date back to the very distant era of the ancient Stone Age - the Paleolithic. For hundreds of thousands of years of this period, people lived in caves, in conditions little different from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and during the late Paleolithic era they embellished their existence by creating works of art, figurines and drawings that reveal a remarkable sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity entered a new Stone Age, the Neolithic.
Neolithic man had a keen sense of geometric form. Firing and painting clay vessels, making reed mats, baskets, fabrics, and later metal processing developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
– Where does symmetry occur in nature?

Suggested answer: wings of butterflies, beetles, tree leaves...

– Symmetry can also be observed in architecture. When constructing buildings, builders strictly adhere to symmetry.

That's why the buildings turn out so beautiful. Also an example of symmetry is humans and animals.

Homework:

1. Come up with your own ornament, draw it on an A4 sheet (you can draw it in the form of a carpet).
2. Draw butterflies, note where elements of symmetry are present.

Scientific and practical conference

Municipal educational institution "Secondary school No. 23"

city ​​of Vologda

section: natural science

design and research work

TYPES OF SYMMETRY

The work was completed by an 8th grade student

Kreneva Margarita

Head: higher mathematics teacher

year 2014

Project structure:

1. Introduction.

2. Goals and objectives of the project.

3. Types of symmetry:

3.1. Central symmetry;

3.2. Axial symmetry;

3.3. Mirror symmetry (symmetry about a plane);

3.4. Rotational symmetry;

3.5. Portable symmetry.

4. Conclusions.

Symmetry is the idea through which man has tried for centuries to comprehend and create order, beauty and perfection.

G. Weil

Introduction.

The topic of my work was chosen after studying the section “Axial and central symmetry” in the course “8th grade Geometry”. I was very interested in this topic. I wanted to know: what types of symmetry exist, how they differ from each other, what are the principles for constructing symmetrical figures in each type.

Goal of the work : Introduction to different types of symmetry.

Tasks:

Study the literature on this issue.

Summarize and systematize the studied material.

Prepare a presentation.

In ancient times, the word “SYMMETRY” was used to mean “harmony”, “beauty”. Translated from Greek, this word means “proportionality, proportionality, sameness in the arrangement of parts of something on opposite sides of a point, straight line or plane.

There are two groups of symmetries.

The first group includes symmetry of positions, shapes, structures. This is the symmetry that can be directly seen. It can be called geometric symmetry.

The second group characterizes the symmetry of physical phenomena and laws of nature. This symmetry lies at the very basis of the natural scientific picture of the world: it can be called physical symmetry.

I'll stop studyinggeometric symmetry .

In turn, there are also several types of geometric symmetry: central, axial, mirror (symmetry relative to the plane), radial (or rotary), portable and others. Today I will look at 5 types of symmetry.

Central symmetry

Two points A and A 1 are called symmetrical with respect to point O if they lie on a straight line passing through point O and are on opposite sides of it at the same distance. Point O is called the center of symmetry.

The figure is said to be symmetrical about the pointABOUT , if for each point of the figure there is a point symmetrical to it relative to the pointABOUT also belongs to this figure. DotABOUT called the center of symmetry of a figure, the figure is said to have central symmetry.

Examples of figures with central symmetry are a circle and a parallelogram.

The figures shown on the slide are symmetrical relative to a certain point

2. Axial symmetry

Two pointsX And Y are called symmetrical about a straight linet , if this line passes through the middle of the segment XY and is perpendicular to it. It should also be said that each point is a straight linet is considered symmetrical to itself.

Straightt – axis of symmetry.

The figure is said to be symmetrical about a straight linet, if for each point of the figure there is a point symmetrical to it relative to the straight linet also belongs to this figure.

Straighttcalled the axis of symmetry of a figure, the figure is said to have axial symmetry.

An undeveloped angle, isosceles and equilateral triangles, a rectangle and a rhombus have axial symmetry.letters (see presentation).

Mirror symmetry (symmetry about a plane)

Two points P 1 And P are called symmetrical relative to the plane a if they lie on a straight line perpendicular to the plane a and are at the same distance from it

Mirror symmetry well known to every person. It connects any object and its reflection in a flat mirror. They say that one figure is mirror symmetrical to another.

On a plane, a figure with countless axes of symmetry was a circle. In space, a ball has countless planes of symmetry.

But if a circle is one of a kind, then in the three-dimensional world there is a whole series of bodies with an infinite number of planes of symmetry: a straight cylinder with a circle at the base, a cone with a circular base, a ball.

It is easy to establish that every symmetrical plane figure can be aligned with itself using a mirror. It is surprising that such complex figures as a five-pointed star or an equilateral pentagon are also symmetrical. As this follows from the number of axes, they are distinguished by high symmetry. And vice versa: it is not so easy to understand why such a seemingly regular figure, like an oblique parallelogram, is asymmetrical.

4. P rotational symmetry (or radial symmetry)

Rotational symmetry - this is symmetry, the preservation of the shape of an objectwhen rotating around a certain axis through an angle equal to 360°/n(or a multiple of this value), wheren= 2, 3, 4, … The indicated axis is called the rotary axisn-th order.

Atn=2 all points of the figure are rotated through an angle of 180 0 ( 360 0 /2 = 180 0 ) around the axis, while the shape of the figure is preserved, i.e. each point of the figure goes to a point of the same figure (the figure transforms into itself). The axis is called the second-order axis.

Figure 2 shows a third-order axis, Figure 3 - 4th order, Figure 4 - 5th order.

An object can have more than one rotation axis: Fig. 1 - 3 axes of rotation, Fig. 2 - 4 axes, Fig. 3 - 5 axes, Fig. 4 – only 1 axis

The well-known letters “I” and “F” have rotational symmetry. If you rotate the letter “I” 180° around an axis perpendicular to the plane of the letter and passing through its center, the letter will align with itself. In other words, the letter “I” is symmetrical with respect to a rotation of 180°, 180°= 360°: 2,n=2, which means it has second-order symmetry.

Note that the letter “F” also has second-order rotational symmetry.

In addition, the letter has a center of symmetry, and the letter F has an axis of symmetry

Let's return to examples from life: a glass, a cone-shaped pound of ice cream, a piece of wire, a pipe.

If we take a closer look at these bodies, we will notice that all of them, in one way or another, consist of a circle, through an infinite number of symmetry axes there are countless symmetry planes. Most of these bodies (they are called bodies of rotation) also have, of course, a center of symmetry (the center of a circle), through which at least one rotational axis of symmetry passes.

For example, the axis of the ice cream cone is clearly visible. It runs from the middle of the circle (sticking out of the ice cream!) to the sharp end of the funnel cone. We perceive the totality of symmetry elements of a body as a kind of symmetry measure. The ball, without a doubt, in terms of symmetry, is an unsurpassed embodiment of perfection, an ideal. The ancient Greeks perceived it as the most perfect body, and the circle, naturally, as the most perfect flat figure.

To describe the symmetry of a particular object, it is necessary to indicate all the rotation axes and their order, as well as all planes of symmetry.

Consider, for example, a geometric body composed of two identical regular quadrangular pyramids.

It has one rotary axis of the 4th order (axis AB), four rotary axes of the 2nd order (axes CE,DF, MP, NQ), five planes of symmetry (planesCDEF, AFBD, ACBE, AMBP, ANBQ).

5 . Portable symmetry

Another type of symmetry isportable With symmetry.

Such symmetry is spoken of when, when moving a figure along a straight line to some distance “a” or a distance that is a multiple of this value, it coincides with itself The straight line along which the transfer occurs is called the transfer axis, and the distance “a” is called the elementary transfer, period or symmetry step.

A

A periodically repeating pattern on a long strip is called a border. In practice, borders are found in various forms (wall painting, cast iron, plaster bas-reliefs or ceramics). Borders are used by painters and artists when decorating a room. To make these ornaments, a stencil is made. We move the stencil, turning it over or not, tracing the outline, repeating the pattern, and we get an ornament (visual demonstration).

The border is easy to build using a stencil (the starting element), moving or turning it over and repeating the pattern. The figure shows five types of stencils:A ) asymmetrical;b, c ) having one axis of symmetry: horizontal or vertical;G ) centrally symmetrical;d ) having two axes of symmetry: vertical and horizontal.

To construct borders, the following transformations are used:

A ) parallel transfer;b ) symmetry about the vertical axis;V ) central symmetry;G ) symmetry about the horizontal axis.

You can build sockets in the same way. To do this, the circle is divided inton equal sectors, in one of them a sample pattern is made and then the latter is sequentially repeated in the remaining parts of the circle, rotating the pattern each time by an angle of 360°/n .

A clear example of the use of axial and portable symmetry is the fence shown in the photograph.

Conclusion: Thus, there are different types of symmetry, symmetrical points in each of these types of symmetry are constructed according to certain laws. In life, we encounter one type of symmetry everywhere, and often in the objects that surround us, several types of symmetry can be noted at once. This creates order, beauty and perfection in the world around us.

LITERATURE:

Handbook of Elementary Mathematics. M.Ya. Vygodsky. – Publishing house “Nauka”. – Moscow 1971 – 416 pages.

Modern dictionary of foreign words. - M.: Russian language, 1993.

History of mathematics in schoolIX - Xclasses. G.I. Glaser. – Publishing house “Prosveshcheniye”. – Moscow 1983 – 351 pages.

Visual geometry 5th – 6th grades. I.F. Sharygin, L.N. Erganzhieva. – Publishing house “Drofa”, Moscow 2005. – 189 pages

Encyclopedia for children. Biology. S. Ismailova. – Avanta+ Publishing House. – Moscow 1997 – 704 pages.

Urmantsev Yu.A. Symmetry of nature and the nature of symmetry - M.: Mysl arxitekt / arhkomp2. htm, , ru.wikipedia.org/wiki/

For centuries, symmetry has remained a subject that has fascinated philosophers, astronomers, mathematicians, artists, architects and physicists. The ancient Greeks were completely obsessed with it - and even today we tend to encounter symmetry in everything from furniture arrangement to haircuts.

Just keep in mind that once you realize this, you'll probably feel an overwhelming urge to look for symmetry in everything you see.

(Total 10 photos)

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1. Broccoli Romanesco

Perhaps you saw Romanesco broccoli in the store and thought it was another example of a genetically modified product. But in fact, this is another example of the fractal symmetry of nature. Each broccoli floret has a logarithmic spiral pattern. Romanesco is similar in appearance to broccoli, and in taste and consistency - to cauliflower. It is rich in carotenoids, as well as vitamins C and K, which makes it not only beautiful, but also healthy food.

For thousands of years, people have marveled at the perfect hexagonal shape of honeycombs and asked themselves how bees could instinctively create a shape that humans could only reproduce with a compass and ruler. How and why do bees have a passion for creating hexagons? Mathematicians believe this is an ideal shape that allows them to store the maximum amount of honey possible using the minimum amount of wax. Either way, it's all a product of nature, and it's damn impressive.

3. Sunflowers

Sunflowers boast radial symmetry and an interesting type of symmetry known as the Fibonacci sequence. Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, etc. (each number is determined by the sum of the two previous numbers). If we took our time and counted the number of seeds in a sunflower, we would find that the number of spirals grows according to the principles of the Fibonacci sequence. There are many plants in nature (including Romanesco broccoli) whose petals, seeds and leaves correspond to this sequence, which is why it is so difficult to find a clover with four leaves.

But why do sunflowers and other plants follow mathematical rules? Like the hexagons in a hive, it's all a matter of efficiency.

4. Nautilus Shell

In addition to plants, some animals, such as the Nautilus, follow the Fibonacci sequence. The shell of the Nautilus twists into a Fibonacci spiral. The shell tries to maintain the same proportional shape, which allows it to maintain it throughout its life (unlike humans, who change proportions throughout life). Not all Nautiluses have a Fibonacci shell, but they all follow a logarithmic spiral.

Before you envy the math clams, remember that they don’t do this on purpose, it’s just that this form is the most rational for them.

5. Animals

Most animals have bilateral symmetry, which means they can be split into two identical halves. Even humans have bilateral symmetry, and some scientists believe that a person's symmetry is the most important factor that influences the perception of our beauty. In other words, if you have a one-sided face, you can only hope that it is compensated by other good qualities.

Some go to complete symmetry in an effort to attract a mate, such as the peacock. Darwin was positively annoyed by the bird, and wrote in a letter that "The sight of the tail feathers of a peacock, whenever I look at it, makes me sick!" To Darwin, the tail seemed cumbersome and made no evolutionary sense, as it did not fit with his theory of “survival of the fittest.” He was furious until he came up with the theory of sexual selection, which states that animals evolve certain features to increase their chances of mating. Therefore, peacocks have various adaptations to attract a partner.

There are about 5,000 types of spiders, and they all create a nearly perfect circular web with radial supporting threads at nearly equal distances and spiral webs for catching prey. Scientists aren't sure why spiders like geometry so much, as tests have shown that a round web doesn't lure food any better than an irregularly shaped web. Scientists theorize that radial symmetry evenly distributes the impact force when prey is caught in the net, resulting in fewer breaks.

Give a couple of tricksters a board, mowers, and the safety of darkness, and you'll see that people create symmetrical shapes, too. Due to the complexity of the design and incredible symmetry of crop circles, even after the creators of the circles confessed and demonstrated their skills, many people still believe that they were made by space aliens.

As the circles become more complex, their artificial origin becomes increasingly clear. It's illogical to assume that aliens will make their messages increasingly difficult when we couldn't even decipher the first ones.

Regardless of how they came to be, crop circles are a pleasure to look at, mainly because their geometry is impressive.

Even tiny formations like snowflakes are governed by the laws of symmetry, since most snowflakes have hexagonal symmetry. This occurs in part because of the way water molecules line up when they solidify (crystallize). Water molecules become solid by forming weak hydrogen bonds, they align in an orderly arrangement that balances the forces of attraction and repulsion, forming the hexagonal shape of a snowflake. But at the same time, each snowflake is symmetrical, but not one snowflake is similar to the other. This happens because as each snowflake falls from the sky, it experiences unique atmospheric conditions that cause its crystals to arrange themselves in a certain way.

9. Milky Way Galaxy

As we have already seen, symmetry and mathematical models exist almost everywhere, but are these laws of nature limited to our planet? Obviously not. A new section at the edge of the Milky Way Galaxy has recently been discovered, and astronomers believe that the galaxy is an almost perfect mirror image of itself.

10. Sun-Moon Symmetry

Considering that the Sun has a diameter of 1.4 million km and the Moon is 3,474 km in diameter, it seems almost impossible that the Moon can block sunlight and provide us with about five solar eclipses every two years. How does this work? Coincidentally, while the Sun is about 400 times wider than the Moon, the Sun is also 400 times farther away. Symmetry ensures that the Sun and Moon are the same size when viewed from Earth, so the Moon can obscure the Sun. Of course, the distance from the Earth to the Sun can increase, which is why we sometimes see annular and partial eclipses. But every one to two years, a precise alignment occurs and we witness a spectacular event known as a total solar eclipse. Astronomers don't know how common this symmetry is among other planets, but they think it's quite rare. However, we should not assume that we are special, as it is all a matter of chance. For example, every year the Moon moves about 4 cm away from the Earth, meaning that billions of years ago every solar eclipse would have been a total eclipse. If things continue like this, total eclipses will eventually disappear, and this will be accompanied by the disappearance of annular eclipses. It turns out that we are simply in the right place at the right time to see this phenomenon.