How to draw axial and central symmetry. How to draw a symmetrical object

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

© Sukhacheva Elena Vladimirovna, 2008-2009.

Axial symmetry and the concept of perfection

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

to comprehend the meaning of perfection. This concept was first substantiated by artists, philosophers and mathematicians of Ancient Greece. And the word “symmetry” itself was invented by them. It denotes proportionality, harmony and identity of the parts of the whole. The ancient Greek thinker Plato argued that only an object that is symmetrical and proportionate can be beautiful. Indeed, those phenomena and forms that are proportional and complete “please the eye.” We call them correct.

Axial symmetry as a concept

Symmetry in the world of living beings is manifested in the regular arrangement of identical parts of the body relative to the center or axis. More often in

Axial symmetry occurs in nature. It determines not only the general structure of the organism, but also the possibilities of its subsequent development. The geometric shapes and proportions of living beings are formed by “axial symmetry”. Its definition is formulated as follows: this is the property of objects to be combined under various transformations. The ancients believed that the sphere possesses the principle of symmetry to the fullest extent. They considered this form harmonious and perfect.

Axial symmetry in living nature

If you look at any living creature, the symmetry of the structure of the body immediately catches your eye. Human: two arms, two legs, two eyes, two ears and so on. Each animal species has a characteristic color. If a pattern appears in the coloring, then, as a rule, it is mirrored on both sides. This means that there is a certain line along which animals and people can be visually divided into two identical halves, that is, their geometric structure is based on axial symmetry. Nature creates any living organism not chaotically and senselessly, but according to the general laws of the world order, because nothing in the Universe has a purely aesthetic, decorative purpose. The presence of various forms is also due to natural necessity.

Axial symmetry in inanimate nature

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

In this lesson we will look at another characteristic of some figures - axial and central symmetry. We encounter axial symmetry every day when we look in the mirror. Central symmetry is very common in living nature. At the same time, figures that have symmetry have a number of properties. In addition, we subsequently learn that axial and central symmetries are types of movements with the help of which a whole class of problems is solved.

This lesson is devoted to axial and central symmetry.

Definition

The two points are called symmetrical relatively straight if:

In Fig. 1 shows examples of points symmetrical with respect to a straight line and , and .

Rice. 1

Let us also note the fact that any point on a line is symmetrical to itself relative to this line.

Figures can also be symmetrical relative to a straight line.

Let us formulate a strict definition.

Definition

The figure is called symmetrical relative to straight, if for each point of the figure the point symmetrical to it relative to this straight line also belongs to the figure. In this case the line is called axis of symmetry. The figure has axial symmetry.

Let's look at a few examples of figures that have axial symmetry and their axes of symmetry.

Example 1

The angle has axial symmetry. The axis of symmetry of the angle is the bisector. Indeed: let’s lower a perpendicular to the bisector from any point of the angle and extend it until it intersects with the other side of the angle (see Fig. 2).

Rice. 2

(since - the common side, (property of a bisector), and triangles are right-angled). Means, . Therefore, the points are symmetrical with respect to the bisector of the angle.

It follows from this that an isosceles triangle also has axial symmetry with respect to the bisector (altitude, median) drawn to the base.

Example 2

An equilateral triangle has three axes of symmetry (bisectors/medians/altitudes of each of the three angles (see Fig. 3).

Rice. 3

Example 3

A rectangle has two axes of symmetry, each of which passes through the midpoints of its two opposite sides (see Fig. 4).

Rice. 4

Example 4

A rhombus also has two axes of symmetry: straight lines that contain its diagonals (see Fig. 5).

Rice. 5

Example 5

A square, which is both a rhombus and a rectangle, has 4 axes of symmetry (see Fig. 6).

Rice. 6

Example 6

For a circle, the axis of symmetry is any straight line passing through its center (that is, containing the diameter of the circle). Therefore, a circle has infinitely many axes of symmetry (see Fig. 7).

Rice. 7

Let us now consider the concept central symmetry.

Definition

The points are called symmetrical relative to the point if: - the middle of the segment.

Let's look at a few examples: in Fig. 8 shows the points and , as well as and , which are symmetrical with respect to the point , and the points and are not symmetrical with respect to this point.

Rice. 8

Some figures are symmetrical about a certain point. Let us formulate a strict definition.

Definition

The figure is called symmetrical about the point, if for any point of the figure the point symmetrical to it also belongs to this figure. The point is called center of symmetry, and the figure has central symmetry.

Let's look at examples of figures with central symmetry.

Example 7

For a circle, the center of symmetry is the center of the circle (this is easy to prove by recalling the properties of the diameter and radius of a circle) (see Fig. 9).

Rice. 9

Example 8

For a parallelogram, the center of symmetry is the point of intersection of the diagonals (see Fig. 10).

Rice. 10

Let's solve several problems on axial and central symmetry.

Task 1.

How many axes of symmetry does the segment have?

A segment has two axes of symmetry. The first of them is a line containing a segment (since any point on a line is symmetrical to itself relative to this line). The second is the perpendicular bisector to the segment, that is, a straight line perpendicular to the segment and passing through its middle.

Answer: 2 axes of symmetry.

Task 2.

How many axes of symmetry does a straight line have?

A straight line has infinitely many axes of symmetry. One of them is the line itself (since any point on the line is symmetrical to itself relative to this line). And also the axes of symmetry are any lines perpendicular to a given line.

Answer: there are infinitely many axes of symmetry.

Task 3.

How many axes of symmetry does the beam have?

The ray has one axis of symmetry, which coincides with the line containing the ray (since any point on the line is symmetrical to itself relative to this line).

Answer: one axis of symmetry.

Task 4.

Prove that the lines containing the diagonals of a rhombus are its axes of symmetry.

Proof:

Consider a rhombus. Let us prove, for example, that the straight line is its axis of symmetry. It is obvious that the points are symmetrical to themselves, since they lie on this line. In addition, the points and are symmetrical with respect to this line, since . Let us now choose an arbitrary point and prove that the point symmetric with respect to it also belongs to the rhombus (see Fig. 11).

Rice. eleven

Draw a perpendicular to the line through the point and extend it until it intersects with . Consider triangles and . These triangles are right-angled (by construction), in addition, they have: - a common leg, and (since the diagonals of a rhombus are its bisectors). So these triangles are equal: . This means that all their corresponding elements are equal, therefore: . From the equality of these segments it follows that the points and are symmetrical with respect to the straight line. This means that it is the axis of symmetry of the rhombus. This fact can be proven similarly for the second diagonal.

Proven.

Task 5.

Prove that the point of intersection of the diagonals of a parallelogram is its center of symmetry.

Proof:

Consider a parallelogram. Let us prove that the point is its center of symmetry. It is obvious that the points and , and are pairwise symmetrical with respect to the point , since the diagonals of a parallelogram are divided in half by the point of intersection. Let us now choose an arbitrary point and prove that the point symmetric with respect to it also belongs to the parallelogram (see Fig. 12).

« Symmetry" - a word of Greek origin. It means proportionality, the presence of a certain order, patterns in the arrangement of parts.

Since ancient times, people have used symmetry in drawings, ornaments, and household items.
Symmetry is widespread in nature. It can be observed in the form of leaves and flowers of plants, in the arrangement of various organs of animals, in the form of crystalline bodies, in a fluttering butterfly, a mysterious snowflake, a mosaic in a temple, a starfish.
Symmetry is widely used in practice, in construction and technology. This is strict symmetry in the form of ancient buildings, harmonious ancient Greek vases, the Kremlin building, cars, airplanes and much more. (slide 4) Examples of using symmetry are parquet and borders. (see hyperlink on the use of symmetry in borders and parquet floors) Let's look at several examples where you can see symmetry in various objects using a slide show (include icon).

Definition: – is symmetry about a point.
Definition: Points A and B are symmetrical about some point O if point O is the midpoint of segment AB.
Definition: Point O is called the center of symmetry of the figure, and the figure is called centrally symmetrical.
Property: Figures that are symmetrical about a certain point are equal.
Examples:

Algorithm for constructing a centrally symmetrical figure
1. Let’s construct a triangle A 1B 1 C 1, symmetrical to the triangle ABC, relative to the center (point) O. To do this, connect the points A, B, C with the center O and continue these segments;
2. Measure the segments AO, BO, CO and lay off on the other side of point O, segments equal to them (AO=A 1 O 1, BO=B 1 O 1, CO=C 1 O 1);

3. Connect the resulting points with segments A 1 B 1; A 1 C 1; B1 C 1.
We got ∆A 1 B 1 C 1 symmetrical ∆ABC.

– this is symmetry about the drawn axis (straight line).
Definition: Points A and B are symmetrical about a certain line a if these points lie on a line perpendicular to this one and at the same distance.
Definition: An axis of symmetry is a straight line when bent along which the “halves” coincide, and a figure is called symmetrical about a certain axis.
Property: Two symmetrical figures are equal.
Examples:

Algorithm for constructing a figure symmetrical with respect to some straight line
Let's construct a triangle A1B1C1, symmetrical to triangle ABC with respect to straight line a.
For this:
1. Let us draw straight lines from the vertices of triangle ABC perpendicular to straight line a and continue them further.
2. Measure the distances from the vertices of the triangle to the resulting points on the straight line and plot the same distances on the other side of the straight line.
3. Connect the resulting points with segments A1B1, B1C1, B1C1.

We obtained ∆A1B1C1 symmetrical ∆ABC.

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 won'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.