If you think for a minute and imagine any object in your mind, then in 99% of cases the figure that comes to mind will be of the correct shape. Only 1% of people, or rather their imagination, will draw an intricate object that looks completely wrong or disproportionate. This is rather an exception to the rule and refers to unconventionally thinking individuals with a special view of things. But returning to the absolute majority, it is worth saying that a significant proportion of correct items still prevails. The article will talk exclusively about them, namely about symmetrical drawing of them.

Drawing the right objects: just a few steps to the finished drawing

Before you start drawing a symmetrical object, you need to select it. In our version it will be a vase, but even if it doesn’t in any way resemble what you decided to depict, don’t despair: all the steps are absolutely identical. Follow the sequence and everything will work out:

1. All objects of regular shape have a so-called central axis, which should definitely be highlighted when drawing symmetrically. To do this, you can even use a ruler and draw a straight line down the center of the landscape sheet.
2. Next, look carefully at the item you have chosen and try to transfer its proportions onto a sheet of paper. This is not difficult to do if you mark light strokes on both sides of the line drawn in advance, which will later become the outlines of the object being drawn. In the case of a vase, it is necessary to highlight the neck, bottom and the widest part of the body.
3. Do not forget that symmetrical drawing does not tolerate inaccuracies, so if there are some doubts about the intended strokes, or you are not sure of the correctness of your own eye, double-check the laid down distances with a ruler.
4. The last step is connecting all the lines together.

Symmetrical drawing is available to computer users

Due to the fact that most of the objects around us have the correct proportions, in other words, they are symmetrical, computer application developers have created programs in which you can easily draw absolutely everything. You just need to download them and enjoy the creative process. However, remember, a machine will never be a substitute for a sharpened pencil and a sketchbook.

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Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested in this work, please download the full version.

Lesson type: combined.

Lesson objectives:

• Consider axial, central and mirror symmetries as properties of some geometric figures.
• Teach to construct symmetrical points and recognize figures with axial symmetry and central symmetry.
• Improve problem solving skills.

Lesson objectives:

• Formation of spatial representations of students.
• Developing the ability to observe and reason; developing interest in the subject through the use of information technology.
• Raising a person who knows how to appreciate beauty.

Lesson equipment:

• Use of information technology (presentation).
• Drawings.
• Homework cards.

During the classes

I. Organizational moment.

Inform the topic of the lesson, formulate the objectives of the lesson.

II. Introduction.

What is symmetry?

The outstanding mathematician Hermann Weyl highly appreciated the role of symmetry in modern science: “Symmetry, no matter how broadly or narrowly we understand this word, is an idea with the help of which man tried to explain and create order, beauty and perfection.”

We live in a very beautiful and harmonious world. We are surrounded by objects that please the eye. For example, a butterfly, a maple leaf, a snowflake. Look how beautiful they are. Have you paid attention to them? Today we will touch on this wonderful mathematical phenomenon - symmetry. Let's get acquainted with the concept of axial, central and mirror symmetries. We will learn to build and identify figures that are symmetrical relative to the axis, center and plane.

The word “symmetry” translated from Greek sounds like “harmony”, meaning beauty, proportionality, proportionality, uniformity in the arrangement of parts. Man has long used symmetry in architecture. It gives harmony and completeness to ancient temples, towers of medieval castles, and modern buildings.

In the most general form, “symmetry” in mathematics is understood as such a transformation of space (plane), in which each point M goes to another point M" relative to some plane (or line) a, when the segment MM" is perpendicular to the plane (or line) a and divides it in half. The plane (straight line) a is called the plane (or axis) of symmetry. The fundamental concepts of symmetry include plane of symmetry, axis of symmetry, center of symmetry. A plane of symmetry P is a plane that divides a figure into two mirror-like equal parts, located relative to each other in the same way as an object and its mirror image.

III. Main part. Types of symmetry.

Central symmetry

Symmetry about a point or central symmetry is a property of a geometric figure when any point located on one side of the center of symmetry corresponds to another point located on the other side of the center. In this case, the points are located on a straight line segment passing through the center, dividing the segment in half.

Practical task.

1. Points are given A, IN And M M relative to the middle of the segment AB.
2. Which of the following letters have a center of symmetry: A, O, M, X, K?
3. Do they have a center of symmetry: a) a segment; b) beam; c) a pair of intersecting lines; d) square?

Axial symmetry

Symmetry about a line (or axial symmetry) is a property of a geometric figure when any point located on one side of the line will always correspond to a point located on the other side of the line, and the segments connecting these points will be perpendicular to the axis of symmetry and divided by it in half.

Practical task.

1. Given two points A And IN, symmetrical with respect to some line, and a point M. Construct a point symmetrical to the point M relative to the same line.
2. Which of the following letters have an axis of symmetry: A, B, D, E, O?
3. How many axes of symmetry does: a) a segment have? b) straight; c) beam?
4. How many axes of symmetry does the drawing have? (see Fig. 1)

Mirror symmetry

Points A And IN are called symmetrical relative to the plane α (plane of symmetry) if the plane α passes through the middle of the segment AB and perpendicular to this segment. Each point of the α plane is considered symmetrical to itself.

Practical task.

1. Find the coordinates of the points to which points A (0; 1; 2), B (3; -1; 4), C (1; 0; -2) go with: a) central symmetry relative to the origin; b) axial symmetry relative to the coordinate axes; c) mirror symmetry relative to coordinate planes.
2. Does the right glove go into the right or left glove in mirror symmetry? axial symmetry? central symmetry?
3. The figure shows how the number 4 is reflected in two mirrors. What will be visible in place of the question mark if the same is done with the number 5? (see Fig. 2)
4. The picture shows how the word KANGAROO is reflected in two mirrors. What happens if you do the same with the number 2011? (see Fig. 3)

Rice. 2

This is interesting.

Symmetry in living nature.

Almost all living beings are built according to the laws of symmetry; it is not for nothing that the word “symmetry” means “proportionality” when translated from Greek.

Among flowers, for example, there is rotational symmetry. Many flowers can be rotated so that each petal takes the position of its neighbor, the flower aligns with itself. The minimum angle of such rotation is not the same for different colors. For the iris it is 120°, for the bellflower – 72°, for the narcissus – 60°.

There is helical symmetry in the arrangement of leaves on plant stems. Positioned like a screw along the stem, the leaves seem to spread out in different directions and do not obscure each other from the light, although the leaves themselves also have an axis of symmetry. Considering the general plan of the structure of any animal, we usually notice a certain regularity in the arrangement of body parts or organs, which are repeated around a certain axis or occupy the same position in relation to a certain plane. This regularity is called body symmetry. The phenomena of symmetry are so widespread in the animal world that it is very difficult to indicate a group in which no symmetry of the body can be noticed. Both small insects and large animals have symmetry.

Symmetry in inanimate nature.

Among the infinite variety of forms of inanimate nature, such perfect images are found in abundance, whose appearance invariably attracts our attention. Observing the beauty of nature, you can notice that when objects are reflected in puddles and lakes, mirror symmetry appears (see Fig. 4).

Crystals bring the charm of symmetry to the world of inanimate nature. Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have rotational symmetry and, in addition, mirror symmetry.

One cannot help but see symmetry in faceted gemstones. Many cutters try to give diamonds the shape of a tetrahedron, cube, octahedron or icosahedron. Since the garnet has the same elements as the cube, it is highly prized by gemstone connoisseurs. Artistic items made from garnets were discovered in the graves of Ancient Egypt dating back to the pre-dynastic period (over two millennia BC) (see Fig. 5).

In the Hermitage collections, gold jewelry of the ancient Scythians receives special attention. The artistic work of gold wreaths, tiaras, wood and decorated with precious red-violet garnets is unusually fine.

One of the most obvious uses of the laws of symmetry in life is in architectural structures. This is what we see most often. In architecture, axes of symmetry are used as means of expressing architectural design (see Fig. 6). In most cases, patterns on carpets, fabrics, and indoor wallpaper are symmetrical about the axis or center.

Another example of a person using symmetry in his practice is technology. In engineering, symmetry axes are most clearly designated where it is necessary to estimate the deviation from the zero position, for example, on the steering wheel of a truck or on the steering wheel of a ship. Or one of the most important inventions of mankind that has a center of symmetry is the wheel; the propeller and other technical means also have a center of symmetry.

"Look in the mirror!"

Should we think that we only see ourselves in a “mirror image”? Or, at best, can we only find out from photos and films what we “really” look like? Of course not: it is enough to reflect the mirror image a second time in the mirror to see your true face. Trellis come to the rescue. They have one large main mirror in the center and two smaller mirrors on the sides. If you place such a side mirror at right angles to the middle one, then you can see yourself exactly in the form in which others see you. Close your left eye, and your reflection in the second mirror will repeat your movement with your left eye. Before the trellis, you can choose whether you want to see yourself in a mirror image or in a direct image.

It is easy to imagine what kind of confusion would reign on Earth if the symmetry in nature were broken!

Rice. 4	Rice. 5	Rice. 6

IV. Physical education minute.

• « Lazy Eights» – activate structures that ensure memorization, increase stability of attention.
  Draw the number eight in the air in a horizontal plane three times, first with one hand, then with both hands at once.
• « Symmetrical drawings » – improve hand-eye coordination and facilitate the writing process.
  Draw symmetrical patterns in the air with both hands.

V. Independent testing work.

Ι option

ΙΙ option

1. In the rectangle MPKH O is the point of intersection of the diagonals, RA and BH are perpendiculars drawn from the vertices P and H to the straight line MK. It is known that MA = OB. Find the angle POM.
2. In the rhombus MPKH the diagonals intersect at the point ABOUT. On the sides MK, KH, PH points A, B, C are taken, respectively, AK = KV = RS. Prove that OA = OB and find the sum of the angles POC and MOA.
3. Construct a square along the given diagonal so that the two opposite vertices of this square lie on opposite sides of the given acute angle.

VI. Summing up the lesson. Assessment.

• What types of symmetry did you learn about in class?
• Which two points are called symmetrical with respect to a given line?
• Which figure is called symmetrical with respect to a given line?
• Which two points are said to be symmetrical about a given point?
• Which figure is called symmetrical about a given point?
• What is mirror symmetry?
• Give examples of figures that have: a) axial symmetry; b) central symmetry; c) both axial and central symmetry.
• Give examples of symmetry in living and inanimate nature.

VII. Homework.

1. Individual: complete the structure using axial symmetry (see Fig. 7).

Rice. 7

2. Construct a figure symmetrical to the given one with respect to: a) a point; b) straight (see Fig. 8, 9).

Rice. 8	Rice. 9

3. Creative task: “In the animal world.” Draw a representative from the animal world and show the axis of symmetry.

VIII. Reflection.

• What did you like about the lesson?
• What material was most interesting?
• What difficulties did you encounter when completing this or that task?
• What would you change during the lesson?

You will need

• - properties of symmetrical points;
• - properties of symmetrical figures;
• - ruler;
• - square;
• - compass;
• - pencil;
• - paper;
• - a computer with a graphics editor.

Instructions

Draw a straight line a, which will be the axis of symmetry. If its coordinates are not specified, draw it arbitrarily. Place an arbitrary point A on one side of this line. You need to find a symmetrical point.

Helpful advice

Symmetry properties are used constantly in AutoCAD. To do this, use the Mirror option. To construct an isosceles triangle or isosceles trapezoid, it is enough to draw the lower base and the angle between it and the side. Reflect them using the specified command and extend the sides to the required size. In the case of a triangle, this will be the point of their intersection, and for a trapezoid, this will be a given value.

You constantly come across symmetry in graphic editors when you use the “flip vertically/horizontally” option. In this case, the axis of symmetry is taken to be a straight line corresponding to one of the vertical or horizontal sides of the picture frame.

Sources:

• how to draw central symmetry

Constructing a cross section of a cone is not such a difficult task. The main thing is to follow a strict sequence of actions. Then this task will be easily accomplished and will not require much labor from you.

You will need

• - paper;
• - pen;
• - circle;
• - ruler.

Instructions

When answering this question, you must first decide what parameters define the section.
Let this be the straight line of intersection of the plane l with the plane and the point O, which is the intersection with its section.

The construction is illustrated in Fig. 1. The first step in constructing a section is through the center of the section of its diameter, extended to l perpendicular to this line. The result is point L. Next, draw a straight line LW through point O, and construct two guide cones lying in the main section O2M and O2C. At the intersection of these guides lie point Q, as well as the already shown point W. These are the first two points of the desired section.

Now draw a perpendicular MS at the base of the cone BB1 ​​and construct generatrices of the perpendicular section O2B and O2B1. In this section, through point O, draw a straight line RG parallel to BB1. Т.R and Т.G are two more points of the desired section. If the cross section of the ball were known, then it could be built already at this stage. However, this is not an ellipse at all, but something elliptical that has symmetry with respect to the segment QW. Therefore, you should build as many section points as possible in order to connect them later with a smooth curve to obtain the most reliable sketch.

Construct an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and construct the corresponding guides O2A and O2N. Through t.O, draw a straight line passing through PQ and WG until it intersects with the newly constructed guides at points P and E. These are two more points of the desired section. Continuing in the same way, you can find as many points as you want.

True, the procedure for obtaining them can be slightly simplified using symmetry with respect to QW. To do this, you can draw straight lines SS’ in the plane of the desired section, parallel to RG until they intersect with the surface of the cone. The construction is completed by rounding the constructed polyline from chords. It is enough to construct half of the desired section due to the already mentioned symmetry with respect to QW.

Video on the topic

Tip 3: How to graph a trigonometric function

You need to draw schedule trigonometric functions? Master the algorithm of actions using the example of constructing a sinusoid. To solve the problem, use the research method.

You will need

• - ruler;
• - pencil;
• - knowledge of the basics of trigonometry.

Instructions

Video on the topic

note

If the two semi-axes of a single-strip hyperboloid are equal, then the figure can be obtained by rotating a hyperbola with semi-axes, one of which is the above, and the other, different from the two equal ones, around the imaginary axis.

Helpful advice

When examining this figure relative to the Oxz and Oyz axes, it is clear that its main sections are hyperbolas. And when this spatial figure of rotation is cut by the Oxy plane, its section is an ellipse. The neck ellipse of a single-strip hyperboloid passes through the origin of coordinates, because z=0.

The throat ellipse is described by the equation x²/a² +y²/b²=1, and the other ellipses are composed by the equation x²/a² +y²/b²=1+h²/c².

Sources:

• Ellipsoids, paraboloids, hyperboloids. Rectilinear generators

The shape of a five-pointed star has been widely used by man since ancient times. We consider its shape beautiful because we unconsciously recognize in it the relationships of the golden section, i.e. the beauty of the five-pointed star is mathematically justified. Euclid was the first to describe the construction of a five-pointed star in his Elements. Let's join in with his experience.

You will need

• ruler;
• pencil;
• compass;
• protractor.

Instructions

The construction of a star comes down to the construction and subsequent connection of its vertices to each other sequentially through one. In order to build the correct one, you need to divide the circle into five.
Construct an arbitrary circle using a compass. Mark its center with point O.

Mark point A and use a ruler to draw line segment OA. Now you need to divide the segment OA in half; to do this, from point A, draw an arc of radius OA until it intersects the circle at two points M and N. Construct the segment MN. The point E where MN intersects OA will bisect segment OA.

Restore the perpendicular OD to the radius OA and connect points D and E. Make a notch B on OA from point E with radius ED.

Now, using line segment DB, mark the circle into five equal parts. Label the vertices of the regular pentagon sequentially with numbers from 1 to 5. Connect the dots in the following sequence: 1 with 3, 2 with 4, 3 with 5, 4 with 1, 5 with 2. Here is the regular five-pointed star, into a regular pentagon. This is exactly the way I built it

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

Mathematics	In algebra lessons we studied the graphs of the functions y=x and y=x The pictures show various pictures depicted using the branches of parabolas. (a) Octahedron, (b) rhombic dodecahedron, (c) hexagonal octahedron.
Russian language	The printed letters of the Russian alphabet also have different types of symmetries. There are “symmetrical” words in the Russian language - palindromes, which can be read equally in both directions.	A D L M P T F W– vertical axis V E Z K S E Y - horizontal axis F N O X- both vertical and horizontal B G I Y R U C CH SCHY- no axis Radar hut Alla Anna
Literature	Sentences can also be palindromic. Bryusov wrote a poem “The Voice of the Moon”, in which each line is a palindrome. Look at the quadruples by A.S. Pushkin “The Bronze Horseman”. If we draw a line after the second line we can notice elements of axial symmetry	And the rose fell on Azor's paw. I come with the sword of the judge. (Derzhavin) "Search for a taxi" "Argentina beckons the Negro" “The Argentine appreciates the black man,” “Lesha found a bug on the shelf.” The Neva is dressed in granite; Bridges hung over the waters; Dark green gardens Islands covered it...

Biology

The human body is built on the principle of bilateral symmetry. Most of us view the brain as a single structure; in reality, it is divided into two halves. These two parts - 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 Control of the basic movements of the human body and its sensory functions is evenly distributed between the two hemispheres of the brain. The left hemisphere controls the right side of the brain, and the right hemisphere controls the left side.

Botany

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 monocotyledonous plants, fivefold - for dicotyledonous plants. A characteristic feature of the structure of plants and their development is spirality. Pay attention to the leaf arrangement of the shoots - this is also a peculiar type of spiral - a helical one. Even Goethe, who was not only a great poet, but also a natural scientist, considered spirality to be one of the characteristic features of all organisms, a manifestation of the innermost essence of life. The tendrils of plants twist in a spiral, the growth of tissues in tree trunks occurs in a spiral, the seeds in a sunflower are arranged in a spiral, and spiral movements are observed during the growth of roots and shoots.

A characteristic feature of the structure of plants and their development is spirality. Look at the pine cone. The scales on its surface are arranged strictly regularly - along two spirals that intersect approximately at a right angle. The number of such spirals in pine cones is 8 and 13 or 13 and 21.

Zoology

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. 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.

Axial symmetry

	Various types of symmetry of physical phenomena: symmetry of electric and magnetic fields (Fig. 1) In mutually perpendicular planes, the propagation of electromagnetic waves is symmetrical (Fig. 2)	Fig.1 Fig.2
Art	Mirror symmetry can often be observed in works of art. Mirror" symmetry is widely found in works of art of primitive civilizations and in ancient paintings. Medieval religious paintings are also characterized by this type of symmetry. One of Raphael’s best early works, “The Betrothal of Mary,” was created in 1504. Under a sunny blue sky lies a valley topped by a white stone temple. In the foreground is the betrothal ceremony. The High Priest brings Mary and Joseph's hands together. Behind Mary is a group of girls, behind Joseph is a group of young men. Both parts of the symmetrical composition are held together by the counter-movement of the characters. For modern tastes, the composition of such a painting is boring, since the symmetry is too obvious.

Chemistry	A water molecule has a plane of symmetry (straight vertical line). DNA molecules (deoxyribonucleic acid) play an extremely important role in the world of living nature. It is a double-chain high-molecular polymer, the monomer of which is nucleotides. DNA molecules have a double helix structure built on the principle of complementarity.
Architeculture	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)

© Sukhacheva Elena Vladimirovna, 2008-2009.

Today we will talk about a phenomenon that each of us constantly encounters in life: symmetry. What is symmetry?

We all roughly understand the meaning of this term. The dictionary says: symmetry is proportionality and complete correspondence of the arrangement of parts of something relative to a straight line or point. There are two types of symmetry: axial and radial. Let's look at the axial one first. This is, let’s say, “mirror” symmetry, when one half of an object is completely identical to the second, but repeats it as a reflection. Look at the halves of the sheet. They are mirror symmetrical. The halves of the human body are also symmetrical (front view) - identical arms and legs, identical eyes. But let’s not be mistaken; in fact, in the organic (living) world, absolute symmetry cannot be found! The halves of the sheet copy each other far from perfectly, the same applies to the human body (take a closer look for yourself); The same is true for other organisms! By the way, it is worth adding that any symmetrical body is symmetrical relative to the viewer only in one position. It’s worth, say, turning a sheet of paper, or raising one hand, and what happens? – you see for yourself.

People achieve true symmetry in the works of their labor (things) - clothes, cars... In nature, it is characteristic of inorganic formations, for example, crystals.

But let's move on to practice. You shouldn’t start with complex objects like people and animals; let’s try to finish drawing the mirror half of the sheet as the first exercise in a new field.

Drawing a symmetrical object - lesson 1

We make sure that it turns out as similar as possible. To do this, we will literally build our soul mate. Don’t think that it’s so easy, especially the first time, to draw a mirror-corresponding line with one stroke!

Let's mark several reference points for the future symmetrical line. We proceed like this: with a pencil, without pressing, we draw several perpendiculars to the axis of symmetry - the midrib of the leaf. Four or five is enough for now. And on these perpendiculars we measure to the right the same distance as on the left half to the line of the edge of the leaf. I advise you to use a ruler, don’t rely too much on your eye. As a rule, we tend to reduce the drawing - this has been observed from experience. We do not recommend measuring distances with your fingers: the error is too large.

Let's connect the resulting points with a pencil line:

Now let’s look meticulously at whether the halves are really the same. If everything is correct, we will circle it with a felt-tip pen and clarify our line:

The poplar leaf has been completed, now you can take a swing at the oak leaf.

Let's draw a symmetrical figure - lesson 2

In this case, the difficulty lies in the fact that the veins are marked and they are not perpendicular to the axis of symmetry and not only the dimensions but also the angle of inclination will have to be strictly observed. Well, let’s train our eye:

So a symmetrical oak leaf has been drawn, or rather, we built it according to all the rules:

How to draw a symmetrical object - lesson 3

And let’s consolidate the theme - we’ll finish drawing a symmetrical lilac leaf.

It also has an interesting shape - heart-shaped and with ears at the base, you'll have to puff:

This is what they drew:

Take a look at the resulting work from a distance and evaluate how accurately we were able to convey the required similarity. Here's a tip: look at your image in the mirror and it will tell you if there are any mistakes. Another way: bend the image exactly along the axis (we have already learned how to bend it correctly) and cut out the leaf along the original line. Look at the figure itself and at the cut paper.