How to build symmetrical figures. Constructing symmetrical points relative to the center

TRIANGLES.

§ 17. SYMMETRY RELATIVELY TO THE RIGHT STRAIGHT.

1. Figures that are symmetrical to each other.

Let's draw some figure on a sheet of paper with ink, and with a pencil outside it - an arbitrary straight line. Then, without allowing the ink to dry, we bend the sheet of paper along this straight line so that one part of the sheet overlaps the other. This other part of the sheet will thus produce an imprint of this figure.

If you then straighten the sheet of paper again, then there will be two figures on it, which are called symmetrical relative to a given line (Fig. 128).

Two figures are called symmetrical with respect to a certain straight line if, when bending the drawing plane along this straight line, they are aligned.

The straight line with respect to which these figures are symmetrical is called their axis of symmetry.

From the definition of symmetrical figures it follows that all symmetrical figures are equal.

You can get symmetrical figures without using bending of the plane, but with the help geometric construction. Let it be necessary to construct a point C" symmetrical to a given point C relative to straight line AB. Let us drop a perpendicular from point C
CD to straight line AB and as its continuation we will lay down the segment DC" = DC. If we bend the drawing plane along AB, then point C will align with point C": points C and C" are symmetrical (Fig. 129).

Suppose now we need to construct a segment C "D", symmetrical this segment CD relative to straight AB. Let's construct points C" and D", symmetrical to points C and D. If we bend the drawing plane along AB, then points C and D will coincide, respectively, with points C" and D" (Drawing 130). Therefore, segments CD and C "D" will coincide , they will be symmetrical.

Let us now construct a figure symmetrical given polygon ABCDE relative to this axis of symmetry MN (Fig. 131).

To solve this problem, let’s drop the perpendiculars A A, IN b, WITH With, D d and E e to the axis of symmetry MN. Then, on the extensions of these perpendiculars, we plot the segments
A A" = A A, b B" = B b, With C" = Cs; d D"" =D d And e E" = E e.

The polygon A"B"C"D"E" will be symmetrical to the polygon ABCDE. Indeed, if you bend the drawing along a straight line MN, then the corresponding vertices of both polygons will align, and therefore the polygons themselves will align; this proves that the polygons ABCDE and A" B"C"D"E" are symmetrical about the straight line MN.

2. Figures consisting of symmetrical parts.

Often there are geometric figures that are divided by some straight line into two symmetrical parts. Such figures are called symmetrical.

So, for example, an angle is a symmetrical figure, and the bisector of the angle is its axis of symmetry, since when bent along it, one part of the angle is combined with the other (Fig. 132).

In a circle, the axis of symmetry is its diameter, since when bending along it, one semicircle is combined with another (Fig. 133). The figures in drawings 134, a, b are exactly symmetrical.

Symmetrical figures are often found in nature, construction, and jewelry. The images placed on drawings 135 and 136 are symmetrical.

It should be noted that symmetrical figures can be combined simply by moving along a plane only in some cases. To combine symmetrical figures, as a rule, it is necessary to turn one of them with the opposite side,

People's lives are filled with symmetry. It’s convenient, beautiful, and there’s no need to invent new standards. But what is it really and is it as beautiful in nature as is commonly believed?

Symmetry

Since ancient times, people have sought to organize the world around them. Therefore, some things are considered beautiful, and some are not so much. From an aesthetic point of view, the golden and silver ratios are considered attractive, as well as, of course, symmetry. This term has Greek origin and literally means “proportionality”. Of course we're talking about not only about coincidence on this basis, but also on some others. IN in a general sense symmetry is a property of an object when, as a result of certain formations, the result is equal to the original data. This occurs both in living and in inanimate nature, as well as in objects made by man.

First of all, the term "symmetry" is used in geometry, but finds application in many scientific fields, and its meaning remains generally unchanged. This phenomenon occurs quite often and is considered interesting, since several of its types, as well as elements, differ. The use of symmetry is also interesting, because it is found not only in nature, but also in patterns on fabric, borders of buildings and many other man-made objects. It is worth considering this phenomenon in more detail, because it is extremely fascinating.

Use of the term in other scientific fields

In what follows, symmetry will be considered from a geometric point of view, but it is worth mentioning that given word used not only here. Biology, virology, chemistry, physics, crystallography - all this is an incomplete list of areas in which this phenomenon studied with various sides and in different conditions. For example, the classification depends on what science this term refers to. Thus, the division into types varies greatly, although some basic ones, perhaps, remain unchanged throughout.

Classification

There are several main types of symmetry, of which three are the most common:

In addition, in geometry there are also following types, they are much less common, but no less interesting:

sliding;
rotational;
point;
progressive;
screw;
fractal;
etc.

In biology, all species are called slightly differently, although in essence they may be the same. Division into certain groups occurs on the basis of the presence or absence, as well as the quantity of certain elements, such as centers, planes and axes of symmetry. They should be considered separately and in more detail.

Basic elements

The phenomenon has certain features, one of which is necessarily present. So called basic elements include planes, centers and axes of symmetry. It is in accordance with their presence, absence and quantity that the type is determined.

The center of symmetry is the point inside a figure or crystal at which the lines connecting everything in pairs converge parallel friend to the other side. Of course, it does not always exist. If there are sides to which there is no parallel pair, then such a point cannot be found, since it does not exist. According to the definition, it is obvious that the center of symmetry is that through which a figure can be reflected onto itself. An example would be, for example, a circle and a point in its middle. This element is usually designated as C.

The plane of symmetry, of course, is imaginary, but it is precisely it that divides the figure into two parts equal to each other. It can pass through one or more sides, be parallel to it, or divide them. For the same figure, several planes can exist at once. These elements are usually designated as P.

But perhaps the most common is what is called “axis of symmetry”. This is a common phenomenon that can be seen both in geometry and in nature. And it is worthy of separate consideration.

Axles

Often the element in relation to which a figure can be called symmetrical is

a straight line or segment appears. In any case, we are not talking about a point or a plane. Then the figures are considered. There can be a lot of them, and they can be located in any way: dividing the sides or being parallel to them, as well as intersecting corners or not doing so. Axes of symmetry are usually designated as L.

Examples include isosceles and In the first case there will be vertical axis symmetry, on both sides of which equal faces, and in the second the lines will intersect each angle and coincide with all bisectors, medians and heights. Ordinary triangles do not have this.

By the way, the totality of all the above elements in crystallography and stereometry is called the degree of symmetry. This indicator depends on the number of axes, planes and centers.

Examples in geometry

Conventionally, we can divide the entire set of objects of study by mathematicians into figures that have an axis of symmetry and those that do not. All circles, ovals, as well as some special cases automatically fall into the first category, while the rest fall into the second group.

As in the case when it was said about the axis of symmetry of a triangle, this element for a quadrilateral does not always exist. For a square, rectangle, rhombus or parallelogram it is, and for irregular figure, accordingly, no. For a circle, the axis of symmetry is the set of straight lines that pass through its center.

In addition, it is interesting to consider volumetric figures from this point of view. At least one axis of symmetry in addition to all regular polygons and the ball will have some cones, as well as pyramids, parallelograms and some others. Each case must be considered separately.

Examples in nature

In life it is called bilateral, it occurs most
often. Any person and many animals are an example of this. Axial is called radial and is much less common, usually in flora. And yet they exist. For example, it is worth thinking about how many axes of symmetry a star has, and does it have any at all? Of course, we are talking about marine life, and not about the subject of study by astronomers. And the correct answer would be: it depends on the number of rays of the star, for example five, if it is five-pointed.

In addition, radial symmetry is observed in many flowers: daisies, cornflowers, sunflowers, etc. There are a huge number of examples, they are literally everywhere around.

Arrhythmia

This term, first of all, reminds most of medicine and cardiology, but it initially has a slightly different meaning. IN in this case a synonym would be “asymmetry”, that is, the absence or violation of regularity in one form or another. It can be found as an accident, and sometimes it can become a wonderful technique, for example in clothing or architecture. After all, there are a lot of symmetrical buildings, but the famous one is slightly tilted, and although it is not the only one, it is the most famous example. It is known that this happened by accident, but this has its own charm.

In addition, it is obvious that the faces and bodies of people and animals are not completely symmetrical either. There have even been studies that show that “correct” faces are judged to be lifeless or simply unattractive. Still, the perception of symmetry and this phenomenon in itself are amazing and have not yet been fully studied, and therefore are extremely interesting.

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Lesson type: combined.

Lesson objectives:

Consider axial, central and mirror symmetries as properties of some geometric figures.
To teach how to construct symmetrical points and recognize figures that have 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 a subject through use information technologies.
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 goals 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 the word, is an idea with the help of which man has 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 Maple Leaf, 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 view"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 straight 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 such a property 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.

Points are given A, IN And M M relative to the middle of the segment AB.
Which of the following letters have a center of symmetry: A, O, M, X, K?
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.

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

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 coordinate axes; c) mirror symmetry relative to coordinate planes.
Does the right glove go into the right or left glove with mirror symmetry? axial symmetry? central symmetry?
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)
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, not without reason translated from Greek word"symmetry" means "proportionality".

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 a rotation for various colors not the same. 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 overall plan 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(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. Art products of garnets were found in graves Ancient Egypt, dating back to the predynastic period (over two millennia BC) (see Fig. 5).

In the Hermitage collections special attention used gold jewelry of the ancient Scythians. Extraordinarily thin artwork golden wreaths, tiaras, wood and decorated with precious red-violet garnets.

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 expression 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 most important inventions of humanity, having a center of symmetry is the wheel, and the propeller and other technical means also have a center of symmetry.

"Look in the mirror!"

Should we consider that we see ourselves only in “ mirror image"? Or in best case scenario Only in photographs and film can we find out 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

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.
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.
Construct a square along the given diagonal so that two opposite vertices of this square lay on different sides of this 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 it by applying 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?

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 correct form. 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 the right items still prevails. In the article we'll talk exclusively about them, namely about the 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:

All objects of regular shape have a so-called central axis, which is definitely worth highlighting when drawing symmetrically. To do this, you can even use a ruler and draw a straight line down the center of the landscape sheet.
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.
Don't 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 marked distances using a ruler.
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, the developers computer applications created programs in which you can easily draw absolutely everything. Just download them and enjoy creative process. However, remember, a machine will never be a substitute for a sharpened pencil and a sketchbook.

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 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 products of their labor (things) - clothes, cars... In nature, it is characteristic 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.

He has too 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.