Draw a picture with central symmetry. Central and axial symmetry

Consider axial and central symmetries as properties of some geometric figures; Consider axial and central symmetries as properties of some geometric figures; Be able to build symmetrical points and be able to recognize figures that are symmetrical about a point or a line; Be able to build symmetrical points and be able to recognize figures that are symmetrical about a point or a line; Improving problem solving skills; Improving problem solving skills; Continue work on the accuracy of recording and performing a geometric drawing; Continue work on the accuracy of recording and performing a geometric drawing;

Oral work "Gentle poll" Oral work "Gentle poll" What point is called the midpoint of the segment? Which triangle is called an isosceles triangle? What property do the diagonals of a rhombus have? Formulate the property of the bisector of an isosceles triangle. Which lines are called perpendicular? What is an equilateral triangle? What property do the diagonals of a square have? What figures are called equal?

What new concepts did you learn in class? What new concepts did you learn in class? What have you learned about geometric shapes? What have you learned about geometric shapes? Give examples of geometric figures with axial symmetry. Give examples of geometric figures with axial symmetry. Give an example of figures with central symmetry. Give an example of figures with central symmetry. Give examples of objects from the surrounding life that have one or two types of symmetry. Give examples of objects from the surrounding life that have one or two types of symmetry.

I . Symmetry in mathematics :

Basic concepts and definitions.

Axial symmetry (definitions, construction plan, examples)

Central symmetry (definitions, construction plan, withmeasures)

Summary table (all properties, features)

II . Symmetry Applications:

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 n R runs throughout the 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 as early as the 5th century BC. e. The word "symmetry" is Greek, it means "proportionality, proportionality, the sameness in the arrangement of parts." It is widely used by all areas of modern science without exception. Many great people 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 really pleasing to the eye. Who has not admired the symmetry of nature's creations: leaves, flowers, birds, animals; or human creations: buildings, technology, - all that surrounds us from childhood, that strives for beauty and harmony. Hermann Weyl said: "Symmetry is the idea through which man has tried for centuries to comprehend and create order, beauty and perfection." Hermann Weyl is a German mathematician. Its activity falls on the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what signs to see the presence or, conversely, the absence of symmetry in a particular case. Thus, a mathematically rigorous representation was formed relatively recently - at the beginning of the 20th century. It is rather complicated. We will turn and once again recall the definitions that are 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 the line a if this line passes through the midpoint of the segment AA 1 and is perpendicular to it. Each point of the line a is considered symmetrical to itself.

Definition. The figure is said to be symmetrical with respect to a straight line. a, if for each point of the figure the point symmetrical to it with respect 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 build a symmetrical figure relative to a straight line from each point, we draw a perpendicular to this straight line and extend it by the same distance, mark the resulting point. We do this with each point, we get the symmetrical vertices of the new figure. Then we connect them in series and get a symmetrical figure of this 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 the point O if O is the midpoint of the segment AA 1. Point O is considered symmetrical to itself.

Definition. A figure is called symmetric with respect to the point O if for each point of the figure the point symmetric to it with respect to the point O also belongs to this figure.

3.2 Construction plan

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

To construct a point symmetrical to a point BUT relative to the point O, it suffices to draw a straight line OA(Fig. 46 ) and on the other side of the point O set aside a segment equal to a segment OA. In other words , points A and ; In and ; C and are symmetrical with respect to some point O. In fig. 46 built a triangle symmetrical to a triangle ABC relative to the point O. These triangles are equal.

Construction of symmetrical points about the center.

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

In general, figures that are symmetrical about some point are equal to .

3.3 Examples

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

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

The figures show an angle symmetrical about the vertex, a segment symmetrical to another segment about the center BUT 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. Summary of the lesson

Let's summarize the knowledge gained. Today in the lesson we got acquainted with 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 with respect to some straight line.	All points of the figure must be symmetrical about the point chosen as the center of symmetry.
Properties	1. Symmetric points lie on perpendiculars to the line. 3. Straight lines turn into straight lines, angles into equal angles. 4. The sizes and shapes of the figures are saved.	1. Symmetrical points lie on a straight line passing through the center and the 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 saved.

II. Application of symmetry

Mathematics

In algebra lessons, we studied the graphs of the functions y=x and y=x

The figures show various pictures depicted with the help of 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 Russian - palindromes, which can be read the same way in both directions.

A D L M P T V- vertical axis

B E W K S E Yu - horizontal axis

W N O X- both vertical and horizontal

B G I Y R U C W Y Z- no axis

Radar hut Alla Anna

Literature

Sentences can also be palindromic. Bryusov wrote the poem "Voice of the Moon", in which each line is a palindrome.

Look at the quadruplets of A.S. Pushkin's "The Bronze Horseman". If we draw a line after the second line, we can see the elements of axial symmetry

And the rose fell on Azor's paw.

I go with the judge's sword. (Derzhavin)

"Look for a taxi"

"Argentina beckons a black man",

"Appreciates the Negro Argentine",

"Lesha found a bug on the shelf."

The Neva is dressed in granite;

Bridges hung over the waters;

Dark green gardens

The islands were covered with it ...

Biology

The human body is built on the principle of bilateral symmetry. Most of us think of the brain as a single structure, in fact it is divided into two halves. These two parts - two hemispheres - fit snugly together. In full accordance with the general symmetry of the human body, each hemisphere is an almost exact mirror image of the other.

The 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, while 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 monocots, five - for dicots. A characteristic feature of the structure of plants and their development is helicity.

Pay attention to the leaf arrangement shoots - this is also a kind of spiral - helical. Even Goethe, who was not only a great poet, but also a naturalist, considered helicity 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, tissue grows in a spiral in tree trunks, seeds in a sunflower are arranged in a spiral, spiral movements are observed during the growth of roots and shoots.

A characteristic feature of the structure of plants and their development is helicity.

Look at the pine cone. The scales on its surface are arranged in a strictly regular manner - 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 is understood as correspondence in size, shape and outline, as well as the relative location of body parts located on opposite sides of the dividing line. With radial or radiative symmetry, the body has the form of a short or long cylinder or a vessel with a central axis, from which parts of the body extend in a radial order. These are coelenterates, echinoderms, starfish. With bilateral symmetry, there are three axes of symmetry, but only one pair of symmetrical sides. Because the other two sides - the abdominal and dorsal - are not similar to each other. This kind of symmetry is characteristic of most animals, including insects, fish, amphibians, reptiles, birds, and mammals.

Axial symmetry

Different 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 the works of art of primitive civilizations and in ancient painting. Medieval religious paintings are also characterized by this kind of symmetry.

One of Raphael's best early works, The Betrothal of Mary, was created in 1504. A valley topped with a white-stone temple stretches out under the sunny blue sky. In the foreground is the betrothal ceremony. The High Priest brings the hands of Mary and Joseph closer 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 oncoming movement of the characters. For modern tastes, the composition of such a picture is boring, because the symmetry is too obvious.

Chemistry

The water molecule has a plane of symmetry (straight vertical line). DNA molecules (deoxyribonucleic acid) play an extremely important role in the world of wildlife. It is a double-stranded high molecular weight polymer whose monomer is nucleotides. DNA molecules have a double helix structure built on the principle of complementarity.

architewho

Since ancient times, man has used symmetry in architecture. Ancient architects used symmetry especially brilliantly in architectural structures. Moreover, the ancient Greek architects were convinced that in their works they are guided by the laws that govern nature. Choosing symmetrical forms, the artist thus 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 - park - a complex of landscape gardening sculpture, which was created over 40 years.

Pashkov House Louvre (Paris)

Axial symmetry. With axial symmetry, each point of the figure goes to a point symmetrical to it with respect to a fixed line.

Picture 35 from the presentation "Ornament" to geometry lessons on the topic "Symmetry"

Dimensions: 360 x 260 pixels, format: jpg. To download a picture for a geometry lesson for free, right-click on the image and click "Save Image As...". To show pictures in the lesson, you can also download the entire presentation “Ornament.ppt” with all the pictures in a zip archive for free. The size of the archive is 3324 KB.

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Symmetry

"Point of symmetry" - Central symmetry. A a A1. Axial and central symmetry. Point C is called the center of symmetry. Symmetry in life. The round cone is axially symmetrical; the axis of symmetry is the axis of the cone. Shapes that have more than two axes of symmetry. The parallelogram has only central symmetry.

"Mathematical Symmetry" - What is symmetry? physical symmetry. Symmetry in biology. The history of symmetry. However, complex molecules, as a rule, lack symmetry. palindromes. Symmetry. In x and m and and. HAS A LOT IN COMMON WITH TRANSLATIONAL SYMMETRY IN MATHEMATICS. And actually, how would we live without symmetry? Axial symmetry.

"Ornament" - b) On the strip. Parallel translation Central symmetry Axial symmetry Rotation. Linear (layout options): Create an ornament using central symmetry and parallel translation. Planar. One of the varieties of ornament is a mesh ornament. Transforms used to create the ornament:

"Symmetry in nature" - One of the main properties of geometric shapes is symmetry. The topic was not chosen by chance, because next year we have to start studying a new subject - geometry. The phenomenon of symmetry in living nature was noticed even in ancient Greece. We are in the school scientific society because we love to learn something new and unknown.

"Movement in Geometry" - Mathematics is beautiful and harmonious! Give examples of movement. Movement in geometry. What is called movement? To what sciences is movement applied? How is movement used in various areas of human activity? group of theorists. Concept of motion Axial symmetry Central symmetry. Can we see movement in nature?

"Symmetry in Art" - Levitan. RAPHAEL. II.1. Proportion in architecture. Rhythm is one of the main elements of the expressiveness of a melody. R. Descartes. Ship Grove. A. V. Voloshinov. Velasquez Surrender of Breda. Outwardly, harmony can manifest itself in melody, rhythm, symmetry, proportionality. II.4. Proportion in literature.

Total in the topic 32 presentations

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

Approximately we all understand the meaning of this term. The dictionary says: symmetry is the proportionality and full correspondence of the arrangement of parts of something relative to a line or point. There are two types of symmetry: axial and radial. Let's look at the axis first. This is, let's say, "mirror" symmetry, when one half of the 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 (full face) are also symmetrical - the same arms and legs, the same eyes. But let's not be mistaken, in fact, in the organic (living) world, absolute symmetry cannot be found! The halves of the sheet do not copy each other perfectly, the same applies to the human body (look at it for yourself); the same is true of other organisms! By the way, it is worth adding that any symmetrical body is symmetrical relative to the viewer in only one position. It is necessary, say, to turn the sheet, or raise one hand, and what? - see for yourself.

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

But let's move on to practice. It’s not worth starting with complex objects like people and animals, let’s try to finish the mirror half of the sheet as the first exercise in a new field.

Draw a symmetrical object - lesson 1

Let's try to make it as similar as possible. To do this, we will literally build our soul mate. Do not think that it is 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 act like this: we draw with a pencil without pressure several perpendiculars to the axis of symmetry - the middle vein of the sheet. Four or five is enough. 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 the ruler, do not really rely on the eye. As a rule, we tend to reduce the drawing - it has been noticed in experience. We do not recommend measuring distances with your fingers: the error is too large.

Connect the resulting points with a pencil line:

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

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

Let's draw a symmetrical figure - lesson 2

In this case, the difficulty lies in the fact that the veins are indicated 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 exactly observed. Well, let's train the eye:

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

How to draw a symmetrical object - lesson 3

And we will fix the topic - we will finish drawing a symmetrical leaf of lilac.

He also has an interesting shape - heart-shaped and with ears at the base you have to puff:

Here is what they drew:

Look at the resulting work from a distance and evaluate how accurately we managed to convey the required similarity. Here's a tip for you: 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 correctly) and cut the leaf along the original line. Look at the figure itself and at the cut paper.

You will need

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

Instruction

Draw a line a, which will be the axis of symmetry. If its coordinates are not given, draw it arbitrarily. On one side of this line, put an arbitrary point A. you need to find a symmetrical point.

Helpful advice

Symmetry properties are constantly used in the AutoCAD program. For this, the Mirror option is used. To build an isosceles triangle or an isosceles trapezoid, it is enough to draw the lower base and the angle between it and the side. Mirror them with 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, a straight line corresponding to one of the vertical or horizontal sides of the picture frame is taken as the axis of symmetry.

Sources:

how to draw central symmetry

Constructing a 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 easy to do and will not require much effort from you.

You will need

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

Instruction

When answering this question, you first need to decide what parameters the section is set to.
Let this be the line of intersection of the plane l with the plane and the point O, which is the point of 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. As a result, point L is obtained. Further, through point O, draw a straight line LW, and build two directing cones lying in the main section O2M and O2C. At the intersection of these guides lie the point Q, as well as the already shown point W. These are the first two points of the required section.

Now draw a perpendicular MC at the base of the cone BB1 and build the generators of the perpendicular section O2B and O2B1. In this section, draw a straight line RG through t.O, parallel to BB1. T.R and t.G - two more points of the desired section. If the cross section of the ball were known, then it could be constructed already at this stage. However, this is not an ellipse at all, but something elliptical, having symmetry with respect to the segment QW. Therefore, you should build as many points of the section as possible in order to connect them in the future with a smooth curve to get the most reliable sketch.

Construct an arbitrary section point. To do this, draw an arbitrary diameter AN at the base of the cone and build the corresponding guides O2A and O2N. Through PO 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 and further, you can arbitrarily desired points.

True, the procedure for obtaining them can be slightly simplified using symmetry with respect to QW. To do this, it is possible to draw straight lines SS' parallel to RG 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 suffices to construct half of the required section due to the already mentioned symmetry with respect to QW.

Tip 3: How to Graph a Trigonometric Function

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

You will need

- ruler;
- pencil;
- Knowledge of the basics of trigonometry.

Instruction

Portal for the student. Self-training

1. Basic concepts of symmetry and its types.

2. Axial symmetry.

3. Central symmetry

4. Summary of the lesson

II. Application of symmetry

Symmetry

Draw a symmetrical object - lesson 1

Let's draw a symmetrical figure - lesson 2

How to draw a symmetrical object - lesson 3

Tip 3: How to Graph a Trigonometric Function

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