Draw a symmetrical shape relative to a straight line. Symmetrical drawing of objects of the correct shape

Goals:

educational:
- give an idea of symmetry;
- to acquaint with the basic types of symmetry on the plane and in space;
- develop strong skills in building symmetrical figures;
- expand the understanding of known figures, introducing the properties associated with symmetry;
- show the possibilities of using symmetry when solving different tasks;
- consolidate the knowledge gained;
general educational:
- teach yourself to set yourself up for work;
- teach to control yourself and your neighbor on your desk;
- teach how to evaluate yourself and your deskmate;
developing:
- to intensify independent activity;
- develop cognitive activity;
- teach to generalize and systematize the information received;
educational:
- to instill in students a "sense of the shoulder";
- educate communication;
- instill a culture of communication.

DURING THE CLASSES

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

Exercise 1(3 min).

“Let's take a sheet of paper, fold it into pieces and cut out some figurine. Now expand the sheet and look at the fold line.

Question: What is the function of this line?

Supposed answer: This line divides the shape in half.

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

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

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

Assignment 2 (2 minutes).

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

Assignment 3 (5 minutes).

- Draw a circle in a notebook.

Question: Determine how the axis of symmetry runs?

Supposed answer: Differently.

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

Supposed answer: Lot.

- That's right, a circle has many axes of symmetry. The same remarkable figure is the ball (spatial figure)

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

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

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

I distribute to students the halves of plasticine figures.

Assignment 4 (3 min).

- Using the information received, fill in the missing part of the figure.

Note: the figure can be both planar and volumetric. It is important that the students determine how the axis of symmetry goes and complete the missing piece. The correctness of the execution is determined by the neighbor on the desk, assesses how correctly the work has been done.

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

Assignment 5 (group work 5 min).

- Determine visually the axis of symmetry and build the second part from a lace of a different color relative to it.

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

The elements of the drawings are presented to the students

Assignment 6 (2 minutes).

Find the symmetrical parts of these patterns.

To consolidate the material covered, I propose the following tasks, provided for 15 minutes:

Name all equal elements of the triangle KOR and KOM. What is the appearance of these triangles?

2. Draw in a notebook several isosceles triangles with a common base equal to 6 cm.

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

- Our initial ideas about the form date back to a very distant era of the ancient Stone Age - the Paleolithic. For hundreds of millennia of this period, people lived in caves, in conditions that did not differ much from the life of animals. Humans made tools for hunting and fishing, developed languages to communicate with each other, and in the late Paleolithic era adorned their existence, creating works of art, figurines and drawings that reveal a wonderful sense of form.
When there was a transition from simple gathering of food to active production, from hunting and fishing to agriculture, humanity enters a new stone Age, in the Neolithic.
Neolithic man had a keen sense of geometric shape. The burning and painting of earthen vessels, the making of reed mats, baskets, fabrics, and later - the processing of metals developed ideas about planar and spatial figures. Neolithic ornaments were pleasing to the eye, revealing equality and symmetry.
- Where does symmetry occur in nature?

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

“Symmetry can be seen in architecture as well. When constructing buildings, builders adhere to symmetry.

That is why the buildings are so beautiful. Also, an example of symmetry is a person, animals.

Home assignment:

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

TRIANGLES.

§ 17. SYMMETRY REGARDING THE LINE.

1. Shapes symmetrical to each other.

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

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

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

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.

It is possible to obtain symmetrical figures without using the bending of the plane, but with the help of a geometric construction. Suppose it is required to construct a point C "symmetric to a given point C with respect to line AB. Let us drop from point C the perpendicular
CD on line AB and on its continuation set aside the segment DC "= DC. If we bend the plane of the drawing along AB, then point C will be combined with point C": points C and C "are symmetric (Fig. 129).

Let it now be required to construct a segment C "D" symmetric to a given segment CD relative to the straight line AB. Let's construct points C "and D", symmetric to points C and D. If we bend the plane of the drawing along AB, then points C and D will be aligned with points C "and D", respectively (Fig. 130). Therefore, the segments CD and C "D" will match , they will be symmetrical.

Let us now construct a figure symmetric to the given polygon ABCDE with respect to the given axis of symmetry MN (Fig. 131).

To solve this problem, we drop the perpendiculars А a, V b, WITH With, D d and E e on the axis of symmetry МN. Then, on the extensions of these perpendiculars, we postpone the segments
a A "= A a, b B "= B b, With C "= Cc; d D "" = D d and e E "= E e.

Polygon A "B" C "D" E "will be symmetric to the polygon ABCDE. Indeed, if you bend the drawing along the straight line MN, then the corresponding vertices of both polygons will coincide, which means that the polygons themselves will combine; 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 shapes that are divided by some straight line into two symmetrical parts. Such figures are called symmetrical.

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

In a circle, the axis of symmetry is its diameter, since when bending along it, one semicircle is aligned with the other (Fig. 133). In the same way, the figures in drawings 134, a, b are symmetrical.

Symmetrical shapes are often found in nature, construction, and jewelry. The images shown in drawings 135 and 136 are symmetrical.

It should be noted that symmetrical figures can be combined by simple movement on a plane only in some cases. To combine symmetrical shapes, as a rule, you need to turn one of them with the back side,

Central symmetry
Axial symmetry
Conclusion

Definition

Symmetry (from the Greek. Symmetria - proportionality), in a broad sense - the immutability of the structure of a material object relative to its transformations. Symmetry plays a huge role in art and architecture. But it can be seen both in music and poetry. Symmetry is widespread in nature, especially in crystals, plants and animals. Symmetry can also be found in other areas of mathematics, for example, when plotting functions.

Central symmetry

Two points A and A 1 are called symmetric about the point O, if O - midpoint AA 1.point O considered symmetrical to itself.

Draws a point that is centrally symmetric to a given

Build AO Beam
Measure the length of the segment AO
Point A1 is symmetrical to point A relative to center O.

A 1

Draws a line segment that is centrally symmetric to a given

Build AO Beam
Measure the length of the segment AO
To put aside on the ray AO on the other side of the point O the segment OA 1, equal to the segment OA.
Construct ray VO
Measure the length of the segment IN
Put on the VO beam on the other side of the point O the segment ОВ 1, equal to the segment ОВ.
Connect points A 1 and B 1 with a segment

A 1

V 1

A 1

WITH 1

V 1

Centrally symmetrical figures are equal

Constructing a figure that is centrally symmetric to a given

Rotate point A around the center of rotation O by 90 °

A 1

90 °

Rotating points to different angles

A 1

135 °

45 °

A 2

90 °

A 3

Axial symmetry

Shape transformation F in figure F 1, at which each of its points goes to a point symmetric with respect to a given straight line, is called a transformation of symmetry with respect to a straight line a... Straight a called the axis of symmetry.

Draw a point symmetric to a given

2. AO = OA '

Constructing a line segment symmetric to a given

AA ' s, AO = OA'.
BB ' s, BO' = O 'B'.

3. A ’B’ is the required segment.

Draw a triangle symmetrical to a given

1. AA ’ c AO = OA’

2. BB ' c BO' = O'B '

3. SS ’ c C O” = O ”C’

4. "A'B" C "- the required triangle.

Constructing a figure that is symmetric about the axis of symmetry

Shapes with one axis of symmetry

Injection

Isosceles

triangle

Isosceles trapezoid

Shapes with two axes of symmetry

Rectangle

Rhombus

Shapes with more than two axes of symmetry

Square

Equilateral triangle

A circle

Shapes that are not axially symmetrical

Arbitrary triangle

Parallelogram

Irregular polygon

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

If you think for a minute and imagine an object in your imagination, then in 99% of cases the figure that comes to mind will be 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 non-traditional thinking individuals with a special outlook on things. But returning to the absolute majority, it is worth saying that a significant proportion of correct subjects still prevails. The article will focus exclusively on them, namely, symmetrical drawing of those.

Drawing the correct items: just a few steps to a 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 does not in any way resemble what you decided to portray, do not despair: all steps are absolutely identical. Stick to the sequence and everything will work out:

All objects of the correct shape have a so-called central axis, which, when drawing symmetrically, should definitely be highlighted. To do this, you can even use a ruler and draw a straight line in the center of the album sheet.
Next, take a close look at the item you have chosen and try to transfer its proportions to a sheet of paper. It is not difficult to do this if, on both sides of the line drawn in advance, outline light strokes, 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.
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 about the correctness of your own eye, double-check the marked distances with a ruler.
The last step is to connect all the lines together.

Symmetrical drawing is available to computer users

Due to the fact that most of the objects around us have correct proportions, in other words, symmetric, the developers of computer applications have created programs in which you can easily draw absolutely everything. You just need to download them and enjoy the creative process. Remember, though, a machine will never replace a sharpened pencil and sketchbook.

I ... Symmetry in mathematics :

Basic concepts and definitions.

Axial symmetry (definitions, construction plan, examples)

Central symmetry (definitions, construction plan, formeasures)

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 goes 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 a person. And it was used by sculptors as early as the 5th century BC. e. The word "symmetry" is Greek, it means "proportionality, proportionality, uniformity 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, LN 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 indeed pleasing to the eye. Who has not admired the symmetry of nature's creations: leaves, flowers, birds, animals; or human creations: buildings, technology, - everything that surrounds us from childhood, those that strive for beauty and harmony. Hermann Weil said: "Symmetry is the idea through which man, for centuries, has tried to comprehend and create order, beauty and perfection." Hermann Weil is a German mathematician. His activity falls on the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what criteria to perceive the presence or, conversely, the absence of symmetry in one or another case. Thus, a mathematically rigorous concept was formed relatively recently - at the beginning of the twentieth century. It's quite complicated. We will 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 symmetric with respect to the straight line a if this straight line passes through the middle of the segment AA 1 and is perpendicular to it. Each point of the straight line a is considered symmetrical to itself.

Definition. The figure is called symmetrical about a straight line. a if, for each point of the figure, a point symmetric to it with respect to a straight line a also belongs to this figure. Straight a is called the axis of symmetry of the figure. The figure is also said to have axial symmetry.

2.2 Building plan

And so, to build a symmetrical figure with respect 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 shape. 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 symmetric 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 called symmetric about point O if for each point of the figure the point symmetric to it about point O also belongs to this figure.

3.2 Build plan

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

To draw a point symmetrical to a point A relative to point O, it is enough to draw a straight line OA(fig. 46 ) and on the other side of the point O postpone a segment equal to the segment OA. In other words , points A and ; In and ; With and are symmetric with respect to some point O. In Fig. 46 built a triangle symmetrical to the triangle ABC relative to point O. These triangles are equal.

Draws symmetrical points about the center.

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

In general, figures symmetrical about some point are equal .

3.3 Examples

Here are some 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 infinitely many of them - any point of 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 A and a quadrilateral symmetric about its vertex M.

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

4. Lesson summary

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.

Summarizing table

	Axial symmetry	Central symmetry
Peculiarity	All points of the figure must be symmetrical about some straight line.	All points of the shape must be symmetrical about the point selected as the center of symmetry.
Properties	1. Symmetrical points lie on perpendiculars to a straight line. 3. Straight lines turn into straight lines, angles into equal angles. 4. Sizes and shapes of 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 preserved.

II. Applying symmetry

Mathematics

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

The figures show various pictures depicted using the branches of parabolas.

(a) Octahedron,

(b) rhombic dodecahedron, (c) hexagonal octahedron.

Russian language

Printed letters The Russian alphabet also has different types of symmetries.

There are "symmetrical" words in Russian - palindromes that can be read the same way in two directions.

A D L M P T V W- vertical axis

V E Z K S E Y - horizontal axis

J N O X- both vertical and horizontal

B G I Y R U Y Z- no axis

Radar hut Alla Anna

Literature

Can be palindromic and sentences. Bryusov wrote a poem "The Voice of the Moon", in which each line is a palindrome.

Look at the quatrains of 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 go with the sword of the judge. (Derzhavin)

"Search for a taxi"

"Argentina Manit Negro",

"The Argentinean appreciates the negro",

"Lesha found a bug on the shelf."

The Neva was dressed in granite;

Bridges hung over the waters;

Dark green gardens

The islands were covered with it ...

Biology

The human body is built according to the principle of bilateral symmetry. Most of us view the brain as a single structure; in fact, it is divided into two halves. These two parts - the 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, and the right side controls the left side.

Botany

A flower is considered symmetrical when each perianth is composed of an equal number of parts. Flowers, having paired parts, are considered to be flowers with double symmetry, etc. Triple symmetry is common for monocotyledonous plants, quintuple symmetry for dicots Characteristic feature the structure of plants and their development is helicity.

Pay attention to the shoots of the leaf arrangement - this is also a kind of spiral - helical. Even Goethe, who was not only a great poet, but also a natural scientist, considered helicity one of the characteristic features of all organisms, a manifestation of the most intimate essence of life. The antennae of plants are spirally twisted, the tissues grow in the trunks of trees in a spiral, the seeds in the 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 pinecone. The scales on its surface are arranged in a strictly regular way - along two spirals, which intersect at approximately right angles. 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 shape, as well as the relative position of body parts located on opposite sides of the dividing line. With radial or radiant 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 radiate out in a radial order. These are coelenterates, echinoderms, starfish. With bilateral symmetry, there are three axes of symmetry, but there is only one pair of symmetrical sides. Because the other two sides - the ventral and dorsal - are not alike. This type of symmetry is typical for most animals, including insects, fish, amphibians, reptiles, birds, and mammals.

Axial symmetry

Different kinds symmetry physical phenomena: symmetry of electric and magnetic fields (fig. 1)

In mutually perpendicular planes, the distribution is symmetric electromagnetic waves(fig. 2)

fig. 1 fig. 2

Art

Mirror symmetry can often be observed in works of art. Mirror "symmetry is widespread 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 the best early works Raphael - "The Betrothal of Mary" - created in 1504. A valley crowned with a white-stone temple stretches under the sunny blue sky. Foreground: the betrothal ceremony. The high priest brings the hands of Mary and Joseph closer. Behind Mary - a group of girls, behind Joseph - young men. Both parts of the symmetrical composition are held together by the oncoming movement of the characters. For modern taste, the composition of such a picture is boring, since 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 living world. It is a double-stranded high molecular weight polymer, the monomer of which is nucleotides. DNA molecules have a double helix structure built on the principle of complementarity.

Architeculture

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

Pashkov House Louvre (Paris)