Axial and central. Perfection of lines - axial symmetry in life

Since ancient times, man has developed ideas about beauty. All creations of nature are beautiful. People are beautiful in their own way, animals and plants are amazing. The sight of a precious stone or a salt crystal pleases the eye; it is difficult not to admire a snowflake or a butterfly. But why does this happen? It seems to us that the appearance of objects is correct and complete, the right and left halves of which look the same, as if in a mirror image.

Apparently, people of art were the first to think about the essence of beauty. Ancient sculptors who studied the structure of the human body, back in the 5th century BC. The concept of “symmetry” began to be used. This word is of Greek origin and means harmony, proportionality and similarity in the arrangement of the constituent parts. Plato argued that only that which is symmetrical and proportionate can be beautiful.

In geometry and mathematics, three types of symmetry are considered: axial symmetry (relative to a straight line), central (relative to a point) and mirror symmetry (relative to a plane).

If each of the points of an object has its own exact mapping within it relative to its center, there is central symmetry. Its example is such geometric bodies as a cylinder, a sphere, a regular prism, etc.

The axial symmetry of points relative to a straight line provides that this straight line intersects the middle of the segment connecting the points and is perpendicular to it. Examples are the bisector of an undeveloped angle of an isosceles triangle, any line drawn through the center of a circle, etc. If axial symmetry is characteristic, the definition of mirror points can be visualized by simply bending it along the axis and putting equal halves “face to face.” The desired points will touch each other.

With mirror symmetry, the points of an object are located equally relative to the plane that passes through its center.

Nature is wise and rational, therefore almost all of its creations have a harmonious structure. This applies to both living beings and inanimate objects. The structure of most life forms is characterized by one of three types of symmetry: bilateral, radial or spherical.

Most often, axial can be observed in plants developing perpendicular to the soil surface. In this case, symmetry is the result of rotation of identical elements around a common axis located in the center. The angle and frequency of their location may be different. Examples are trees: spruce, maple and others. In some animals, axial symmetry also occurs, but this is less common. Of course, nature is rarely characterized by mathematical precision, but the similarity of the elements of an organism is still striking.

Biologists often consider not axial symmetry, but bilateral (bilateral) symmetry. An example of this is the wings of a butterfly or dragonfly, plant leaves, flower petals, etc. In each case, the right and left parts of the living object are equal and are mirror images of each other.

Spherical symmetry is characteristic of the fruits of many plants, some fish, mollusks and viruses. Examples of radial symmetry are some types of worms and echinoderms.

In human eyes, asymmetry is most often associated with irregularity or inferiority. Therefore, in most creations of human hands, symmetry and harmony can be traced.

The concept of “central symmetry” of a figure presupposes the existence of a certain point - the center of symmetry. On both sides of it there are points belonging to each of them. Each of them has a symmetrical point to itself.

It should be said that the concept of a center is absent in Euclidean geometry. Moreover, in the eleventh book, in the thirty-eighth sentence, there is a definition of the spatial symmetrical axis. The concept of a center first appeared in the 16th century.

Central symmetry is present in such well-known figures as a parallelogram and a circle. Both the first and second figures have the same center. The center of symmetry of a parallelogram is located at the point of intersection of lines emerging from opposite points; in a circle is the center of itself. A straight line is characterized by the presence of an infinite number of such sections. Each of its points can be a center of symmetry. A right parallelepiped has nine planes. Of all the symmetrical planes, three are perpendicular to the ribs. The other six pass through the diagonals of the faces. However, there is a figure that does not have it. It is an arbitrary triangle.

In some sources, the concept of “central symmetry” is defined as follows: a geometric body (figure) is considered symmetrical with respect to the center C if each point A of the body has a point E lying within the same figure, such that the segment AE, passing through center C, cut in half in it. For corresponding pairs of points there are equal segments.

The corresponding angles of the two halves of a figure in which central symmetry is present are also equal. Two figures lying on either side of the central point can in this case be superimposed on each other. However, it must be said that the imposition is carried out in a special way. Unlike mirror symmetry, central symmetry involves rotating one part of the figure one hundred and eighty degrees around the center. Thus, one part will be in a mirror position relative to the second. Two parts of the figure can thus be superimposed on each other without being removed from the common plane.

In algebra, odd and even functions are studied using graphs. For the graph is constructed symmetrically with respect to the coordinate axis. For an odd function, it is relative to the point of origin, that is, O. Thus, an odd function is characterized by central symmetry, and an even function is characterized by axial symmetry.

Central symmetry suggests that a flat figure has a second order. In this case, the axis will lie perpendicular to the plane.

The central one is quite common. Among the variety of forms, the most perfect examples can be found in abundance. Such specimens that attract the eye include various types of plants, mollusks, insects, and many animals. A person admires the beauty of individual flowers and petals, he is surprised by the ideal structure of the honeycomb, the arrangement of seeds on the sunflower cap, and leaves on the plant stem. Central symmetry is found everywhere in life.

Definition of symmetry: Two points A and A1 are called symmetrical relative to point O if O is the middle of the segment AA1. Point O is considered symmetrical to itself. Definition of symmetry: Two points A and A1 are called symmetrical relative to point O if O is the middle of the segment AA1. Point O is considered symmetrical to itself.

For example: In the figure, points M and M1, N and N1 are symmetrical relative to point O, and points P and Q are not symmetrical relative to this point. For example: In the figure, points M and M1, N and N1 are symmetrical relative to point O, but points P and Q are not symmetrical about this point

Definition of central symmetry: A figure is called symmetrical relative to point O if, for each point of the figure, a point symmetrical to it relative to point O also belongs to this figure. Point O is called the center of symmetry of the figure. The figure is also said to have central symmetry. 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. Point O is called the center of symmetry of the figure. The figure is also said to have central symmetry.

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Presentation for the lesson

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Goals and objectives:

• improving knowledge about axial symmetry;
• introduce the concept of central symmetry;
• teach to recognize figures with axial symmetry and central symmetry;
• improving knowledge and skills when working with drawing and measuring instruments;
• develop spatial imagination, design skills and creativity;
• promote the development of interest in technical creativity;
• broadening your horizons.

Materials and tools:

• Teacher's computer (laptop), multimedia projector, screen; slide presentation for the lesson; compass for the board; student's compasses, triangles, colored cardboard and paper, scissors, glue.

Lesson plan:

Organizational part (preparation for work).

Updating basic knowledge.

Repetition of geometric material.

Practical work, explanation and demonstration of basic methods of performing work, competitions.

Summing up the lesson, discussing the work done.

Cleaning workplaces.

Progress of the lesson

Organizing time. Checking readiness for class.

Task No. 1. "Divide the triangle" Slide 2

ANSWER (Fig. 2):

rice. 2

Divide the equilateral triangle shown in the figure as follows:

1. Three lines into four equal parts.

2. Three lines into six equal parts.

3. Three lines into three equal parts.

4. One line into four arbitrary parts

Task No. 2. Slide 3

In a square of 6 by 6 cells, draw a geometric pattern, after 2 two columns of cells, repeat it to the end of the sheet.

In ancient times, the word "SYMMETRY" was used to mean "harmony", "beauty". Indeed, translated from Greek this word means “proportionality, proportionality, uniformity in the arrangement of parts.”

We encounter symmetry everywhere - in nature, technology, art, science. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human development. Man has long used symmetry in architecture. It gives harmony and completeness to ancient temples, towers of medieval castles, and modern buildings. What is symmetry? Why does symmetry literally permeate the entire world around us?

We will consider the symmetry that can be directly seen - the symmetry of positions, shapes, structures. It can be called geometric symmetry.

AXIAL SYMMETRY Slide 4

An isosceles (but not equilateral) triangle also has one line symmetry. A equilateral triangle - three lines symmetry.

U unexpanded of an angle there is one line of symmetry - a straight line on which the bisector of the angle is located.

A rectangle and a rhombus that are not squares have two lines of symmetry, A square - four lines of symmetry.

Speech "Mirror (axial) symmetry" Appendix No. 1

Find figures that have a line of symmetry (Task No. 1) Appendix No. 2

CENTRAL SYMMETRY Slide 8

The simplest figures with central symmetry are the circle and parallelogram.

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, a straight line has an infinite number of them - any point on a straight line is its center of symmetry.

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

Find figures with central symmetry (Task No. 2) Appendix No. 2

Find figures that have both axes of symmetry (Task No. 3) Appendix No. 2

Speech "Symmetry in Letters" Appendix No. 3

Once - hands waved up
And at the same time we sighed
Two or three bent down and reached the floor
And four - stood up straight and repeat first.
The air we breathe in is strong
When bending, exhale in a friendly manner
But you don't need to bend your knees.
So that your hands don't get tired,
We'll put them on our belts.
We jump like balls
Girls and boys.

Practical work "Flying saucer" Appendix No. 5

What geometric body does a flying saucer resemble? (cylinder)

What tool will we use? (compass)

Safety rules when working with compasses.

Now we begin practical work (Fig. 10):

1. To make a flying saucer we use cardboard of any color.
2. On the wrong side of the cardboard we draw a circle R55 (1 piece) and R36 (2 pieces).
3. Along the length of the cardboard we lay out a rectangle 220 mm long and 12 mm wide (we mark the valves along the length).
4. Cut out all the details.
5. We glue parts No. 2 and No. 3, we get a cylinder.
6. Glue the cylinder to part No. 1
7. The result was "Flying Saucer".
8. Design according to your own design.
9. Competitions.
10. Summarizing

Lesson summary

Today in class we repeated and studied axial and central symmetries.

• How many axes of symmetry does a line segment have? (2 each).
• Do a line segment, a straight line, or a square have a center of symmetry? (2 each)
• Which of these letters have an axis of symmetry? (M, A, N, E)
• Which of these letters have a center of symmetry? (BUT) Appendix No. 6

Everything is correct.

Today everyone did a good job and figured out symmetry, but if anyone still doubts, I have prepared this hint for you

Awarding and congratulations to the winners of the competition.

Cleaning workplaces.

Literature.

1. Tarasov L. This amazing symmetrical world. M., 1982
2. Sharygin I.F., Erganzhieva L.N. Visual geometry. M., 1995
3. Internet resources.

CENTER OF SYMMETRY - 1. According to Fedorov (1901), the point of intersection of the elements of symmetry in a given figure. 2. Syn. term inversion center.

Geological Dictionary: in 2 volumes. - M.: Nedra. Edited by K. N. Paffengoltz et al.. 1978 .

See what "CENTER OF SYMMETRY" is in other dictionaries:

center of symmetry- - [English-Russian gemological dictionary. Krasnoyarsk, KrasBerry. 2007.] Topics: gemology and jewelry production EN center of symmetry ... Technical Translator's Guide

center of symmetry- simetrijos centras statusas T sritis Standartizacija ir metrologija apibrėžtis Figūros taškas, iš kurio išeinantis bet kuris vektorius gali turėti priešingą vektorių. atitikmenys: engl. symmetry center vok. Symmetriezentrum, n rus. center… … Penkiakalbis aiškinamasis metrologijos terminų žodynas

center of symmetry- simetrijos centras statusas T sritis chemija apibrėžtis Figūros taškas, iš kurio išeinantis bet kuris vektorius gali turėti priešingą vektorių. atitikmenys: engl. symmetry center rus. center of symmetry ryšiai: sinonimas – inversijos centras… … Chemijos terminų aiškinamasis žodynas

center of symmetry- simetrijos centras statusas T sritis fizika atitikmenys: engl. center of symmetry; symmetry center vok. Symmetriezentrum, n rus. center of symmetry, m pranc. center de symétrie, m … Fizikos terminų žodynas

A point invariably associated with a solid body through which the resultant of the gravitational forces acting on the particles of this body passes at any position of the body in space. For a homogeneous body that has a center of symmetry (circle, ball, cube, etc.),... ... encyclopedic Dictionary

A; m. [from Greek. kentron tip, focus] 1. Math., physics. The point of intersection of which l. axes, lines in a figure, the point of concentration of which l. relationships, forces in the body. C. lenses. C. circle. C. symmetry. C. gravity (also; the most basic, essence).... ... encyclopedic Dictionary

Geom. a point invariably associated with a solid body through which the resultant force of all gravitational forces acting on the particles of the body passes through it at any position in space; it may not coincide with any of the points of a given body (for example, at ... ... Physical encyclopedia

A point invariably associated with a solid body through which the resultant of the gravitational forces acting on the particles of this body passes at any position of the body in space. For a homogeneous body that has a center of symmetry (circle, ball, cube, etc.),... ... Big Encyclopedic Dictionary

Center of gravity- CENTER OF GRAVITY, the point through which the resultant of the forces of gravity acting on the particles of a solid body passes at any position of the body in space. For a homogeneous body that has a center of symmetry (circle, ball, cube, etc.), the center of gravity is ... Illustrated Encyclopedic Dictionary

In a circle, a special point inside a figure, characterized by the fact that any straight line drawn through it on both sides of it and at equal distances will meet identical (corresponding) points of the figure. If there is C. and. each face corresponds to another face,... ... Geological encyclopedia

Books

• , S.A. Chaplygin. In 1939, it was 50 years since the Paris Academy of Sciences awarded S.V. Kovalevskaya’s memoir about the motion of a rigid body with a fixed point. As you know, for the first time the task was...
• The motion of a rigid body around a fixed point. , S.A. Chaplygin. “In 1939, it was 50 years since the Paris Academy of Sciences awarded S.V. Kovalevskaya’s memoir about the motion of a rigid body with a fixed point. As you know, for the first time the task...