Points X And X" are called symmetrical relatively straight a, and each of them is symmetrical to the other, if a is the seridine perpendicular of the segment XX". Each point of the line a is considered symmetrical to itself (relative to the line a). If a line a is given, then each point X corresponds to a single point X", symmetrical to X relative to a.

Symmetry plane relatively straight a called such display, at which each point this plane is put V correspondence dot, symmetrical to her relatively straight a.

Let's prove that axial symmetry is a movement using the coordinate method: take straight line a as the x axis Cartesian coordinates. Then, with symmetry about it, a point with coordinates (x;y) will be transformed into a point with coordinates (x, -y).

Let's take any two points A(x1, y1) and B(x2, y2) and consider the points A"(x1,- y1) and B"(x2, -y2) that are symmetrical to them with respect to the x-axis. Calculating the distances A"B" and AB, we get

Thus, axial symmetry preserves distance, therefore it is movement.

Turn

Turn plane relatively center O on the corner () V given direction is defined as follows: each point X of the plane is assigned a corresponding point X" such that, firstly, OX" = OX, secondly and thirdly, the ray OX" is delayed from the ray OX in in this direction. Point O is called center turning, and the angle is angle turning.

Let us prove that rotation is a movement:

Let, when rotating around point O, points X and Y be associated with points X" and Y". Let us show that X"Y"=XY.

Let's consider general case, when points O, X, Y do not lie on the same line. Then angle X"OY" equal to angle XOY. Indeed, let the angle XOY from OX to OY be measured in the direction of rotation. (If this is not the case, then consider the YOX angle). Then the angle between OX and OY" equal to the sum XOY angle and rotation angle (from OY to OY"):

on the other side,

Since (as rotation angles), therefore. In addition, OX"=OX, and OY"=OY. Therefore - on two sides and the angle between them. Therefore X"Y"=XY.

If points O, X, Y lie on the same line, then the segments XY and X"Y" will be either the sum or the difference equal segments OX, OY and OX", OY". Therefore, in this case X"Y"=XY. So turning is movement.

Concept symmetry runs through the entire history of mankind. It is found already at the origins human knowledge. It arose in connection with the study of a living organism, namely man. And it was used by sculptors back in the 5th century BC. Word " symmetry "Greek, it means " proportionality, proportionality, uniformity in the arrangement of parts”.

1.1. Axial symmetry

Two points A and A1 are called symmetrical with respect to line a if this line passes through the middle of segment AA1 and is perpendicular to it (Figure 2.1). Each point of a line a is considered symmetrical to itself.

A figure is called symmetrical with respect to line a if, for each point of the figure, a point symmetrical with respect to line a also belongs to this figure (Figure 2.2).

Straight line a is called the axis of symmetry of the figure.


The figure is also said to have axial symmetry.

The following have axial symmetry geometric figures like a corner isosceles triangle, rectangle, rhombus (Figure 2.3).

A figure may have more than one axis of symmetry. A rectangle has two, a square has four, an equilateral triangle has three, a circle has any straight line passing through its center.

If you look closely at the letters of the alphabet (Figure 2.4), then among them you can find those that have horizontal or vertical, and sometimes both, axes of symmetry. Objects with axes of symmetry are quite often found in living and inanimate nature.

There are figures that do not have a single axis of symmetry. Such figures include a parallelogram, different from a rectangle, and a scalene triangle.

In his activity, a person creates many objects (including ornaments) that have several axes of symmetry.

1.2 Central symmetry

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

The simplest figures with central symmetry are the circle and parallelogram (Figure 2.6).

Point O is called the center of symmetry of the figure. IN similar cases the figure has central symmetry. The center of symmetry of a circle is the center of the circle, and the center of symmetry of a parallelogram is the point of intersection of its diagonals.

A straight line also has central symmetry, but unlike a circle and a parallelogram, which have only one center of symmetry, 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 a triangle.

1.3. Rotational symmetry

Suppose that an object is aligned with itself when rotated around a certain axis through an angle equal to 360°/n (or a multiple of this value), where n = 2, 3, 4, ... In this case, about rotational symmetry, and the specified axis is called rotary nth order axis.

Let's look at examples with all the known letters " AND" And " F" Regarding the letter " AND", then it has the so-called rotational symmetry. If you turn the letter " AND» 180° around an axis perpendicular to the plane of the letter and passing through its center, then the letter will align with itself.

In other words, the letter " AND» symmetrical with respect to 180° rotation. notice, that rotational symmetry also has the letter " F».

In Figure 2.7. examples of simple objects with rotary axes of different orders are given - from 2nd to 5th.

Scientific and practical conference

Municipal educational institution "Secondary" comprehensive school No. 23"

city ​​of Vologda

section: natural science

design and research work

TYPES OF SYMMETRY

The work was completed by an 8th grade student

Kreneva Margarita

Head: higher mathematics teacher

year 2014

Project structure:

1. Introduction.

2. Goals and objectives of the project.

3. Types of symmetry:

3.1. Central symmetry;

3.2. Axial symmetry;

3.3. Mirror symmetry(symmetry relative to the plane);

3.4. Rotational symmetry;

3.5. Portable symmetry.

4. Conclusions.

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

G. Weil

Introduction.

The topic of my work was chosen after studying the section “Axial and central symmetry” in the course “8th grade Geometry”. I was very interested in this topic. I wanted to know: what types of symmetry exist, how they differ from each other, what are the principles for constructing symmetrical figures in each type.

Goal of the work : Introduction to different types of symmetry.

Tasks:

Study the literature on this issue.

Summarize and systematize the studied material.

Prepare a presentation.

In ancient times, the word “SYMMETRY” was used to mean “harmony”, “beauty”. Translated from Greek, this word means “proportionality, proportionality, uniformity in the arrangement of parts of something according to opposite sides from a point, line or plane.

There are two groups of symmetries.

The first group includes symmetry of positions, shapes, structures. This is the symmetry that can be directly seen. It can be called geometric symmetry.

The second group characterizes symmetry physical phenomena and the laws of nature. This symmetry lies at the very core natural science picture world: it can be called physical symmetry.

I'll stop studyinggeometric symmetry .

In turn, there are also several types of geometric symmetry: central, axial, mirror (symmetry relative to the plane), radial (or rotary), portable and others. Today I will look at 5 types of symmetry.

Central symmetry

Two points A and A 1 are called symmetrical with respect to point O if they lie on a straight line passing through point O and are located along different sides at the same distance from it. Point O is called the center of symmetry.

The figure is said to be symmetrical about the pointABOUT , if for each point of the figure there is a point symmetrical to it relative to the pointABOUT also belongs to this figure. DotABOUT called the center of symmetry of a figure, the figure is said to have central symmetry.

Examples of figures with central symmetry are a circle and a parallelogram.

The figures shown on the slide are symmetrical relative to a certain point

2. Axial symmetry

Two pointsX And Y are called symmetrical about a straight linet , if this line passes through the middle of the segment XY and is perpendicular to it. It should also be said that each point is a straight linet is considered symmetrical to itself.

Straightt – axis of symmetry.

The figure is said to be symmetrical about a straight linet, if for each point of the figure there is a point symmetrical to it relative to the straight linet also belongs to this figure.

Straighttcalled the axis of symmetry of a figure, the figure is said to have axial symmetry.

An undeveloped angle, an isosceles angle, and an angle have axial symmetry. equilateral triangles, rectangle and rhombus,letters (see presentation).

Mirror symmetry (symmetry about a plane)

Two points P 1 And P are called symmetrical relative to the plane a if they lie on a straight line perpendicular to the plane a and are at the same distance from it

Mirror symmetry well known to every person. It connects any object and its reflection in flat mirror. They say that one figure is mirror symmetrical to another.

On a plane, a figure with countless axes of symmetry was a circle. In space, a ball has countless planes of symmetry.

But if a circle is one of a kind, then in the three-dimensional world there is a whole series of bodies with an infinite number of planes of symmetry: a straight cylinder with a circle at the base, a cone with a circular base, a ball.

It is easy to establish that each is symmetrical flat figure can be aligned with itself using a mirror. It is surprising that such complex figures as a five-pointed star or an equilateral pentagon are also symmetrical. As this follows from the number of axes, they are distinguished by high symmetry. And vice versa: it is not so easy to understand why such a seemingly correct figure, like an oblique parallelogram, is asymmetrical.

4. P rotational symmetry (or radial symmetry)

Rotational symmetry - this is symmetry, the preservation of the shape of an objectwhen rotating around a certain axis through an angle equal to 360°/n(or a multiple of this value), wheren= 2, 3, 4, … The indicated axis is called the rotary axisn-th order.

Atn=2 all points of the figure are rotated through an angle of 180 0 ( 360 0 /2 = 180 0 ) around the axis, while the shape of the figure is preserved, i.e. each point of the figure goes to a point of the same figure (the figure transforms into itself). The axis is called the second-order axis.

Figure 2 shows a third-order axis, Figure 3 - 4th order, Figure 4 - 5th order.

An object can have more than one rotation axis: Fig. 1 - 3 axes of rotation, Fig. 2 - 4 axes, Fig. 3 - 5 axes, Fig. 4 – only 1 axis

Everyone famous letters“I” and “F” have rotational symmetry. If you rotate the letter “I” 180° around an axis perpendicular to the plane of the letter and passing through its center, the letter will align with itself. In other words, the letter “I” is symmetrical with respect to a rotation of 180°, 180°= 360°: 2,n=2, which means it has second-order symmetry.

Note that the letter “F” also has second-order rotational symmetry.

In addition, the letter has a center of symmetry, and the letter F has an axis of symmetry

Let's return to examples from life: a glass, a cone-shaped pound of ice cream, a piece of wire, a pipe.

If we take a closer look at these bodies, we will notice that all of them, one way or another, consist of a circle, through infinite set whose axes of symmetry pass through countless planes of symmetry. Most of these bodies (they are called bodies of rotation) also have, of course, a center of symmetry (the center of a circle), through which at least one rotational axis of symmetry passes.

For example, the axis of the ice cream cone is clearly visible. It runs from the middle of the circle (sticking out of the ice cream!) to the sharp end of the funnel cone. We perceive the totality of symmetry elements of a body as a kind of symmetry measure. The ball, without a doubt, in terms of symmetry, is an unsurpassed embodiment of perfection, an ideal. The ancient Greeks perceived it as the most perfect body, and the circle, naturally, as the most perfect flat figure.

To describe the symmetry of a particular object, it is necessary to indicate all the rotation axes and their order, as well as all planes of symmetry.

Consider, for example, geometric body, composed of two identical regular quadrangular pyramids.

It has one rotary axis of the 4th order (axis AB), four rotary axes of the 2nd order (axes CE,DF, MP, NQ), five planes of symmetry (planesCDEF, AFBD, ACBE, AMBP, ANBQ).

5 . Portable symmetry

Another type of symmetry isportable With symmetry.

Such symmetry is spoken of when, when moving a figure along a straight line to some distance “a” or a distance that is a multiple of this value, it coincides with itself The straight line along which the transfer occurs is called the transfer axis, and the distance “a” is called the elementary transfer, period or symmetry step.

A

A periodically repeating pattern on a long strip is called a border. In practice, borders are found in various forms (wall painting, cast iron, plaster bas-reliefs or ceramics). Borders are used by painters and artists when decorating a room. To make these ornaments, a stencil is made. We move the stencil, turning it over or not, tracing the outline, repeating the pattern, and we get an ornament (visual demonstration).

The border is easy to build using a stencil (the starting element), moving or turning it over and repeating the pattern. The figure shows five types of stencils:A ) asymmetrical;b, c ) having one axis of symmetry: horizontal or vertical;G ) centrally symmetrical;d ) having two axes of symmetry: vertical and horizontal.

To construct borders, the following transformations are used:

A ) parallel transfer;b ) symmetry about the vertical axis;V ) central symmetry;G ) symmetry about the horizontal axis.

You can build sockets in the same way. To do this, the circle is divided inton equal sectors, in one of them a sample pattern is made and then the latter is sequentially repeated in the remaining parts of the circle, rotating the pattern each time by an angle of 360°/n .

A clear example The fence shown in the photograph can serve as an application of axial and portable symmetry.

Conclusion: Thus, there are different kinds symmetries, symmetrical points in each of these types of symmetry are constructed according to certain laws. In life, we encounter one type of symmetry everywhere, and often in the objects that surround us, several types of symmetry can be noted at once. This creates order, beauty and perfection in the world around us.

LITERATURE:

Guide to elementary mathematics. M.Ya. Vygodsky. – Publishing house “Nauka”. – Moscow 1971 – 416 pages.

Modern dictionary foreign words. - M.: Russian language, 1993.

History of mathematics in schoolIX - Xclasses. G.I. Glaser. – Publishing house “Prosveshcheniye”. – Moscow 1983 – 351 pages.

Visual geometry 5th – 6th grades. I.F. Sharygin, L.N. Erganzhieva. – Publishing house “Drofa”, Moscow 2005. – 189 pages

Encyclopedia for children. Biology. S. Ismailova. – Avanta+ Publishing House. – Moscow 1997 – 704 pages.

Urmantsev Yu.A. Symmetry of nature and the nature of symmetry - M.: Mysl arxitekt / arhkomp2. htm, , ru.wikipedia.org/wiki/

Class hour in 9th grade, strategy "Advanced lecture»

Axial and central symmetry, parallel translation,
rotation - like plane movements

Buyakova Elena Valerievna

Target: show different ways to define the equation of a line and general equation straight.

Tasks:

1) become familiar with concepts such as direction vector and line normal vector;

2) show four different ways to specify the equation of a line;

3) show interchangeability in various ways direct tasks.

During the classes.

1. Lesson topic. Dividing the class into pairs.

2. Instructions on reading the text (Appendix 1) and doing the work

Reading and filling are carried out individually. The text is divided into two parts.

The first number of the pair checks the correspondence of the written words to the readable text.

The second number of the pair remembers the basic facts in order to explain to the first number.

The pairs read the second part of the text, switching roles.

3. Question for the first part: What do you remember about axial and central symmetry? ?

4. Question for the second part of the text: What associations do you have with the topic “parallel transfer, rotation?” »?

Words are written on the board - associations found by each pair (without repetitions); in notebooks, students add to their lists of these words. Then the corresponding text is read.

5. Discussion in pairs.

6. Reflection - 10 minute essay on the topic “Plane movements: types and their differences”

Annex 1

Central and axial symmetry

Definition. Symmetry (means “proportionality”) is the property of geometric objects to combine with themselves under certain transformations. Under symmetry understand every correctness in internal structure bodies or figures.

Symmetry about a point is central symmetry (Fig. 23 below), and symmetry about a straight line- This is axial symmetry (Fig. 24 below).

Symmetry about a point assumes that there is something on both sides of a point at equal distances, such as other points or locus points (straight lines, curved lines, geometric shapes).

If you connect symmetrical points (points of a geometric figure) with a straight line through a symmetry point, then the symmetrical points will lie at the ends of the straight line, and the symmetry point will be its middle. If you fix the symmetry point and rotate the straight line, then the symmetrical points will describe curves, each point of which will also be symmetrical to the point of the other curved line.

Symmetry about a straight line(axis of symmetry) assumes that along a perpendicular drawn through each point of the axis of symmetry, two symmetrical points are located at the same distance from it. The same geometric figures can be located relative to the axis of symmetry (straight line) as relative to the point of symmetry.

An example would be a sheet of notebook that is folded in half if a straight line is drawn along the fold line (axis of symmetry). Each point on one half of the sheet will have a symmetrical point on the second half of the sheet if they are located at the same distance from the fold line and perpendicular to the axis.

The line of axial symmetry, as in Figure 24, is vertical, and the horizontal edges of the sheet are perpendicular to it. That is, the axis of symmetry serves as a perpendicular to the midpoints of the horizontal straight lines bounding the sheet. Symmetrical points(R and F, C and D) are located at the same distance from the axial line - perpendicular to the straight lines connecting these points. Consequently, all points of the perpendicular (axis of symmetry) drawn through the middle of the segment are equidistant from its ends; or any point perpendicular (axis of symmetry) to the middle of a segment is equidistant from the ends of this segment.

Parallel transfer

Parallel translation is a movement in which all points of the plane move in the same direction by the same distance.

Read more: parallel transfer arbitrary points planes X and Y puts in correspondence such points X" and Y" such that XX"=YY" or you can also say this: parallel translation is a mapping in which all points of the plane are moved to the same vector - translation vector. Parallel translation is specified by the translation vector: knowing this vector, you can always say to which point any point on the plane will go.

Parallel translation is a movement that preserves directions. Indeed, let the parallel transfer of points X and Y move to points X" and Y" respectively. Then the equality XX"=YY" holds. But from this equality by attribute equal vectors It follows that XY=X"Y", from which we obtain that, firstly, XY=X"Y", that is, parallel transfer is a movement, and secondly, that XY X"Y", that is, parallel transfer preserves the directions.

This property parallel transfer- his characteristic property, that is, the statement is true: movement that preserves directions is parallel translation.

Turn

Rotate the plane relative to the center O by given angle () in this direction is defined as follows: each point X of the plane is associated with a point X" such that, firstly, OX"=OX, secondly and thirdly, the ray OX" is laid off from the ray OX in a given direction. Point O is called center of rotation, and the angle is rotation angle.

Let us prove that rotation is a movement:

Let, when rotating around point O, points X and Y be associated with points X" and Y". Let us show that X"Y"=XY.

Let's consider the general case when points O, X, Y do not lie on the same line. Then angle X"OY" is equal to angle XOY. Indeed, let the angle XOY from OX to OY be measured in the direction of rotation. (If this is not the case, then consider the YOX angle). Then the angle between OX and OY" is equal to the sum of the angle XOY and the angle of rotation (from OY to OY"):

on the other side,

Since (as rotation angles), therefore . In addition, OX"=OX, and OY"=OY. Therefore - on two sides and the angle between them. Therefore X"Y"=XY.

If the points O, X, Y lie on the same line, then the segments XY and X"Y" will be either the sum or the difference of equal segments OX, OY and OX", OY". Therefore, in this case X"Y"=XY. So turning is movement.

(means “proportionality”) - the property of geometric objects to be combined with themselves under certain transformations. By “symmetry” we mean any regularity in the internal structure of the body or figure.

Central symmetry— symmetry about a point.

relative to the point O, if for each point of a 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.

IN one-dimensional space (on a straight line) central symmetry is mirror symmetry.

On a plane (in 2-dimensional space) symmetry with center A is a rotation of 180 degrees with center A. Central symmetry on a plane, like rotation, preserves orientation.

Central symmetry in three-dimensional space is also called spherical symmetry. It can be represented as a composition of reflection relative to a plane passing through the center of symmetry, with a rotation of 180° relative to a straight line passing through the center of symmetry and perpendicular to the above-mentioned plane of reflection.

IN 4-dimensional space, central symmetry can be represented as a composition of two 180° rotations around two mutually perpendicular planes, passing through the center of symmetry.

Axial symmetry- symmetry relative to a straight line.

The figure is called symmetrical relatively straight a, if for each point of a figure a point symmetrical to it relative to the straight line and also belongs to this figure. Straight line a is called the axis of symmetry of the figure.

Axial symmetry has two definitions:

- Reflective symmetry.

In mathematics, axial symmetry is a type of motion ( mirror reflection), in which the set fixed points is a straight line called the axis of symmetry. For example, a flat rectangle is asymmetrical in space and has 3 axes of symmetry, if it is not a square.

- Rotational symmetry.

IN natural sciences By axial symmetry we mean rotational symmetry, relative to rotations around a straight line. In this case, bodies are called axisymmetric if they transform into themselves at any rotation around this straight line. In this case, the rectangle will not be an axisymmetric body, but the cone will be.

Images on a plane of many objects in the world around us have an axis of symmetry or a center of symmetry. Many tree leaves and flower petals are symmetrical about the average stem.

We often encounter symmetry in art, architecture, technology, and everyday life. The facades of many buildings have axial symmetry. In most cases, patterns on carpets, fabrics, and indoor wallpaper are symmetrical about the axis or center. Many parts of mechanisms, such as gears, are symmetrical.