Video lesson “Rotation and central symmetry. Central and axial symmetry

Concept symmetry runs 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 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 an angle, isosceles triangle, rectangle, rhombus (Figure 2.3).

A figure can 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 said to be 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 so-called rotational symmetry. If you rotate 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. Note that the letter “” also has rotational symmetry. 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 "Grade 8 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, sameness in the arrangement of parts of something on opposite sides of a point, straight 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 basis of the natural scientific picture of the 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 on opposite sides of it at the same distance. 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, isosceles and equilateral triangles, a rectangle and a rhombus have axial symmetry.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 a 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 every symmetrical plane figure can be aligned with itself using a mirror. It is surprising that such complex figures as 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

The well-known 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, in one way or another, consist of a circle, through an infinite number of symmetry axes there are countless symmetry planes. 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, N.Q.), 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 of the use of axial and portable symmetry is the fence shown in the photograph.

Conclusion: Thus, there are different types of symmetry, 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:

Handbook of 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/

§ 1. Rotation and central symmetry - Textbook on Mathematics, grade 6 (Zubareva, Mordkovich)

Short description:

In this section we move on to study new topic in geometry: rotation and central symmetry. Which will help us understand what rotation is in a geometric sense, how to rotate points, segments or entire figures, as well as which points of segments or figures can be considered symmetrical.
Rotation of a point can be considered the movement of a point in a circle around another point on a plane, while the other point remains motionless. The rotation can be made to any distance; such a distance is measured in degrees; it can be measured using a protractor. In addition to points, entire figures and pictures can be moved. So, we can see many examples of the use of turns in real life– symmetrical plants, flowers, fruits, cut in half, building elements, For example, spiral staircases, shoes - right and left shoes. So, the stars rotate around the pole, changing their position only relative to one point. To geometrically construct a rotation, it is convenient to use a compass and a protractor. Symmetry can be defined as the equally distant arrangement of points relative to one center. IN Everyday life we often encounter symmetrical objects. But it is worth noting that perfect symmetry does not exist in nature; even a person’s face cannot be perfectly symmetrical. But the objects that we use for everyday activities, cooking, preparing homework, playing, are most often symmetrical. Interesting? We invite you to familiarize yourself with the material in the paragraph in the textbook in more detail!

When studying the topic “Rotation”, students are given the task: draw a figure on a landscape sheet, select the center of rotation and the angle of rotation. Construct a new figure. The working technique may be different. For example, children often use appslication. At our virtual exhibition, the second work was made using this technique.But in picture 3, the student used a ready-made image (applique) and drew the second moving figure independently.

Particularly interesting are works done with pencils, felt-tip pens or paints. Of course, when compiling these works, children firstmade a template. This stencil template helped them complete creative works on other topics "Symmetry about a line", "Symmetry about a point", "Parallel transfer".

Children especially enjoy making dynamic models. They can be twisted and rotated clockwise and counterclockwise. At the presented exhibition there is only one static work in the first drawing. The rest of the work is dynamic.

To make a dynamic model, one figure must be drawn on a landscape sheet. Cut out the second figure using a template from white cardboard. Some guys also covered the second movable figure with colorless film for greater reliability. For example, a beautiful fish in the top row. She is already more than 10 years old, but she looks like new. The bright colors did not fade or fade. To mark the center, students use a small round dot made of cardboard, attach the movable figure to the album sheet using ordinary sewing threads. Some children used metal nuts. True, this option does not look very aesthetically pleasing.

Among the best works on the topic “Turning” there are works made on plywood using a burning device. Among them there are moving models and static drawings. For dynamic models, a much larger amount of work needs to be done, because the moving figure needs to be cut out. What labor-intensive work!

Best works are displayed on a stand in the classroom. And works on plywood are in cabinets. After the Exhibition in the office, I archive creative works in thematic folders; they replenish the methodological base of the office. This folder is presented at Exhibitions in the gymnasium, held as part of various methodological events and seminars. For example, an exhibition of creative works by students as part of the Day open doors in the gymnasium, to which parents of students are traditionally invited.

Back forward

Attention! Slide previews are for informational purposes only and may not represent all the features of the presentation. If you are interested this work, please download the full version.

Lesson objectives:

Equipment: interactive whiteboard, lesson presentation.

Lesson plan.

1. Organizing time.
2. Repeating operations with decimals.
3. Studying new material, initial consolidation.
4. Lesson summary, homework.

During the classes

1. Organizational moment.

A message about the requirements for the lesson, the necessary tools and aids.

What does mathematics study in 6th grade?

2. Repetition.

1) Remember the rules for working with decimal fractions, give examples.

2) Mental arithmetic (using the “Mathematical simulator”, grade 6, p. 10, task for ID).

3) Paperwork No. 14, 15 on the first line in each issue (1 student at the board, if desired, works for a grade).

№14 a) 2, 31+ 15, 7= 18, 01

c) 4, 327 – 2, 05 = 2, 277

e) 15.6 + 0.671 = 16, 271

№15 a) 91.05 3.2 = 291, 36

c) 268.8: 5.6 = 48

e) 7.02 0.0055 = 0, 03861

3. Studying new material.

The topic of our lesson is “Rotation and central symmetry” (Slide 1)

Geometry deals with issues related to the movement of figures. Today we will learn about rotation and central symmetry.

1) Take points O and A on the plane. Rotate point A around point O by a certain angle. Point A will go to point A 1. (Slide 2). Let's make the same construction in a notebook, fill in the gaps in the text.

In this case, point O (fixed point) will be the center of rotation, point A will be a moving point, and the angle of rotation will be angle AOA 1. Rotation can be either clockwise or counterclockwise.

Thus we can give the definition of rotation:

Def. Rotation (rotation) - a movement in which at least one point of the plane remains stationary (mouse click).

2) Look at the drawing (mouse click). The rotations of the points are also shown here. Describe this drawing and determine by what angle the point rotates in each case. For which point can the angle of rotation be determined without a protractor? Describe the location of the start and end points relative to the center. (Oral work on Figure 2 from the textbook)

3) Rotation is a natural process occurring in nature, the world around us.

Look at the pictures and characterize each turn. (Slide 3, 4)

4) Let’s complete task No. 1 in writing. (Slide 5)

Construct the image of a segment MN = 4 cm when rotated through an angle of 90° around point O clockwise.

(The algorithm for performing the rotation is discussed and the construction in notebooks is carried out step by step along with the animation. The teacher monitors the completion of tasks and provides the necessary assistance).

Compare the segments MN and M 1 N 1.

5) On the next slide you see various ornaments (Slide 6). They all consist of identically repeating elements. List these items. Pay attention to fragments of ornaments b), d), f), g). What do they have in common? (Each of them can be obtained from another part by rotating 180° relative to some point).

6) Consider the following turn. (Slide 7)

Let's mark points O and A on the plane and draw a straight line AO. On this line, let us plot from point O a segment OA 1, equal to the segment AO, but on the other side of point O. We obtain the unfolded angle AOA 1. This means that point A 1 can be obtained by rotating point A 180° around point O. Points A and A 1 are called symmetrical relative to point O, and point O is called the center of symmetry.

Consider a drawing of yellow and red fish. They are symmetrical about point O.

Def. Figures that are symmetrical about a point are called centrally symmetrical figures.

How are centrally symmetrical points located relative to the center of symmetry?

(Lie on the same straight line with the center of symmetry)

7) Orally No. 1 page 7 fig. 7. (Slide 8). Indicate the center of symmetry and some pairs of centrally symmetrical points.

(The slide runs as usual or the picture is displayed on interactive whiteboard, so that the necessary construction can be performed).

8) Orally ( Slide 9). Indicate which figures in the pictures have a center of symmetry.

4. Lesson summary.

Answer the questions:

• How did you understand what a turn is?
• How to use rotation to obtain centrally symmetrical points?
• How to construct centrally symmetric points?