Axial symmetry is an unusual, complex pattern. Project "Types of Symmetry"

Axial symmetry. With axial symmetry, each point of the figure goes to a point that is symmetrical to it relative to a fixed straight line.

Dimensions: 360 x 260 pixels, format: jpg.

To display pictures in the lesson, you can also download the entire presentation “Ornament.ppt” with all the pictures in a zip archive for free. The archive size is 3324 KB.

Download presentation

Symmetry

“Point of symmetry” - Central symmetry. A a A1. Axial and central symmetry. Point C is called the center of symmetry. Symmetry in everyday life. A circular cone has axial symmetry; the axis of symmetry is the axis of the cone. Figures that have more than two axes of symmetry. A parallelogram has only central symmetry.

“Mathematical symmetry” - What is symmetry? Physical symmetry. Symmetry in biology. History of symmetry. However, complex molecules generally lack symmetry. Palindromes. Symmetry. In x and m and i. HAS A LOT IN COMMON WITH PROGRESSAL SYMMETRY IN MATHEMATICS. But actually, how would we live without symmetry? Axial symmetry. “Ornament” - b) On the strip. Parallel translation Central symmetry Axial symmetry Rotation. Linear (arrangement options): Creating a pattern using central symmetry and parallel translation. Planar. One of the varieties of ornament is a mesh ornament. Transformations used to create an ornament:“Symmetry in Nature” - One of the main properties of geometric shapes is symmetry. The topic was not chosen by chance, because in next year We have to start studying a new subject - geometry. The phenomenon of symmetry in living nature was noticed back in

Ancient Greece . We study 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 movement? What sciences does motion apply to? How is movement used in

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

There are a total of 32 presentations in the topic

Goals:

• educational:
  • give an idea of ​​symmetry;
  • introduce the main types of symmetry on the plane and in space;
  • develop strong skills in constructing symmetrical figures;
  • expand your understanding of famous figures by introducing properties associated with symmetry;
  • show the possibilities of using symmetry in solving various problems;
  • consolidate acquired knowledge;
• general education:
  • teach yourself how to prepare yourself for work;
  • teach how to control yourself and your desk neighbor;
  • teach to evaluate yourself and your desk neighbor;
• developing:
  • intensify independent activity;
  • develop cognitive activity;
  • learn to summarize and systematize the information received;
• educational:
  • develop a “shoulder sense” in students;
  • cultivate communication skills;
  • instill a culture of communication.

DURING THE CLASSES

In front of each person 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 figure. Now let's unfold the sheet and look at the fold line.

Question: What function does this line serve?

Suggested answer: This line divides the figure in half.

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

Suggested answer: All points of the halves are at an equal 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 relative to it are at the same distance), this line is an axis of symmetry.

Task 2 (2 minutes).

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

Task 3 (5 minutes).

– Draw a circle in your notebook.

Question: Determine how the axis of symmetry goes?

Suggested answer: Differently.

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

Suggested answer: A lot of.

– That’s right, a circle has many axes of symmetry. An equally remarkable figure is a ball (spatial figure)

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

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

– Let’s consider volumetric figures: cube, pyramid, cone, cylinder, etc. These figures also have an axis of symmetry. Determine how many axes of symmetry do the square, rectangle, equilateral triangle and the proposed three-dimensional figures have?

I distribute halves of plasticine figures to students.

Task 4 (3 min).

– Using the information received, complete the missing part of the figure.

Note: the figure can be both planar and three-dimensional. It is important that students determine how the axis of symmetry runs and complete the missing element. The correctness of the work is determined by the neighbor at the desk and evaluates how correctly the work was done.

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

Task 5 (group work 5 minutes).

– Visually determine the axis of symmetry and, relative to it, complete the second part from a lace of a different color.

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

Elements of drawings are presented to students

Task 6 (2 minutes).

– Find the symmetrical parts of these drawings.

To consolidate the material covered, I suggest the following tasks, scheduled for 15 minutes:

Name all equal elements of the triangle KOR and KOM. What type of triangles are these?

2. Draw several isosceles triangles in your notebook with common ground equal to 6 cm.

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

– Our initial ideas about form date back to the very distant era of the ancient Stone Age - the Paleolithic. For hundreds of thousands of years of this period, people lived in caves, in conditions little different from the life of animals. People made tools for hunting and fishing, developed a language to communicate with each other, and during the late Paleolithic era they embellished their existence by creating works of art, figurines and drawings that reveal a remarkable sense of form.
When there was a transition from simple gathering of food to its active production, from hunting and fishing to agriculture, humanity entered a new Stone Age, the Neolithic.
Neolithic man had a keen sense of geometric form. Firing and painting clay vessels, making reed mats, baskets, fabrics, and later metal processing developed ideas about planar and spatial figures. Neolithic patterns were pleasing to the eye, revealing equality and symmetry.
– Where does symmetry occur in nature?

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

– Symmetry can also be observed in architecture. When constructing buildings, builders strictly adhere to symmetry.

That's why the buildings turn out so beautiful. Also an example of symmetry is humans and animals.

Homework:

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

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

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 of 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, axial symmetry is understood as 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.

Homothety and similarity.Homothety is a transformation in which each point M (plane or space) is assigned to a point M", lying on OM (Fig. 5.16), and the ratio OM":OM= λ the same for all points other than ABOUT. Fixed point ABOUT called the center of homothety. Attitude OM": OM considered positive if M" and M lie on one side of ABOUT, negative - by different sides. Number X called the homothety coefficient. At X< 0 homothety is called inverse. Atλ = - 1 homothety turns into a symmetry transformation about a point ABOUT. With homothety, a straight line goes into a straight line, the parallelism of straight lines and planes is preserved, angles (linear and dihedral) are preserved, each figure goes into it similar (Fig. 5.17).

The converse is also true. A homothety can be defined as an affine transformation in which the lines connecting the corresponding points pass through one point - the center of the homothety. Homothety is used to enlarge images (projection lamp, cinema).

Central and mirror symmetries.Symmetry (in the broad sense) is a property of a geometric figure F, characterizing a certain correctness of its shape, its invariability under the action of movements and reflections. A figure Φ has symmetry (symmetrical) if there are non-identical orthogonal transformations that take this figure into itself. The set of all orthogonal transformations that combine the figure Φ with itself is the group of this figure. So, a flat figure (Fig. 5.18) with a point M, transforming-

looking into yourself in the mirror reflection, symmetrical about the straight axis AB. Here the symmetry group consists of two elements - a point M converted to M".

If the figure Φ on the plane is such that rotations relative to any point ABOUT to an angle of 360°/n, where n > 2 is an integer, translate it into itself, then the figure Ф has nth-order symmetry with respect to the point ABOUT - center of symmetry. An example of such figures is regular polygons, for example star-shaped (Fig. 5.19), which has eighth-order symmetry relative to its center. The symmetry group here is the so-called nth order cyclic group. The circle has symmetry of infinite order (since it is compatible with itself by rotating through any angle).

The simplest types of spatial symmetry are central symmetry (inversion). In this case, relative to the point ABOUT the figure Ф is combined with itself after successive reflections from three mutually perpendicular planes, i.e. a point ABOUT - the middle of the segment connecting the symmetrical points F. So, for a cube (Fig. 5.20) the point ABOUT is the center of symmetry. Points M and M" cube

Scientific and practical conference

Municipal educational institution "Secondary 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 8th grade Geometry course. 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 is called the center of symmetry of a figure, they say that the figure has 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 regular 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, a 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:

Handbook of Elementary Mathematics. M.Ya. Vygodsky. – Publishing house “Nauka”. – Moscow 1971 – 416 pages.

Modern dictionary of 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/