Axial symmetry in living and inanimate nature. Symmetry in needlework

(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, indoor wallpaper. Many parts of mechanisms, such as gears, are symmetrical.

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 about a 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, 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 science 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 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, 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 whole line 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 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

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, 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 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/

MBOU "Tyukhtet Secondary School No. 1"

Scientific association of students “We want to learn actively”

physico-mathematical and technical direction

Arvinti Tatyana,

Lozhkina Maria,

MBOU "TSOSH No. 1"

5 "A" class

MBOU "TSOSH No. 1"

mathematic teacher

Introduction………………………………………………………………………………...3

I. 1. Symmetry. Types of symmetry..…………………………………………......4

I. 2. Symmetry around us………………………………………………………....6

I. 3. Axial and centrally symmetrical ornaments ….…………………………… 7

II. Symmetry in needlework

II. 1. Symmetry in knitting………………………………………………………...10

II. 2. Symmetry in origami…..……………………………………………………11

II. 3. Symmetry in beading…………………………………………………………….12

II. 4. Symmetry in embroidery………………………………………………………13

II. 5. Symmetry in crafts made from matches………………………………………………………...14

II. 6. Symmetry in Macrame weaving……………………………………………………….15

Conclusion……………………………………………………………………………….16

Bibliography………………………………………………………..17

Introduction

One of the fundamental concepts of science, which, along with the concept of “harmony”, relates to almost all structures of nature, science and art, is “symmetry”.

The outstanding mathematician Hermann Weyl highly appreciated the role of symmetry in modern science:

“Symmetry, no matter how broadly or narrowly we understand the word, is an idea with the help of which man has tried to explain and create order, beauty and perfection.”

We all admire beauty geometric shapes, their combination, looking at the pillows, knitted napkins, clothes with embroidery.

Many centuries different peoples wonderful types of decorative and applied arts were created. Many people believe that mathematics is not interesting and consists only of formulas, problems, solutions and equations. We want to show with our work that mathematics is a diverse science, and the main objective– show that mathematics is very amazing and unusual item for study, closely related to human life.

This work examines handicraft items for their symmetry.

The types of needlework we are considering are closely related to mathematics, since the works use various geometric figures that are subject to mathematical transformations. In this regard, such mathematical concepts as symmetry and types of symmetry were studied.

Purpose of the study: studying information about symmetry, searching for symmetrical handicraft items.

Research objectives:

· Theoretical: study the concepts of symmetry and its types.

· Practical: find symmetrical crafts, determine the type of symmetry.

Symmetry. Types of symmetry

Symmetry(means "proportionality") - the property of geometric objects to combine with themselves under certain transformations. Symmetry is understood as any regularity in the internal structure of the body or figure.

Symmetry about a point is central symmetry, and symmetry about a line is axial symmetry.

Symmetry about a point (central symmetry) assumes that there is something on both sides of a point at equal distances, for example other points or the locus of points (straight lines, curved lines, geometric figures). 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.

A rotation around a given point O is a movement in which each ray emanating from this point rotates through the same angle in the same direction.

Symmetry relative to 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 axis of symmetry serves as a perpendicular to the midpoints of the horizontal lines bounding the sheet. Symmetrical points 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 and equidistant from the ends of this segment.

Koll" href="/text/category/koll/" rel="bookmark">Hermitage collections special attention used gold jewelry of the ancient Scythians. The artistic work of gold wreaths, tiaras, wood and decorated with precious red-violet garnets is unusually fine.

One of the most obvious uses of the laws of symmetry in life is in architectural structures. This is what we see most often. In architecture, axes of symmetry are used as means of expressing architectural design.

Another example of a person using symmetry in his practice is technology. In engineering, symmetry axes are most clearly designated where it is necessary to estimate the deviation from the zero position, for example, on the steering wheel of a truck or on the steering wheel of a ship. Or one of the most important inventions of mankind that has a center of symmetry is the wheel; the propeller and other technical means also have a center of symmetry.

Axial and centrally symmetrical ornaments

Compositions built on the principle of a carpet ornament can have a symmetrical structure. The drawing in them is organized according to the principle of symmetry relative to one or two axes of symmetry. Carpet patterns often contain a combination of several types of symmetry - axial and central.

Figure 1 shows a diagram for marking the plane for a carpet ornament, the composition of which will be built along the axes of symmetry. On the plane along the perimeter, the location and size of the border are determined. The central field will be occupied by the main ornament.

Options for various compositional solutions of the plane are shown in Figure 1 b-d. In Figure 1 b, the composition is built in the central part of the field. Its outline may vary depending on the shape of the field itself. If the plane has the shape of an elongated rectangle, the composition is given the outline of an elongated rhombus or oval. The square shape of the field would be better supported by a composition outlined by a circle or an equilateral rhombus.

Figure 1. Axial symmetry.

Figure 1c shows the composition diagram discussed in the previous example, which is supplemented with small corner elements. In Figure 1d, the composition diagram is built along the horizontal axis. It includes central element with two side ones. The considered schemes can serve as the basis for composing compositions that have two axes of symmetry.

Such compositions are perceived equally by viewers from all sides; they, as a rule, do not have a clearly defined top and bottom.
Carpet ornaments can contain in their central part compositions that have one axis of symmetry (Figure 1e). Such compositions have a pronounced orientation; they have a top and a bottom.

The central part can not only be made in the form of an abstract ornament, but also have a theme.
All examples of the development of ornaments and compositions based on them discussed above were related to rectangular planes. Rectangular shape surfaces are a common, but not the only type of surface.

Boxes, trays, plates can have surfaces in the shape of a circle or an oval. One of the options for their decor can be centrally symmetrical ornaments. The basis for creating such an ornament is the center of symmetry, through which an infinite number of axes of symmetry can pass (Figure 2a).

Let's consider an example of developing an ornament limited by a circle and having central symmetry (Figure 2). The structure of the ornament is radial. Its main elements are located along the radius lines of the circle. The border of the ornament is decorated with a border.

Figure 2. Centrally symmetrical ornaments.

II. Symmetry in needlework

II. 1. Symmetry in knitting

We found knitted crafts with central symmetry:

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Today we will talk about a phenomenon that each of us constantly encounters in life: symmetry. What is symmetry?

We all roughly understand the meaning of this term. The dictionary says: symmetry is proportionality and complete correspondence of the arrangement of parts of something relative to a straight line or point. There are two types of symmetry: axial and radial. Let's look at the axial one first. This is, let’s say, “mirror” symmetry, when one half of an object is completely identical to the second, but repeats it as a reflection. Look at the halves of the sheet. They are mirror symmetrical. The halves of the human body are also symmetrical (front view) - identical arms and legs, identical eyes. But let’s not be mistaken, in fact, absolute symmetry cannot be found in the organic (living) world! The halves of the sheet copy each other far from perfectly, the same applies to human body(take a closer look for yourself); The same is true for other organisms! By the way, it is worth adding that any symmetrical body is symmetrical relative to the viewer only in one position. It’s worth, say, turning a sheet of paper, or raising one hand, and what happens? – you see for yourself.

People achieve true symmetry in the works of their labor (things) - clothes, cars... In nature, it is characteristic of inorganic formations, for example, crystals.

But let's move on to practice. You shouldn’t start with complex objects like people and animals; let’s try to finish drawing the mirror half of the sheet as the first exercise in a new field.

Drawing a symmetrical object - lesson 1

We make sure that it turns out as similar as possible. To do this, we will literally build our soul mate. Don’t think that it’s so easy, especially the first time, to draw a mirror-corresponding line with one stroke!

Let's mark several reference points for the future symmetrical line. We proceed like this: with a pencil, without pressing, we draw several perpendiculars to the axis of symmetry - the midrib of the leaf. Four or five is enough for now. And on these perpendiculars we measure to the right the same distance as on the left half to the line of the edge of the leaf. I advise you to use a ruler, don’t rely too much on your eye. As a rule, we tend to reduce the drawing - this has been observed from experience. We do not recommend measuring distances with your fingers: the error is too large.

Let's connect the resulting points with a pencil line:

Now let’s look meticulously to see if the halves are really the same. If everything is correct, we will circle it with a felt-tip pen and clarify our line:

The poplar leaf has been completed, now you can take a swing at the oak leaf.

Let's draw a symmetrical figure - lesson 2

In this case, the difficulty lies in the fact that the veins are marked and they are not perpendicular to the axis of symmetry and not only the dimensions but also the angle of inclination will have to be strictly observed. Well, let’s train our eye:

So a symmetrical oak leaf has been drawn, or rather, we built it according to all the rules:

How to draw a symmetrical object - lesson 3

And let’s consolidate the theme - we’ll finish drawing a symmetrical lilac leaf.

It also has an interesting shape - heart-shaped and with ears at the base, you'll have to puff:

Here's what they drew:

Take a look at the resulting work from a distance and evaluate how accurately we were able to convey the required similarity. Here's a tip: look at your image in the mirror and it will tell you if there are any mistakes. Another way: bend the image exactly along the axis (we have already learned how to bend it correctly) and cut out the leaf along the original line. Look at the figure itself and at the cut paper.

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 download a free image for a geometry lesson, right-click on the image and click “Save image as...”. 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.

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 various areas of human activity? A group of theorists. The concept of movement Axial symmetry Central symmetry. Can we see movement in nature?

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

There are a total of 32 presentations in the topic