Scientific and practical conference
MOU "Secondary school number 23"
the city of Vologda
section: natural - scientific
design and research work
SYMMETRY TYPES
Completed the work of a student of grade 8 "a"
Kreneva Margarita
Supervisor: 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 the plane);
3.4. Rotational symmetry;
3.5. Portable symmetry.
4. Conclusions.
Symmetry is the idea through which man, over the centuries, has tried 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 "Geometry of the 8th grade". 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 of constructing symmetrical figures in each of the types.
purpose of work : Introduce different kinds of symmetry.
Tasks:
Study the literature on this issue.
Summarize and systematize the material studied.
Prepare a presentation.
In ancient times, the word "SYMMETRY" was used in the meaning of "harmony", "beauty". Translated from Greek, this word means “proportionality, proportionality, the 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 the symmetry of positions, forms, structures. This is the symmetry that you can see directly. It can be called geometric symmetry.
The second group characterizes symmetry physical phenomena and the laws of nature. This symmetry lies at the very foundation of the natural-scientific picture of the world: it can be called physical symmetry.
I will focus on 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. I will look at 5 types of symmetry today.
Central symmetry
Two points A and A 1 are called symmetric with respect to the point O if they lie on a straight line passing through m O and are located on opposite sides of it at the same distance. Point O is called the center of symmetry.
The figure is called symmetrical about a point.O if for each point of the figure a point symmetrical to it relative to the pointO also belongs to this figure. PointO is called the center of symmetry of the figure, the figure is said to have central symmetry.
Examples of shapes that have central symmetry are a circle and a parallelogram.
The figures shown on the slide are symmetrical about some point
2. Axial symmetry
Two pointsX and Y are called symmetric about a straight linet , if this straight line passes through the middle of the segment XY and is perpendicular to it. It should also be said that each point of the straight linet is considered symmetrical to itself.
Straightt - axis of symmetry.
The figure is called symmetrical about a straight line.t, if for each point of the figure there is a point symmetrical to it relative to a straight linet also belongs to this figure.
Straighttcalled the axis of symmetry of the figure, the figure is said to have axial symmetry.
Axial symmetry is possessed by an undeveloped angle, isosceles and equilateral triangles, a rectangle and a rhombus,letters (see presentation).
Mirror symmetry (symmetry about a plane)
Two points P 1 and P are called symmetric about the plane and if they lie on a straight line perpendicular to the plane a, and are at the same distance from it
Mirror symmetry is 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, the figure with countless axes of symmetry was a circle. In space, an infinite number of planes of symmetry have a ball.
But if the circle is one of a kind, then in the three-dimensional world there is whole line bodies with an infinite set 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 symmetrical flat figure can be aligned with itself using a mirror. It is surprising that such complex shapes as a five-pointed star or an equilateral pentagon are also symmetrical. As it follows from the number of axes, they are distinguished precisely by their high symmetry. And vice versa: it is not so easy to understand why such a seemingly regular figure as an oblique parallelogram is asymmetrical.
4. P rotational symmetry (or radial symmetry)
Rotational symmetry - this is symmetry, the preserved shape of the objectwhen rotating around a certain axis at an angle equal to 360 ° /n(or multiple of this value), wheren= 2, 3, 4, ... The specified axis is called the pivot axisnth order.
Atn = 2 all points of the figure are rotated at 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 a shape goes to a point of the same shape (the shape transforms into itself). The axis is called the second-order axis.
Figure 2 shows the axis of the third order, in Figure 3-4 - 4th order, in Figure 4 - 5th order.
An object can have more than one pivot axis: fig. 1 - 3 pivot axes, 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, then the letter will be aligned with itself. In other words, the letter "I" is symmetrical about 180 ° rotation, 180 ° = 360 °: 2,n= 2, so it has second-order symmetry.
Note that the letter "Ф" also possesses rotational symmetry of the second order.
In addition, the letter and has a center of symmetry, and the letter F has an axis of symmetry
Let's go back to real life examples: 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 an infinite set of axes of symmetry of which an infinite number of planes of symmetry pass. Most of these bodies (they are called bodies of revolution) have, of course, a center of symmetry (the center of a circle), through which at least one rotary axis of symmetry passes.
It is clearly visible, for example, the axis at the cone of the pound with ice cream. It runs from the middle of the circle (sticking out of the ice cream!) To the sharp end of the funky 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 him 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 rotary axes and their order, as well as all the planes of symmetry.
Consider, for example, a geometric body made up of two identical regular quadrangular pyramids.
It has one 4th order rotary axis (AB axis), four 2nd order rotary axes (CE axes,DF, MP, NQ), five planes of symmetry (planesCDEF, AFBD, ACBE, AMBP, ANBQ).
5 . Portable symmetry
Another kind of symmetry isportable with immetry.
Such symmetry is spoken of when, when moving a figure along a straight line for some distance "a" or a distance that is a multiple of this value, it is combined with itself The straight line along which the transfer is made is called the transfer axis, and the distance "a" is called the elementary transfer, period or step of symmetry.
a
A periodically repeating pattern on a long ribbon is called a border. In practice, curbs are found in various forms (murals, cast iron, plaster bas-reliefs or ceramics). Borders are used by painters and artists when decorating a room. To complete these ornaments, a stencil is made. We move the stencil, turning it over or without turning it over, draw the outline, repeating the drawing, and we get an ornament (visual demonstration).
The border can be easily constructed using a stencil (original element), sliding or flipping it and repeating the pattern. The figure shows stencils of five types:a ) asymmetrical;b, c ) having one axis of symmetry: horizontal or vertical;G ) centrally symmetrical;d ) having two axes of symmetry: vertical and horizontal.
The following transformations are used to build curbs:
a
) parallel transfer;b
) symmetry about the vertical axis;v
) central symmetry;G
) symmetry about the horizontal axis.
Similarly, you can build sockets. To do this, divide the circle byn equal sectors, in one of them a sample of the pattern is performed and then the latter is sequentially repeated in the remaining parts of the circle, each time rotating the pattern through an angle of 360 ° /n .
An illustrative example the use of axial and portable symmetry can be the fence shown in the photograph.
Conclusion: Thus, there are different types of symmetry, symmetric points in each of these types of symmetry are built according to certain laws. In life, we meet everywhere in one form or another of symmetry, and often in the objects that surround us, several types of symmetry can be noted at once. It creates order, beauty and perfection in the world around us.
LITERATURE:
Reference for elementary mathematics... M. Ya. Vygodsky. - Publishing house "Science". - Moscow 1971. - 416p.
Modern vocabulary foreign words... - M .: Russian language, 1993.
History of mathematics at schoolIX - Xclasses. G.I. Glazer. - Publishing house "Education". - Moscow 1983 - 351 p.
Visual geometry 5 - 6 grades. I.F. Sharygin, L.N. Erganzhieva. - Publishing house "Drofa", Moscow 2005. - 189p.
Encyclopedia for children. Biology. S. Ismailova. - Publishing house "Avanta +". - Moscow 1997. - 704p.
Urmantsev Yu.A. Symmetry of nature and the nature of symmetry - M .: Thought arxitekt / arhkomp2. htm, , ru.wikipedia.org/wiki/
I ... Symmetry in mathematics :
Basic concepts and definitions.
Axial symmetry (definitions, construction plan, examples)
Central symmetry (definitions, construction plan, formeasures)
Summary table (all properties, features)
II ... Symmetry Applications:
1) in mathematics
2) in chemistry
3) in biology, botany and zoology
4) in art, literature and architecture
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1. Basic concepts of symmetry and its types.
The concept of symmetry n R goes 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 a person. And it was used by sculptors as early as the 5th century BC. NS. The word "symmetry" is Greek, it means "proportionality, proportionality, uniformity in the arrangement of parts." It is widely used by all areas of modern science without exception. Many great people thought about this pattern. For example, LN Tolstoy said: “Standing in front of a black board and drawing different figures on it with chalk, I was suddenly struck by the thought: why is symmetry clear to the eye? What is symmetry? This is an innate feeling, I answered myself. What is it based on? " The symmetry is indeed pleasing to the eye. Who has not admired the symmetry of nature's creations: leaves, flowers, birds, animals; or human creations: buildings, technology, - everything that surrounds us from childhood, those that strive for beauty and harmony. Hermann Weil said: "Symmetry is the idea through which man, for centuries, has tried to comprehend and create order, beauty and perfection." Hermann Weil is a German mathematician. His activity falls on the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what criteria to perceive the presence or, conversely, the absence of symmetry in one or another case. Thus, a mathematically rigorous concept was formed relatively recently - at the beginning of the twentieth century. It's quite complicated. We will turn and once again remember the definitions that were given to us in the textbook.
2. Axial symmetry.
2.1 Basic definitions
Definition. Two points A and A 1 are called symmetric with respect to the straight line a if this straight line passes through the middle of the segment AA 1 and is perpendicular to it. Each point of the straight line a is considered symmetrical to itself.
Definition. The figure is called symmetrical about a straight line. a if, for each point of the figure, a point symmetric to it with respect to a straight line a also belongs to this figure. Straight a is called the axis of symmetry of the figure. The figure is also said to have axial symmetry.
2.2 Building plan
And so, to build a symmetrical figure with respect to a straight line from each point, we draw a perpendicular to this straight line and extend it by the same distance, mark the resulting point. We do this with each point, we get the symmetrical vertices of the new shape. Then we connect them in series and get a symmetrical figure of this relative axis.
2.3 Examples of figures with axial symmetry.
3. Central symmetry
3.1 Basic definitions
Definition. Two points A and A 1 are called symmetric with respect to point O if O is the middle of the segment AA 1. Point O is considered symmetrical to itself.
Definition. A figure is called symmetric about point O if for each point of the figure the point symmetric to it about point O also belongs to this figure.
3.2 Build plan
Construction of a triangle symmetrical to a given one about the center O.
To draw a point symmetrical to a point A relative to point O, it is enough to draw a straight line OA(fig. 46 )
and on the other side of the point O postpone a segment equal to the segment OA.
In other words ,
points A and 
; In and 
; With and 
are symmetric with respect to some point O. In Fig. 46 built a triangle symmetrical to the triangle ABC
relative to point O. These triangles are equal.
Draws symmetrical points about the center.
In the figure, points M and M 1, N and N 1 are symmetric about point O, and points P and Q are not symmetric about this point.
In general, figures symmetrical about some point are equal .
3.3 Examples
Here are some examples of figures with central symmetry. The simplest figures with central symmetry are the circle and the parallelogram.
Point O is called the center of symmetry of the figure. In such 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.
The straight line also has central symmetry, however, unlike the circle and the parallelogram, which have only one center of symmetry (point O in the figure), the straight line has infinitely many of them - any point of the straight line is its center of symmetry.
The figures show an angle symmetrical about the vertex, a segment symmetrical to another segment about the center A and a quadrilateral symmetric about its vertex M.
An example of a shape that does not have a center of symmetry is a triangle.
4. Lesson summary
Let's summarize the knowledge gained. Today in the lesson we got acquainted with two main types of symmetry: central and axial. Let's look at the screen and systematize the knowledge gained.
Summarizing table
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Axial symmetry |
Central symmetry |
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Peculiarity |
All points of the figure must be symmetrical about some straight line. |
All points of the shape must be symmetrical about the point selected as the center of symmetry. |
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Properties |
1. Symmetrical points lie on perpendiculars to a straight line. 3. Straight lines turn into straight lines, angles into equal angles. 4. Sizes and shapes of figures are saved. |
1. Symmetric points lie on a straight line passing through the center and this point figures. 2. The distance from a point to a straight line is equal to the distance from a straight line to a symmetrical point. 3. The sizes and shapes of the figures are preserved. |
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II. Applying symmetry
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Maths |
In algebra lessons, we studied the graphs of the functions y = x and y = x The figures show various pictures depicted using the branches of parabolas. (a) Octahedron, (b) rhombic dodecahedron, (c) hexagonal octahedron. |
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Russian language |
The printed letters of the Russian alphabet also have different types of symmetries. There are "symmetrical" words in Russian - palindromes that can be read the same way in two directions. |
A D L M P T V W- vertical axis V E Z K S E Y - horizontal axis J N O X- both vertical and horizontal B G I Y R U Y Z- no axis Radar hut Alla Anna |
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Literature |
Can be palindromic and sentences. Bryusov wrote a poem "The Voice of the Moon", in which each line is a palindrome. Look at the quatrains of A.S. Pushkin " Bronze Horseman". If we draw a line after the second line, we can notice elements of axial symmetry |
And the rose fell on Azor's paw. I go with the sword of the judge. (Derzhavin) "Search for a taxi" «Аргентина манит негра», "The Argentinean appreciates the negro", "Lesha found a bug on the shelf." The Neva was dressed in granite; Bridges hung over the waters; Dark green gardens The islands were covered with it ... |
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Biology |
The human body is built according to the principle of bilateral symmetry. Most of us view the brain as a single structure; in fact, it is divided into two halves. These two parts - the two hemispheres - fit snugly together. In full accordance with the general symmetry of the human body, each hemisphere is an almost exact mirror image of the other. The control of the basic movements of the human body and its sensory functions is evenly distributed between the two hemispheres of the brain. The left hemisphere controls the right side of the brain, and the right side controls the left side. |
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Botany |
A flower is considered symmetrical when each perianth is composed of an equal number of parts. Flowers, having paired parts, are considered to be flowers with double symmetry, etc. Triple symmetry is common for monocotyledonous plants, quintuple symmetry for dicots Characteristic feature the structure of plants and their development is helicity. Pay attention to the shoots of the leaf arrangement - this is also a kind of spiral - helical. Even Goethe, who was not only a great poet, but also a natural scientist, considered helicity to be one of the characteristic features of all organisms, a manifestation of the innermost essence of life. The antennae of plants are spirally twisted, the tissues grow in the trunks of trees in a spiral, the seeds in the sunflower are arranged in a spiral, spiral movements are observed during the growth of roots and shoots. |
A characteristic feature of the structure of plants and their development is helicity.
Look at the pinecone. The scales on its surface are arranged in a strictly regular way - along two spirals, which intersect at approximately right angles. The number of such spirals in pine cones is 8 and 13 or 13 and |
21.|
Zoology |
Symmetry in animals means correspondence in size, shape and shape, as well as the relative position of body parts located on opposite sides of the dividing line. With radial or radiant symmetry, the body has the form of a short or long cylinder or a vessel with a central axis, from which parts of the body radiate out in a radial order. These are coelenterates, echinoderms, sea stars... With bilateral symmetry, there are three axes of symmetry, but there is only one pair of symmetrical sides. Because the other two sides - the ventral and dorsal - are not alike. This type of symmetry is typical for most animals, including insects, fish, amphibians, reptiles, birds, and mammals. |
Axial symmetry
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Different kinds symmetry of physical phenomena: symmetry of electric and magnetic fields (Fig. 1) In mutually perpendicular planes, the propagation of electromagnetic waves is symmetric (Fig. 2) |
fig. 1 fig. 2 |
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Art |
Mirror symmetry can often be observed in works of art. Mirror "symmetry is widespread in the works of art of primitive civilizations and in ancient painting. Medieval religious paintings are also characterized by this kind of symmetry. One of Raphael's best early works, The Betrothal of Mary, was created in 1504. A valley crowned with a white-stone temple stretches under the sunny blue sky. Foreground: the betrothal ceremony. The high priest brings the hands of Mary and Joseph closer. Behind Mary - a group of girls, behind Joseph - young men. Both parts of the symmetrical composition are held together by the oncoming movement of the characters. For modern taste, the composition of such a picture is boring, since the symmetry is too obvious. |
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Chemistry |
The water molecule has a plane of symmetry (straight vertical line). DNA molecules (deoxyribonucleic acid) play an extremely important role in the living world. It is a double-stranded high molecular weight polymer, the monomer of which is nucleotides. DNA molecules have a double helix structure built on the principle of complementarity. |
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Architeculture |
Since ancient times, man has used symmetry in architecture. The ancient architects used the symmetry in architectural structures especially brilliantly. Moreover, the ancient Greek architects were convinced that in their works they were guided by the laws that govern nature. Choosing symmetrical forms, the artist thereby expressed his understanding of natural harmony as stability and balance. The city of Oslo, the capital of Norway, has an expressive ensemble of nature and art. This is Frogner - park - a complex of landscape gardening sculptures, which was created over 40 years. |
Pashkov House Louvre (Paris) |
© Elena Vladimirovna Sukhacheva, 2008-2009.
« Symmetry"- a word of Greek origin. It means proportionality, the presence of a certain order, patterns in the arrangement of parts.
Since ancient times, people have used symmetry in drawings, ornaments, and household items.
Symmetry is widespread in nature. It can be observed in the form of leaves and flowers of plants, in the arrangement of various organs of animals, in the form of crystalline bodies, in a fluttering butterfly, a mysterious snowflake, a mosaic in a temple, a starfish.
Symmetry is widely used in practice, in construction and engineering. This is a strict symmetry in the form of antique buildings, harmonious ancient Greek vases, the Kremlin building, cars, airplanes and much more. (slide 4) Examples of the use of symmetry are parquet and curb. (see hyperlink on using symmetry in curbs and parquets) Let's look at a few examples where you can see symmetry in various subjects using a slideshow (include icon).
Definition: is symmetry about a point.
Definition: Points A and B are symmetric about some point O if point O is the midpoint of segment AB.
Definition: Point O is called the center of symmetry of the figure, and the figure is called centrally symmetric.
Property: Shapes that are symmetrical about some point are equal.
Examples:
Algorithm for constructing a centrally symmetric figure
1. Construct a triangle A 1B 1 C 1, symmetrical to a triangle ABC, relative to the center (point) O. To do this, we connect points A, B, C with center O and continue these segments;
2. Measure the segments AO, BO, CO and set aside on the other side of the point O, equal segments (AO = A 1 O 1, BO = B 1 O 1, CO = C 1 O 1);
3. Connect the resulting points with segments A 1 B 1; A 1 C 1; B1 C 1.
Received ∆А 1 В 1 С 1 symmetric ∆ABS.
- this is symmetry about the drawn axis (straight line).
Definition: Points A and B are symmetric with respect to some straight line a, if these points lie on a straight line perpendicular to the given one, and at the same distance.
Definition: The axis of symmetry is called a straight line when bending along which the "halves" will coincide, and the figure is called symmetrical about some axis.
Property: Two symmetrical shapes are equal.
Examples:
Algorithm for constructing a figure symmetric with respect to some straight line
Let's construct a triangle А1В1С1, symmetrical to a triangle ABC with respect to a straight line a.
For this:
1. Draw from the vertices of the triangle ABC straight lines perpendicular to the straight line a and continue them further.
2. We measure the distance from the vertices of the triangle to the resulting points on the straight line and postpone the same distances on the other side of the straight line.
3. Connect the resulting points with segments A1B1, B1C1, B1C1.
Received ∆ A1B1C1 symmetric ∆ABS.
Today we will talk about a phenomenon that each of us has to constantly meet in life: about symmetry. What is symmetry?
Approximately we all understand the meaning of this term. The dictionary says: symmetry is proportionality and full correspondence of the arrangement of parts of something relative to a straight line or point. Symmetry is of two types: axial and radial. Let's consider axial first. This is, let's say, "mirror" symmetry, when one half of the 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 (full face) are also symmetrical - the same arms and legs, the same eyes. But let's not be mistaken, in fact, in the organic (living) world, you cannot find absolute symmetry! The halves of the leaf copy each other far from perfect, the same applies to the human body (take a closer look); it is the same with other organisms! By the way, it should be added that any symmetrical body is symmetrical with respect to the viewer in only one position. It is worth, say, turning the sheet, or raising one hand, and what? - you can 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 get down to practice. It's not worth starting with complex objects like people and animals, let's try as the first exercise in a new field to finish drawing a mirror half of the sheet.
How to draw a symmetrical object - lesson 1
We make sure that it turns out as similar as possible. For this, we will literally build our soul mate. Do not think that it is so easy, especially the first time, to draw a mirror-corresponding line with one stroke!
Let's mark some anchor points for the future symmetrical line. We proceed as follows: we draw several perpendiculars to the axis of symmetry - the midrib of the leaf with a pencil without pressing. 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, do not rely on the eye too much. As a rule, we tend to reduce the drawing - it has been noticed from experience. We do not recommend measuring distances with your fingers: the error is too large.
We connect the resulting points with a pencil line:
Now we are meticulously looking - are the halves really the same. If everything is correct, we will circle it with a felt-tip pen, we will clarify our line:
The poplar leaf has been finished, now you can swing at the oak one.
How to draw a symmetrical shape - lesson 2
In this case, the difficulty lies in the fact that the veins are indicated 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 accurately observed. Well, we train the eye:
So a symmetrical oak leaf was drawn, or rather, we built it according to all the rules:
How to draw a symmetrical object - lesson 3
And let's fix the theme - draw a symmetrical lilac leaf.
He also has an interesting shape - heart-shaped and with ears at the base you will have to pant:
So they drew:
Take a look at the resulting work from a distance and see how accurately we managed 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 the leaf along the original line. Look at the figure itself and the cut paper.
MBOU "Tyukhtetsk secondary school No. 1"
Scientific association of students "We want to study actively"
Physics, Mathematics and Engineering
Arvinti Tatiana,
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 handicrafts made from matches …………………………………………… ... 14
II. 6. Symmetry in "Macrame" weaving ……………………………………………… .15
Conclusion …………………………………………………………………………… .16
Bibliographic list ……………………………………………………… ..17
Introduction
One of the fundamental concepts of science, which, along with the concept of "harmony" is related to almost all structures of nature, science and art, is "symmetry".
Prominent mathematician Hermann Weil praised the role of symmetry in modern science:
"Symmetry, no matter how broadly or narrowly we understand this word, is an idea with the help of which a person tried to explain and create order, beauty and perfection."
We all admire the beauty of geometric shapes, their combination, considering pillows, knitted napkins, embroidered clothes.
Many centuries different nations created wonderful types of arts and crafts. Many people think 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 goal is to show that mathematics is a very amazing and unusual subject 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 various geometric figures that obey mathematical transformations. In this regard, such mathematical concepts as symmetry, 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, its types.
· Practical: find symmetrical crafts, determine the type of symmetry.
Symmetry. Symmetry types
Symmetry(means "proportionality") - the property of geometric objects to be combined with themselves under certain transformations. Symmetry is understood as any correctness in internal structure body or figure.
Point symmetry is central symmetry, and line symmetry is axial symmetry.
Symmetry about a point (central symmetry) means that there is something on either side of a point at equal distances, such as other points or a locus of points (straight lines, curved lines, geometric shapes). If you connect a straight line symmetric points (points of a geometric figure) through a point of symmetry, then the symmetrical points will lie at the ends of the line, and the point of symmetry will be its midpoint. If you fix a point of symmetry and rotate a straight line, then the symmetrical points will describe curves, each point of which will also be symmetrical to a point on another curved line.
A rotation around a given point O is a movement in which each ray emanating from this point rotates by the same angle in the same direction.
Symmetry about a straight line (axis of symmetry) implies that two symmetrical points are located along the perpendicular drawn through each point of the axis of symmetry. The same geometric figures can be located relative to the axis of symmetry (straight line) as relative to the point of symmetry. An example is a sheet of a notebook that is folded in half if you draw a straight line (axis of symmetry) along the fold line. Each point of one half of the sheet will have a symmetrical point on the other half of the sheet if they are located at the same distance from the fold line perpendicular to the axis. The axis of symmetry serves as a perpendicular to the midpoints of the horizontal lines bounding the leaf. Symmetrical points are located at the same distance from the center line - the 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 line segment and equidistant from the ends of this segment.
Coll "href =" / text / category / koll / "rel =" bookmark "> Hermitage collections special attention enjoy the gold jewelry of the ancient Scythians. The artwork of gold wreaths, tiaras, wood and precious red-violet garnets is extraordinarily delicate.
One of the most obvious uses of the laws of symmetry in life is the building of architecture. This is what we see most often. In architecture, axes of symmetry are used as a means of expressing architectural intent.
Another example of human use of symmetry in his practice is technique. In engineering, the axes of symmetry are most clearly indicated where it is required to estimate the deviation from the zero position, for example, at the steering wheel of a truck or at the steering wheel of a ship. Or one of the most important inventions of mankind, having a center of symmetry, is a wheel, and a propeller and other technical means also have a center of symmetry.
Axial and centrally symmetrical ornaments
Compositions built on the principle of carpet ornament can have a symmetrical structure. The drawing in them is organized according to the principle of symmetry about one or two axes of symmetry. In carpet ornaments, there is often a combination of several types of symmetry - axial and central.
Figure 1 shows a scheme for marking a plane for a carpet ornament, the composition of which will be built along the axes of symmetry. On the plane along the perimeter, the place and size of the border are determined. Central field will occupy the main ornament.
Variants of various compositional solutions of the plane are shown in Figure 1 b-e. In Figure 1b, the composition is built in the central part of the field. Its shape can vary depending on the shape of the field itself. If the plane has the shape of an elongated rectangle, the compositions are given the shape of an elongated rhombus or oval. Square shape margins are better supported by a composition outlined by a circle or an equilateral rhombus.
Figure 1. Axial symmetry.
Figure 1c shows the composition diagram from the previous example, supplemented with small corner pieces. In Figure 1d, the composition diagram is plotted along the horizontal axis. It includes a center piece with two side pieces. The considered schemes can serve as a basis for composing compositions that have two axes of symmetry.
Such compositions are equally perceived by viewers from all sides; as a rule, they do not have a pronounced top and bottom.
Carpet ornaments may contain compositions in their central part 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 the examples of the development of ornaments and compositions based on them considered above were related to planes with a rectangular shape. The rectangular shape of the surface is a common, but not the only type of surface.
Caskets, trays, plates can have planes in the form of a circle or 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).
Consider an example of developing an ornament bounded by a circle and having a central symmetry (Figure 2). The structure of the ornament is radial. Its main elements are located along the lines of the radius of the circle. The border of the ornament is framed 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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