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, in the organic (living) world, absolute symmetry cannot be found! 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 products of their labor (things) - clothes, cars... In nature, it is characteristic 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 at whether 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.
He has too interesting shape- heart-shaped and with ears at the base, you’ll have to puff:
This is 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 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.
I . Symmetry in mathematics :
Basic concepts and definitions.
Axial symmetry (definitions, construction plan, examples)
Central symmetry (definitions, construction plan, whenmeasures)
Summary table (all properties, features)
II . Applications of symmetry:
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 R goes back 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. e. The word “symmetry” is Greek and means “proportionality, proportionality, sameness in the arrangement of parts.” It is widely used by all areas of modern science without exception. Many great people have thought about this pattern. For example, L.N. 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 truly pleasing to the eye. Who hasn’t admired the symmetry of nature’s creations: leaves, flowers, birds, animals; or human creations: buildings, technology, everything that surrounds us since childhood, everything that strives for beauty and harmony. Hermann Weyl said: “Symmetry is the idea through which man throughout the ages has tried to comprehend and create order, beauty and perfection.” Hermann Weyl is a German mathematician. His activities span the first half of the twentieth century. It was he who formulated the definition of symmetry, established by what criteria one can determine the presence or, conversely, absence of symmetry in a given case. Thus, a mathematically rigorous concept was formed relatively recently - at the beginning of the twentieth century. It's quite complicated. Let us 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 symmetrical with respect to line a if this line passes through the middle of segment AA 1 and is perpendicular to it. Each point of a line a is considered symmetrical to itself.
Definition. The figure is said to be symmetrical about a straight line A, if for each point of the figure there is a point symmetrical to it relative to the straight line A also belongs to this figure. Straight A called the axis of symmetry of the figure. The figure is also said to have axial symmetry.
2.2 Construction plan
And so, to construct a symmetrical figure relative to a straight line, from each point we draw a perpendicular to this straight line and extend it to the same distance, mark the resulting point. We do this with each point and get symmetrical vertices of a new figure. Then we connect them in series and get a symmetrical figure of a given 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 symmetrical 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 said to be symmetrical with respect to point O if, for each point of the figure, a point symmetrical with respect to point O also belongs to this figure.
3.2 Construction plan
Construction of a triangle symmetrical to the given one relative to the center O.
To construct a point symmetrical to a point A relative to the point ABOUT, it is enough to draw a straight line OA(Fig. 46 )
and on the other side of the point ABOUT set aside the segment equal to the segment OA.
In other words ,
points A and 
; In and 
; C and 
symmetrical about some point O. In Fig. 46 a triangle is constructed that is symmetrical to a triangle ABC
relative to the point ABOUT. These triangles are equal.
Construction of symmetrical points relative to the center.
In the figure, points M and M 1, N and N 1 are symmetrical relative to point O, but points P and Q are not symmetrical relative to this point.
In general, figures that are symmetrical about a certain point are equal .
3.3 Examples
Let us give examples of figures that have central symmetry. The simplest figures with central symmetry are the circle and 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.
A straight line also has central symmetry, but unlike a circle and a parallelogram, which have only one center of symmetry (point O in the figure), a straight line has an infinite number of them - any point on the straight line is its center of symmetry.
The pictures show an angle symmetrical relative to the vertex, a segment symmetrical to another segment relative to the center A and a quadrilateral symmetrical about its vertex M.
An example of a figure that does not have a center of symmetry is a triangle.
4. Lesson summary
Let us summarize the knowledge gained. Today in class we learned about two main types of symmetry: central and axial. Let's look at the screen and systematize the knowledge gained.
Summary table
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Axial symmetry |
Central symmetry |
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Peculiarity |
All points of the figure must be symmetrical relative to some straight line. |
All points of the figure must be symmetrical relative to the point chosen as the center of symmetry. |
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Properties |
1. Symmetrical points lie on perpendiculars to the line. 3. Straight lines turn into straight lines, angles into equal angles. 4. The sizes and shapes of the figures are preserved. |
1. Symmetrical points lie on a 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. Application of symmetry
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Mathematics |
In algebra lessons we studied the graphs of the functions y=x and y=x The pictures show various pictures depicted using the branches of parabolas. (a) Octahedron, (b) rhombic dodecahedron, (c) hexagonal octahedron. |
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Russian language |
Printed letters The Russian alphabet also has different types of symmetries. There are “symmetrical” words in the Russian language - palindromes, which can be read equally in both directions. |
A D L M P T F W– vertical axis V E Z K S E Y - horizontal axis F N O X- both vertical and horizontal B G I Y R U C CH SCHY- no axis Radar hut Alla Anna |
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Literature |
Sentences can also be palindromic. Bryusov wrote a poem “The Voice of the Moon”, in which each line is a palindrome. Look at the quadruples 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 come with the sword of the judge. (Derzhavin) "Search for a taxi" "Argentina beckons the Negro" “The Argentine appreciates the black man,” “Lesha found a bug on the shelf.” The Neva is dressed in granite; Bridges hung over the waters; Dark green gardens Islands covered it... |
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Biology |
The human body is built on the principle of bilateral symmetry. Most of us view the brain as a single structure; in reality, it is divided into two halves. These two parts - two hemispheres - fit tightly to each other. In full accordance with the general symmetry of the human body, each hemisphere is an almost exact mirror image of the other 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 hemisphere controls the left side. |
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Botany |
A flower is considered symmetrical when each perianth consists of an equal number of parts. Flowers having paired parts are considered flowers with double symmetry, etc. Triple symmetry is common in monocotyledons, and quintuple symmetry in dicotyledons. Characteristic feature The structure of plants and their development is helicity. Pay attention to the leaf arrangement of the shoots - this is also a peculiar type of spiral - a helical one. Even Goethe, who was not only a great poet, but also a natural scientist, considered helicity one of characteristic features of all organisms, a manifestation of the innermost essence of life. The tendrils of plants twist in a spiral, the growth of tissues in tree trunks occurs in a spiral, the seeds in a sunflower are arranged in a spiral, and spiral movements are observed during the growth of roots and shoots. |
A characteristic feature of the structure of plants and their development is spirality.
Look at the pine cone. The scales on its surface are arranged strictly regularly - along two spirals that intersect approximately at a right angle. 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 outline, as well as the relative arrangement of body parts located on opposite sides of the dividing line. With radial or radial symmetry, the body has the shape of a short or long cylinder or vessel with a central axis, from which parts of the body extend radially. These are coelenterates, echinoderms, and starfish. With bilateral symmetry, there are three axes of symmetry, but only one pair of symmetrical sides. Because the other two sides - abdominal and dorsal - are not similar to each other. This type of symmetry is characteristic of most animals, including insects, fish, amphibians, reptiles, birds, and mammals. |
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Different kinds symmetry physical phenomena: symmetry of electric and magnetic fields (Fig. 1) The distribution is symmetrical in mutually perpendicular planes electromagnetic waves(Fig. 2) |
Fig.1 Fig.2 |
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Art |
Mirror symmetry can often be observed in works of art. Mirror" symmetry is widely found in works of art of primitive civilizations and in ancient paintings. Medieval religious paintings are also characterized by this type of symmetry. One of the best early works Raphael - “The Betrothal of Mary” - created in 1504. Under a sunny blue sky lies a valley topped by a white stone temple. In the foreground is the betrothal ceremony. The High Priest brings Mary and Joseph's hands together. Behind Mary is a group of girls, behind Joseph is a group of young men. Both parts symmetrical composition secured by the counter-movement of the characters. For modern tastes, the composition of such a painting is boring, since the symmetry is too obvious. |
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Chemistry |
A water molecule has a plane of symmetry (straight vertical line). DNA molecules (deoxyribonucleic acid) play an extremely important role in the world of living nature. It is a double-chain high-molecular polymer, the monomer of which is nucleotides. DNA molecules have a structure double helix, built on the principle of complementarity. |
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Architeculture |
Man has long used symmetry in architecture. The ancient architects made especially brilliant use of symmetry in architectural structures. Moreover, the ancient Greek architects were convinced that in their works they were guided by the laws that govern nature. Choosing symmetrical shapes, 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 that was created over the course of 40 years. |
Pashkov House Louvre (Paris) |
© Sukhacheva Elena Vladimirovna, 2008-2009
Central symmetry. Central symmetry is movement.
Picture 9 from the presentation “Types of symmetry” for geometry lessons on the topic “Symmetry”
Dimensions: 1503 x 939 pixels, format: jpg. To download a picture for free 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 “Types of symmetry.ppt” with all the pictures in a zip archive for free. Archive size - 1936 KB.
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Axial symmetry and the concept of perfection
Axial symmetry is inherent in all forms in nature and is one of fundamental principles beauty. Since ancient times, man has tried
to comprehend the meaning of perfection. This concept was first substantiated by artists, philosophers and mathematicians Ancient Greece. And the word “symmetry” itself was invented by them. It denotes proportionality, harmony and identity of the parts of the whole. The ancient Greek thinker Plato argued that only an object that is symmetrical and proportionate can be beautiful. Indeed, those phenomena and forms that are proportional and complete “please the eye.” We call them correct.
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Axial symmetry occurs in nature. It not only determines general structure organism, but also the possibilities of its subsequent development. Geometric shapes and the proportions of living beings are formed by “axial symmetry”. Its definition is formulated in the following way: this is the property of objects to be combined when various transformations. The ancients believed that the principle of symmetry in the most in full has a sphere. They considered this form harmonious and perfect.
Axial symmetry in living nature
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Axial symmetry in inanimate nature
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Picture 35 from the presentation “Ornament” for geometry lessons on the topic “Symmetry”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.
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