Where symmetry is applied in life. Central symmetry is the source of life. Central symmetry in life

"Mathematical symmetry" - Types of symmetry. Symmetry in mathematics. HAS A LOT IN COMMON WITH AXIAL SYMMETRY IN MATH. In poetry, rhyme is a translational symmetry. Symmetry in chemistry and physics. physical symmetry. In x and m and and. Bilateral symmetry. The role of symmetry in the world. Spiral symmetry. Symmetry in chemistry.

"Ornament" - Types of ornament. Geometric. a) Inside the band. 1 2 3. Creating an ornament using axial symmetry and parallel translation. 2011. Transforms used to create an ornament: Planar. c) On both sides of the strip. Turn.

"Movement in geometry" - Movement in geometry. To what sciences is movement applied? Concept of motion Axial symmetry Central symmetry. What figure does a segment, angle, etc., pass into when moving? Give examples of movement. What is called movement? How is movement used in various areas of human activity? Mathematics is beautiful and harmonious!

"Symmetry in nature" - We are engaged in the school scientific society because we love to learn something new and unknown. In the 19th century, in Europe, there were single works devoted to the symmetry of plants. Symmetry in nature and in life. One of the main properties of geometric shapes is symmetry. The work was completed by: Zhavoronkova Tanya Nikolaeva Lera Supervisor: Artyomenko Svetlana Yurievna.

"Symmetry around us" - Rotations (rotary). Center point. Rotations. Symmetry on the plane. Axial symmetry relative to a straight line. Around us. Symmetry in space. Horizontal. Symmetry rules. Mirror. Two kinds of symmetry. All kinds of axial symmetry. The Greek word symmetry means “proportionality”, “harmony”.

"Point of symmetry" - Examples of the above types of symmetry. Such figures include a parallelogram other than a rectangle, a scalene triangle. We encounter symmetry in nature, everyday life, architecture and technology. Symmetry in architecture. Symmetry in nature. Symmetry of plane figures. A rectangle and a rhombus, which are not squares, have two axes of symmetry.

Total in the topic 32 presentations

The theme of this work is the concept of symmetry. There is an opinion that symmetry plays a leading, although not always conscious, role in modern science, art, technology and the life around us.

What is symmetry? Why does symmetry literally permeate the entire world around us?

There are, in principle, two groups of symmetries. The first group includes the symmetry of positions, shapes, structures. This is the symmetry that can be directly seen. It can be called geometric symmetry.

The second group characterizes the symmetry of 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.

Target : To study the manifestations of symmetry in various areas of human life and society.

Tasks:

1. Determine the main features of the concept of symmetry.

2. Determine the presence of symmetry in living and non-living nature, in linguistics, in art.

3. To study the advantages of symmetrical objects in the figurative perception of a person.

Relevance due to the fact that symmetry surrounds a person, finding its manifestation both in living and non-living nature, as well as in most of human creations: in architecture, in art, etc. Explaining the laws of symmetry is important for understanding beauty and harmony. The results of the project will be of interest to secondary school students.

In this work, I explore geometric symmetry and show that geometric symmetry is present in everything that surrounds us, which we constantly encounter in everyday life.

2. The value of symmetry in our lives.

The concept of symmetry runs through the entire centuries-old history of human creativity. Since ancient times, many peoples have owned the idea of ​​symmetry in a broad sense - as the equivalent of balance and harmony.

Forms of perception and expression in many areas of science and art are ultimately based on symmetry, which is used and manifested in specific concepts and means inherent in certain areas of science and art.

Symmetry (from the Greek symmetria - "proportionality") is a concept meaning the persistence, repeatability, "invariance" of any features of the structure of the object under study when certain transformations are carried out with it.

Truly symmetrical objects surround us literally from all sides, we are dealing with symmetry wherever there is any order. Symmetry resists chaos, disorder. It turns out that symmetry is balance, orderliness, beauty, perfection.

The whole world can be considered as a manifestation of the unity of symmetry and asymmetry. A structure that is asymmetrical as a whole can be a harmonious composition of symmetrical elements.

Symmetry is manifold, ubiquitous. She creates beauty and harmony.

Over the course of millennia, in the course of social practice and knowledge of the laws of objective reality, mankind has accumulated numerous data indicating the presence of two tendencies in the surrounding world: on the one hand, towards strict orderliness, harmony, on the other hand, towards their violation. People have long paid attention to the correctness of the shape of crystals, flowers, honeycombs and other natural objects and reproduced this proportionality in works of art, in the objects they create, through the concept of symmetry.

“Symmetry,” writes the famous scientist J. Newman, “establishes a funny and amazing relationship between objects, phenomena and theories that seem to be outwardly unrelated: terrestrial magnetism, female veil, polarized light, natural selection, group theory, work habits bees in a hive, the structure of space, drawings of vases, quantum physics, flower petals, cell division sea ​​urchins, equilibrium configurations of crystals, Romanesque cathedrals, snowflakes, music, the theory of relativity…”

Consider examples of symmetry in various areas of our lives.

1. Symmetry in nature.

3.1. Symmetry in inanimate nature.

A snowflake is a crystal of frozen water.

The world of crystals is a special world of symmetry, with which great discoveries are associated both in the field of mathematics and in the field of crystallography. In crystals, symmetry axes of 1,2,3,4 and 6 orders are possible.

Snowflakes are the most striking example of the beauty of axial symmetry forms. Any snowflake has a rotational axis of symmetry, and in addition, each snowflake is mirror symmetrical. (pic 1)

Fig.1 Symmetry of snowflakes: axial symmetry.

Reflection in water is the only example of horizontal symmetry in nature. (fig.2)

Fig.2 Symmetry of the lake: horizontal symmetry.

3.2 . symmetry in plants.

The symmetry of the cone characteristic of plants is clearly visible on the example of any tree (Fig. 3).

Rice. 3 Cone symmetry: axis and plane of symmetry.

The specificity of the structure of plants is determined by the characteristics of the habitat to which they adapt, the characteristics of their lifestyle. The tree absorbs moisture and nutrients from the soil through the root system, that is, below, and the rest of the vital functions are performed by the crown, that is, at the top. Therefore, the directions "up" and "down" for the tree are significantly different. And the directions in the plane perpendicular to the vertical are practically indistinguishable for the tree: air, light, and moisture are equally supplied to the tree in all these directions. As a result, a vertical rotary axis and a vertical plane of symmetry appear.

Most flowering plants exhibit radial and bilateral symmetry. 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 for monocot plants, five - for dicots (Fig. 4).

Fig.4 Flower - radial symmetry (double, triple, quintuple)

Perhaps when you saw Romanesco broccoli in the store, you thought it was another example of a genetically modified product. But in fact, this is another example of the fractal symmetry of nature. Each broccoli inflorescence has a pattern of the same logarithmic spiral as the entire head (Fig. 5).

Fig.5 Brocolli - fractal symmetry

Sunflowers (Fig. 6)boast radial symmetry and an interesting type of symmetry known as the Fibonacci sequence. Fibonacci sequence: 1, 2, 3, 5, 8, 13, 21, 24, 55, 89, 144, etc. (each number is determined by the sum of the two previous numbers). If we took our time and counted the number of seeds in a sunflower, we would find that the number of spirals grows according to the principles of the Fibonacci sequence. In nature, there are so many plants (including romanesco broccoli) whose petals, seeds and leaves correspond to this sequence, which is why it is so difficult to find a four-leaf clover

Fig.6 Sunflower - radial symmetry

Conclusion: In plants, we observe the following types symmetry:

• Tree - has an axis and a plane of symmetry
• Flower - radial symmetry (coincides with itself when rotated, has many planes of symmetry passing through the center of the flower)
• Leaves of flowers - bilateral symmetry (have only one plane of symmetry)
• Broccoli - fractal symmetry

3.3 Symmetry in animals

Symmetry in animals is understood as correspondence in size, shape and outline, as well as the relative location of body parts located on opposite sides of the dividing line.

Most animals are bilaterally symmetrical, which means they can be split into two identical halves. Some reach complete symmetry in an effort to attract a partner, such as a peacock (Fig. 7).

Rice. 7 Peacock - mirror symmetry

Darwin was positively annoyed by this bird, and wrote in a letter that "The sight of the peacock's tail feathers, whenever I look at it, makes me sick!" To Darwin, the tail seemed cumbersome and made no evolutionary sense, as it did not fit with his theory of "survival of the fittest". He was furious until he came up with the theory of sexual selection, which claims that animals develop certain features to increase their chances of mating. Therefore, peacocks have various adaptations to attract a partner.

Mirror symmetry is clearly visible in the butterfly; the symmetry of left and right appears here with almost mathematical rigor (Fig. 8).

Fig. 8 Butterfly - mirror symmetry

The symmetry of the Nautilus shell is very interesting (Fig. 9).

Rice. nine Nautilus shell - Fibonacci spiral

Nautilus shell twists into a "Fibonacci spiral". The shell tries to maintain the same proportional shape, which allows it to maintain it throughout its life (unlike people who change proportions throughout their lives). Not all Nautiluses have a Fibonacci shell, but they all follow a logarithmic spiral.

Conclusion: We see that bilateral (mirror) symmetry is a characteristic symmetry of all representatives of the animal world.

3.4. Symmetry in humans.

The human body also has bilateral symmetry (appearance and skeletal structure) (Fig. 10).

Fig.10 Bilateral symmetry

This symmetry has always been and is the main source of our aesthetic admiration for the well-built human body. Our own mirror symmetry is very convenient, it allows a person to move in a straight line and turn right and left with equal ease.

Conclusion: Man, as well as representatives of the animal world, has mirror symmetry.

4. Symmetry in Russian.

Symmetry can also be observed in the Russian language.

For example:

The letters A, M, T, W, P have a vertical axis of symmetry

B, Z, K, C, E, B, E - horizontal.

And the letters Zh, N, O, F, X have two axes of symmetry.

Symmetry can also be seen in the words: Cossack, hut.

There are whole phrases with this property (if you do not take into account the spaces between words):

“Look for a taxi”, “Argentina beckons a black man”, “Argentine appreciates a black man”,

"Lesha walked on a valve stick." And the rose fell on Azor's paw.

Such words are called palindromes.

Many poets were fond of them.

SEARCH TAXI

ARGENTINA MANIT NEGRA

LYOSHA ON THE STICK OF THE VALVE FOUND

A ROSE FALLED ON AZOR'S PAWS

Conclusion: Thus, we see an example of axial symmetry in letters, symmetry in whole phrases.

5. Symmetry in art.

5.1. Symmetry in architecture.

How long a person lives, so much he builds.

In ancient times, residential buildings were usually built symmetrically around a certain central point. Regardless of whether their shape was round,

square or rectangular, it was quite easy to determine the location of such a point in them. Very often, the hearth was placed at such a point. He was the focal point around which the life of the whole family passed.

The role of symmetry and proportions in architecture is great. It gives harmony and completeness to ancient temples, towers of medieval castles, modern buildings. Only by relentlessly following the laws of geometry, the architects of antiquity could create their masterpiece s.

Fine examples of symmetry are demonstrated by works of architecture. The general plans of buildings, facades, ornaments, cornices, columns show proportionality and harmony.

The most famous monuments are: St. Isaac's Cathedral, the Bolshoi Theater, the Winter Palace (Russia); Arc de Triomphe, Notre Dame Cathedral (France); Gugong Museum, Temple of Heaven (China); Pantheon, Milan Cathedral (Italy) (Fig. 11).

St. Isaac's Cathedral Bolshoi Theater

Winter Palace Notre Dame Cathedral

Gougong Museum Milan Cathedral

Fig.11

These architectural structures demonstrate mirror symmetry, but if we consider the individual walls of these buildings, we will see that they all have an axis of symmetry.

Symmetrical objects and buildings are more stable. Symmetry is widely used in building construction and decorative elements. This makes architectural structures more beautiful, harmonious, solemn and reliable.

Conclusion: Thus, we found out that there is mirror and axial symmetry in the buildings that surround us.

5.2. Symmetry in poetry and music.

In poetry we are dealing with the unity of symmetry and asymmetry. “The soul of music - rhythm - consists in the correct periodic repetition of parts of a musical work,” wrote the famous Russian physicist G.V. Wulf. - The correct repetition of the same parts as a whole is the essence of symmetry. We are all the more justified in applying the concept of symmetry to a piece of music in that this piece is written with the help of notes, i.e. receives a spatial geometric image, parts of which we can survey. He also wrote: “Like musical works, verbal works, especially poems, can also be symmetrical.”

The poems imply the symmetry of the alternation of rhymes, stressed syllables, that is, again, rhythm. The composer in his symphony may return to the same theme several times, gradually developing it.

Preservation of the theme and its change (development, development) - this is the unity of symmetry and asymmetry. And the more successfully the composer or poet solves the problem of the relationship between symmetry and asymmetry, the higher the artistic value of the created work of art.

Conclusion: The rhyme of poetry and the rhythm of music is one example of symmetry.

5.3. Symmetry in painting.

In art there is a mathematical theory of painting. This is perspective theory. Perspective is the doctrine of how to convey on a flat sheet of paper a sense of the depth of space, that is, to convey to others the world as we see it. It is based on the observance of several laws. The laws of perspective lie in the fact that the farther an object is from us, the smaller it seems to us, completely fuzzy, it has fewer details, its base is higher (Fig. 12).

Fig.12 Perspective.

If we follow all the rules, then the pictures will be harmonious, they will have a sense of stability, balance. If we break some rules, then the image will immediately become original, original and interesting.

Thus, the beauty of painting is determined, first of all, by the laws of mathematics.

To analyze the symmetry of the image, one can refer to the painting “Madonna Litta” by the brilliant Italian artist and scientist Leonardo da Vinci kept in the Hermitage (Fig. 13).

Fig. 13 Madonna Litta

You can pay attention: the figures of the Madonna and the child fit into a regular triangle, which, due to its symmetry, is especially clearly perceived by the eye of the viewer. Thanks to this, the mother and child immediately find themselves in the center of attention, as if brought to the fore. The head of the Madonna is exactly, but at the same time naturally placed between two symmetrical windows in the background.

paintings. Calm horizontal lines of gentle hills and clouds are visible in the windows. All this creates a feeling of peace and tranquility, enhanced by the harmonious combination of blue with yellowish and reddish tones.

The internal symmetry of the picture is well felt.

It turns out that whenever we admire this or that work of art, we talk about harmony, beauty, emotional impact, we thereby touch on the same inexhaustible problem - the problem of the relationship between symmetry and asymmetry. As a rule, being in a museum or in a concert hall, we do not think about this problem. After all, it is impossible to feel and analyze sensation at the same time.

Conclusion: So, we see that works of art are also subject to the laws of symmetry.

6. Symmetry in mathematics.

The idea of ​​symmetry is often the starting point in the hypotheses and theories of scientists of past centuries who believed in the mathematical harmony of the universe and saw in this harmony a manifestation of the divine principle. Since ancient times, man has been actively using the idea of ​​symmetry in his reflections on the picture of the universe.

The ancient Greeks believed that the universe is symmetrical simply because symmetry is beautiful. Based on symmetry considerations, they made a number of conjectures.

So, Pythagoras (5th century BC), considering the sphere as the most symmetrical and perfect form, concluded that the Earth is spherical and moves around the sphere. At the same time, he believed that the Earth moves along the sphere of a certain “central fire”. Around the same "fire", according to Pythagoras, the six planets known at that time, as well as the Moon, the Sun, and the stars, were supposed to circulate.

Using the idea of ​​symmetry extensively, scientists liked to refer not only to the spherical shape, but also to regular convex polyhedra. Even in the days of the ancient Greeks, an amazing fact was established - there are only five correct

convex polyhedra of various shapes. Symmetries of geometric bodies attached great importance to the Greek thinkers of the Pythagorean era. They believed that in order for a body to be "perfectly symmetrical", it must have an equal number of faces meeting at the corners, and these faces must be regular polygons, that is, figures with equal sides and angles. First explored by the Pythagoreans, these five regular polyhedra were later described in detail by Plato. The ancient Greek philosopher Plato attached particular importance to regular polyhedra, considering them to be the personification of four natural elements: fire-tetrahedron (the top is always turned upwards), earth-cube (the most stable body), air-octahedron, water-icosahedron (the most "rolling" body). The dodecahedron was presented as an image of the entire universe. That is why regular polyhedra are also called Platonic solids.

geometric symmetry- this is the most famous type of symmetry for many people. A geometric object is said to be symmetrical if, after it has been transformed geometrically, it retains some of its original properties. For example, a circle rotated around its center will have the same shape and size as the original circle. Therefore, the circle is called symmetric with respect to rotation (has axial symmetry).

The simplest types of spatial symmetry are central, axial, mirror-rotation and transfer symmetry.

central symmetry.

Two points A and A1 are called symmetric with respect to the point O, if O -middle of segmentAA 1 . Point O is considered symmetrical to itself.

Axial symmetry.

Convert F shape to F shape 1 , at which each of its points goes to a point symmetric with respect to the given line, is called a symmetry transformation with respect to the line a. The line a is called the axis of symmetry.

Mirror-rotation symmetry.

If another square is inscribed inside a square with a rotation, then this will be an example of mirror-rotation symmetry.

Portable symmetry.

If, during the transfer of a flat figure F along a given straight line AB by a distance a (or a multiple of this value), the figure is combined with itself, then they speak of translational symmetry. The straight line AB is called the transfer axis, the distance is called the elementary transfer or period.

7. Conclusion

We meet with symmetry everywhere - in nature, technology, art, science. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human development. Since ancient times, man has used symmetry in architecture. It gives harmony and completeness to ancient temples, towers of medieval castles, modern buildings. Symmetry literally permeates the whole world around us.

Knowledge of the geometric laws of nature is of great practical importance. We must not only learn to understand these laws, but also make them serve us for our benefit.

The principles of symmetry play an important role in physics and mathematics, chemistry and biology, engineering and architecture, painting and sculpture, poetry and music. The laws of nature that govern the picture of phenomena, inexhaustible in its diversity, in turn, obey the principles of symmetry.

There are many types of symmetry in both the plant and animal kingdoms, but with all its diversity of living organisms, the principle of symmetry always works, and this fact once again emphasizes the harmony of our world.

8.List of literature, Internet resources.

• L. Tarasov. This amazing symmetrical world. M., 1982
• Vulf G.V. Symmetry and its manifestations in nature. M., ed. Dep. Nar. com. Enlightenment, 1991
• Editor-in-Chief Maria Aksenova. Encyclopedia for children volume 2. M., "Avanta +" 2001.
• http://bse.scilib.com/
• http://vitim-school.edusite.ru
• http://obraz.volganet.ru
• http://elhow.ru
• http://365facts.ru

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Definition of symmetry; Central symmetry; Axial symmetry; Symmetry about the plane; rotational symmetry; Mirror symmetry; Symmetry of similarity; Symmetry of plants; Animal symmetry; Symmetry in architecture; Is man a symmetrical being? Symmetry of words and numbers;

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Definition of symmetry

SYMMETRY - proportionality, the sameness in the arrangement of parts of something on opposite sides of a point, line or plane. (Ozhegov's Explanatory Dictionary) So, a geometric object is considered symmetrical if something can be done with it, after which it will remain unchanged.

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Central symmetry

A figure is called symmetric with respect to the point O if for each point of the figure the point symmetric to it with respect to the point O also belongs to this figure. Point O is called the center of symmetry of the figure.

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Examples of figures with central symmetry are the circle and the parallelogram. 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. Any line also has central symmetry (any point on the line is its center of symmetry). The graph of an odd function is symmetrical with respect to the origin. An example of a figure that does not have a center of symmetry is an arbitrary triangle.

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Axial symmetry

A figure is called symmetric with respect to the line a if for each point of the figure the point symmetric to it with respect to the line a also belongs to this figure. The line a is called the axis of symmetry of the figure.

Slide 7

An undeveloped angle has one axis of symmetry - a straight line on which the bisector of the angle is located. An isosceles triangle also has one axis of symmetry, and an equilateral triangle has three axes of symmetry. A rectangle and a rhombus, which are not squares, each have two axes of symmetry, and a square has four axes of symmetry. A circle has an infinite number of them. The graph of an even function when plotted is symmetrical about the y-axis. There are figures that do not have any axis of symmetry. Such figures include a parallelogram other than a rectangle, a scalene triangle.

Slide 8

Symmetry about the plane

Points A and A1 are called symmetrical with respect to the plane a (plane of symmetry) if the plane a passes through the midpoint of the segment AA1 and is perpendicular to this segment. Each point of the plane is considered symmetrical to itself. Two figures are said to be symmetric with respect to a plane (or mirror-symmetric with respect to) if they consist of pairwise symmetrical points. This means that for each point of one figure, a (relatively) symmetrical point to it lies in another figure.

Slide 9

Rotational symmetry

A body (or figure) has rotational symmetry if, when rotated through an angle of 360º / n, where n is an integer, it completely coincides with its original position near some straight line AB (axis of symmetry). Radial symmetry is a form of symmetry that is preserved when an object rotates around a certain point or line. Often this point coincides with the center of gravity of the object, that is, the point at which an infinite number of axes of symmetry intersect. Such objects can be a circle, a sphere, a cylinder or a cone.

Slide 10

Mirror symmetry

Mirror symmetry connects any object and its reflection in a flat mirror. One figure (or body) is said to be mirror symmetrical to another if together they form a mirror symmetrical figure (or body). Symmetrically mirrored figures, for all their similarities, differ significantly from each other. Two mirror-symmetric flat figures can always be superimposed on each other. However, for this it is necessary to remove one of them (or both) from their common plane.

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Similarity symmetry

Similarity symmetry is a kind of analogue of the previous symmetries with the only difference that they are associated with a simultaneous decrease or increase in similar parts of the figure and the distances between them. The simplest example of such symmetry is nesting dolls. Sometimes figures can have different types of symmetry. For example, some letters have rotational and mirror symmetry: Zh, N, M, O, A.

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There are many other kinds of symmetries that are abstract in nature. For example: Permutation symmetry, which consists in the fact that if identical particles are interchanged, then no changes occur; Gauge symmetries are related to scale change. In inanimate nature, symmetry first of all arises in such a natural phenomenon as crystals, of which almost all solid bodies are composed. It is she who determines their properties. The most obvious example of the beauty and perfection of crystals is the well-known snowflake.

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We meet with symmetry everywhere: in nature, technology, art, science. The concept of symmetry runs through the entire centuries-old history of human creativity. The principles of symmetry play an important role in physics and mathematics, chemistry and biology, engineering and architecture, painting and sculpture, poetry and music. The laws of nature also obey the principles of symmetry.

Slide 14

plant symmetry

Many flowers have an interesting property: they can be rotated so that each petal takes the position of its neighbor, while the flower is combined with itself. Such a flower has an axis of symmetry. Helical symmetry is observed in the arrangement of leaves on the stems of most plants. Being located by a screw along the stem, the leaves seem to spread out in all directions and do not obscure each other from the light, which is essential for plant life. Bilateral symmetry is also possessed by plant organs, for example, the stems of many cacti. In botany, radially symmetrically built flowers are often found.

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animal symmetry

Symmetry in animals is understood as correspondence in size, shape and outline, as well as the relative location of body parts located on opposite sides of the dividing line. The main types of symmetry are radial (radial) - it is possessed by echinoderms, coelenterates, jellyfish, etc.; or bilateral (two-sided) - we can say that every animal (be it an insect, fish or bird) consists of two halves - right and left. Spherical symmetry takes place in radiolarians and sunflowers. Any plane drawn through the center divides the animal into equal halves.

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Symmetry in architecture

The symmetry of a structure is associated with the organization of its functions. The projection of the plane of symmetry - the axis of the building - usually determines the location of the main entrance and the beginning of the main traffic flows. Each detail in a symmetrical system exists as a twin of its obligatory pair located on the other side of the axis, and due to this, it can only be considered as part of the whole. Mirror symmetry is the most common in architecture. The buildings of Ancient Egypt and the temples of ancient Greece, amphitheatres, baths, basilicas and triumphal arches of the Romans, palaces and churches of the Renaissance, as well as numerous buildings of modern architecture are subordinate to it.

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For a better reflection of symmetry, accents are placed on the structures - especially significant elements (domes, spiers, tents, front entrances and stairs, balconies and bay windows). To design the decoration of architecture, an ornament is used - a rhythmically repeating pattern based on the symmetrical composition of its elements and expressed by line, color or relief. Historically, several types of ornaments have developed based on two sources - natural forms and geometric figures. But an architect is first and foremost an artist. And therefore, even the most “classical” styles often used dissymmetry - a nuanced deviation from pure symmetry or asymmetry - a deliberately asymmetrical construction.

Slide 18

Is man a symmetrical being?

No one will doubt that outwardly a person is built symmetrically: the left hand always corresponds to the right and both hands are exactly the same. But the similarity between our hands, ears, eyes, and other parts of the body is the same as between an object and its reflection in a mirror. The asymmetry of the face of the statue of Venus de Milo is expressed by the displacement of the nose to the right from the midline, in a higher position of the left auricle and left eye socket and a smaller distance from the midline of the left eye socket than the right one. Supporters of symmetry believed that the face of Venus would be much more beautiful if it were symmetrical.

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Numerous measurements of facial parameters in men and women have shown that its right half, compared to the left, has more pronounced transverse dimensions, which gives the face coarser features inherent in male gender. The left half of the face has more pronounced longitudinal dimensions, which gives it smooth lines and femininity. This fact explains the predominant desire of females to pose for artists on the left side of the face, and males on the right.

Slide 20

Symmetry of words and numbers

A palindrome (from Gr. Palindromos - running back) is some object in which the symmetry of the components is specified from the beginning to the end and from the end to the beginning. For example, a phrase or text. The forward text of a palindrome, read in accordance with the normal direction of reading in a given script (usually from left to right), is called forward, the reverse is called a crab or reverse (from right to left). Some numbers also have symmetry. The path led to the left, to the port of Lesha on a shelf of a bug found Argentina beckoning a black man 101 2002 6996

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Axial symmetry is inherent in all forms in nature and is one of the fundamental principles of 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 very word "symmetry" was coined by them. It denotes the 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. And indeed, those phenomena and forms that have proportionality and completeness are “pleasant to the eye”. We call them correct.

Axial symmetry occurs in nature. It determines not only the general structure of the organism, but also the possibilities of its subsequent development. The geometric shapes and proportions of living beings are formed by "axial symmetry". Its definition is formulated as follows: it is the property of objects to be combined under various transformations. The ancients believed that the sphere possesses the principle of symmetry to the fullest extent. They considered this form harmonious and perfect. Axial Symmetry in Living Nature If you look at any living being, the symmetry of the body structure immediately catches your eye. Man: two arms, two legs, two eyes, two ears, and so on. Each type of animal has a characteristic color. If a pattern appears in the coloring, then, as a rule, it is mirrored on both sides. This means that there is a certain line along which animals and people can be visually divided into two identical halves, that is, their geometric structure is based on axial symmetry. Nature creates any living organism not chaotically and senselessly, but according to the general laws of the world order, because nothing in the Universe has a purely aesthetic, decorative purpose. The presence of various forms is also due to a natural need.

In the world, we are surrounded everywhere by such phenomena and objects as: a typhoon, a rainbow, a drop, leaves, flowers, etc. Their mirror, radial, central, axial symmetry are obvious. To a large extent, it is due to the phenomenon of gravity. Often, the concept of symmetry is understood as the regularity of the change of any phenomena: day and night, winter, spring, summer and autumn, and so on. In practice, this property exists wherever there is order. And the very laws of nature - biological, chemical, genetic, astronomical - are subject to the principles of symmetry common to all of us, since they have an enviable consistency. Thus, balance, identity as a principle has a universal scope. Axial symmetry in nature is one of the "cornerstone" laws on which the universe as a whole is based.

"SYMMETRY - A SYMBOL OF BEAUTY, HARMONY AND PERFECTION"

FROM symmetry(ancient Greek - "proportionality") - the regular arrangement of similar (identical) parts of the body or forms of a living organism, the totality of living organisms relative to the center or axis of symmetry. This implies that proportionality is part of harmony, the correct combination of parts of the whole.

G armonia- a Greek word meaning "consistency, proportion, unity of parts and whole." Outwardly, harmony can manifest itself in melody, rhythm, symmetry and proportion. The law of harmony reigns in everything, And everything in the world is rhythm, chord and tone. J. Dryden

FROM perfection- the highest degree, the limit of any positive quality, ability, or skill.

“Freedom is the main inner sign of every being, created in the image and likeness of God; in this sign lies the absolute perfection of the plan of creation.” N. A. Berdyaev Symmetry is the fundamental principle of the structure of the world.

Symmetry is a common phenomenon, its universality serves as an effective method of understanding nature. Symmetry in nature is needed to maintain stability. Inside the external symmetry lies the internal symmetry of the construction, which guarantees balance.

Symmetry is a manifestation of the desire of matter for reliability and strength.

Symmetrical forms provide repeatability of successful forms, therefore they are more resistant to various influences. Symmetry is multifaceted.

In nature and, in particular, in living nature, symmetry is not absolute and always contains some degree of asymmetry. Asymmetry - (Greek α- - "without" and "symmetry") - lack of symmetry.

Symmetry in nature

Symmetry, like proportion, was considered a necessary condition for harmony and beauty.

Looking closely at nature, you can see the common even in the most minor things and details, find manifestations of symmetry. The shape of a tree leaf is not random: it is strictly regular. The leaf is, as it were, glued together from two more or less identical halves, one of which is mirrored relative to the other. The symmetry of the leaf is persistently repeated, whether it be a caterpillar, a butterfly, a bug, etc.

There is a very complex multilevel classification of symmetry types. Here we will not consider these difficulties of classification, we will note only the fundamental provisions and recall the simplest examples.

At the highest level, three types of symmetry are distinguished: structural, dynamic, and geometric. Each of these types of symmetry at the next level is divided into classical and non-classical.

Below are the following hierarchical levels. A graphic representation of all levels of subordination gives a branched dendrogram.

In everyday life, we most often encounter the so-called mirror symmetry. This is the structure of objects when they can be divided into right and left or upper and lower halves by an imaginary axis, called the axis of mirror symmetry. In this case, the halves located on opposite sides of the axis are identical to each other.

Reflection in the plane of symmetry. Reflection is the most well-known and most commonly occurring type of symmetry in nature. The mirror reproduces exactly what it "sees", but the order considered is reversed: your double's right hand will actually be left, since the fingers are placed on it in reverse order. Mirror symmetry can be found everywhere: in the leaves and flowers of plants. Moreover, mirror symmetry is inherent in the bodies of almost all living beings, and such a coincidence is by no means accidental. Mirror symmetry has everything that can be divided into two mirror equal halves. Each of the halves serves as a mirror image of the other, and the plane separating them is called the plane of mirror reflection, or simply the mirror plane.

rotational symmetry. The appearance of the pattern will not change if it is rotated by some angle around the axis. The symmetry that arises in this case is called rotational symmetry. The leaves and flowers of many plants exhibit radial symmetry. This is such a symmetry in which a leaf or flower, turning around the axis of symmetry, passes into itself. On cross sections of the tissues that form the root or stem of a plant, radial symmetry is clearly visible. The inflorescences of many flowers also have radial symmetry.

Flowers, mushrooms, trees have radial-beam symmetry. Here it can be noted that on unplucked flowers and mushrooms, growing trees, the symmetry planes are always oriented vertically. Determining the spatial organization of living organisms, the right angle organizes life by the forces of gravity. The biosphere (layer of being of living beings) is orthogonal to the vertical line of the earth's gravity. Vertical stems of plants, tree trunks, horizontal surfaces of water spaces and in general Earth's crust make a right angle. The right angle underlying the triangle governs the space of symmetry of similarities, and similarity, as already mentioned, is the goal of life. Both nature itself and the original part of man are in the power of geometry, subject to symmetry both as essences and as symbols. No matter how the objects of nature are built, each has its own main feature, which is displayed by the form, whether it is an apple, a grain of rye or a person.

Examples of radial symmetry.

The simplest type of symmetry is mirror (axial), which occurs when a figure rotates around the axis of symmetry.

In nature, mirror symmetry is characteristic of plants and animals that grow or move parallel to the surface of the Earth. For example, the wings and body of a butterfly can be called the standard of mirror symmetry.

Axial symmetry this is the result of rotating exactly the same elements around a common center. Moreover, they can be located at any angle and with different frequencies. The main thing is that the elements rotate around a single center. In nature, examples of axial symmetry are most often found among plants and animals that grow or move perpendicular to the Earth's surface.

Also exists screw symmetry.

Translation can be combined with reflection or rotation, and new symmetry operations arise. Rotation by a certain number of degrees, accompanied by translation to a distance along the axis of rotation, generates helical symmetry - the symmetry of a spiral staircase. An example of helical symmetry is the arrangement of leaves on the stem of many plants. If we consider the arrangement of leaves on a tree branch, we will notice that the leaf is separated from the other, but also rotated around the axis of the trunk.

The leaves are arranged on the trunk along a helical line, so as not to obscure sunlight from each other. The head of a sunflower has processes arranged in geometric spirals that unwind from the center outwards. The youngest members of the spiral are in the center. In such systems, one can notice two families of spirals that unwind in opposite directions and intersect at angles close to right. But no matter how interesting and attractive the manifestations of symmetry in the world of plants are, there are still many secrets that control the development processes. Following Goethe, who spoke of the striving of nature towards a spiral, it can be assumed that this movement is carried out along a logarithmic spiral, starting each time from a central, fixed point and combining translational movement (stretching) with a turn of rotation.

Based on this, it is possible to formulate in a somewhat simplified and schematized form (from two points) the general law of symmetry, which is clearly and everywhere manifested in nature:

1. Everything that grows or moves vertically, i.e. up or down relative to earth's surface, obeys radial-beam symmetry in the form of a fan of intersecting planes of symmetry. The leaves and flowers of many plants exhibit radial symmetry. This is such a symmetry in which a leaf or flower, turning around the axis of symmetry, passes into itself. On cross sections of the tissues that form the root or stem of a plant, radial symmetry is clearly visible. The inflorescences of many flowers also have radial symmetry.

2. Everything that grows and moves horizontally or obliquely with respect to the earth's surface is subject to bilateral symmetry, leaf symmetry.

This universal law of two postulates obeys not only flowers, animals, easily mobile liquids and gases, but also hard, unyielding stones. This law affects the changing forms of clouds. On a calm day, they have a dome shape with more or less clearly expressed radial-radial symmetry. The influence of the universal law of symmetry is, in fact, purely external, rough, imposing its stamp only on the external form of natural bodies. Their internal structure and details escape from his power.

Symmetry is based on similarity. It means such a relationship between elements, figures, when they repeat and balance each other.

Similarity symmetry. Another type of symmetry is similarity symmetry, associated with the simultaneous increase or decrease of similar parts of the figure and the distances between them. Matryoshka is an example of this kind of symmetry. Such symmetry is very widespread in wildlife. It is demonstrated by all growing organisms.

The basis of the evolution of living matter is the symmetry of similarity. Consider a rose flower or a head of cabbage. An important role in the geometry of all these natural bodies is played by the similarity of their similar parts. Such parts, of course, are interconnected by some common geometrical law, not yet known to us, which makes it possible to derive them from each other. The symmetry of similarity, realized in space and time, manifests itself everywhere in nature on everything that grows. But it is precisely the growing forms that countless figures of plants, animals and crystals belong to. The shape of the tree trunk is conical, strongly elongated. Branches are usually arranged around the trunk in a helix. This is not a simple helix: it gradually narrows towards the top. And the branches themselves decrease as they approach the top of the tree. Therefore, here we are dealing with a helical axis of symmetry of similarity.

Living nature in all its manifestations reveals the same goal, the same meaning of life: every living object repeats itself in its own kind. The main task of life is life, and the accessible form of being lies in the existence of separate integral organisms. And not only primitive organizations, but also complex cosmic systems, such as man, demonstrate an amazing ability to literally repeat from generation to generation the same forms, the same sculptures, character traits, the same gestures, manners.

Nature discovers similarity as its global genetic program. The key to change also lies in similarity. Similarity governs living nature as a whole. Geometric similarity is the general principle of the spatial organization of living structures. A maple leaf is like a maple leaf, a birch leaf is like a birch leaf. Geometric similarity permeates all branches of the tree of life. Whatever metamorphoses a living cell belonging to whole organism and performing the function of its reproduction into a new, special, single object of being, it is the point of "beginning", which, as a result of division, will be transformed into an object similar to the original one. This unites all types of living structures, for this reason there are stereotypes of life: a person, a cat, a dragonfly, an earthworm. They are endlessly interpreted and varied by division mechanisms, but remain the same stereotypes of organization, form and behavior.

For living organisms, the symmetrical arrangement of parts of the body organs helps them to maintain balance during movement and functioning, ensures their vitality and better adaptation to the outside world, which is also true in the plant world. For example, the trunk of a spruce or pine is most often straight and the branches are evenly spaced relative to the trunk. The tree, developing under the action of gravity, reaches a stable position. Towards the top of the tree, its branches become smaller in size - it takes on the shape of a cone, since light must fall on the lower branches, as well as on the upper ones. In addition, the center of gravity should be as low as possible, the stability of the tree depends on this. The laws of natural selection and universal gravitation have contributed to the fact that the tree is not only aesthetically beautiful, but also expediently arranged.

It turns out that the symmetry of living organisms is associated with the symmetry of the laws of nature. At the everyday level, when we see the manifestation of symmetry in animate and inanimate nature, we involuntarily feel a sense of satisfaction with the universal, as it seems to us, order that reigns in nature.

As the ordering of living organisms, their complication in the course of the development of life, asymmetry more and more prevails over symmetry, displacing it from biochemical and physiological processes. However, a dynamic process also takes place here: symmetry and asymmetry in the functioning of living organisms are closely related. Externally, man and animals are symmetrical, but their internal structure is significantly asymmetrical. If in lower biological objects, for example, lower plants, reproduction proceeds symmetrically, then in higher ones there is a clear asymmetry, for example, the division of sexes, where each sex introduces genetic information peculiar only to it into the process of self-reproduction. Thus, the stable preservation of heredity is a manifestation of symmetry in a certain sense, while asymmetry is manifested in variability. In general, the deep internal connection of symmetry and asymmetry in living nature determines its emergence, existence and development.

The universe is an asymmetric whole, and life as it is presented must be a function of the asymmetry of the universe and its consequences. Unlike the molecules of inanimate nature, the molecules of organic substances have a pronounced asymmetric character (chirality). Attaching great importance to the asymmetry of living matter, Pasteur considered it to be the only, clearly demarcating line that can currently be drawn between animate and inanimate nature, i.e. what distinguishes living matter from non-living matter. Modern science has proved that in living organisms, as in crystals, changes in structure correspond to changes in properties.

It is assumed that the resulting asymmetry occurred abruptly as a result of the Big Biological Bang (by analogy with the Big Bang, which resulted in the formation of the Universe) under the influence of radiation, temperature, electromagnetic fields, etc. and found its reflection in the genes of living organisms. This process is essentially also a process of self-organization.