A message on the topic of symmetry in ornaments. Symmetry in Ornament: An Introduction

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TYPES AND ROLE OF SYMMETRY IN THE APPEARANCE OF THE ORNAMENTAL IMAGE

GREEK WORD SYMMETRY MEANS PROPORTIONALITY, AND THE WORD ASYMMETRY - ITS ABSENCE. SYMMETRY IS A CERTAIN SPATIAL ORDER, A MATHEMATICALLY EXACT REGULARITY IN THE ARRANGEMENT OF OBJECTS OR THEIR PARTS. EXACTLY THANKS TO THE SYMMETRY OF ORNAMENT, IT CAN PLAY AN ORGANIZING ROLE IN WORKS OF DECORATIVE ART.

EXISTS SEVERAL TYPES OF SYMMETRY. MOST OFTEN WE MEET WITH MIRROR SYMMETRY. IT IS INSTALLED IN THE HUMAN BODY AND MOST ANIMALS AND MANY OBJECTS THAT WE USE. IT IS WIDELY USED IN ART.

SYMMETRY ATTRACTS PEOPLE'S ATTENTION FROM ANCIENT TIMES. IN THE OBSERVED SYMMETRY OF NATURAL FORMS, THE CONFIRMATION OF THE INTELLIGENT DEVICE OF THE UNIVERSE WAS FOUND, THE ESSENCE OF BEAUTY, THE EVIDENCE OF HARMONY, WERE SEARCHED IN IT.

TO THE CONCEPTS OF SYMMETRY ARE THE PLANE OF SYMMETRY, THE AXIS OF SYMMETRY, THE CENTER OF SYMMETRY.

A PLANE OF SYMMETRY IS A PLANE THAT DIVIDES A FIGURE INTO TWO MIRROR EQUAL PARTS LOCATED RELATIVELY TO EACH OTHER AS AN OBJECT AND ITS MIRROR REFLECTION.

AXIS OF SYMMETRY IT IS CALLED A STRAIGHT LINE AROUND WHICH A SYMMETRIC FIGURE CAN BE TURNED SEVERAL TIMES IN SUCH WAY THAT EVERY TIME THE FIGURE IS "SELF-COMBINING" WITH ITSELF IN SPACE.

THE CENTER OF SYMMETRY SUCH A SPECIAL POINT INSIDE THE FIGURE IS CALLED, CHARACTERIZED BY THE THAT ANY DIRECT LINE CARRIED OUT THROUGH THE POINT ON BOTH SIDES OF IT AND AT EQUAL DISTANCES ENCOUNTERS THE SAME (CORRESPONDING) POINTS OF THE FIGURE. "IDEAL" EXAMPLE OF SUCH FIGURE IS A BALL, THE CENTER OF WHICH IS ITS CENTER OF SYMMETRY.

TYPES OF SYMMETRY, USED IN ORNAMENTAL STRUCTURES:

* AXIAL (SYMMETRY RELATIVELY DIRECT)

* CENTRAL (SYMMETRY REGARDING THE POINT)

* MIRROR (SYMMETRY REGARDING THE PLANE)

EXAMPLE(s)

A COMMON TYPE OF SYMMETRY IN ORNAMENT IS BORDER. A BORDER IS A LINEAR ORNAMENT OBTAINED BY THE PARALLEL TRANSFER OF A CERTAIN FIGURE. LINEAR ORNAMENT IS USUALLY USED WHERE IT IS NECESSARY TO LIMIT SOME SURFACE OR TO BREAK IT INTO PARTS.

EXAMPLE(s)

IN PRACTICE, THE BORDER CAN BE BUILT NOT ONLY IN A STRAIGHT, BUT ALSO ALONG A BROKEN OR CURVED LINE. IN ANY CASE, PARALLEL TRANSFER IS PERFORMED FOLLOWING THE BENDING AND FRACTURES OF THE AXIS. BORDER, EXCEPT THE PARALLEL TRANSFER SYMMETRY, MAY HAVE OTHER SYMMETRY ELEMENTS. THEY APPEAR IN THE CASES WHEN ONE TYPE OF SYMMETRY OR THAT IS ATTRACTED TO EVERY SEPARATELY TAKEN ELEMENTARY ORNAMENTAL MOTIF.

SOCKET, AS A TYPE OF SYMMETRY, FOUND WIDE APPLICATION IN ORNAMENT. FOUR TYPES OF SOCKETS WHICH DIFFER IN THE APPLICATION OF THE NUMBER OF AXES AND PLANES OF SYMMETRY APPLIED TO THE ELEMENTARY FIGURE. SO WHEN CONSTRUCTING SOCKETS, ONLY THE PLANE OF SYMMETRY OR ONLY THE AXIS OF SYMMETRY, OR THE AXIS AND THE PLANE OF SYMMETRY, OR AND A PLENTY OF PLANES OF SYMMETRY CAN BE USED.

EXAMPLE(s)

*MIRROR SOCKET *AXIAL SOCKET

IN THE ART OF ORNAMENT IS FREQUENTLY USED - NET ORNAMENTS). NET ORNAMENT, AS A KIND OF CENTRAL AND AXIAL SYMMETRY, IS WIDELY COMMON IN EMBROIDERY. IT FILLS THE ENTIRE SURFACE AND IS LOCATED ON AN INVISIBLE GRID WITH THE MOST VARIOUS SHAPES OF CELLS: RHOMBUS, SQUARE, TRIANGLE.

THERE ARE FIVE SYSTEMS OF POINTS (NODES OF THE ORNAMENTAL GRID), WHICH ARE THE BASIS FOR THE CONSTRUCTION OF MOST NET ORNAMENTS: SQUARE; REGULAR TRIANGULAR, THE BASIS OF WHICH IS AN EQUILATERAL TRIANGLE; RECTANGULAR, CONSISTING OF RECTANGLES WITH ANY ASPECT RATIO; RHOMBIC, CONSISTING OF RHOMBOS WITH ANY RATIO OF DIAGONALS; PARALLELOGRAMMATIC, CONSISTING OF PARALLELOGRAMS OF AN ARBITRARY TYPE, AND THE CELL SLOPE CAN BE BOTH LEFT AND RIGHT.

EXAMPLE(s)

THERE ARE 17 TYPES OF SYMMETRY OF NET ORNAMENTS (SHUBNIKOV A.V. SYMMETRY. M.; L., 1940). EACH OF THESE TYPES IS ASSOCIATED WITH ITS OWN VISUAL PERCEPTION, WHICH CAN BE Illustrated ONLY ONLY ONE EXAMPLE, WHEN A SINGLE ELEMENTARY FIGURE IS ACCEPTED AS THE BASIS. THIS IS DONE IN ORDER TO DO NOT OVERLIGHTEN THE PICTURES WITH DETAILS AT THE FIRST TIME, AND TO MAKE IT EASY TO UNDERSTAND THE ESSENCE OF THIS ORDER TYPE OF SYMMETRY OF NETWORK ORNAMENT. FOR THE BASIS WE TAKE THE SAME ELEMENTARY FIGURE AS FOR THE CONSTRUCTION OF ROSETTES AND A RIBBON ORNAMENT.

IN REALITY, IMAGES OF AN ELEMENTARY FIGURE, REPEATED WHEN TRANSFERD ON BOTH AXIS, CAN BE COMPLETELY SEPARATED FROM EACH OTHER, I.E. DO NOT INTERCEPSE BUT MAY THEM ALSO CONSIST OF DISCONNECTED PARTS:

EXAMPLE(s)

Alexandrov Artyom, Makarov Alexander

This work contains theoretical material and a presentation on the types of symmetry in the ancient Russian ornament.

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SYMMETRY OF ANCIENT RUSSIAN ORNAMENT

The phenomenon of symmetry finds a multifaceted and multilevel expression in various sciences and arts. Traditionally, philosophical understanding of the concept of symmetry occurs on the basis of the natural sciences and mathematics. In addition to the specific scientific content (mathematical, physical, etc.), it has a universal ontological meaning, as well as the status of a categorical definition and is used in describing mathematical concepts, physical phenomena and processes, various objects of living and inanimate nature, art objects.

In the process historical development and comprehension of the concept of symmetry, a special stage of the philosophical and aesthetic analysis of symmetry as a measure of the beauty and harmony of the universe is associated with the work of an outstanding mathematician of the 20th centuryHermann Weyl "Symmetry" (1952). By symmetry, Weil understood the immutability (invariance) of an object under a certain kind of transformation: an object is symmetrical when it can be subjected to some operation, after which it will look the same as before the transformation. The author made an important conclusion about ornamental symmetry: “Thus, there are 17 essentially various kinds symmetries that a two-dimensional ornament with a double infinite ratio can have ... The art of ornament contains implicitly the most ancient part of higher mathematics known to us”…

Speaking about the symmetry characteristic of the ornament, Egyptian, Greek, Arabic ornaments are usually cited as an example. Meanwhile, Russian ornament (along with historical and cultural significance) has interesting mathematical features.

Let us briefly consider the state of mathematical significance in Russia in the period of the 9th - 10th centuries. The practical activity of people in Russia, as in other countries of Europe and Asia, made it necessary to develop arithmetic knowledge and ideas about the properties of geometric shapes. Excavations of ancient settlements testify that mathematical knowledge of a utilitarian nature was widespread in Russia. These were skills rather than knowledge that were transmitted orally and included ideas about natural numbers and operations with them (addition, subtraction, multiplication and division), as well as about simple fractions. In addition, such a geometric tool as a compass was well known. This statement is supported by the widespread use of circle ornaments on jewelry and household items.

According to Academician B. A. Rybakov, a well-known archaeologist and historian of world renown, the ancient Slavic ornament was based on universal ideas about the world. Myth and ritual combined elements of magic and totemism, the beginnings of artistic creativity, social norms that regulate people's behavior. All this is reflected in the motifs characteristic of the Russian ornament.

Among them are the following.

Motive " amulet "signs, which were applied to clothes, household items and various details of the home.

The Slavs place their protective symbols on the “vulnerable” areas of the dwelling and yard. With an incantatory ornament, they close all the openings through which evil spirits can penetrate to a person: gates, doors, windows, and gates. The symbol of the sun, placed in the openings, should protect the dwelling from the evil spirits of the night. We also find solar signs on the roof of the house - in the form of a ridge (moreover, the horse, as mentioned above, in itself symbolized the sun), under the roof - in the form of a real image of the sun or its symbol, on the facade, on the window frames. On the facade of the house, the Slavs depict the sky and the course of the sun - in the morning, at noon and in the evening. At the same time, a thunder sign (a circle divided into six sectors) is usually placed on the central towel - a symbol of Rod or Perun, which protected the house from lightning. Next to the symbol of the sun, the Slavs almost always depict the sign of the earth or the sown field, denoting the unity of the heavenly and the earthly, explicit. The world of reality is also symbolized by images of birds, plants, goddesses, intertwined with fantastic creatures.

Ceremonial towels have a special magic and a special meaning among the Slavs. The patterns on them represent nothing more than Slavic calendar, which figuratively reflected family and tribal events or agrarian ritual celebrations characteristic of agricultural culture. On all holidays, bread and salt were the first to be taken out on a towel. Salt is a symbol of the sun, love; bread - earth, mercy; a towel is a symbol of human life, a streak of fate, a part of pure outer space. During the betrothal, the hands of the bride and groom were connected, wrapped in a towel, so that the young would live in abundance; with the same meaning, during childbirth, the grandmother took the baby on a new towel. Ordinary towels were woven and embroidered with a simple charman ornament against pestilence, for asking for rain, etc., moreover, in order for the towel to have miraculous power, the weaver had to do the work in one light day.

Motif of the ancient goddess Mokosha (the only one female image in ancient Russian mythology) as a specific embodiment of the idea of the Great Foremother, common to all peoples at a certain stage of historical existence. Usually her image was placed in the center of a three-part composition.

Images of Lada (goddess of fun, prosperity and harmony in the family; goddess of brides) or Mokosh protecting the world in the house are also located on carved platbands (they are depicted with their hands raised, as if asking for protection from the supreme gods, who give heavenly moisture and light), as well as on outbuildings (they are depicted with their hands down, as if they are addressing the mother-damp earth, which gives the harvest).

Tree of life motif.

Even the most abstract ornament of Russian masters retains its man-made. In traditional national decorations and household items, you can find all kinds of symmetry on the plane: central, axial, rotary, portable.

The earth is represented by an ideogram: a rectangle divided by diagonals into four parts with a pattern repeating in them. This configuration is characterized by axial symmetry in combination with the central one.

These types of symmetry predominate in the images of the plant world.

For example, incantatory signs reflect the dynamics of the development of a sprout (each fragment of the picture symbolizes one of the stages of this process).

There are several types of sun signs, they are characterized by rotational symmetry of different orders. The most common is a circle divided by radii into equal sectors (“Wheel of Jupiter”), as well as a circle with a cross inside. Of particular interest are the signs depicting the "running" sun.

For ancient Russian art, a type of symmetry is characteristic, expressed in doubling (pairing) of one or another image.

Pairing responded to two types of wishes:"in all directions" « everywhere on earth”, “from all sides”, “from everywhere”.

Thus, the description of individual ancient Russian ornamental motifs and the scheme of their location on the details of the dwelling, decoration and household items are the reason for the aesthetic appeal of the ornament.

Head: O. V. Afanasyeva

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"SYMMETRY IN ANCIENT RUSSIAN ORNAMENT" Completed by: students of group No. 7 A. Alexandrov, A. Makarov Leader: O. V. Afanasyeva

Purpose: To demonstrate the possibilities of the laws of geometry when creating various ornaments that adorn many objects of arts and crafts. Tasks: Repeat and expand knowledge about known types of symmetry. Learn to identify and describe symmetry in an ornament pattern. Summarize and systematize the information used to create the ornament.

TYPES OF PLANE ORNAMENT A planar ornament is an ornament made on a plane with the help of lines. Planar ornaments are used to decorate clothes, to paint the walls of buildings in the form of borders.

a) Inside the strip b) On the strip c) On both sides of the strip LINEAR (location options):

MESH It is called mesh because its composition is built using a grid. mesh ornament It is used to decorate the floor, ceiling, walls of the room.

CENTRIC A centric ornament is a pattern whose decorative elements are grouped in such a way that they create a centric movement. A closed ornament is most often used to decorate napkins, tablecloths, plates, etc.

GEOMETRIC In a geometric ornament, geometric figures act as a motif: a circle, a square, a triangle, etc.

VEGETABLE In vegetable - forms are borrowed from the natural world, for example: plant leaves, fruits, flowers, twigs, etc.

OLD RUSSIAN ORNAMENT Ornamental compositions, as decorations for household items, existed in Russia for a very long time. Peasant huts were decorated with carved patterns. Curtains embroidered with intricate ornaments hung on the windows, and the whole house was filled with painted dishes, embroidery and lace. Russian ornament is very diverse. Decorative elements were made up of symbols and reflected the feelings and interests of the peasant. The color in the pattern had special meaning. Red color expressed delight, joy.

EXAMPLES OF RUSSIAN ORNAMENT

THE MOTIVE OF THE “GUARD” SIGNS The Slavs placed their “charm” symbols on the “vulnerable” areas of the dwelling and yard. With a spell ornament, they closed all the openings through which evil spirits could penetrate to a person: gates, doors, windows, and gates. The symbol of the sun, placed in the openings, should protect the dwelling from the evil spirits of the night. On the roof of the house - a skate (symbolized the sun). On the facade of the house, the Slavs depicted the sky and the course of the sun - in the morning, at noon and in the evening.

Ceremonial towels had a special magic and a special meaning among the Slavs. The patterns on them represented nothing more than the Slavic calendar, which figuratively reflected family and tribal events or agrarian ritual celebrations characteristic of agricultural culture. On all holidays, bread and salt were the first to be taken out on a towel. Salt is a symbol of the sun, love; bread - earth, mercy; a towel is a symbol of human life, a streak of fate, a part of pure outer space.

THE MOTIVE OF THE BRIDGE It is typical for the so-called mermaid bracelets as a sign of water and the kingdom of the underground lord Pereplut. MOTIVE TREE OF LIFE

MOTIVE OF THE ANCIENT GODDESS MOKOSHA The only female image in ancient Russian mythology, as a specific embodiment of the idea of the Great Foremother, common to all peoples at a certain stage of historical existence. Usually her image was placed in the center of a three-part composition.

TYPES OF SYMMETRY USED TO CREATE ORNAMENT: Parallel translation Central symmetry Axial symmetry Rotation

PARALLEL TRANSFER Parallel translation moves all points of the figure in the same direction by the same distance.

CENTRAL SYMMETRY With central symmetry, each point goes to a point symmetrical to it with respect to a fixed point.

AXIAL SYMMETRY With axial symmetry, each point of the figure goes to a point symmetrical to it with respect to a fixed line.

ROTATION When rotating, all points of the figure are rotated by the same angle around the same point - the center of rotation.

Creating an Ornament with Axial Symmetry and Parallel Translation AXIAL SYMMETRY

PARALLEL TRANSFER

Creating an Ornament with Central Symmetry and Parallel Translation CENTRAL SYMMETRY

PARALLEL TRANSFER

HOW IS THE ORNAMENT OBTAINED? 1 2 3

ORNAMENT RECEIVED: 1 2 3 With the help of parallel transfer. Using axial symmetry and parallel translation. Using axial symmetry and parallel translation.

USED SOURCES O.V. Slesarev "The Wonderful Mystery of Geometry" Mathematics at school, O.E. Anatsko, E.M. Khanukovich, Binary lesson. http://ru.wikipedia.org Perelman Ya.I. "Entertaining Geometry". http://festival.1september.ru/articles/560632/ G. Weil "Symmetry". B. A. Rybakov "Paganism of Ancient Russia".

Completed:
Gorbunova Natalya Alexandrovna,
Kriulina Anna Alexandrovna,
students of group L-21

Supervisor:
Milchakova Tatyana Mikhailovna,
mathematics teacher
Kirov
2007

1. RATIONALE FOR THE CHOICE OF THE TOPIC.
Currently, there is a decrease in interest in mathematics among students. This is due to the insufficient saturation of the educational content with applied aspects that would reveal the connection between mathematics and the surrounding reality. Today, in order to design models of various processes, create good computer programs, a person needs to think broadly, see the manifestations of mathematics in various fields and be able to use them in their practical activities. That's why topical issue today is the identification of relationships both within mathematics itself and between mathematics and other fields of knowledge.
One of the most favorable topics of the course of mathematics, the study of which can contribute to the integration of knowledge, as well as aesthetic education, the development of creative and intellectual abilities students, is the theme "Symmetry". Classes related to symmetry can be practical and reflect cross-disciplinary integration.
A person sees symmetry, proportionality, periodicity everywhere: in nature, in music, in architecture, in painting. At the present stage of development of science, the principle of symmetry covers more and more new areas. Symmetry, and the theme of proportion associated with it, can be attributed to one of the main laws of a mathematical nature in the structure of the Universe. The phenomenon of symmetry can be considered in absolutely different areas of our life. And knowledge of its laws in one area may allow one to draw conclusions concerning subjects from another area. All this can become an additional factor in the formation of positive motivation in the study of mathematical disciplines, as well as understanding and awareness of the position on the universality of mathematical knowledge.

2. PURPOSE AND BRIEF FORMULATION OF THE PROBLEM.

The purpose of our work is to expand students' ideas about the concept of "Symmetry" and the formation of the ability to identify the principle of symmetry, its various types in the phenomena of the surrounding reality, in particular, in ancient Russian ornaments.
The tasks of the work are as follows:
- to generalize and expand the concept of symmetry in mathematics;
- expand ideas about the areas of application of mathematics (on the examples of the penetration of symmetry into art, in particular, into architecture);
- to teach to see various manifestations of symmetry in art;
- expand the scope of mathematical knowledge of students (types of symmetry);
- to expand the general cultural horizons through acquaintance with examples of art, in particular, with the masterpieces of ancient Russian architecture, located in modern Kirov.

3. FROM THE HISTORY OF ORNAMENT.

Geometric ornament originated at the dawn of human culture. What could be simpler than straight or wavy lines, circles, cells, crosses? It is these motifs that adorn the walls of clay vessels of primitive people, the most ancient products made of stone, metal, wood and bone. For ancient man they were conventional signs with which he could express his concept of the world. A straight horizontal line meant earth, a wavy line meant water, a cross meant fire, a rhombus, circle or square meant the sun. From combinations of the same simplest elements, more complex signs were made - symbols. Often they played the role of a talisman, they were drawn as a spell. Archaeologists and ethnographers put a lot of work and effort into deciphering ancient symbols.
Most of the geometric patterns also came to embroidery from ancient times, when many of them had a certain symbolic meaning. Probably, at one time the arrangement of ornaments on objects was not arbitrary either.
We see the same thing in folk architecture: decorative elements are located on the gates, around the windows; one or another consecrated image crowned the highest point of the house - the spike of the roof.
Decorated with embroidery smooth surface canvases, and on it only a few variants of ornamental compositions were possible - a simple alternation of figures and rows arranged vertically, horizontally and diagonally. Folk ingenuity and here found a way out in the construction of the ornament not only in breadth, but also in depth, creating multifaceted images in one plane. When one geometric motif is repeated, the gaps in the background between the figures also take on the appearance of a pattern. Then the whole ornament seems to us as if two-dimensional: the pattern and the background become equal in rights and can change places. This technique is especially typical for embroideries made using the technique of color interlacing.
Among the various motifs, the following can be distinguished, characteristic of the Russian ornament.

 The motif of "charm" signs that were applied to clothes, household items and various details of the dwelling
 Braid motif, typical for the so-called mermaid bracelets, which were interpreted as a sign of water and the kingdom of the underground lord Pereplut
 The motif of the ancient goddess Mokosh as a specific embodiment of the idea of the great Foremother, common to all peoples at a certain stage of historical existence
 Tree of life motif
Ornament occupied a large place in the folk art craft. One of the largest Czech explorers folk culture Josef Vydra identifies four main functions of the ornament:
1) constructive - it supports the tectonics of the object and affects its spatial perception;
2) operational - it facilitates the use of the object;
3) representational - it increases the impression of the value of the subject;
4) mental - it acts on a person with its symbolism and, thus, excites or calms him.
The oldest motifs depicting the likeness of a pictogram (square, rhombus, circle) were conditional symbols of the sun and were considered sacred images. In addition to the geometric, in the ornament of Ancient Russia, you can often find various ancient pagan plots. For example, a female figure personified the goddess of the earth, fertility; waterfowl - water element. In pagan art, the tree of life embodied the power of living nature, it depicted the divine tree, on which the growth of herbs, cereals, trees and the "growth" of man himself depended. Very often you can find plots of magical calendar rituals that are associated with the main stages of agricultural work.
Of all the richness of the motifs of the ornamental art of Ancient Russia, one can nevertheless single out the most popular motif - a plant motif. The plant motif is present at the basis of all the ornaments of Ancient Russia. The most common of these is the winding vine motif with large flowers enclosed in circles formed by offshoots.
The ornamental art of Ancient Russia plays a very important role both in the development of decorative and applied art in Russia and in art in general. Ornament is a unique tool that deserves to exist as an independent art direction.

4. TYPES OF SYMMETRY.

Symmetry is a certain order in the construction of any spatial form, which allows this form to be combined with itself under certain rotations, shifts or reflections.
In the science of symmetry, two types of symmetry are distinguished: finite (for example, rosettes) and infinite, whose structure can be continued in one ( wavy line, meander, etc.), in two or three directions. Both of these types of symmetrical structures are used in ornamentation.

4.1 Mirror symmetry.
The most common type of symmetry in ornaments. An object or figure is divided into two halves by a plane so that one half, reflected in this plane as in a mirror, coincides with the other. You can find mirror symmetry in pictorial compositions not ornamental at all. The desire, especially inherent in the arts of ancient peoples, to give images a stable immobility, to embody in them not something transient, temporary, but eternal and unchanging, prompted them to resort to strict symmetry, to equalize the right and left parts of the composition, like scales. Mirror symmetry contributes to the impression of balance and peace, as it makes the polar directions - on both sides of the plane of symmetry - equivalent to our gaze. It equalizes his ability to move from this plane in one direction or another. And in the ornament, the use of mirror symmetry is associated with the same sense of balance and peace.

4.2 Axial symmetry.
A fundamentally different kind of symmetry is inherent in those figures that are combined with themselves without the help of mirror reflection, by turning around an axis perpendicular to the image plane. This is axial symmetry, and the number of such combinations during a complete circular revolution of the figure is called the order of the axis. Axial symmetry can have any order expressed by an integer - from the second to infinity (the first order means the combination of the figure with itself in only one, its original position, such a figure is asymmetric). There can be an infinite number of figures with axial symmetry, but they are all very clearly organized, their parts equal to each other are distributed around a single center (points through which the axis of symmetry passes) evenly and in the same relation to it. All rotation angles at which the figure once again coincides with itself must be equal, otherwise complete alignment will not occur, the distances from the same-characteristic points of its contour to the center must also be the same. Axial symmetry is often found in nature and is widely used in ornaments: this is the symmetry of a flower and an ornamental ornament - a rosette. If the figure has only axial symmetry and no other, then it gives the impression of restless mobility, expresses intense rotary motion. But more often we have to see sockets that have not only axes, but also planes of mirror symmetry. Such forms are much more balanced, calmer: the right rotation here is, as it were, balanced by an equivalent left movement. This kind of rosette is also balanced and therefore static inside itself, since here there is no asymmetry not only in general, but also in each individual element of its structure.
From what has been said, it can be clear why it is precisely motifs with this kind of symmetry that have received special distribution and significance in ornamental art. The completeness of their form creates an image of harmonious peace. The integrity and closedness of the form allow you to organize any surface, marking its center, opposed to the periphery. V. A. Favorsky said that rosettes have “face” shapes - like “eyes”. This means that they are turned from the surface to the viewer, and not along it, and therefore are more static, do not participate in the rhythmic run of the pattern. Therefore, it is the rosettes that can give the ornamental surface a kind of activity in the third dimension, perpendicular to it.

4.3 Parallel transfer.
If decorative identical motifs are evenly arranged along the axis, then a ribbon ornament, a border, can be formed in this way, which can be infinitely extended in both directions. Such an ornament is characterized by a special symmetry: if it is shifted along the axis by one link, then each of the figures of the pattern will overlap the middle figure and be combined with it. In practice, a linear, ribbon ornament can be built not only along a straight axis, but also along a broken or curved line in various ways. In any case, this line remains the axis for the ornament, that is, the transfer is thought to be made along it, following any of its bends and fractures.
A linear border is one of the most common and important types of ornament. It is constantly used where it is necessary to limit a surface, to close it or to divide it, to isolate surfaces with different artistic qualities.
The border, in addition to the transfer symmetry, may also have (or not have) other symmetry elements. They arise in those cases when one or another type of symmetry is inherent in each individual elementary motif of the ornament, oriented in a certain way with respect to the axis.
5. ORNAMENTS OF THE KIROV REGION

5.1 Mirror symmetry.

Ornament on the rotunda of the Alexander Garden

5.3 Parallel transfer
Lattice of the square near the Dynamo stadium

6. RESEARCH OF ORNAMENTS IN THE CITY OF KIROV

In the course of our research, we found that there is no certain predominant type of symmetry in the ornaments of our city. Most ornaments combine mirror and axial symmetry. From this we can conclude that the architects of the city adhered to balance and harmony in its design. We also noticed that many ornaments consist of circles, rhombuses, which in ancient times symbolized the sun, sunlight.
Most ornaments are located on small architectural forms: on rotundas, arches, lattices. According to our assumption, ornaments on the gratings were used as amulets. They seem to guard the territory that this lattice protects.
7. SELF-ASSESSMENT OF WORK.

We believe that we have achieved the goal of our project. With the help of this work, students can expand their knowledge on the topic "Symmetry" and form the ability to identify types of symmetry in space.
In the course of our work, we encountered some obstacles, for example, in the search for the ornaments of the city of Kirov, since not all of them are well preserved or have come down to our times. We also learned a lot of new things from the history of ornament: what significance was attached to ornament in antiquity. We have learned to correctly distinguish the types of symmetry in the world around us, directly in the ornaments.
It was interesting for us to do this work, despite the difficulties, to study the history of the ornament, to find it on the facades of buildings and on the gratings of our city.

8. REFERENCES

1. Afanasiev A. N. Mythology of Ancient Russia. – M.: Eksmo, 2006.
2. Klimova N. T. Folk ornament in the composition of art products. – M.: art, 1993.
3. Rybakov B. A. Paganism of Ancient Russia. – M.: Nauka, 1988.
4. Fokina L. V. Ornament
5. Tarasov L. V. This amazingly symmetrical world: a guide for students. – M.: Enlightenment, 1982.
6. Tinsky A. G. Planning and development of the city of Vyatka in the 17-19 centuries. - Kirov, Volga-Vyatka Prince. publishing house, Kirov branch, 1976.

The content of the published material is entirely based on a thorough scientific analysis of many examples of Paleolithic and Neolithic ornamental art. The data obtained after studying their symmetrical and antisymmetrical elements convincingly prove that there are practically no differences in the ornaments of different peoples of the world, and the elementary components of patterns are preserved throughout the history of decorative art and represent decorative archetypes.

Due to the large amount of information, the material is divided into interconnected chapters. The first chapter is the introduction you are currently reading. The remaining three chapters are devoted respectively to the symmetry of rosettes, one-dimensional models of borders and two-dimensional ornamental structures. This article and subsequent chapters talk about some of the results of the symmetry analysis of samples of the decorative arts of the Paleolithic and Neolithic. They are dedicated to the search for "decorative archetypes" - the universal foundation of all decorative art. After all, the development of the ornament has always gone along with the development of mankind and reflects the human desire to understand and express patterns - the factors underlying any scientific knowledge.

The final conclusion that we managed to reach is that most of the ornamental motifs studied from the point of view of the theory of symmetry are much more ancient than we could expect. This fact pushes back the date of the appearance of decorative art by several thousand years before the emergence of the most ancient civilizations.

The connection between visual art and geometry has always existed. This connection becomes especially evident when we apply the theory of symmetry to the study of arts and crafts. Therefore, ornamental art is called by H. Weyl "the oldest aspect of higher mathematics, given in an implicit form", and by A. Speiser "prehistory of group theory".

The idea of researching ornaments different cultures by analogy with the symmetry of crystals on a plane (G. Pohl's theory) and applying the theory of groups of finite order A. Speiser, was supported by the intensive development of symmetry theory in the 20th century. There have been a number of works devoted mainly to decorative arts ancient civilizations that made the greatest contribution to the development of arts and crafts (Egyptian, Arabic, Mauritanian, etc.), and ethnic ornamental art. However, only in some of the latest works, researchers turn to the very roots, to the origins of arts and crafts, to the Paleolithic and Neolithic eras. Antisymmetry - an extension of the classical theory of symmetry - and the science of color symmetry, allowed for a deeper analysis of monochromatic and polychromatic ornamental motifs of the Neolithic era and ancient civilizations.

What was the analysis based on? First of all, the fact that ornamentation is usually limited to a two-dimensional plane. In this vein, symmetrical planar groups were considered: rosettes, borders, etc. The discrete symmetry of the rosettes made up two studied groups: cyclic and dihedral. The cyclic group, expressed by the simplest formula Dn, is created by two reflections on the line of intersection of the invariant point - the center of the displacement of the n-order. Cyclic group patterns are made up by translating the elementary components in a circle by rotating around a fixed point that is a multiple of 360 degrees divided by the n-th order. The dihedral group of ornaments consists of patterns inscribed in regular polyhedra and forming regular polyhedra.

Both groups of symmetry - cyclic and dihedral - are easy to detect in natural forms.

Rice. 1. Cyclic symmetry in ornament and nature

Rice. 2. Dihedral symmetry in ornament and nature

In total, seven discrete groups were identified, which can be expressed in the following sequence: 11, 1g, 12, m1, 1m, mg and mm, where g - denotes a sliding reflection (from the English glide), m - an ordinary reflection. With the variable n we express an n-order rotation. The resulting sequence is interpreted relative to the coordinate plane, which takes into account elements that are perpendicular and parallel to the displacement axis.

Rice. 3. Antisymmetry

When we refer to continuous symmetry groups in subsequent chapters, the presence of a continuous displacement will be denoted by the index 0, and in antisymmetric groups, "anti-displacement" will be denoted by the single quote symbol -".

By the term "pre-scientific period" we mean the period from 25000-10000 BC.

In the absence of written sources, the study of the geometry of the prehistoric period can only be carried out on the basis of the analysis of artifacts, where geometric knowledge is presented in an explicit form. The oldest of the artifacts of the Paleolithic and Neolithic eras are bones, drawings on stone. Later - ceramic painting, engraving, pressing, as well as architectural objects and structures, the so-called megalithic monuments.

The next chapter is entirely devoted to the symmetry of rosettes.

List of all four chapters:

Sources

1. Weyl H., Symmetry, Princeton University Press, Princeton, 1952.
2. Polya G., Uber die Analogie der Kristall symmetrie in der Ebene, Z. Kristall. 60 (1924), 278-282.
3. Speiser A., Die Theorie der Gruppen von endlicher Ordnung, 2nd ed., Berlin, 1927.
4. N. V. Belov, Medieval Moorish Ornament and Symmetry Groups, Soviet Physicists - Crystallography 1 (1956), 482-483.
5. Garido J., Les groupes de symetrie des ornaments employes par les anciennes civilizations du Mexique, C.R. Acad. sci. Paris 235 (1952), 1184-1186.
6. Grunbaum B., The Emperor's New Clothes: Full Regalia, G string, or Nothing, Math. Inteligencer 6, 4 (1984), 47-53.
7. Grunbaum B., Grunbaum Z., Shephard G.C., Symmetry in Moorish and Other Ornaments, Comput. Math. Appl. 12B, 3/4 (1986), 641-653.
8. Muller E., Gruppentheoretische und Strukturanalytische Untersuchungen der Maurischen Ornamente aus der Alhambra in Granada, Ph.D. Thesis, Univ. Zurich and Ruschlikon, 1944.
9. Crowe D.W., The Geometry of African Art I. Bakuba Art, J. Geometry 1 (1971), 169-182.
10. Crowe D.W., The Geometry of African Art, II. A Catalog of Benin Patterns, Historia Math. 2 (1975), 57-71.
11. Crowe D.W., The Geometry of African Art m. The Smoking Pipes of Begho, In The Geometric Vein, ed. C.Davis, B. Grunbaum and F.A. Sherk, Springer Verlag, Berlin, Heidelberg, New York, 1981.
12. Washburn D.K., Symmetry Analysis of Ceramic Design: Two Tests of the Method on Neolithic Material from Greece and the Aegean, In Structure and Cognition in Art, Cambridge University Press, London, 1983.
13. Jablan S.V., Theory of Symmetry and Ornament, Mathematical Institute, Beograd, 1995.
14. Jablan S.V., Antisimetrijska ornamentika I, Dijalektika 1-4 (1985), 107-148.
15. Jablan S.V., Antisimetrijska ornamentika II, Dijalektika 3-4 (1986), 13-56.

Purpose: Demonstrate the possibilities of the laws of geometry when creating various ornaments that adorn many objects of decorative and applied arts Tasks: Repeat and expand knowledge about known types of symmetry Learn to identify and describe symmetry in an ornament pattern Summarize and systematize the information used to create an ornament

The construction of the ornament is based on the mathematical component. "Ornament is the mathematical embodiment of beauty" What is an ornament? An ornament is a pattern built by alternation in a certain order. The word "ornament", from Latin, means "decoration".

Our country is rich in talents. Many types of folk art are widely known and highly valued not only in our country, but also abroad. These are Palekh and Mstera miniatures, painted nesting dolls of Polkhov Maidan, Bogorodsk wooden toys, Zhostovo trays and much, much more. In the products of folk craftsmen - in wood, stone, clay, on fabric - images that come from pagan beliefs and legends, folk epics and fairy tales come to life. Each of us at least once in our lives saw an embroidered towel, a Russian shirt, a girl's sundress embroidered with patterns ... How can we preserve and leave this beauty for posterity? To do this, we must be able to understand a lot in our past. Why ancient Russian ornaments look so elegant, what embroidered incomprehensible birds, trees, outlandish flowers and herbs mean, according to what laws the ornament itself is built - I tried to answer these and some other questions in my work.

In art, symmetry has become widespread as one of the types of harmonious composition. It is characteristic of works of architecture and arts and crafts. Symmetry is a kind of measure of the beauty and harmony of the world around us. The outstanding mathematician of the 20th century Hermann Weil The outstanding mathematician of the 20th century Hermann Weil in his work "Symmetry" paid special attention to ornamental symmetry. He showed that the principles of any ornamental composition are based on general provisions fundamental mathematical structures, and connected ornamental symmetry with discrete groups of motions on the plane.

The main feature of the ornament as a means of decoration is subordination to the image, form and purpose of the object being decorated. The ornament cannot exist independently, outside of this subject. In the construction of an ornament, the principle of symmetry is usually used, the methods of rhythmic repetition of one or more elements (rapport). The ornament can be continued in both directions, even if its original composition is limited, closed. The elements that form an ornament can more or less accurately reproduce reality, but most often they are processed, stylized, and decoratively generalized.

In folk art, where the ornament found the greatest distribution, stable forms and principles of constructing the ornament gradually took shape, which largely determined the national artistic traditions of different peoples. Each epoch, each style in architecture, each national culture has worked out in architecture, each national culture has developed its own system of ornamentation (motifs, forms, location on the surface to be decorated), therefore, it is often possible to determine from the ornament, by what time and by the ornament one can determine , to what time and to what country this or that work of art belongs. Russian ornament (along with historical and art. Russian ornament (along with historical and cultural significance) is especially interesting in this sense, as it has peculiar mathematical, color and symbolic features.

Russian ornament Ornaments, as decorations for household items, existed in Russia for a very long time. Peasant huts were decorated with carved patterns. Curtains embroidered with intricate ornaments hung on the windows, and the whole house was filled with painted dishes, embroidery and lace. Russian ornament is very diverse. Decorative elements were made up of symbols and reflected the feelings and interests of a person. The color in the pattern had a special meaning. Red color expressed delight, joy.

High development ornament reached in medieval Russia. Russian ornamentation is characterized by both geometric and floral forms, as well as images of birds, animals and human figures. In the mutual arrangement of the repeating figures of the Russian ornament, only three types of symmetry are mainly involved: 1 - this is a consistent arrangement in a line - figurative symmetry

It has long been noted that there is nothing random in embroidery. The ancient Slavic ornament was based on the universal ideas of our distant ancestors about the world. These ideas are reflected in the motifs characteristic of Russian ornaments and Russian patterns. These symbols hide a special meaning, which, unfortunately, we can no longer always read. In the ornament, a person expressed his ideas about the cosmos, life, love for nature and man, about happiness. In folk art, ornamental motifs captured a poetic attitude to the world.

For the Russian ornament, motifs are also characteristic: "coastal" signs that were applied to clothes, household items and various details of the dwelling; - "braid" motifs, which were interpreted as signs of water; -- the motif of the ancient goddess Mokosha (as an idea of the Great Foremother); - motif of the "tree of life"; --motives of "solar" (solar) signs. -So, for example, a straight line in the ornaments of many nations means land, and a wavy line means water. The combination of these lines, located horizontally in the Russian ornament, is a symbol of "mother-cheese earth", located - vertically or horizontally - a symbol of rain. A rhombus is a symbol of a field, a dot in it is a grain, and in general - a symbol of a sown field.

Incantatory coastal ornaments were used to decorate all the openings through which people could get in. evil spirits. It was important to protect those places where the enchanted fabric of clothing ended and the human body began - sleeves, shoulders and collars were usually embroidered with red threads. Often decorated with a hem. The fabric itself was considered impenetrable to the spirits of evil, because it was made with items adorned with magical ornaments (rattle, spinning wheel, loom).

Figures with central symmetry, such as a circle and a regular hexagon, often act as amulets. Among the ancient Slavs, these figures were amulets against thunderstorms, they are called "solar" signs. The movement of the sun is conveyed in the ornament with a sign that we know as the swastika. Solar signs are characterized by rotational symmetry of various orders. The most common are a circle divided by radii into equal sectors divided by radii into equal sectors, as well as a circle with a cross inside. The "running" sun depicts a circle with arched lines radiating from the center.

Thus, the description of individual Russian ornamental motifs and the scheme of their location on the details of dwellings, items of decoration, clothing and everyday life clearly demonstrates the presence in them of central, rotary, portable, axial and mirror types of symmetry, which are the reason - noy aesthetic appeal of the Russian ornament.

Unfortunately, only a single object from the pagan past has survived to this day – this is a towel. It is used without interruption of tradition in folk life for its intended purpose to protect holy corners, goddesses, door and window openings, weddings, holidays, funerals, weddings, holidays, funerals with its magical images and charms from the evil eye, " lesson" and any damage and be a haven for the souls of dead relatives. In many ways, it is thanks to the towels with ancient embroidery that have been preserved in Russian villages that we can make an excursion into our distant past.

To folk art alive, it needs to be in sensitive hands. The threads that connect us with the art of distant ancestors are getting thinner and thinner. So the ancient Russian ornament can leave our life, depriving it of something important. Indifference inevitably leads to losses, and losing spirits. Indifference inevitably leads to losses, and we have no right to lose the spiritual, artistic traditions of the people. It is respect for these traditions that is called patriotism; it is from this that the feeling of the Motherland begins in us.

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