Start in science. T.I

Completed:
Gorbunova Natalia 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 caused by the insufficient saturation of educational content with applied aspects that would reveal the connection of mathematics with the surrounding reality. Today, in order to construct models of various processes and create good computer programs, a person needs to think broadly, see the manifestations of mathematics in a variety of areas and be able to use them in his practical activities. Therefore, a pressing issue today is to identify relationships both within mathematics itself and between mathematics and other areas of knowledge.
One of the most favorable topics in a mathematics course, the study of which can contribute to the integration of knowledge, as well as aesthetic education, development of creative and intellectual abilities of students, is the topic “Symmetry”. Activities related to symmetry can be practical and reflect interdisciplinary integration.
Man sees symmetry, proportionality, and 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 related theme of proportion can be considered one of the main mathematical laws in the structure of the Universe. The phenomenon of symmetry can be considered in completely different areas of our lives. And knowledge of its laws in one area can allow one to draw conclusions regarding 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 about the universality of mathematical knowledge.

2. GOAL AND BRIEF FORMULATION OF THE TASK.

The goal of our work is to expand students’ understanding of the concept of “Symmetry” and to develop 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 objectives of the work are as follows:
- generalize and expand the idea of symmetry in mathematics;
- expand ideas about the areas of application of mathematics (using examples of the penetration of symmetry into art, in particular, into architecture);
- teach to see various manifestations of symmetry in art;
- expand the scope of students’ mathematical knowledge (types of symmetry);
- expand your general cultural horizons through acquaintance with examples of art, in particular, with masterpieces of ancient Russian architecture located in modern Kirov.

3. FROM THE HISTORY OF ORNAMENT.

Geometric patterns arose at the dawn of human culture. What could be simpler than straight or wavy lines, circles, cells, crosses? It is these motifs that decorate 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 the help of 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. Combinations of the same simple elements were used to create more complex signs - symbols. Often they played the role of a talisman, they were drawn like a spell. Archaeologists and ethnographers put a lot of work and effort into deciphering ancient symbols.
Most of the geometric patterns in embroidery also come 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.
We see the same thing in folk architecture: decorative elements are located on gates, around windows; one or another consecrated image crowned the highest point of the house - the roof spike.
The smooth surface of the canvas was decorated with embroidery, and only a few variants of ornamental compositions were possible on it - a simple alternation of figures and rows located vertically, horizontally and diagonally. Here, too, folk ingenuity found a way out in constructing an ornament not only in breadth, but also in depth, creating multifaceted images in one plane. When repeating one geometric motif, the background spaces 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 and can change places. This technique is especially typical for embroidery done using the colored weaving technique.
Among the various motifs, the following can be distinguished, characteristic of Russian ornament.

 The motif of “amulet” signs that were applied to clothing, household items and various details of the home
 Braided motif, characteristic of the so-called Rusal bracelets, which were interpreted as a sign of water and the kingdom of the underground ruler Pereplut
 The motif of the ancient goddess Mokoshi 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 folk art. One of the largest Czech researchers of folk culture, Josef Vydra, identifies four main functions of ornament:
1) constructive - it supports the tectonics of the object and affects its spatial perception;
2) operational - it facilitates the use of the item;
3) representational - it increases the impression of the value of the object;
4) mental - it affects a person with its symbolism and, thus, excites or calms him.
The most ancient motifs, depicting something like a pictogram (square, rhombus, circle) were conventional symbols of the sun and were considered sacred images. In addition to the geometric one, in the ornaments of Ancient Rus', one can often find various ancient pagan themes. For example, the female figure personified the goddess of the earth and fertility; waterfowl - the 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 motifs in the ornamental art of Ancient Rus', one can still single out the most popular motif - the plant one. The plant motif is present at the basis of all the ornaments of Ancient Rus'. The most common of them is the motif of a climbing vine with large flowers enclosed in circles formed by shoots.
The ornamental art of Ancient Rus' plays a very important role both in the development of decorative and applied art in Rus', and 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, allowing this form to combine 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.), two or three directions. The ornament uses both of these types of symmetrical structures.

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 also find mirror symmetry in pictorial compositions that are not at all ornamental in nature. Especially inherent in the arts of ancient peoples, the desire to give images stable immobility, to embody in them not something transitory, temporary, but eternal and unchanging, prompted the resort to strict symmetry, equalizing the right and left parts of the composition, like scales. Mirror symmetry contributes to the impression of balance and calm, since it makes the polar directions - on either side of the plane of symmetry - equal to our gaze. It equalizes his ability to move from this plane in one direction or another. And in ornament, the use of mirror symmetry is associated with the same feeling of balance and peace.

4.2 Axial symmetry.
A fundamentally different type of symmetry is inherent in those figures that are combined with themselves without the help of mirror reflection, through rotation around a certain axis perpendicular to the image plane. This is axial symmetry, and the number of such alignments throughout a full circular revolution of the figure is called the axis order. Axial symmetry can have any order expressed by an integer - from the second to infinity (the first order means the combination of a figure with itself only in one, its original position, such a figure is asymmetrical). There can be countless figures with axial symmetry, but they are all very clearly organized, their equal parts are distributed around a single center (the point 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 design - a rosette. If a figure has only axial symmetry and no other, then it gives the impression of restless mobility and expresses intense rotational movement. But more often we see rosettes 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 balanced and therefore static within itself, since there is no asymmetry not only as a whole, but also in each individual element of its structure.
From what has been said, it can be clear why exactly motifs with symmetry of this kind have received particular distribution and significance in ornamental art. The completeness of their form creates an image of harmonious peace. The integrity and closedness of the form make it possible to organize any surface, marking its center, opposed to the periphery. V. A. Favorsky said that rosettes are “face-shaped” - like “eyes”. This means that they are facing away from the surface towards the viewer, and not along it, and therefore are more static and do not participate in the rhythmic running of the pattern. Therefore, it is the rosettes that can give the ornamental surface a peculiar activity in the third dimension, perpendicular to it.

4.3 Parallel transfer.
If decorative identical motifs are evenly placed along the axis, then a ribbon ornament or border is formed, which can be endlessly 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 align with it. In practice, a linear, ribbon ornament can be built not only along a straight axis, but also along a broken or variously curved line. In any case, this line remains an axis for the ornament, i.e., the transfer is imagined to be performed along it, following any of its bends and fractures.
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, close it, or divide it, to isolate surfaces with different artistic qualities.
A border, in addition to transfer symmetry, may also have (or not have) other symmetry elements. They arise in cases where one or another type of symmetry is inherent in each individual elementary motif of an 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. STUDY OF ORNAMENTS OF THE CITY OF KIROV

In the course of our research, we discovered that there is no specific 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 and rhombuses, which in ancient times symbolized the sun, sunlight.
Most of the ornaments are located on small architectural forms: on rotundas, arches, trellises. According to our assumption, the ornaments on the gratings were used as amulets. They seem to guard the territory that this lattice encloses.
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 develop the ability to identify types of symmetry in space.
In the course of our work, we encountered some obstacles, for example, in finding the ornaments of the city of Kirov, since not all of them were well preserved or survived to our times. We also learned a lot of new things from the history of ornament: what importance was attached to ornament in ancient times. We have learned to correctly distinguish types of symmetry in the surrounding world, directly in 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 grilles of our city.

8. REFERENCES

1. Afanasyev A. N. Mythology of Ancient Rus'. – M.: Eksmo, 2006.
2. Klimova N. T. Folk ornament in the composition of artistic products. – M.: Fine Arts, 1993.
3. Rybakov B. A. Paganism of Ancient Rus'. – M.: Nauka, 1988.
4. Fokina L.V. Ornament
5. Tarasov L.V. This surprisingly symmetrical world: a manual for students. – M.: Education, 1982.
6. Tinsky A. G. Planning and development of the city of Vyatka in the 17-19 centuries. - Kirov, Volgo-Vyatka Prince. publishing house, Kirov department, 1976.

Symmetry, perceived as a manifestation of order, has aesthetic value, that is, it is perceived as something beautiful.

A simple example convinces us of this. An ink blot in itself is not beautiful, but if you fold a sheet of paper with a wet blot in half, you will get a blot that already makes a different impression. The mirror symmetry of the new blot will determine its beauty.

The pattern in the figure is obtained using mirror symmetry (Fig. 28). However, the law of its construction is too simple and obvious, and therefore the aesthetic value of such a pattern is low.

Rice. 28. Pattern obtained using mirror symmetry

Portable symmetry (copying a figure and shifting it horizontally) is the simplest technique for creating an ornamental row - a border (Fig. 29).

By creating different spaces between copies, you can achieve different rhythms within a row - for example, moving figures in pairs (Fig. 30).

You can alternate pairs with a single image (Fig. 31).

The next border has a more complex construction law. This technique is often used when creating multi-row ornaments. By moving the entire row vertically up or down, we get two absolutely identical rows (Fig. 32).

Rice. 30. Examples of figurative symmetry with the addition of rhythm within a row

Rice. 31. Examples of alternating a pair with a single image

Rice. 32. Examples of row shifting

Rice. 33. Examples of symmetry with spacing between images

Rice. 34. Combination examples

It can be continuous or with certain intervals between images (Fig. 33).

Combined. Vertical and horizontal shift with mirror symmetry (Fig. 34).

The resulting voids can be filled with other elements (Fig. 35).

Rice. 35. Example of filling voids

The alternation of horizontal and vertical shifts, made in a certain rhythm, creates the basis for a winding line that unites the elements of the ornament. The gaps that arise can also be filled with other ornaments (Fig. 36).

Repeated shifting of horizontal rows vertically (or vertical rows horizontally) allows you to fill the entire decorated plane with images (Fig. 37).

Rice. 36. Example of filling voids

Rice. 37. Examples of multiple vertical shifts of horizontal rows

A different picture is obtained when using another method of spatial transfer - rotation, we get a figure with radial symmetry, the so-called “rosette”. Rosettes are obtained by rotating the figure around a vertical axis through an angle of 360 degrees / n (n = 2, 3, 4......), that is, they have rotational symmetry of the nth order (Fig. 38).

Rice. 38. Examples of radial symmetry obtained by rotation

There are several options for constructing a socket. For example, the center of rotation may be at one of the edges of the figure (Fig. 39).

The center of rotation is within the element (Fig. 40).

The center of rotation is located outside the element (Fig. 41).

In the center of the resulting rosette there is free space, which can be filled with another image, or another rosette can be inserted there (Fig. 42).

Rice. 39. An example of radial symmetry with the center of rotation at the edge of the figure

Rice. 40. Example of radial symmetry with the center of rotation within the element

Rice. 41. An example of radial symmetry with the center of rotation outside the figure

Rice. 42. Example of radial symmetry with filled free space

It is assumed that when creating rosettes, we rotate the image element so that all angles are equal, and when dividing 360 degrees (the rotated angle) by the rotation angle, we get an integer natural number - 3, 5, 8, 12, etc. In other words, the circle is divided into a certain number of sectors, each of which contains a rosette element.

Let's return to another technique discussed above - mirror reflection. It is easy to verify that the resulting copy cannot be obtained either by rotation or by lateral transfer. It is mirror symmetrical relative to the original one.

The position of the plane in which the ornamental element is reflected can be arbitrary. It is necessary to obtain one - the only mirror copy of the element (Fig. 43). All other reflections produced by differently positioned planes can be obtained by rotating the first mirror copy about some center.

Rice. 43. Examples of mirror reflection

You can mirror a whole row (Fig. 44).

Rice. 44. Examples of mirror reflection of a row

Having received one “mirror” pair, you can get its mirror image (Fig. 45).

Rice. 45. Examples of mirror reflection of a “mirror” pair

By combining shift and mirror reflection, it is possible to obtain an interesting solution to a linear ornament (Fig. 46).

Rice. 46. Examples of a combination of shift and mirror reflection

The mirror reflection technique can be used when creating rosettes. In this case, it is necessary to obtain a pair of mirror-image sectors, and then rotate them around the center (in this case, the number of sectors into which the circle of the rosette is divided must be even) (Fig. 47).

Rice. 47. Examples of rosettes obtained using mirror reflection

Finally, ornamental symmetry is built on one of five possible flat lattices. Preliminary drawing of lattices is a useful auxiliary technique when constructing an ornament.

The simplest lattice is created by vertical and horizontal shifts of a square. In this case, the elements of the ornament can be located in different squares of the grid, which greatly facilitates drawing (Fig. 48).

An ornament constructed using a square lattice (Fig. 49).

Rice. 48. An example of ornamental symmetry based on a flat lattice

Rice. 49. An example of ornamental symmetry based on a square lattice

The ornament is also constructed using a triangular and rhombic lattice (Fig. 50).

Six equilateral, adjacent triangles form a hexagonal (hexagonal) lattice.

Rice. 50. Examples of ornaments based on triangular and rhombic lattices

(Ornaments are often built on the basis of regular hexagons in some Islamic countries).

Ornamental symmetry is the basic principle of constructing any ornament.

Goal: Demonstrate the possibilities of the laws of geometry in creating various ornaments that decorate many objects of decorative and applied art. Objectives: Repeat and expand knowledge about known types of symmetry. Learn to identify and describe symmetry in the design of an ornament. Summarize and systematize the information used in creating an ornament.

The basis for constructing an ornament is a mathematical component. “Ornament is the mathematical embodiment of beauty” What is ornament? An ornament is a pattern constructed by alternation in a certain order. The word "ornament" from Latin means "decoration".

Our country is rich in talent. Many types of folk art are widely known and highly valued not only here, but also abroad. These are Palekh and Mstera miniatures, painted dolls from 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 has seen 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 need to be able to understand a lot about our past. Why do ancient Russian ornaments look so elegant, what do the embroidered strange birds, trees, strange flowers and herbs represent, what are the laws by which 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 decorative arts. Symmetry is a kind of measure of the beauty and harmony of the world around us. Outstanding mathematician of the 20th century Hermann Weyl Outstanding mathematician of the 20th century Hermann Weyl in his work “Symmetry” paid special attention to ornamental symmetry. He showed that the principles of any ornamental composition are based on the general provisions of fundamental mathematical structures, and connected ornamental symmetry with discrete groups of movements on the plane.

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

In folk art, where ornament was most widespread, stable forms and principles of ornament construction gradually emerged, which largely determined the national artistic traditions of different peoples. Each era, each style in architecture, each national culture, development in architecture, each national culture has developed its own system of ornamentation (motifs, forms, location on the surface to be decorated), therefore, one can often determine by the ornament 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, have existed in Rus' for a very long time. Peasant huts were decorated with carved patterns. There were curtains embroidered with fancy patterns 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. The red color expressed delight and joy.

Ornament reached a high level of development in medieval Rus'. Russian ornament is characterized by both geometric and floral forms, as well as images of birds, animals and human figures. In the relative arrangement of repeating figures of Russian ornament, only three types of symmetry are mainly involved: 1 - this is a sequential arrangement in a line - figurative symmetry

It has long been noted that there is nothing accidental in embroidery. The basis of the ancient Slavic ornament was 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 space, life, love for nature and man, and happiness. In folk art, ornamental motifs capture a poetic attitude towards the world.

The following motifs are also characteristic of Russian ornament: “coastal” signs, which were applied to clothing, household items and various details of the home; - “braid” motifs, which were interpreted as signs of water; - motif of the ancient goddess Mokoshi (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 peoples means earth, and a wavy line means water. The combination of these lines, located horizontally in a Russian ornament, is a symbol of the “cheese mother earth”; those located vertically or horizontally are a symbol of rain. A rhombus is a symbol of a field, a dot in it is a grain, and in general it is a symbol of a sown field.

All openings through which evil spirits could penetrate to a person were decorated with incantatory coastal ornaments. 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. The hem was also often decorated. The fabric itself was considered impenetrable to the spirits of evil, because... its production involved objects richly decorated with magical ornaments (ravel, spinning wheel, loom).

Figures with central symmetry, such as a circle and a regular hexagon, are often used 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 by a sign that we know as the swastika. Solar signs are characterized by rotational symmetry of different orders. The most common are a circle divided by radii into equal sectors, and a circle with a cross inside. The “running” sun depicts a circle with arcuate lines diverging from the center.

Thus, the description of individual Russian ornamental motifs and the scheme of their arrangement on the details of the home, decoration, clothing and household items clearly demonstrates the presence in them of central, rotational, portable, axial and mirror types of symmetry, which are the causes - new aesthetic appeal of Russian ornament.

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

For folk art to live, it needs to be in sensitive hands. The threads connecting us with the art of distant ancestors are becoming thinner and thinner. So the ancient Russian ornament can leave our lives, 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; this is where our sense of the Motherland begins.

Alexandrov Artyom, Makarov Alexander

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

Download:

Preview:

SYMMETRY OF ANCIENT RUSSIAN ORNAMENT

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

In the process of historical development and understanding of the concept of symmetry, a special stage in 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). Weyl understood symmetry as the immutability (invariance) of an object under a certain kind of transformation: an object is symmetrical in the case 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 different types of symmetry that a two-dimensional ornament with a double infinite relation can have... The art of ornament contains in an implicit form the most ancient part of the higher mathematics known to us”…

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

Let us briefly consider the state of mathematical significance in Rus' during the period of the 9th – 10th centuries. The practical activities of people in Rus', as in other countries of Europe and Asia, made it necessary to develop arithmetic knowledge and ideas about the properties of geometric figures. Excavations of ancient settlements indicate that mathematical knowledge of a utilitarian nature was widespread in Rus'. 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 patterns on jewelry and household items.

According to Academician B. A. Rybakov, a famous archaeologist and world-famous historian, the basis of 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, and social norms regulating people's behavior. All this is reflected in the motifs characteristic of Russian ornament.

Among them are the following.

Motive “ amulet signs, which were applied to clothing, household items and various parts of the home.

The Slavs place their protective symbols in “vulnerable” areas of their homes and yard. With an incantatory ornament they cover all openings through which evil spirits can penetrate to a person: gates, doors, windows, bars. The symbol of the sun, placed in the openings, should protect the home from the night evil spirits. We also find solar signs on the roof of the house - in the form of a ridge (and the horse, as mentioned above, itself symbolized the sun), under the roof - in the form of a real image of the sun or its symbol, on the facade, on 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. In this case, a thunder sign (a circle divided into six sectors) is usually placed on the central towel - a symbol of Rod or Perun, who protected the house from lightning striking it. Next to the symbol of the sun, the Slavs almost always depict the sign of the earth or a sown field, denoting the unity of the heavenly and earthly, obvious. The waking world is also symbolized by images of birds, plants, goddesses, intertwined with fantastic creatures.

Ritual towels have a special magic and special meaning among the Slavs. The patterns on them represent nothing more than the Slavic calendar, which figuratively reflected family and tribal events or agrarian and ritual celebrations characteristic of agricultural culture. On all holidays, bread and salt were the first to be brought out on a towel. Salt is a symbol of the sun, love; bread - earth, mercy; a towel is a symbol of human life, a stripe of fate, a part of pure outer space. During the betrothal, the hands of the bride and groom were joined, wrapped in a towel, so that the young people could live in abundance; with the same meaning, during childbirth, the grandmother accepted the baby onto a new towel. Everyday towels were woven and embroidered with simple protective care.a decoration against pestilence, to ask for rain, etc., and in order for the towel to have miraculous powers, the weaver had to complete the work in one daylight.

Motif of the ancient goddess Mokoshi (the only female image in ancient Russian mythology) as a specific embodiment of the idea of the Great Mother, 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 Mokoshi, who protect the world in the house, are also located on the carved frames (they are depicted with their hands raised up, 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 turning to the mother, the raw earth, which gives a harvest).

Tree of life motif.

Even the most abstract ornament by Russian masters retains its handiwork. In traditional national jewelry and household items you can find all types of symmetry on a plane: central, axial, rotary, portable.

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

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

For example, spell 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 are 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.

Ancient Russian art is characterized by a type of symmetry, expressed in doubling (pairing) of one or another image.

The pairing answered two types of wishes:"in all directions" « everywhere on earth”, “from all sides”, “from everywhere”.

Thus, the description of individual Old Russian ornamental motifs and the scheme of their arrangement on the details of the home, decorations and household items are the reason for the aesthetic appeal of the ornament.

Head: O. V. Afanasyeva

Preview:

To use presentation previews, create a Google account and log in to it: https://accounts.google.com

Slide captions:

“SYMMETRY IN ANCIENT RUSSIAN ORNAMENT” Performed by: students of group No. 7 A. Alexandrov, A. Makarov Supervisor: O. V. Afanasyeva

Purpose: To demonstrate the possibilities of the laws of geometry in creating various ornaments that decorate many objects of decorative and applied art. Objectives: Repeat and expand knowledge about known types of symmetry. Learn to identify and describe symmetry in an ornamental design. Summarize and systematize information used to create an ornament.

TYPES OF FLAT ORNAMENT A planar ornament is a decoration made on a plane using lines. Planar ornaments are used when decorating clothes and painting 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. The mesh pattern is used to decorate the floor, ceiling, and walls of the room.

CENTRICAL A centric design is a pattern whose decorative elements are grouped so as to create a centric movement. A closed pattern is most often used to decorate napkins, tablecloths, plates, etc.

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

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

ANCIENT RUSSIAN ORNAMENT Ornamental compositions, as decorations of household items, existed in Rus' for a very long time. Peasant huts were decorated with carved patterns. There were curtains embroidered with fancy patterns 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 a special meaning. The red color expressed delight and joy.

EXAMPLES OF RUSSIAN ORNAMENT

MOTIF OF “PROTECTION” SIGNS The Slavs placed their “protection” symbols on “vulnerable” areas of the home and yard. With incantatory ornaments they covered all openings through which evil spirits could penetrate to a person: gates, doors, windows, bars. The symbol of the sun, placed in the openings, should protect the home from the night evil spirits. There is a ridge on the roof of the house (symbolizing 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.

Ritual towels had a special magic and 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 and ritual celebrations characteristic of agricultural culture. On all holidays, bread and salt were the first to be brought out on a towel. Salt is a symbol of the sun, love; bread - earth, mercy; a towel is a symbol of human life, a stripe of fate, a part of pure outer space.

BRAID MOTIF Characteristic of the so-called Rusal bracelets, as a sign of water and the kingdom of the underground ruler Pereplut. TREE OF LIFE MOTIF

MOTIF OF THE ANCIENT GODDESS MOKOSH The only female image in ancient Russian mythology, as a specific embodiment of the idea of the Great Mother, 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 With parallel translation, all points of the figure are shifted in the same direction by the same distance.

CENTRAL SYMMETRY With central symmetry, each point goes to a point symmetric to it relative to a fixed point.

AXIAL SYMMETRY With axial symmetry, each point of the figure goes to a point that is symmetrical to it relative to a fixed straight line.

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

Creating a pattern using axial symmetry and parallel translation AXIAL SYMMETRY

PARALLEL TRANSFER

Creating a pattern using central symmetry and parallel translation CENTRAL SYMMETRY

PARALLEL TRANSFER

HOW WAS THE ORNAMENT OBTAINED? 1 2 3

ORNAMENT OBTAINED: 1 2 3 Using parallel transfer. Using axial symmetry and parallel translation. Using axial symmetry and parallel translation.

SOURCES USED O.V. Slesarev "The wonderful secret of geometry." Mathematics at school, O.E. Anatsko, E.M. Hanukovich, 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 Rus'”.

The task is to select illustrations for texts (insert illustrations

To text)

And learn the terms

Pages 2,3

TYPES AND ROLE OF SYMMETRY IN THE EMERGENCE OF ORNAMENTAL IMAGE

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

EXISTS SEVERAL TYPES OF SYMMETRY. MOST OFTEN WE MEET WITH MIRROR SYMMETRY. IT IS INHERENT IN THE HUMAN BODY AND MOST ANIMALS, AS WELL AS MANY OBJECTS WE USE. IT IS WIDELY APPLIED IN ART.

SYMMETRY HAS ATTRACTED PEOPLE'S ATTENTION SINCE ANCIENT TIMES. IN THE OBSERVED SYMMETRY OF NATURAL FORMS, THEY FOUND CONFIRMATION OF THE REASONABLE ORGANIZATION OF THE UNIVERSE, IN IT THEY SEEKED THE ESSENCE OF BEAUTY, EVIDENCE OF HARMONY.

ON THE CONCEPTS OF SYMMETRY THERE ARE PLANE OF SYMMETRY, AXIS OF SYMMETRY, CENTER OF SYMMETRY.

A PLANE OF SYMMETRY IS CALLED A PLANE THAT DIVIDES A FIGURE INTO TWO MIRROR EQUAL PARTS, LOCATED RELATIVE TO EACH OTHER, LIKE AN OBJECT AND ITS MIRROR REFLECTION.

AXIS OF SYMMETRY IT IS CALLED A STRAIGHT LINE AROUND WHICH A SYMMETRICAL FIGURE CAN BE ROTATED SEVERAL TIMES SO THAT EACH TIME THE FIGURE “SELF-ALIGNS” WITH ITSELF IN SPACE.

CENTER OF SYMMETRY THIS SPECIAL POINT INSIDE A FIGURE IS CALLED, CHARACTERIZED BY THE FACT THAT ANY STRAIGHT LINE DRAWN THROUGH THE POINT ON BOTH SIDE OF IT AND AT EQUAL DISTANCES MEETS THE SAME (CORRESPONDING) POINTS OF THE FIGURE. AN "IDEAL" EXAMPLE OF SUCH A FIGURE IS A BALL, THE CENTER OF WHICH IS ITS CENTER OF SYMMETRY.

TYPES OF SYMMETRY, USED IN ORNAMENTAL STRUCTURES:

* AXIAL (SYMMETRY RELATIVELY STRAIGHT)

* CENTRAL (SYMMETRY RELATIVE TO A POINT)

* MIRROR (SYMMETRY RELATIVE TO PLANE)

EXAMPLE(s)

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

EXAMPLE(s)

IN PRACTICE, A CURB CAN BE BUILT NOT ONLY IN A STRAIGHT, BUT ALSO ALONG A BROKEN OR CURVED LINE. IN ANY CASE, PARALLEL TRANSFER IS ACCOMPLISHED BY FOLLOWING THE BENDS AND FRACTURES OF THE AXIS. A CURB, IN ADDITION TO PARALLEL TRANSPORTATION SYMMETRY, MAY HAVE OTHER SYMMETRY ELEMENTS. THEY APPEAR IN THOSE CASES WHEN ONE OR ANOTHER TYPE OF SYMMETRY IS INHERENT IN EACH INDIVIDUAL ELEMENTARY MOTIF OF THE ORNAMENT.

SOCKET AS A KIND OF SYMMETRY, IT HAS BEEN WIDELY APPLIED IN ORNAMENT. THERE ARE FOUR TYPES OF ROSETS, WHICH DIFFER IN THE NUMBER OF AXES AND PLANES OF SYMMETRY APPLIED TO THE ELEMENTARY FIGURE. SO WHEN CONSTRUCTION OF ROSETS, ONLY THE SYMMETRY PLANE, OR ONLY THE SYMMETRY AXIS, OR THE AXIS AND THE SYMMETRY PLANE, OR MULTIPLE SYMMETRY PLANES CAN BE USED.

EXAMPLE(s)

*MIRROR SOCKET *AXIAL SOCKET

IN THE ART OF ORNAMENT THEY ARE OFTEN USED - MESH ORNAMENTS). MESH 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 A VARIOUS CELL SHAPES: RHOMBUS, SQUARE, TRIANGLE.

THERE ARE FIVE SYSTEMS OF POINTS (NODES OF THE ORNAMENTAL NET), WHICH ARE THE BASIS OF THE CONSTRUCTION OF MOST NET ORNAMENTS: SQUARE; A REGULAR TRIANGULAR, THE BASIS OF WHICH IS AN EQUILATERAL TRIANGLE; RECTANGULAR, CONSISTED OF RECTANGLES WITH ANY ASPECT RATIO; RHOMBIC, CONSISTED OF RHOMBICS WITH ANY RATIO OF DIAGONALS; PARALLELOGRAMMATIC, CONSISTED OF PARALLELOGRAMS OF AN ARBITRARY TYPE, AND THE TILT OF THE CELL 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 BY JUST ONE EXAMPLE, WHEN A SINGLE ELEMENTARY FIGURE IS TAKEN AS THE BASIS. THIS IS DONE SO THAT THE DRAWINGS ARE NOT EXTREMELY cluttered with DETAILS UNNEEDED AT FIRST AND SO THAT IT IS EASY TO UNDERSTAND THE ESSENCE OF ONE OR ANOTHER TYPE OF SYMMETRY OF THE NET ORNAMENT. WE WILL TAKE AS A BASIS THE SAME ELEMENTARY FIGURE AS FOR THE CONSTRUCTION OF ROSETTES AND RIBBON ORNAMENT.

IN ACTUALITY, THE SAME IMAGES OF AN ELEMENTARY FIGURE, REPEATING WHEN TRANSFERRED ON BOTH AXES, CAN BE COMPLETELY SEPARATED FROM EACH OTHER, I.E. DO NOT CROSS, BUT MAY THEMSELVES CONSISTE OF DISCONTINUED PARTS:

EXAMPLE(s)