"symmetry in ornaments." Rules for constructing an ornamental composition

Makarova T.I. Symmetry in the floral ornament of Ancient Rus' / in the book “Ancient Rus' and the Slavs”. - M., 1978. - P. 370.

Ornament in works of ancient Russian art has rarely been the subject of special study. Meanwhile, observation of its development makes it possible to establish certain patterns that facilitate its understanding. The purpose of this article is to trace the formal patterns of ornament construction in art monuments of the 11th-13th centuries. and determine the reasons for their stylistic commonality. Most of all, this question was developed in written monuments.

Paleographer V. Shchepkin developed the methodological foundations for the study of Old Russian ornament. V. Shchepkin laid the foundations for the genetic analysis of an ornament, identifying its original element (motif) and the nature of its changes (doubling, division, etc.). Further, he formulated a mechanism for creating compositions from individual elements: “the combination of motifs in the ornament occurs ... based on the instincts of symmetry and rhythm.” Finally, he gave a description of the ornament as a whole, determining that the ornaments differ from each other
“1) according to the content of their motives,
2) by the way they are combined,
3) by the nature of its frame.”
However, in the literature they more often recall another aspect of V. Shchepkin’s work - that classification of Russian book ornament from the 11th century. until modern times, which in its main features remains in force to this day. It seems to us that the methodology for studying book ornament proposed by V. Shchepkin deserves attention for the study of ornament in various areas of ancient Russian art. Let's try to show this using the example of analyzing the floral patterns of Ancient Rus'.

N. Shchepkin, based on the principles listed above, came to the conclusion that the ornamentation of the oldest Russian handwritten books represents a well-known stylistic unity. This idea was no longer new in his time, but it was V. Shchepkin who attempted to reveal the essence of this unity. In his opinion, it is due to the fact that the basis of the floral ornament is the same branch motif. In the text, V. Shchepkin says nothing about the origin of this motif, but the table graphically shows that it is formed by dividing a three-petal sprout, which is most correctly called crin 4 (Fig. 1, 1, 2). It is this element that underlies the floral ornamentation the oldest Russian books: Ostromir Gospel (1056), Izbornik Svyatoslav (1073), Mstislav Gospel (before 1117), Service Book of Vyarlmam Khutynsny (XII century), Yuriev Gospel (1119-1123).

At the beginning of our century N.P. Kondakov noted the similarity of the ornamentation of the oldest Russian miniatures with cloisonné enamel. The same circumstance prompted B.A. Rybakov call manuscripts with ornaments similar to cloisonne enamels, enamel. A study of the ornamentation of products with cloisonné enamels showed that this closeness involves something more than simple imitation. The basis of the floral ornament in this case was the same element krin 8 (Fig. 1, 15, 6). Sometimes it is identical to the handwritten krin, sometimes not quite, but the general scheme of their design is the same: in the simplest version it is a three-petal flower.

Currently, the Middle Eastern origin of the krill sprout is recognized by all researchers. The emergence of this symbol in the art of the most ancient center of agriculture is quite logical, just as its revival in the agricultural cultures of Europe and Byzantium is logical. The symbolism of the image itself did not exclude attempts (perhaps intuitive) to capture in it the features of the surrounding nature. But the content of his image, as a symbol of the ever-regenerating forces of nature, remained unchanged. We will not dwell on the question of when this symbol and the idea it expresses first penetrated into Eastern Europe. Probably, the study of this problem would take us back to the times of the Indo-European community. What is more important is how this idea, in the specific embodiment that the most ancient manuscripts give us, participated in the construction of ornamental compositions of a plant nature. We will deal with this aspect of the problem, leaving aside questions of the semantics of the ornament. Krin in its simplest form is not found very often in ornamentation. As a rule, we deal with its numerous modifications. Here we are faced with a second, no less important factor in the similarity of the floral patterns of manuscripts and enamels: in the changes in krin when constructing compositions from it, there is a certain similarity, some kind of unified program. Let us try to analyze the nature of the changes in krin first using the example of a handwritten ornament. To do this, based on the principles formulated by V. Shchepkin, we outline its most general typology.

We divide the entire composition into two large sections: the rosette and the border. In each of them, different groups of compositions depend on the changes that the main element of the floral ornament of the Krin manuscripts or its derivative, the branch, undergoes. The laws of symmetry show that the infinite variety of ornaments is an apparent phenomenon. It comes down to three main categories: rosette, border and grid, and each of them can be reduced to several types of symmetry.

Let us consider the main compositions formed by krin in the floral ornament of manuscripts, and from the point of view of the types of symmetry.

The handwritten floral ornament uses two categories of symmetry: rosette and border. They will give the two largest divisions in our typology. Most of the compositions that form the basis of the rich carpet of floral patterns of handwritten headpieces belong to rosettes with the so-called mirror symmetry. They are characterized by strict equality of the right and left halves.

Its simplest example is demonstrated by a krin enclosed in a circle or a heart-shaped figure (Fig. 1, 3, 4). The second type of composition is a rosette, in which a similar pattern is repeated four times (Fig. 1.5).

The third type of composition is due to further changes in the krin due to the endless complication of the very scheme of its design (Fig. 1, 6, 7). In this case, mirror symmetry is constantly preserved. It is also present when the number of petals increases, some bend down, others break away from the common stem, and the initial flare is difficult to discern in a lush flower (Fig. 1, 6); it is strictly observed even when the krin retains more strict outlines, turning into a tree-like figure with a pronounced trunk (Fig. 1, 7).

The fourth type of composition is built from a branch formed by dividing the krin in half (Fig. 1, 7, 2; 8, L). Lush trees arise from half-crine branches, the design of which often includes a crin (Fig. 1, 11).

Mirror symmetry, equally characteristic of all four groups of compositions we have described, gives them a balanced, static character.

Let us now consider the second category of ornamental compositions of manuscripts - a border, that is, a ribbon consisting of figures stretched along one straight line. Along this straight line, the main figure seems to be transferred endlessly. In the borders of a handwritten ornament, such a basic figure turns out to be a crin in its simplest form (Fig. 1, 13) or a complicated form (Fig. 1, 14) or a semi-crin branch (Fig. 1, 12). Depending on these basic figures, borders can be divided into two types. One of them forms borders of repeating or alternating krins of different shapes (Fig. 1, 13, 14)

In another type of borders, we distinguish an endless shoot formed by half-crown branches (Fig. 1, 12). The branch, as the curb moves, seems to move first down and then up and creates the impression of continuous movement. This favorite motif in medieval art, often called the runner or Byzantine vine, is characterized by a strictly defined type of symmetry. Now let’s compare the compositions of floral patterns.

Despite the fact that on products with cloisonné enamel the krin sometimes takes on shapes that are not typical for the krin of manuscripts (Fig. 1, 18, 20), the nature of its changes during the construction of compositions turns out to be the same as in the floral ornament of manuscripts. Here we find rosettes with fourfold repetitions of krin (Fig. 1, 19, 20), complicated tree krins of different shapes (Fig. 1, 23, 27).

The exception is the rosette with “half-crown branches”, which is not found in the handwritten ornament (Fig. 1, 28, 29). From the point of view of the theory of symmetry, they represent a new type of symmetry - central symmetry, significantly different from the mirror symmetry of the rosettes discussed above. It is interesting for us to note that the main element, repeated many times in this rosette, is not the symmetrical crin, but its half branch, itself asymmetrical. The resulting figure creates the impression of rotation, which differs from static rosettes with craniums (Fig. 1, 5; 19, 20).

Borders in enamels give us compositions that are similar in meaning and mechanism of construction. There is no place for a border on miniature gold items, so we see the variant of a ribbon made of rosettes with krins only twice (Fig. 1). As we see, in the floral patterns of manuscripts and enamels, in addition to the similarity of the main krin motif, the mechanism for constructing compositions turned out to be equally similar. It is determined by general types of symmetry. Researcher of Uzbekistan ornament L.I. Rempel came to the conclusion that the most stable in an ornament are not favorite motifs, but “a system of the simplest typical rhythms” that form, as it were, its “framework” 16.

It was these characteristic types of symmetry that made up the “framework,” which turned out to be the same both in manuscripts and in enamel work. Therefore, we can speak in this case about a unified system of ornamentation, since the order determined by the correct arrangement of elements in a certain connection means a system.

The question arises, in what relation was the considered system of plant ornamentation in manuscripts and enamels with other types of art of Ancient Rus'? For an answer, let us turn to floral patterns in monumental art. This will be logical because painting and sculpture represent the pinnacle in the development of Old Russian ornament as a whole. It is also of considerable importance that it has been published, studied and therefore visible. In addition, those monuments that we will take for comparison: mosaics and frescoes of St. Sophia of Kyiv and the wall sculpture of three cathedrals of Vladimir-Suzdal Rus' - Dmitrievsky, Rozhdestvensky and Georgievsky - give us the opportunity to trace the evolution of floral ornament in Rus' from the 11th to the 13th centuries. This choice is also determined by the scope of the article, since the ornament in the key monuments of monumental art must inevitably reflect the general patterns of the development of ornament in Rus'.

So, let's turn to the floral ornament, which was made using the technique of mosaic and fresco in the interior of St. Sophia of Kyiv, and analyze its structure, following the method used above.

The simplest element here also turns out to be krin, but with slightly different outlines (Fig. 1, 32, 33). It forms a rosette, similar in composition to rosettes in manuscripts and enamels (Fig. 1, 34, 35, 46). We observe the same compositional similarity in the often uniquely designed lush krinahs (Fig. 1, 36, 37, 47, 48), which always remain figures in mirror symmetry. Trees of branches are formed using the same principle (Fig. 1, 55, 59, 49, 50).

Finally, the central symmetry of the half-crines is also found on the frescoes, creating the impression of rapid rotation (Fig. 1, 51).
Borders in mosaics and frescoes in Sofia also provide familiar compositional solutions. In the complicated drawing of the vine one can easily discern the clearer flow of the manuscripts (Fig. 1, 40, 52). In addition to the already familiar compositions of krins alternating in a horizontal ribbon (Fig. 1, 41, 42), a vertical ribbon of krins is also found here (Fig. 1, 55). This is the only composition new from the point of view of symmetry, which has not been found either in manuscripts or in enamels. Its compositional closeness to horizontal ribbons is undeniable. The emergence of such a border solution is easily explained: its vertical strip was supposed to decorate the huge planes of the temple interior; there is no place for it in small things and small designs.

So, not in all versions the screen of mosaics and frescoes of Sophia is completely identical to the screen of manuscripts and enamels. Sometimes it resembles acanthus leaves, sometimes winged palmettes, but at the same time the character of their variants remains unchanged, the frame that is so clearly outlined in the enamel ornament. This allows us to assert that the system of early ornamentation of manuscripts, enamels and the interior of Sophia of Kyiv is uniform.

If we compare this system of ornamentation with the floral ornament covering the walls of the temples of Vladimir-Suzdal Rus', then its simplest element here also turns out to be krin. In some cases, it repeats the familiar three-petalled figures (Fig. 1, 71), but more often it takes on the outlines of a deciduous tree in a forest belt (Fig. 1, 54, 62, 70). The presence of peculiar variants (Fig. 1, 64, 73) still does not violate its typological similarity with the krins that adorned the headpiece of the manuscript, the gold colt with enamel or the painting of Sophia. However, the rosette with a four-fold repeated creep, a favorite in these multi-color patterns for white stone plastic, is not found. Obviously, in labor-intensive stone carving, craftsmen simply avoided small forms. Maybe that’s why the evolution of the kreen seemed to take a different path here. The sprout turns into a lush krin (Fig. 1, 65, 67) or a tree, in the design of which the krin is repeated many times (Fig. 1, 56, 66). But the most favorite composition was an overgrown multi-tiered tree, the symmetrically spread branches of which end in either krins or half-crins (Fig. 1, 78). Its simplified versions (Fig. 1, 57, 76, 77) often repeat trees we have already encountered (Fig. 1, 11, 26, 39). In all these cases, the principle of mirror symmetry is strictly observed, and it is precisely this that the multi-meter trees of the Vladimir-Suzdal temples are related to the krins of manuscripts and enamels.

More similarity in borders. Curly shoots are not a common plot here, but they still occur in both simple (Fig. 1, 58) and complicated versions (Fig. 1, 79). But the borders of another group, consisting of alternating or literally repeating krins, are extremely diverse. Some of them are very close to handwritten borders (Fig. 1, 81), others represent their vertical version (Fig. 1, 55, 68), and some are complicated by additional figures and are more elaborate in the design of the border itself (Fig. 1, 80 ). Their compositional similarity with the previously discussed borders is undeniable.

So, over the course of three centuries, from the 11th to the 13th centuries, in the floral patterns that adorned handwritten books, precious items with cloisonné enamel, the majestic interior of the main temple of Rus', Sophia of Kiev, and the famous white-stone churches of the Vladimir-Suzdal land, the craving for favorite compositions has been steadily preserved . It is not disturbed by variations in the design of the symbol of the Krin tree of life, dictated both by the peculiarities of technology and by the numerous impulses that the artistic creativity of ancient Russian masters from different parts of the civilized world received. The framework of the compositions created by their imagination remained essentially unchanged. It was determined by a certain set of types of symmetry established in different areas of Russian art.

A study of the ornaments of the peoples of Siberia, carried out by S.V. Ivanov, showed that such a set of favorite types of symmetry turns out to be different for each people, stable over many centuries.

In the floral ornamentation of the urban art of Rus', the most constant was also the compositional frame determined by the laws of symmetry and the simplest element krin, which among different peoples symbolized the eternal forces of nature.

B.L. Rybakov, in his works on the decorative art of Ancient Rus', convincingly showed that it was “surprisingly uniform in its spirit and images.” Ornament, which permeated all aspects of everyday life and reigned equally in all areas of art, largely contributed to this unity, “defining their common style and era.”

We tried to reveal the foundations of this unity, which was steadfastly preserved in the floral ornamentation of “women’s patterns, the ornamentation of books and architectural decor” throughout the three brightest centuries of Russian history.

Project for mastering the theme “This wonderful world of symmetry”

1. Main idea.

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. The topic “Symmetry” aroused great interest among students and encouraged them to study this material in more depth from different points of view (historical, mathematical, physical, biological and others).

2. Goals.

1. Teach to distinguish the diverse manifestations of symmetry in the surrounding world.

2. Show the important, exclusive role of the principle of symmetry in the scientific knowledge of the world and in human creativity.

3. To develop the creative activity of students, the ability to make generalizations based on data obtained as a result of research.

4. Develop students’ cognitive activities that contribute to the development of a versatile personality.

5. To instill in students a desire for self-improvement and satisfaction of cognitive needs.

3. Working groups and research questions.

Group "Mathematicians"

Mirror reflection. Experiments with mirrors.

Symmetry.

Curbs.

Conclusion.

Group "Historians"

Symmetry of Old Russian ornament.

Draw a conclusion about the presence of symmetry in the ornaments of ancient Russian motifs.

Group "Biologists"

Symmetry in biology.

Formulate a conclusion about the variety of structures that exist in nature.

Physicists group

Symmetry in physics.

Conclusion.

Group "Researchers of the existence of symmetry in music and literature"

Symmetry in music and literature.

Conclusion.

Group "Experts"

During the working groups’ reports, monitor their conclusions, enter assessments (in points) into the designer’s individual card, and at the end of the lesson, evaluate the work of each group.

4. Reporting materials.

Preparing messages.

Create presentations in PowerPoint.

Lesson type: mastering new knowledge.

Working methods and techniques: implementation of design and research technology.

Equipment:

Scissors, paper

Presentations.

During the classes

Teacher's opening remarks:

Dear guys! Our lesson takes place within the framework of design and research technologies and is devoted to such a versatile topic as “Symmetry”.

It is difficult to find a person who does not have some idea of ​​symmetry. Since ancient times, people have used symmetry in drawings, ornaments, and household items. You probably paid attention to how strictly symmetrical the forms of ancient buildings are, how harmonious ancient Greek vases are, and how proportionate their ornaments are. We encounter one or another manifestation of symmetry literally at every step. Look at a fluttering butterfly, a mysterious snowflake, a mosaic in a temple, a starfish, a garnet crystal - all these are examples of symmetry.

The famous mathematician of the last century, Hermann Weyl, said: “Symmetry... is an idea with the help of which man has tried for centuries to explain and create order, beauty and perfection.” These words will serve as an epigraph to our lesson. And we will try to explain and reveal order, beauty and perfection with the help of your research. 5 working groups took part in preparing for the lesson: mathematicians, historians, biologists, physicists, researchers of the existence of symmetry in music and literature. They will introduce us to the materials of their research. The sixth group - experts, will monitor the work in the lesson and evaluate your answers, based on the results of which each student will be graded.

So here we go. Write down the number, class work, lesson topic “This wonderful world of symmetry” in your notebook.

The floor is given to a group of mathematicians.

First student. Every day, each of us sees his reflection in the mirror several times a day. It is so common that we are not surprised, we do not ask questions, we do not make discoveries. And only philosophers and mathematicians do not lose the ability to be surprised. Here's what the German philosopher Immanuel Kant wrote about mirror reflection: “What could be more like my hand or my ear than their own reflection in a mirror? And yet the hand that I see in the mirror cannot be put in the place of a real hand ... "

What changes in an object when it is reflected in a mirror? Let's conduct experiments with mirrors. Try to notice the features of mirror reflection and draw conclusions from each experience, which we will write down in a notebook.

Tasks:

Write your name in block letters and look at its reflection in the mirror. Does the mirror turn your name?

What is the difference between the entries MASHA and YURA (place the strips with names parallel to the surface of the mirror)?

The words TEA and COFFEE are written horizontally in block letters on a strip of paper. Place this strip in front of the mirror on the table. Why didn't the mirror turn the word COFFEE upside down and change the word TEA beyond recognition?

Second student: Experiments with mirrors allowed us to touch upon an amazing mathematical phenomenon - symmetry. In ancient times, the word “symmetry” was used as “harmony”, “beauty”. Indeed, in Greek the word means “proportionality, proportionality, uniformity in the arrangement of parts.”

Let's draw a line along the spelling of the word COFFEE. If you now place the mirror along the drawn straight line, then the half of the figure reflected in the mirror will complement it to the whole. Therefore, such symmetry is called mirror (or axial, if we are talking about a plane). The straight line along which the mirror is placed is called the axis of symmetry. If a symmetrical figure is folded in half along the axis of symmetry, then its parts will coincide.

Look at the picture: it shows a blot and an openwork paper napkin. The blot turned out like this: they dropped paint onto a sheet of paper, folded the sheet in half and then straightened it out. The fold line is the axis of symmetry of the blot. In a similar way, an openwork napkin was made, only a sheet of paper was bent several times, a piece was cut out of this “puff” sheet, and then the sheet was unbent. The napkin has several fold lines, and all of them are axes of symmetry. Geometric figures may have one or more axes of symmetry, or may not have them at all.

Exercise:

Mentally bending the paper, determine how many axes of symmetry each of the figures shown in the figure has. (Rectangle, rhombus, square, parallelogram, regular hexagon, circle, triangles: arbitrary, isosceles, regular).

Which of the figures is “the most symmetrical”?

Which is the most “asymmetrical”?

Third student: Do you think only openwork napkins can be cut out of paper? Not only. Very beautiful symmetrical ribbons are also cut out of paper (ribbon demonstration).

How to get such ribbons? Take a strip of paper 5 cm wide and 20 cm long. Fold it like an accordion and draw a girl with her arms spread to the side so that her “arms” touch the fold line. Cut out the shape, leaving the areas on the fold lines uncut; Let’s unfold the resulting “accordion”. We got lace. If the tape is first folded in half lengthwise and then like an accordion, you will get a tape that is symmetrical about the horizontal axis (demonstration).

Ornaments in the form of ribbons (borders) are used by painters and artists when decorating rooms and buildings. To make these ornaments, a stencil is made. A stencil represents a design cut out on a sheet of cardboard or some other dense material. The painter moves the stencil, turning it over or not turning it over, traces the outline, repeating the design, and gets an ornament.

Exercise: Using a ready-made stencil, get symmetrical patterns using:

parallel transfer;

mirror symmetry;

turn 180 0 around point O;

symmetry about the horizontal axis plus parallel translation.

Expert conclusion.

Teacher: 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's give the floor to our historians.

First student: Before turning to Slavic ornamentation, let us briefly consider the state of mathematical knowledge in Rus' during the period of the 9th – 10th centuries.

In practice, the 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 was widespread in Rus' already in the 9th – 10th centuries. According to B.V. Gnedenko, these were skills rather than knowledge that were transmitted orally and included ideas about natural numbers and operations with them, as well as the simplest fractions. In addition, such a geometric tool as a compass was well known in Ancient Rus'. Therefore, the ornament of circles on jewelry and household items is widespread.

According to Academician B. A. Rybakov, a famous archaeologist and historian with a worldwide reputation, the basis of the ancient Slavic ornament was based on universal ideas about the world. The consciousness of the ancient Slav was determined by the mythological perception of reality. Myth and ritual combined elements of magic and totemism (a set of beliefs and rituals of a tribal society associated with the idea of ​​kinship between groups of people), artistic creativity, and social norms regulating people's behavior. All this is reflected in the motifs of Russian ornament.

Second student: In clothing, collars, shirt cuffs, hem, and slits on a shirt or sundress were covered with a magical protective pattern. The fabric itself was considered impenetrable to the spirits of evil, so its production involved objects richly equipped with magical ornaments (ruffles, spinning wheels, weaving mills). It was important to protect those places where the enchanted fabric of clothing ended and the human body began.

The same is true in folk architecture: decorative elements are located on the gates, around the windows; one or another consecrated image (a horse, a deer's head with antlers, a goddess and birds, the sun) crowned the highest point of the house - the roof gable. Figures with “good” symmetry, such as a circle and a regular hexagon, were often used as amulets. Here is what Rybakov writes about the amulet against thunderstorms: “The same amulet against thunderstorms was widespread among all Eastern Slavs - a hexagon or a circle, but always with six radii, which forces us to distinguish this figure from the general mass of signs, conventionally called solar, and recognize the wheel as a special “thunder” sign.”

Third student: Old Russian ornament usually combined ideograms of water, rain, sun and flora in its above-ground and underground (root) parts.

The water element was represented by rows of dots and dashes, reproducing raindrops, as well as zigzag lines, which serves as an example of figurative symmetry in the simplest ancient Russian ornament. This motif is typical for window frames.

The earth was presented in the form of 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.

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, as well as a circle with a cross inside.

Thus, the description of individual Old Russian ornamental motifs (for example, themes of fertility, rain, sun, etc.) and the diagram of their arrangement on the details of the home, decorations and household items clearly demonstrates the presence in them of central, rotary, portable, axial and mirror types symmetries, which are the reason for the aesthetic appeal of the ornament.

Expert conclusion.

Teacher: Let's listen to a report from a biologist who will tell us about symmetry in the plant and animal worlds.

First student: The cone symmetry characteristic of plants is clearly visible in the example of virtually any tree. A tree, with the help of its root system, absorbs moisture and nutrients from the soil, i.e., from below, and the remaining vital functions are performed by the crown, i.e., above.

The vertical orientation of the body axis characterizes the symmetry of the tree. Leaves, flowers, branches, and fruits have pronounced symmetry. The figures shown show examples in which only mirror symmetry is observed. This situation is typical for leaves and flowers.

Most flowers are characterized by rotational symmetry. For example, the St. John's wort flower has a rotary axis and does not have mirror symmetry; an acacia branch has mirror and figurative symmetry; A hawthorn branch has a sliding axis of symmetry.

Second student: Rotational symmetry is also found in the living world. Examples include the starfish and sea urchin shell.

The phrase “mirror bilateral” is more often used in biology instead of the phrase “mirror symmetry.” This symmetry is clearly visible in the butterfly's left and right wings and appears with almost mathematical rigor.

We can say that each animal consists of two enatimors - the right and left halves. Let us finally note the bilateral symmetry of the human body (we are talking about the appearance and structure of the skeleton). This symmetry has always been and is the main source of our aesthetic admiration for the well-proportioned human body.

Thus, symmetry limits the variety of structures that can exist in nature.

Expert conclusion.

Teacher: Next we’ll talk about symmetry in inanimate nature. It is probably no coincidence that the lifeless castle of the Snow Queen from Andersen’s famous fairy tale is often depicted as a highly symmetrical structure. Word to physicists.

First student: The stones lying at the foot of the mountain are very disorderly; however, each stone is a huge colony of crystals, which are highly symmetrical structures of atoms and molecules. It is crystals that bring the charm of symmetry to the world of inanimate nature. Who among you has not admired snowflakes? Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have symmetry, rotational symmetry and mirror symmetry.

All solids are made of crystals. Look at crystals of topaz, beryl, smoky quartz.

The symmetry of the external form is clearly visible in crystals of rock salt, quartz, and ebonite. And on the next slide you see three forms of diamond crystals: octahedron, dodecahedron, hexagonal octahedron.

Thus, the symmetry of the external shape of a crystal is a consequence of its internal symmetry - the ordered relative arrangement in space of atoms (molecules).

Teacher: In other words, the symmetry of a crystal is associated with the existence of a spatial lattice of atoms, the so-called crystal lattice.

Music sounds... But where is the symmetry in music? The floor is given to researchers of the existence of symmetry in music and literature.

First student: The soul of music, rhythm, consists of the correct periodic repetition of parts of a musical work. The correct repetition of identical parts as a whole is the essence of music. We can rightfully apply the concept of symmetry to a musical work, because this work is written using notes. Composition has the most direct relation to symmetry. The great German poet I.V. Goethe argued that: “Every composition is based on hidden symmetry. To master the laws of composition means to master the laws of symmetry.”

If we take a simple example: “The Squirrel’s Song” from the musical fairy tale “Twice two is four.”

Every day without any rush
I'm gnawing nuts in a hollow:
Click-click-click
Click-click-click

Chorus:

I'm never sad
I have fun and sing:
La-la-la
La-la-la

Everyone can see my dexterity,
I jump along the branches deftly
Skok-skok-skok
Skok-skok-skok

Chorus:

Very red, like autumn,
I flash between the pines:
Jump-jump-jump
Jump-jump-jump.

Chorus:

This song alternates between a verse and a chorus. Symmetry can be seen in poems - this is the alternation of rhymes, stressed syllables, i.e. rhythm.

For example:

A.S. Pushkin.

This year the weather is autumn
I stood in the yard for a long time
Winter was waiting, nature was waiting
Snow fell only in January.

The alternation of rhymes and reading by intonation makes you feel the charm of the symmetry of Pushkin’s poem.

Conclusion of the expert group.

Teacher: Guys, I thank you for the work you did in selecting the material for our lesson. Today we looked at various manifestations of symmetry. We saw that symmetry patterns live a full life in music, in architectural styles, in household items, and in ornaments. Models with symmetrical shapes give us real pleasure. After all, they talk about beauty and harmony.

I wish you great success and harmony in your relationships with your family and friends. Be healthy and happy.

Goodbye. Thank you for the lesson!

LITERATURE

1. First of September. Mathematics No. 2 2004 E. Nesterov Symmetry around us, grades 5-6.

2. Podkhodova N. S., Ovodova E. G. Geometry in space.

3. Weil G. Symmetry. M: Nauka, 1966

4. Voloshilov A.V. Mathematics and Art. M: Education 1992.

5. Gardner M. This right, left world. M.: Mir 1967.

6. Loshanov M. Elements of symmetry in music. Sat Musical Art" Issue 1.M: Music, 1970.

7. Tarasov L.V. This amazing symmetrical world. M.: Education, 1982

8. Shafranovsky I.I. Symmetry in nature. L: Nedra 1968

9. Shubnikov A.V. Koptsik V.A. Symmetry in science and art. M.: Nauka, 1972.

10. I. F. Sharygin, L. N. Erganzhieva Visual geometry. Textbook for grades V – VI. – M.; MIROS, CPC “Marta”, 1992.

Gnedenko B.V. Essays on the history of mathematics in Russia. – 2nd ed., rev. and additional – M.: KomKniga, 2005.

11. Rybakov B. A. Paganism of Ancient Rus'. – M.: Nauka, 1988.

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Reasonableness of work goals

Work planning

Ability to select information

Ability to organize team work

Availability of division of duties

Awareness of the group about the results of the work

Determining the contribution of each group member

Search and information activities

Matching the content to the topic

Logic and consistency of presentation

Clarity of statements and conclusions

Easy to understand

Results and conclusions

Aesthetics of results presentation

Compliance of design with standard requirements

Presentation

Quality of the report

Volume and depth of knowledge on the topic

A culture of speech

Sense of time

Ability to hold the audience's attention

Discussion skills

Evaluation of the process and results of work

The results obtained and their assessment

Level of independence in the design of all stages

Grading Criteria

Total points

Points

110 - 90

89 - 65

64 or less

Grade

Great

Fine

satisfactorily

final grade

State educational institution "Bobrovskaya special (correctional) boarding school for orphans and children left without parental care with disabilities"

MATHEMATICS PROJECT

ON THE TOPIC

"THIS AMAZING WORLD OF SYMMETRY"

prepared by VKK mathematics teacher

N. A. Polubavkina

Regional research conference "Junior"

Research

Symmetry in the world around us

(section of exact sciences)

Performed: Merizanova Anna,

Eliseenko Vera,

8th grade student

Supervisor: Kolesnikova

Lyudmila Alexandrovna,

mathematic teacher

Introduction. . 2

1.1. ..................................................... . 3

1.2. ................................................................... . 4

1.3. Symmetry through the centuries . 7

Chapter 2. Symmetry around us. 8

.. 8

2.2. .......................................................... . 9

Conclusion. 11

Bibliography. 12

Introduction

This school year we discussed this topic in mathematics lessons. We were interested in the topic “Symmetry”. And we decided to create a project on this topic, because in the geometry textbook little attention is paid to studying the topic “Symmetry”, while students often ask the question: why is it needed, where is it found, why is it studied at all.

But symmetry is found in nature, and in science, and in art - the unity and opposition of symmetry is found in everything.

Symmetry is characteristic of various phenomena that underlie all things; it describes many phenomena of life and many sciences

As a result of our work, we asked ourselves the following questions:

Why do you need to know symmetry, where in the world around you does it occur?

We have set ourselves a goal:

form ideas about symmetry , through the systematization of knowledge about symmetry, as well as through the analysis of natural phenomena and human activity.

To reveal the topic of our research work, the following tasks were set:

Learn to recognize symmetrical figures among others.

Get acquainted with the use of symmetry in nature, everyday life, art, and technology.

Demonstrate the varied applications of mathematics in real life.

Realize the degree of your interest in the subject and evaluate the possibilities of mastering it from the point of view of a future perspective (show the possibilities of applying the acquired knowledge in your future profession as an artist, architect, biologist, civil engineer).

To write the work, I used various methods:

2) the method of inductive generalization and specification;

3) use of computer equipment.

Chapter 1. First ideas about symmetry

In this chapter we describe the first ideas about symmetry, historical information on this topic; some examples of symmetrical figures are given; examples of a research nature on the topic: “Symmetry” are considered.

1.1. Historical development and understanding of the concept of symmetry

In the process of historical development and understanding of symmetry, a special stage of symmetry as a measure of beauty and harmony is associated with the work of the outstanding mathematician Hermann Weyl “Symmetry” (1952). G. Weil understood symmetry as the immeasurability (invariance) of any object during transformations: an object is symmetrical in the case when it is subjected to some operation, after which it will look the same as before the transformation.

The Greek word "symmetry" means "proportionality", "proportionality", "sameness in the arrangement of parts." However, the word “symmetry” is often understood as a broader concept: the regularity of changes in certain phenomena (seasons, day and night, etc.), the balance of left and right, the equality of natural phenomena. In fact, we are dealing with symmetry wherever any order is observed. The concept of symmetry was widely used in psychology and morality. Thus, the great Aristotle believed that symmetry has the meaning of a certain average measure to which a virtuous person should strive in his actions. The Roman physician Galen (2nd century AD) understood symmetry as a state of mind equally distant from both extremes, for example, from grief and joy, apathy and excitement. Symmetry, understood as peace and balance, is opposed to chaos and disorder. This is evidenced by the engraving of Marius Escher “Order and Chaos” (Fig. 196), where, as the artist himself wrote, “a stellated dodecahedron, a symbol of beauty and order, is surrounded by a transparent sphere. It reflects a meaningless collection of useless things."

1.2. Mathematical idea of ​​symmetry

The ideas about symmetry outlined above are of a general nature and are not accurate and strict for mathematics.

Definition 1. Symmetry – this is proportionality, the sameness in the arrangement of parts of something on opposite sides of a point, straight line or plane.

A mathematically rigorous definition of symmetry was formed relatively recently - in the 19th century, when the concepts of mirror and rotational symmetry were introduced.

Rosettes and snowflakes are symmetrical and very beautiful figures.

In planimetry, there are axial (symmetry relative to a straight line), central symmetry (symmetry relative to a point), as well as rotational, mirror, and portable.

Definition 2. Two points A and A1 are called symmetrical relative to straight line a, if this line passes through the middle of segment AA1 and is perpendicular to it.

Every point is straight A

Definition 2 . The figure is said to be symmetrical about a straight line A, if for each point of the figure there is a point symmetrical to it relative to the straight line A also belongs to this figure. Straight A called axis of symmetry figures. They say the figure has axial symmetry. Shapes that have an axis of symmetry: rectangle, rhombus, square, equilateral triangle, isosceles triangle, circle, etc.

Definition 3. Two points A and A1 are called symmetrical about point O, if O is the middle of segment AA1. Dot ABOUT is considered symmetrical to itself.

Definition 4. The figure is called symmetrical about point O, if for each point of the figure there is a point symmetrical to it relative to the point ABOUT also belongs to this figure. Dot ABOUT, called center of symmetry of the figure. They say the figure has central symmetry. Examples of figures that have central symmetry: circle, parallelogram, triangle, etc.

Mathematics studies many figures that have both axial and central symmetry (circle, square, etc.), only axial symmetry (for example, an isosceles triangle), and only central symmetry (for example, a general parallelogram).

To understand this topic, we carried out a number of research tasks.

Research assignments.

Exercise 1. On a straight line AB find a point whose distance is the sum of two given points M And N would be the smallest.

Discussion. 1 case. Let M And N lie on opposite sides of , the shortest distance between them is , therefore, the required point X lies at the intersection and https://pandia.ru/text/79/046/images/image024_13.jpg" align="left hspace=12" width ="187" height="132">Any other point on a straight line AB does not have this property, since .gif" width="36" height="23"> Build M1, symmetrical M regarding https://pandia.ru/text/79/046/images/image023_17.gif" width="36 height=27" height="27">.gif" width="36" height="23 src=" >, then the required point X is the point of intersection of the lines MN And AB.

Task 2. Given straight lines AB and dots M And N. Find it at https://pandia.ru/text/79/046/images/image028_8.jpg" align="left hspace=12" width="207" height="140"> Discussion. 1 case. Points M And N lie on one side of the line AB (and, moreover, at different distances from it. Then point X of the line AB, for which the difference in distances from the points M And N the largest, is the point of intersection of the line AB with the continuation of the segment MN. Then any other point X1 of the line AB does not have this property, since (a consequence of the triangle axiom). If M And N is located at the same distance from https://pandia.ru/text/79/046/images/image031_8.jpg" align="left hspace=12" width="207" height="148"> Case 2. Points M And N lie on opposite sides of . Then the required point , Where .

If points M And N are on opposite sides of and at the same distance from it, then the problem has no solutions.

Task 3. Investigate whether the following have a center of symmetry: 1) a segment; 2) beam; 3) square.

Discussion. 1) yes; 2) no; 3 yes

Task 4. Investigate which of the following points of the Latin alphabet have a center of symmetry: A, O, M, X.

Discussion. O and X

Discussion. 1) two; 2) “infinite set”: any line perpendicular to a given one, as well as the line itself; 3) one.

Task 6. Explore which of the following letters have an axis of symmetry: A, B, d, E, O in the alphabet.

Discussion. A, E, O

Conclusion: These examples show us that even points in the alphabet have a symmetrical position. Various geometric shapes have an axis of symmetry.

1.3. Symmetry of Old Russian ornament

Russian ornament is characterized by both floral and geometric forms, as well as images of birds, animals and fantastic animals. Russian ornament is especially clearly expressed in wood carving and embroidery. The most commonly used were so-called braids - interlacing of ribbons, belts, and flower stems. In the 17th century The architect Stepan Ivanov created his famous “Peacock Eye” ornament.

According to the academician, a famous archaeologist and world-famous historian, the basis of the ancient Russian ornament included universal, different ideas about the world. The consciousness of the ancient Slav was conditioned by mythological perceptions of reality. All this was reflected in the motifs characteristic of Russian ornament.

· Motif of “amulet” signs, which were applied to clothing, household items and various details of the home..jpg" width="300" height="239 src=">

· Motive braids, characteristic of Rusal bracelets, which was interpreted as a sign of water and the kingdom of the underground ruler Pereplut.

· Ancient motif goddess Mokoshi as a specific embodiment of the idea of ​​the Great Foremother, common to all peoples at a certain stage of historical existence. Mokosha (Makosh) is the only female image in ancient Russian mythology. Her name brings to mind phlegm, moisture, water. Mokosh patronized all women's activities, especially spinning, and was revered mainly by women.

https://pandia.ru/text/79/046/images/image041_6.jpg" width="324" height="211">

Since ancient times, Russian ornament has developed a special system of arrangement of symbols representing the movement of the Sun around the Earth. There are several types of sun signs; they are characterized by rotational symmetry. The most common is a circle divided by radii into different sectors (“Wheel of Jupiter”), as well as a circle with a cross inside.

Conclusion: Having analyzed the literature on this issue, we came to the conclusion that symmetrical symbols are often found in ancient Russian ornaments. In traditional national jewelry and household items you can find all types of symmetry on a plane: central, axial, rotary, portable.

1.4. Symmetry through the centuries

In his reflections on the picture of the world, people have been actively using the idea of ​​symmetry for a long time. According to legend, the term “symmetry” was coined by the sculptor Pythagoras of Rhegium, who lived in the city of Regulus. He defined deviation from symmetry by the term “asymmetry”. The ancient Greeks believed that the universe was symmetrical simply because it was beautiful. Considering the sphere to be the most symmetrical and perfect form, they concluded that the Earth was spherical and that it moved on a sphere around a certain “central fire”, where the 6 then known planets also moved along with the moon, the Sun, and the stars.

Representatives of the first scientific school in human history, followers of Pythagoras of Samoa, tried to connect symmetry with number.

Widely using the idea of ​​harmony and symmetry, ancient scientists loved to turn not only to spherical forms, but also to regular polyhedra, for the construction of which they used the “golden ratio”. Regular polyhedra have faces that are regular polygons of the same type, and the angles between the faces are equal. The ancient Greeks established an astonishing fact: there are only five regular convex polyhedra, the names of which are associated with the number of faces - tetrahedron, octahedron, icosahedron, cube, dodecahedron.

Chapter 2. Symmetry around us

This chapter describes a theory that indicates various representations of symmetry in nature; in this chapter we prove that structures created by man also have symmetrical figures.

2.1. The role of symmetry in knowledge of nature

The symmetry of crystals is a consequence of their internal structure: their atoms and molecules have an ordered mutual arrangement, forming a symmetrical lattice of atoms - the so-called crystal lattice.

The missing elements of symmetry were determined by academician Axel Vilgelmovich Gadolin (). The famous professor of mineralogy from the German city of Marburg Johann Hessel in 1830. Published his work on the symmetry of crystals. For some reason, his work went unnoticed. But in 1897 Hessel's work was republished, and since then his name has gone down in the history of science.

So, we learned to study and compare the symmetry of crystals. There are 9 symmetry elements and only 32 different sets of symmetry elements - symmetry groups, which determine the external shape of crystals. But since the number of symmetry elements of crystals is finite, then the number of their sets - combinations that describe the symmetry of the external form - is finite. It follows that symmetry is a strict and comprehensive law governing the kingdom of crystals. It determines the shape of the crystal, the number of its faces and edges, and it also dictates its internal structure.

Symmetry can be found in sea creatures such as starfish, sea urchins and some jellyfish.

Leaves, branches, flowers and fruits of plants have pronounced symmetry. Some of them are characterized by only mirror symmetry, or only rotational symmetry, sliding.

It is interesting that among plants of the same species there are those that have both left and right leaf structures.

Living nature is characterized not only by well-known types of symmetry. Thus, the curved stem of a plant and the twisted shape of a mollusk are no less symmetrical than a crystal. But this is a different symmetry - curvilinear, which was discovered in 1926.

And in 1960 The academician introduced the symmetry of similarity into consideration. Similar figures are considered to be of the same shape. Similarity symmetry consists of transferring (rotating) a figure while simultaneously decreasing or increasing its size.

2.2. Symmetry in architectural structures

Symmetry dominates not only in nature, but also in human creativity. Works of architecture demonstrate excellent examples of symmetry. Old Russian buildings are interesting, in particular wooden churches. Slender and expressive, cut into an octagon, that is, with symmetrical octagonal tents, they perfectly corresponded to the concept of beauty in medieval Rus'.

An example is St. Basil's Cathedral on Red Square in Moscow. The temple consists of ten different temples, each of which is strictly symmetrical, but as a whole it has neither mirror nor rotational symmetry.

There are many examples of the use of symmetry and asymmetry in sculpture. For example, the sculpture of the Peloponnesian master from the school of Pythagoras “The Delphic Charioteer”, which depicts the winner in horse-drawn chariot competitions. The figure of a young man in a long chiton is generally symmetrical, but a slight rotation of the torso and head breaks the mirror symmetry, which creates the illusion of movement, and the statue seems alive.

Louis Pasteur believed that it was asymmetry that distinguishes living from non-living, believing that symmetry is the guardian of peace, and asymmetry is the engine of life. An example of how the paradox of symmetry serves not only to convey movement, but also to enhance impression is the image of a Greek vase from the Kamares Cave on the island of Crete.

Conclusion

Symmetry is something common, characteristic of different phenomena, underlying all things, and asymmetry expresses certain individual characteristics of things and phenomena. In nature, in science, and in art, the unity and opposition of symmetry and asymmetry is revealed in everything. The world exists thanks to the unity of these two opposites.

After analyzing the work, we came to the conclusion that symmetry is often found in art, architecture, technology, and everyday life. Thus, the facades of many buildings have axial symmetry. In most cases, patterns on carpets, fabrics, and indoor wallpaper are symmetrical about the axis or center. Many parts of mechanisms, such as gears, are symmetrical.

As a result of the project:

u expanded knowledge about symmetry;

u learned what phenomena from life and

some sciences are described by symmetry;

u new practical techniques: work with educational, scientific and educational literature;

u generalized the concepts, ideas, knowledge that the project result is aimed at obtaining: we looked at where symmetry occurs in life.

Bibliography

1. Afanasyev A. N., Mythology of Ancient Rus'. – M.: Eksmo, 2006.

2. Weil G. Symmetry. – Ed. 2nd, erased – M.: Unified URSS, 2003.

3. Gnedengo on the history of mathematics in Russia. – 2nd ed., rev. and additional – M.: KomKniga, 2005.

4. Fine motifs in Russian folk embroidery. Museum of Folk Art. – M.: Soviet Russia, 1990.

5. Klimova ornament in the composition of artistic products. – M.: Fine Arts, 1993.

Target: explore the transformation of symmetry when constructing ornaments

Tasks:

• Educational: systematization of knowledge in the transformation of figures
• Educating: hard work, patience; promote the development of research skills, skills in constructing beautiful figures and artistic creativity when tiling a plane.
• Developmental: development of logical thinking, development of attention, artistic creativity, development of aesthetic culture, outlook and curiosity of students, ability to highlight, “see” figures in grids

Various examples of shape transformations are considered.

Rice. 1

A name is given to three types of transformations performed according to certain rules. In this case, each point of the figure F is transferred to another point of the figure F’.

The teacher introduces students to examples of centrally symmetrical figures.

Rice. 2

Questions for students:

1. Show the center of symmetry of the figures.
2. Name the figures that have more than one center of symmetry (a figure consisting of two parallel lines A And V, have more than one center of symmetry)
3. Name other examples of centrally symmetrical figures. ( parallelogram)
4. Name a figure other than a table one, which has more than one center of symmetry ( straight)
5. Does a figure consisting of two intersecting lines have a center of symmetry?

Consider Figure 3.

Rice. 3

1. How many axes of symmetry do these figures have?
2. Name the numbers of figures that have one, two, three, four, an infinite number of axes of symmetry.
3. Draw a figure, different from those in the picture, symmetrical about a certain axis.

Consider the following symmetry transformations

Rotational symmetry

Rotating a plane around point O by an angle is called mapping the plane onto itself,

in which each point M is mapped to a point M 1 such that OM = OM 1 and angle MOM 1 is equal to . In this case, point O remains in place, and all other longings rotate around point O in the same direction - clockwise or counterclockwise.

Rice. 5

Mirror symmetry

The geometric figure is called symmetrical relative to the plane S (Figure 16), if for each point E of this figure a point E’ of the same figure can be found, so that the segment EE’ is perpendicular to the plane S and is bisected by this plane (EA = AE’). The plane S is called plane of symmetry. Symmetrical figures, objects and bodies are not equal to each other in the narrow sense of the word (for example, the left glove does not fit the right hand and vice versa). They're called mirror equal.

Rice. 6

Examples of figures with mirror symmetry:

Rice. 7

Let's consider the use of symmetry transformations in ornaments.

What is an ornament?

Ornament (from the Latin ornamentum - decoration) is a pattern consisting of rhythmically repeating elements to decorate any objects or architectural structures. The ornament can be found almost everywhere. Ornament is very often found in embroidery, wood carving, architecture, and even in nature you can find ornaments. It is impossible to imagine ancient Chuvash clothing without ornament. Women have always embroidered all kinds of ornaments on their clothes. Whenever guests were greeted, they were always presented with a towel decorated with ornaments. Ornament has always been present in fabric products. If you were in a village, you would notice that all the houses have very beautiful repeating carvings. The Russian people have always decorated their houses with carved frames, cornices, and platbands. Ornament is used to decorate many buildings. Ornament makes buildings more beautiful. Beautiful columns with ornaments will make any building very beautiful. The ornament will decorate any product, be it a fabric product or a building.

Let's look at several types of ornaments.

Rice. 8

What types of symmetry transformations are given here?

Based on the transformations, ornaments can be divided into three types

• Linear
• Mesh.
• Closed.

Linear ornaments - an ornament in a strip with a linear vertical or horizontal alternation of the motif (ribbon).

Mesh, or rappor, ornament. The motif is repeated both vertically and horizontally; this ornament is endless in all directions. Rapport is the minimum area, including the motif and the distance to the adjacent motif. Usually they use rectangular rapport.

Closed ornament. It is arranged in a rectangle, square or circle (rosette). The motif in it either does not have a repetition, or is repeated with a rotation on the plane.

On rice. 8 highlight linear, mesh, closed patterns. By studying methods for constructing mesh and closed patterns, you can start tiling the plane. You can pave the plane using mesh ornaments. And how this is done, you can see the presentation of the works of my student Andreeva V, a 7th grade student.

So, let's draw conclusions.

Today we repeated the transformation of symmetry and their application in the construction of ornaments, examined the methods and construction of linear, mesh, closed ornaments and methods of tiling a plane with various figures.

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