Ornamental symmetry. Project for mastering the topic “This wonderful world of symmetry Mirror symmetry in ornament
The basis for the construction of an ornament composed of abstract or figurative motifs is the repeated repetition of these motifs according to the laws 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. Various types of symmetry are studied in special branches of mathematics
In the science of symmetry, two types of symmetry are distinguished: final(for example, sockets) and endless, 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.
Among the most common types of symmetry used in creating ornamental compositions is mirror symmetry. This is when an object or figure is divided by a plane into two halves so that one half, reflected in this plane as in a mirror, coincides with the other. Mirror symmetry is inherent in the human body and the bodies of many animals. It promotes the impression of balance and peace. The ornament retains the same feeling.
Another type of symmetry is axial symmetry, in which the figures are aligned by rotating around an axis perpendicular to the image plane. The number of such combinations throughout the full circular revolution of the figure is called the order of the axis. Axial symmetry can have any order expressed by an integer - from second to infinity.
There can be an infinite number of figures with axial symmetry. They are characterized by a clear organization, when parts equal to each other are distributed around a single center (the point through which the axis of symmetry passes) evenly and in the same relation to it. In this case, all angles of rotation of the coincidence of the figure with itself must be equal, otherwise a complete coincidence will not occur. The distance from the same character points of the figure to the center should also be the same.
Axial symmetry is often found in nature and is widely used in ornaments: the symmetry of a flower and its ornamental analogue - a rosette.
When a figure has a pattern based only on axial symmetry, then this pattern gives the impression of infinite mobility and expresses rotational movement in a certain direction.
ornament style morphological artistic
Tiled frieze. Russia. Second half of the 17th century.
More often there are rosettes that combine axial and mirror symmetry (in this case there are not only axes, but also planes of mirror symmetry). Then the planes necessarily pass through the axis, intersect in it, and their number corresponds to the order of axial symmetry of the figure. This kind of form is much more balanced and calm. The mirror image of such a figure is no different from itself, and can be combined with it not only in a mirror way. This form appears to the eye as the most complete and clear: identical elements that mutually balance each other extend from its center in all directions. Such a rosette is balanced and therefore static within itself, since there is no asymmetry in it not only as a whole, but also in each individual element of its structure (in a rosette without planes of symmetry, such elements were themselves asymmetrical and caused a feeling of rotation).
Therefore, motifs with symmetry of this kind have become especially widespread and important in ornamental art. The completeness of their form creates an image of harmonious peace. The integrity and closedness of the form allows you to organize any surface, marking its center, opposed to the periphery.
All symmetries discussed above refer to limited symmetrical structures of the final figures of the ornament. Introducing a new type of symmetry -- parallel transfer will help you understand how potentially infinite patterns are arranged.
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.
Ribbon (linear) border is one of the most common and important types of ornament. It is constantly used to delimit any surface characterized by a variety of artistic qualities. In practice, a linear 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.
In addition to transfer symmetry, a border may also have other symmetry elements. They arise when one or another type of symmetry is inherent in each individual elementary motif of the ornament. There are seven different types of border symmetry in total, and the impression from them and the artistic possibilities of each type used in the ornament turn out to be different.
The rhythmic movement of the border with an asymmetrical initial motif that does not create additional symmetry is one-sided. If you turn such a pattern over in a mirror, it will “pull” in the opposite direction. In addition, such an ornament is addressed differently to the parts that it separates. Thus, it characterizes these surfaces differently and can create the feeling of their different density and depth.
An ornamental motif with mirror symmetry will impart such symmetry to the border, if only the planes of reflection are located perpendicular or parallel to its axis.
It happens that in such an ornament, motifs that mutually reflect each other are shifted along the transfer axis. To restore mirror symmetry, you need to slightly shift one of the halves of the border along the axis. This type of symmetry is called "grazing reflection". Typically, such a border uses a paired motif, for example, the reflection of a leaf, and the leaf takes the place of the reflected flower. The rhythm of the ornament turns out, for all its clarity, to be richer and more complex than in patterns without sliding reflection.
Borders can also be characterized by axial symmetry, along with or without planes of reflection. This means that the entire border can coincide with itself when rotated 180° around any of an infinite number of equally spaced axes passing through the longitudinal centerline of the pattern. Three types of such ornaments can be distinguished: a border without mirror planes, then both edges are identical in the nature of the pattern, their rhythm leads the eye in opposite directions. This pattern looks restless and tense (for example, a classic meander).
If planes of reflection are also added to the rotary axes, the rhythmic tension of the pattern weakens and it looks calmer. Together with transverse planes, such a pattern is also enriched with sliding reflection.
Another type of border combines transverse planes of reflection with longitudinal ones and has, along with mirror, also axial symmetry. It is strictly static, balanced on all sides. Both edges and both directions of the transfer axis have the same character.
The basis of mesh ornaments (rapports) is a simple mesh. The cells of such a grid can be squares, rhombuses, rectangles, parallelograms or equilateral triangles. Depending on this, the nature of the symmetry of the grid itself changes, and therefore the ornament built on it. In addition, the symmetry of the pattern is influenced, as in borders, by the symmetry elements of the repeated motif itself.
In total, mathematicians count 17 types of symmetry of mesh patterns. Here the types of symmetry already known to us can be implemented in different combinations: rotary - second, third, fourth and sixth order, mirror, sliding reflection. And in each case, a certain set of possible reflections and rotations influences the rhythm of the pattern, creating its own measure of balance and mobility, its own directions.
If the planes of reflection, which give the usual pattern balance and stability, are turned awry, the entire pattern will begin to seem far less calm and constructive.
In the art of ornamentation, filling the plane with rectilinear identical figures is often used. This pattern gives the surface a clear rhythmic organization. Only two types of figures - various parallelograms (including rectangles, squares, rhombuses) and hexagons with pairs of parallel sides - fill the plane completely, without allowances or overlaps, maintaining the same orientation with the help of translations alone.
Symmetry of similarity occurs quite often in ornamentation. In this case, identical or similar shaped pattern elements are not equal in size. They can form increasing or decreasing rows or fill the surface with similar figures diverging from one point and increasing as they move away from it.
Ornaments built on the principle of similarity are always extremely dynamic, actively taking over the surface and creating a feeling of movement.
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The work of a student of class 9 "A" of the Municipal Educational Institution "Gymnasium No. 2" Andrey Ulturgashev. supervisor: Kolycheva Iraida Mikhailovna, mathematics teacherSlide 2
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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 embroidered strange birds, trees, strange flowers and herbs represent, what laws are used to structure the ornament itself - I tried to answer these and some other questions in my work.
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Symmetry in geometry is a property of geometric figures. Points A and A1 are called symmetrical relative to Point O (center of symmetry), if O is the middle of segment AA1. Point O is considered symmetrical to itself. Points A and A1 are called symmetrical with respect to straight line “a” (axis of symmetry) if straight line “a” passes through the middle of segment AA1 and is perpendicular to this segment. Each point of the straight line “a” is considered symmetrical to itself. Points A and A1 are called symmetrical relative to the “alpha” plane (plane of symmetry) if the “alpha” plane passes through the middle of the segment AA1 and is perpendicular to this segment. each point of the “alpha” plane is considered symmetrical to itself.
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A point (straight line, plane) is called a center (axis, plane) of symmetry of a figure if each point of the figure is symmetrical with respect to some point of the same figure. If a figure has a center (axis, plane) of symmetry, then it is said to have central (axial, mirror) symmetry. A figure may have one or more centers (axes, planes) of symmetry. Symmetry is a certain spatial order, a mathematically precise pattern in the arrangement of objects or their parts.
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In art, symmetry has become widespread as one of the types of harmonious composition. It is characteristic of works of architecture and decorative and applied art. Symmetry is a kind of measure of the beauty and harmony of the world around us. The 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.
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There are several types of symmetry. Most often we encounter mirror symmetry. It is inherent in the human body and most animals, in many objects that we use, and is widely used in art. If any object or flat figure can be divided by a plane into two halves in such a way that one half, reflected in this plane, as in a mirror, repeats the other, then they have mirror symmetry. The mirror symmetry of everyday objects is emphasized by ornaments built on the same principle. The planes of symmetry of the object and the ornament coincide.
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For example, this decorative paper cutting created by a folk craftsman. The ornament is cut out of a sheet of colored paper folded in half - the simplest way to achieve mirror symmetry.
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Another type of symmetry that is constantly found in folk art is transference or translation. Symmetrical elements can be combined with each other when moving along a straight axis over a segment of a certain length. Each of the designs of these ornaments consists of a number of identical images. We can say that these drawings are formed by repeating a certain motif or transferring it to equal segments.
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Ornament is a huge and very unique area of artistic culture. By definition, it is a pattern consisting of repeating rhythmically ordered elements. It is intended for decorating various objects (dishes, textiles, weapons, etc.), architectural structures, and works of applied art. It is associated with the surface (plane), which it decorates and visually organizes, and with its shape and color it reveals and emphasizes the features of the object, the natural beauty of the material.
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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.
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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 has developed its own system of ornamentation (motifs, forms, location on the surface to be decorated), so it is often possible to determine by the ornament to what time and to what country a particular work of art belongs . Russian ornament (along with its historical and cultural significance) is especially interesting in this sense, as it has peculiar mathematical, color and symbolic features.
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Color is essential in the ornament. Folk craftsmen were well aware of the effect of color not only on vision, but also on the psyche of people. They took into account that certain phenomena, specific moods and experiences are often associated with one color or another in the human mind. Folk craftsmen give preference to red in the color scheme of the ornament. It is with it that the idea of joy, fun, celebration and happiness is associated. It evokes in our minds associations with the sun, the source of all life on Earth.
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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 the Russian ornament, only three types of symmetry are mainly involved: 1 - this is a sequential arrangement in a line - portable symmetry
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And rotation around a common center is rotational symmetry. The number of figures in a full revolution determines the order of rotational symmetry. In the ornament on the plane there is rotational symmetry of the 2nd, 3rd and 6th orders. In accordance with the order of rotational symmetry, the figures of the ornament relative to each other are rotated by 180, 120, 90 or 60 degrees.
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It is believed that ornaments combining mirror and rotational symmetries of the 4th order induce a state of peace. Right angles, according to people’s experience, correspond to stable equilibrium: a calm surface of water and vertical tree trunks reflected in it, the angle between the floor and walls in a room, rectangular shapes of buildings.
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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 the cosmos, life, love for nature and man, and happiness. In folk art, ornamental motifs imprinted a poetic attitude towards the world.
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The following motifs are also characteristic of Russian ornament: “amulet” signs, which were applied to clothing, household items and various details of the home; "braid" motifs, which were interpreted as signs of water; - the motif of the ancient goddess Mokosha (as an idea of the Great Mother); -motif of the “tree of life”; -motives of “solar” (solar) signs. 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 the Russian ornament, is a symbol of the “raw 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.
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All openings through which evil spirits could penetrate to a person were decorated with incantatory amulet 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 furnished with magical ornaments (ravel, spinning wheel, loom).
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The ensemble of the Russian costume is completed by a headdress. For girls, this can only be a bandage that leaves the top of the head uncovered, while married women wore kokoshniks embroidered with gold thread and freshwater pearls.
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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, as well as a circle with a cross inside. The “running” sun depicts a circle with arcuate lines diverging from the center.
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The earth was depicted as a rectangle, divided by diagonals into 4 parts, with a repeating pattern in them. Here we also observe axial symmetry in combination with central symmetry. These same types of symmetry predominate in images of the plant world.
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The water element is usually represented by rows of dashes and dots, reproducing raindrops and zigzag lines (an example of figurative symmetry). This motif is typical for window frames.
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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MATHEMATICS PROJECT
ON THE TOPIC
"THIS AMAZING WORLD OF SYMMETRY"
prepared by VKK mathematics teacher
N. A. Polubavkina
The text of the work is posted without images and formulas.
The full version of the work is available in the "Work Files" tab in PDF format
Introduction 3
Chapter 2. Rules for constructing an ornamental composition. Ornament and laws of symmetry 7
Chapter 3. Research results 10
Conclusion 14
References 15
Appendix 1 16
Appendix 2 19
Appendix 3 20
Introduction
This school year we looked at the topic “Symmetry” in mathematics lessons. I was very interested in this topic. Symmetry is found in nature, 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.
Having examined my works made in art school, I discovered that most of them are subject to the laws of symmetry. Also at art school, I learned what an ornament is and decided to conduct research on how the ornamental motif of a composition in patterns can be considered from a mathematical point of view.
Hypothesis: perhaps symmetry is the main transformation of figures to create ornamental motifs.
Object of study: ornament
Subject of research: symmetry of patterns
The purpose of the work is to prove that symmetry and ornament are a single whole.
1. Study the content and form of ornaments and ornamental motifs.
2. Describe the rules for constructing an ornamental composition.
3. Explore how symmetry manifests itself in ornaments using the example of your own works, the works of your art school comrades, and objects on display at the Roslavl Historical and Art Museum.
Methods: study, comparison, description, work with information sources.
Chapter I. Content and form of ornaments and ornamental motifs
All ornamental designs, according to their visual capabilities, are divided into three types:
a figurative ornament, including a specific drawing of a person, animals, plants, landscape or architectural motifs, a drawing of inanimate objects or a complex emblem;
non-figurative ornament, formed from geometric elements, abstract forms, devoid of specific subject content;
a combined ornament, which is a combination of figurative motifs or individual elements, on the one hand, and abstract forms on the other.
The most complex non-figurative ornamental pattern, or motif, essentially consists of the simplest geometric elements of regular or irregular shape.
Let's start with a point that doesn't mean much in itself, but with skillful rhythmic repetition you can achieve an interesting decorative effect.
A line most accurately conveys the nature of the outlines of any shape. It is the main formative element. And if the number of abstract non-pictorial forms is very large, then there are only three lines that sharply differ from one another:
straight - vertical, horizontal, inclined;
curves - with a constant radius of curvature - circles and their arcs;
curves - with a variable radius of curvature - parabolas, hyperbolas and their segments.
These lines are the primary elements for all ornamental formations. All three types of lines have a certain visual expressiveness and are not similar to one another.
Straight and curved lines themselves are not carriers of artistic expression. But under certain conditions, they can “discover” the emotional charge hidden in them.
Thus, straight lines and curves with a constant radius of curvature are capable of expressing smooth, calm movement. The nature of these lines is constancy, static, balance; At the same time, it is believed that horizontal lines in the ornament symbolize harmony, severity, and a certain stability, while inclined straight lines create the impression of constant movement.
Line, or ribbon, was widely used in decoration to delimit individual motifs.
A zigzag, or broken line, served as a sign of water in Ancient Egypt. It has found application in the ornaments of many peoples at different times.
Square and rectangle are the main shapes in the ornament. They primarily serve to limit the surface filled with ornamental motifs. Rhombus, hexagon and octagon are also used to limit the surface to be decorated.
The circle is considered one of the most perfect and complete forms. It is used both as an ornamental element itself and to limit the surface, which includes other motifs.
The spiral motif became a favorite motif in the ancient art of Egypt, Assyria, and Greece. It is a symbolic image of the history of the development of the world.
Lines with a variable radius of curvature convey dynamism, tension, unevenness and active movement (increasing and decreasing). Broken lines with sharp angles—the so-called lightning-shaped lines—are particularly capable of expressing dynamism.
Any non-figurative ornamental motif consists of a combination of certain lines or closed figures. In some cases these are only straight or curved lines, in others a combination of various lines or figured elements in one motif. The rhythmic organization of elements in a motif creates conditions that stimulate the manifestation of the color-plastic properties of these elements.
Among the ornaments of the pictorial group, the most common is the floral ornament. When creating this kind of pattern, all nations were inspired by the local flora, borrowing relatively few motifs from it.
Here are the most common ones from ancient times.
The lotus in the Ancient East had a cult significance. The lotus was dedicated to the god Osiris and the goddess Isis and served as a symbol of fertility.
Palm trees, with their fan-shaped and feathery leaves, served as a symbol of peace and victory among the ancient Jews and Egyptians. The ornamental form of the plant is called palmette.
The acanthus leaf and curl, following the palmette and lotus, were especially popular among artists who were attracted by the beauty of the jagged leaves of this wild plant of southern Europe.
Hops is a cultivated plant, the picturesque appearance of which has contributed to the widespread use of plant forms in ornamentation.
Grapevine - clusters and branches were especially revered in antiquity and the Middle Ages.
Ivy is an evergreen climbing shrub, sometimes a tree; like the vine was dedicated to Bacchus.
Oak is the king of forests, a symbol of strength and power. Currently, images of oak leaves along with laurel can be found on medals and coins.
Flowers are widely used in ornamental motifs of all times and styles. They serve as decoration for fabrics, wallpaper, dishes and other types of decorative art.
Rosette - in the proper sense of the word, it is a stylized rose. In ornamentation, a rosette is an ornamental motif in the form of a blossoming flower, mostly round in shape.
Animal ornament is the most common ornament among the pictorial group, after the plant one. But fauna is used in ornamental art much less than flora. This is explained both by the properties of the material itself, which is more difficult to undergo artistic processing, and by the prohibitions of some religions on figured images.
More often than others there were images of a lion, tiger, bull, elephant, horse, vulture, eagle, dolphin, etc., which mostly had a symbolic meaning.
Chapter 2. Rules for constructing an ornamental composition. Ornament and laws of symmetry
The basis for the construction of an ornament composed of abstract or figurative motifs is the repeated repetition of these motifs according to the laws 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. Various types of symmetry are studied in special branches of mathematics.
In the science of symmetry, two types of symmetry are distinguished: finite and infinite, whose structure can be extended in one, two or three directions. The ornament uses both of these types of symmetrical structures.
Among the most common types of symmetry used in creating ornamental compositions is symmetry relative to a straight line. This is when an object or figure is divided by a straight line into two halves so that one half, when bent, coincides with the other. Symmetry relative to a straight line is inherent in the human body and the bodies of many animals. It promotes the impression of balance and peace. The ornament retains the same feeling.
Another type of symmetry is axial symmetry, in which the figures are aligned by rotating around an axis perpendicular to the image plane. The number of such combinations throughout the full circular revolution of the figure is called the order of the axis. Axial symmetry can have any order expressed by an integer - from second to infinity.
There can be an infinite number of figures with axial symmetry. They are characterized by a clear organization, when parts equal to each other are distributed around a single center (the point through which the axis of symmetry passes) evenly and in the same relation to it. In this case, all angles of rotation of the coincidence of the figure with itself must be equal, otherwise a complete coincidence will not occur. The distance from the same characteristic points of the figure to the center should also be the same.
Axial symmetry is often found in nature and is widely used in ornaments: the symmetry of a flower and its ornamental analogue, a rosette.
When a figure has a pattern based only on axial symmetry, then this pattern gives the impression of infinite mobility and expresses rotational movement in a certain direction.
Motifs with this kind of symmetry have become particularly widespread and important in ornamental art. The completeness of their form creates an image of harmonious peace. The integrity and closedness of the form allows you to organize any surface, marking its center, opposed to the periphery.
All symmetries discussed above refer to limited symmetrical structures of the final figures of the ornament. Getting to know a new type of symmetry - parallel translation - will help you understand how potentially infinite patterns are arranged.
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.
Ribbon (linear) border is one of the most common and important types of ornament. It is constantly used to delimit any surface characterized by a variety of artistic qualities. In practice, a linear 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.
In addition to transfer symmetry, a border may also have other symmetry elements. They arise when one or another type of symmetry is inherent in each individual elementary motif of the ornament. There are seven different types of border symmetry in total, and the impression from them and the artistic possibilities of each type used in the ornament turn out to be different.
The basis of mesh ornaments (rapports) is a simple mesh. The cells of such a grid can be squares, rhombuses, rectangles, parallelograms or equilateral triangles. Depending on this, the nature of the symmetry of the grid itself changes, and therefore the ornament built on it. In addition, the symmetry of the pattern is influenced, as in borders, by the symmetry elements of the repeated motif itself.
In total, mathematicians count 17 types of symmetry of mesh patterns. Here the types of symmetry already known to us can be implemented in different combinations: relative to a point, relative to a straight line, parallel translation. And in each case, a certain set of possible reflections and rotations influences the rhythm of the pattern, creating its own measure of balance and mobility, its own directions.
Chapter 3. Research results
Having studied the literature in the local history department of the Roslavl Central Library. N.I. Rylenkov on the topic of the work, I decided to explore what types of ornaments are used more often and how symmetry is manifested in the patterns of ornaments using the example of my own works, the works of my art school comrades and the items on display at the Roslavl Historical and Art Museum.
Types of ornament
Elements of ornament are present in 10 designs, floral ornament - 2, animal ornament - 2, mesh ornament - 2, graphic ornament - 8
Folk costume ornament
Elements of ornament are present in 4 items, floral ornament - 1, animal ornament - 0, mesh ornament - 3, graphic ornament - 3
Ornament of household items
Elements of ornament are present in 4 items, floral ornament - 2, animal ornament - 0, mesh ornament - 2, graphic ornament - 2
Fig. 1 Comparative diagram of research objects by type of ornament (data in%)
Conclusion: most often, graphic ornaments are used in drawings, folk costumes and household items; mesh ornaments are also popular - this is due to the fact that these types of ornaments are easier to depict than animal and plant ones.
Symmetry of the ornament pattern
1. Drawings
10 drawings were taken as objects of study (Appendix 1).
Elements of the ornament are present in 10 patterns, pattern symmetry - 6, symmetry relative to a point - 3, symmetry relative to a straight line - 5, transfer symmetry - 4.
2. Ornament of folk costume
Four objects were taken as objects of research: a craft - a doll in a folk costume, a woman’s costume, a men’s shirt and a sash from the exhibition of the Roslavl Historical and Art Museum (Appendix 2).
3. Ornament of household items
4 items were taken as objects of research: a craft - a towel, the design of which was made using the batik technique, curtains on the windows, a god (icon) and a towel from the exhibition of the Roslavl Historical and Art Museum (Appendix 3)
Elements of ornament are present in 4 objects, symmetry of the pattern - 4, symmetry around a point - 2, symmetry around a straight line - 4, symmetry of transfer - 4.
Fig.2 Comparative diagram of research objects according to the symmetry of the ornament pattern (data in%)
Conclusion: in all the details of the patterns of folk costumes and household items, as well as in most drawings, various types of symmetry are used; symmetry relative to a straight line and symmetry of parallel transfer are more common than symmetry relative to a point.
Conclusion
In this work, I examined symmetry in ornaments using examples of my own works, the works of my art school comrades, and objects on display at the Roslavl Historical and Art Museum. As a result of my research, I saw that:
The complexity of ornamental forms is subject to the laws of symmetry.
The predominant ornamental motifs are graphic and mesh patterns.
Most ornaments are characterized by such types of symmetry as symmetry relative to a straight line and transfer symmetry.
The tasks posed in this work are completely solved. I learned to work with information sources and use research methods such as study, comparison and description.
This study helped me more carefully and accurately depict ornaments and designs that use symmetry, as I thoroughly understood the techniques of displaying the details of an ornament symmetrically about a point, a straight line, and using parallel translation.
I have proven that symmetry and ornament are a single whole and I will continue my research work to study other, more complex types of symmetry used in the construction of ornamental motifs.
List of used literature
Beschastnov N.P. The artistic language of ornament. M.: Humanitarian. ed. VLADOS center, 2010. - 335 p.
Butkevich L.M. History of ornament. - M.: Humanitarian. ed. VLADOS center, 2010. - 267 p.
Fokina L.V. Ornament: tutorial. - Rostov n/d: Phoenix, 2007. - 172 p.
http://yandex.ru
http://900igr.net/kartinki/geometrija/Ornament/054-Povorot.html
http://sphinks-t.ru/page-foto-russkie-uzori
http://podelki.mybabbie.ru/vyazanie/vyazanie-geometricheskie-ornamenty
http://vector-images.com/clipart/clp740303
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 Kiev 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 by both the peculiarities of technology and 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.
