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Research work. Symmetry in the surrounding world (section of exact sciences)

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

Due to the large amount of information, the material is divided into interconnected chapters. The first chapter is the introduction you are reading now. The remaining three chapters deal respectively with rosette symmetry, one-dimensional border patterns, and two-dimensional ornamental structures. This article and subsequent chapters talk about some of the results of symmetry analysis of samples of decorative art of the Paleolithic and Neolithic. They are dedicated to the search for “decorative archetypes” - the universal basis of all decorative art. After all, the development of ornament has always gone hand in hand with the development of humanity and reflects man’s desire to understand and express patterns - the factors underlying any scientific knowledge.

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

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

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

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

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

Rice. 1. Cyclic symmetry in ornament and nature

Rice. 2. Dihedral symmetry in ornament and nature

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

Rice. 3. Antisymmetry

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

By the term "pre-scientific period" we mean the period from 25,000-10,000 BC.

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

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

List of all four chapters:

Sources

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

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.

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.

Individual designer card

Class_____ Project Manager_______________________

Project topic__________________________________________

Start date__________________________________________

Date of project defense __________________________________________

Project stages

Criteria for evaluation

Grade

Maximum

Actual

Immersion in the project

Relevance of the selected topic

Practical significance of the work

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



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