Home Small buildings "symmetry and the world around us" research work. The world around us is a world of symmetry. Many faces of symmetry in the world around us.

"symmetry and the world around us" research work. The world around us is a world of symmetry. Many faces of symmetry in the world around us.

We get used to the concept of symmetry from childhood. We know that a butterfly is symmetrical: its right and left wings are the same; a symmetrical wheel whose sectors are identical; symmetrical patterns of ornaments, stars of snowflakes.

A truly vast literature is devoted to the problem of symmetry. From textbooks and scientific monographs to works that pay attention not so much to drawings and formulas, but to artistic images.

The very term “symmetry” in Greek means “proportionality,” which ancient philosophers understood as a special case of harmony - the coordination of parts within the whole. Since ancient times, many peoples have had the idea of symmetry in the broad sense - as the equivalent of balance and harmony.

Symmetry is one of the most fundamental and one of the most general patterns of the universe: inanimate, living nature and society. We meet her everywhere. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human knowledge; it is widely used by all areas of modern science without exception. Truly symmetrical objects surround us literally on all sides; we are dealing with symmetry wherever any order is observed. It turns out that symmetry is balance, orderliness, beauty, perfection. It is diverse, omnipresent. She creates beauty and harmony. Symmetry literally permeates the entire world around us, which is why the topic I have chosen will always be relevant.

Symmetry expresses the preservation of something despite some changes or the preservation of something despite a change. Symmetry presupposes the invariability not only of the object itself, but also of any of its properties in relation to transformations performed on the object. The immutability of certain objects can be observed in relation to various operations - rotations, translations, mutual replacement of parts, reflections, etc. In this regard, different types of symmetry are distinguished. Let's look at all types in more detail.

AXIAL SYMMETRY.

Symmetry about a straight line is called axial symmetry (mirror reflection about a straight line).

If point A lies on the l axis, then it is symmetrical to itself, i.e. A coincides with A1.

In particular, if, when transforming symmetry with respect to the l axis, the figure F transforms into itself, then it is called symmetric with respect to the l axis, and the l axis is called its symmetry axis.

CENTRAL SYMMETRY.

A figure is called centrally symmetric if there is a point relative to which each point of the figure is symmetrical to some point of the same figure. Namely: movement that changes directions to opposite ones is central symmetry.

Point O is called the center of symmetry and is motionless. This transformation has no other fixed points. Examples of figures that have a center of symmetry are a parallelogram, a circle, etc.

The familiar concepts of rotation and parallel translation are used in the definition of so-called translational symmetry. Let's look at translation symmetry in more detail.

1. TURN

A transformation in which each point A of a figure (body) is rotated by the same angle α around a given center O is called rotation or rotation of the plane. Point O is called the center of rotation, and angle α is called the angle of rotation. Point O is a fixed point of this transformation.

The rotational symmetry of the circular cylinder is interesting. It has an infinite number of 2nd order rotary axes and one infinitely high order rotary axis.

2. PARALLEL TRANSFER

A transformation in which each point of a figure (body) moves in the same direction by the same distance is called parallel translation.

To specify a parallel translation transformation, it is enough to specify the vector a.

3. SLIDING SYMMETRY

Sliding symmetry is a transformation in which axial symmetry and parallel translation are performed sequentially. Sliding symmetry is an isometry of the Euclidean plane. Gliding symmetry is the composition of symmetry with respect to some line l and translation to a vector parallel to l (this vector may be zero).

Sliding symmetry can be represented as a composition of 3 axial symmetries (Chales' theorem).

MIRROR SYMMETRY

What could be more like my hand or my ear than their own reflection in the mirror? And yet the hand that I see in the mirror cannot be put in the place of the real hand.

Immanuel Kant.

If a transformation of symmetry relative to a plane transforms a figure (body) into itself, then the figure is called symmetrical relative to the plane, and this plane is called the plane of symmetry of this figure. This symmetry is called mirror symmetry. As the name itself suggests, mirror symmetry connects an object and its reflection in a plane mirror. Two symmetrical bodies cannot be “nested into each other”, since in comparison with the object itself, its mirror double turns out to be turned out along the direction perpendicular to the plane of the mirror.

Symmetrical figures, for all their similarities, differ significantly from each other. The double observed in the mirror is not an exact copy of the object itself. The mirror does not simply copy the object, but swaps (represents) the front and rear parts of the object in relation to the mirror. For example, if your mole is on your right cheek, then your looking-glass double’s is on your left. Hold a book up to the mirror and you will see that the letters seem to be turned inside out. Everything in the mirror is rearranged from right to left.

Bodies are called mirror-equal bodies if, with proper displacement, they can form two halves of a mirror-symmetrical body.

2. 2 Symmetry in nature

A figure has symmetry if there is a movement (non-identical transformation) that transforms it into itself. For example, a figure has rotational symmetry if it is translated into itself by some rotation. But in nature, with the help of mathematics, beauty is not created, as in technology and art, but is only recorded and expressed. It not only pleases the eye and inspires poets of all times and peoples, but allows living organisms to better adapt to their environment and simply survive.

The structure of any living form is based on the principle of symmetry. From direct observation we can deduce the laws of geometry and feel their incomparable perfection. This order, which is a natural necessity, since nothing in nature serves purely decorative purposes, helps us find the general harmony on which the entire universe is based.

We see that nature designs any living organism according to a certain geometric pattern, and the laws of the universe have a clear justification.

The principles of symmetry underlie the theory of relativity, quantum mechanics, solid state physics, atomic and nuclear physics, and particle physics. These principles are most clearly expressed in the invariance properties of the laws of nature. We are talking not only about physical laws, but also others, for example, biological ones.

Speaking about the role of symmetry in the process of scientific knowledge, we should especially highlight the use of the method of analogies. According to the French mathematician D. Polya, “there are, perhaps, no discoveries either in elementary or higher mathematics, or, perhaps, in any other field that could be made without analogies.” Most of these analogies are based on common roots, general patterns that appear in the same way at different levels of the hierarchy.

So, in the modern understanding, symmetry is a general scientific philosophical category that characterizes the structure of the organization of systems. The most important property of symmetry is the conservation (invariance) of certain features (geometric, physical, biological, etc.) in relation to well-defined transformations. The mathematical apparatus for studying symmetry today is the theory of groups and the theory of invariants.

Symmetry in the plant world

The specific structure of plants is determined by the characteristics of the habitat to which they adapt. Any tree has a base and a top, a “top” and a “bottom” that perform different functions. The significance of the difference between the upper and lower parts, as well as the direction of gravity, determine the vertical orientation of the rotary axis of the “wood cone” and the planes of symmetry. A tree, with the help of its root system, absorbs moisture and nutrients from the soil, that is, from below, and the remaining vital functions are performed by the crown, that is, above. At the same time, directions in a plane perpendicular to the vertical are virtually indistinguishable for a tree; in all these directions, air, light, and moisture enter the tree equally.

The tree has a vertical rotary axis (cone axis) and vertical planes of symmetry.

When we want to draw a leaf of a plant or a butterfly, we have to take into account their axial symmetry. The midrib for the leaf serves as an axis of symmetry. Leaves, branches, flowers, and fruits have pronounced symmetry. The leaves are characterized by mirror symmetry. The same symmetry is also found in flowers, but in them mirror symmetry often appears in combination with rotational symmetry. There are also frequent cases of figurative symmetry (acacia branches, rowan trees).

In the diverse world of colors, there are rotary axes of different orders. However, the most common is 5th order rotational symmetry. This symmetry is found in many wildflowers (bell, forget-me-not, geranium, carnation, St. John's wort, cinquefoil), in the flowers of fruit trees (cherry, apple, pear, tangerine, etc.), in the flowers of fruit and berry plants (strawberries, raspberries, viburnum , bird cherry, rowan, rose hip, hawthorn), etc.

Academician N. Belov explains this fact by the fact that the 5th order axis is a kind of instrument of the struggle for existence, “insurance against petrification, crystallization, the first step of which would be their capture by the grid.” Indeed, a living organism does not have a crystalline structure in the sense that even its individual organs do not have a spatial lattice. However, ordered structures are represented very widely in it.

In his book “This Right, Left World,” M. Gardner writes: “On Earth, life originated in spherically symmetrical forms, and then began to develop along two main lines: the world of plants with cone symmetry was formed, and the world of animals with bilateral symmetry.”

In nature, there are bodies that have helical symmetry, that is, alignment with their original position after rotation by an angle around an axis, with an additional shift along the same axis.

If is a rational number, then the rotary axis also turns out to be the translation axis.

The leaves on the stem are not arranged in a straight line, but surround the branch in a spiral. The sum of all previous steps of the spiral, starting from the top, is equal to the value of the subsequent step A+B=C, B+C=D, etc.

Helical symmetry is observed in the arrangement of leaves on the stems of most plants. Arranging as a screw along the stem, the leaves seem to spread out in all directions and do not block each other from the light, which is extremely necessary for plant life. This interesting botanical phenomenon is called phyllotaxis (literally “leaf arrangement”).

Another manifestation of phyllotaxis is the structure of the inflorescence of a sunflower or the scales of a fir cone, in which the scales are arranged in the form of spirals and helical lines. This arrangement is especially clear in the pineapple, which has more or less hexagonal cells that form rows running in different directions.

Symmetry in the animal world

The significance of the form of symmetry for an animal is easy to understand if it is connected with the way of life and environmental conditions. Symmetry in animals means correspondence in size, shape and outline, as well as the relative arrangement of body parts located on opposite sides of the dividing line.

Rotational symmetry of the 5th order is also found in the animal world. This is a symmetry in which an object aligns with itself when rotated around a rotary axis 5 times. Examples include the starfish and sea urchin shell. The entire skin of starfish is encrusted with small plates of calcium carbonate; needles extend from some of the plates, some of which are movable. An ordinary starfish has 5 planes of symmetry and 1 axis of rotation of the 5th order (this is the highest symmetry among animals). Her ancestors appear to have had lower symmetry. This is evidenced, in particular, by the structure of the star larvae: they, like most living beings, including humans, have only one plane of symmetry. Starfish do not have a horizontal plane of symmetry: they have a “top” and a “bottom.” Sea urchins are like living pincushions; their spherical body bears long and movable needles. In these animals, the calcareous plates of the skin merged and formed a spherical carapace. There is a mouth in the center of the lower surface. The ambulacral legs (water-vascular system) are collected in 5 stripes on the surface of the shell.

However, unlike the plant world, rotational symmetry is rarely observed in the animal world.

Insects, fish, eggs, and animals are characterized by a difference between the directions “forward” and “backward” that is incompatible with rotational symmetry.

The direction of movement is a fundamentally selected direction, with respect to which there is no symmetry in any insect, any bird or fish, any animal. In this direction the animal rushes for food, in the same direction it escapes from its pursuers.

In addition to the direction of movement, the symmetry of living beings is determined by another direction - the direction of gravity. Both directions are significant; they define the plane of symmetry of an animal being.

Bilateral (mirror) symmetry is the characteristic symmetry of all representatives of the animal world. This symmetry is clearly visible in the butterfly. The symmetry of the left and right wings appears here with almost mathematical rigor.

We can say that every animal (as well as insects, fish, birds) consists of two enantiomorphs - the right and left halves. Enantiomorphs are also paired parts, one of which falls into the right and the other into the left half of the animal’s body. Thus, enantiomorphs are the right and left ear, right and left eye, right and left horn, etc.

Simplification of living conditions can lead to a violation of bilateral symmetry, and animals from being bilaterally symmetrical become radially symmetrical. This applies to echinoderms (starfish, sea urchins, crinoids). All marine animals have radial symmetry, in which parts of the body radiate away from a central axis, like the spokes of a wheel. The degree of activity of animals correlates with their type of symmetry. Radially symmetrical echinoderms are usually poorly mobile, move slowly, or are attached to the seabed. The body of a starfish consists of a central disk and 5-20 or more rays radiating from it. In mathematical language, this symmetry is called rotational symmetry.

Let us finally note the mirror 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. Let’s not figure out for now whether an absolutely symmetrical person actually exists. Everyone, of course, will have a mole, a strand of hair or some other detail that breaks the external symmetry. The left eye is never exactly the same as the right, and the corners of the mouth are at different heights, at least for most people. Yet these are only minor inconsistencies. No one will doubt that outwardly a person is built symmetrically: the left hand always corresponds to the right and both hands are exactly the same.

Everyone knows that the similarity between our hands, ears, eyes and other parts of the body is the same as between an object and its reflection in a mirror. It is the issues of symmetry and mirror reflection that are given attention here.

Many artists paid close attention to the symmetry and proportions of the human body, at least as long as they were guided by the desire to follow nature as closely as possible in their works.

In modern schools of painting, the vertical size of the head is most often taken as a single measure. With a certain assumption, we can assume that the length of the body is eight times the size of the head. The size of the head is proportional not only to the length of the body, but also to the size of other parts of the body. All people are built on this principle, which is why we are, in general, similar to each other. However, our proportions are only approximately consistent, and therefore people are only similar, but not the same. In any case, we are all symmetrical! In addition, some artists especially emphasize this symmetry in their works.

Our own mirror symmetry is very convenient for us, it allows us to move straight and turn left and right with equal ease. Mirror symmetry is equally convenient for birds, fish and other actively moving creatures.

Bilateral symmetry means that one side of an animal's body is a mirror image of the other side. This type of organization is characteristic of most invertebrates, especially annelids and arthropods - crustaceans, arachnids, insects, butterflies; for vertebrates - fish, birds, mammals. For the first time, bilateral symmetry appears in flatworms, in which the anterior and posterior ends of the body differ from each other.

Let's consider another type of symmetry that is found in the animal world. This is helical or spiral symmetry. Helical symmetry is symmetry with respect to the combination of two transformations - rotation and translation along the axis of rotation, i.e. there is movement along the axis of the screw and around the axis of the screw.

Examples of natural propellers are: tusk of a narwhal (a small cetacean that lives in the northern seas) - left propeller; snail shell – right screw; The horns of the Pamir ram are enantiomorphs (one horn is twisted in a left-handed spiral, and the other in a right-handed spiral). Spiral symmetry is not ideal, for example, the shell of mollusks narrows or widens at the end. Although external helical symmetry is rare in multicellular animals, many important molecules from which living organisms are built - proteins, deoxyribonucleic acids - DNA have a helical structure.

Symmetry in inanimate nature

Crystal symmetry is the property of crystals to align with themselves in various positions by rotation, reflection, parallel translation, or part or combination of these operations. The symmetry of the external shape (cut) of a crystal is determined by the symmetry of its atomic structure, which also determines the symmetry of the physical properties of the crystal.

Let's take a closer look at the multifaceted shapes of crystals. First of all, it is clear that crystals of different substances differ from each other in their shapes. Rock salt is always cubes; rock crystal - always hexagonal prisms, sometimes with heads in the form of trihedral or hexagonal pyramids; diamond - most often regular octahedrons (octahedrons); ice is hexagonal prisms, very similar to rock crystal, and snowflakes are always six-pointed stars. What catches your eye when you look at crystals? First of all, their symmetry.

Many people think that crystals are beautiful, rare stones. They come in different colors, are usually transparent and, best of all, have a beautiful, regular shape. Most often, crystals are polyhedra, their sides (faces) are perfectly flat, and their edges are strictly straight. They delight the eye with the wonderful play of light in their edges and the amazing correctness of their structure.

However, crystals are not museum rarities at all. Crystals surround us everywhere. The solids from which we build houses and machines, the substances that we use in everyday life - almost all of them belong to crystals. Why don't we see this? The fact is that in nature one rarely comes across bodies in the form of separate single crystals (or, as they say, single crystals). Most often, the substance is found in the form of tightly adhered crystalline grains of a very small size - less than a thousandth of a millimeter. This structure can only be seen through a microscope.

Bodies consisting of crystalline grains are called finely crystalline, or polycrystalline (“poly” - in Greek “many”).

Of course, finely crystalline bodies should also be classified as crystals. Then it will turn out that almost all the solid bodies around us are crystals. Sand and granite, copper and iron, paints - all these are crystals.

There are exceptions; glass and plastics do not consist of crystals. Such solids are called amorphous.

Studying crystals means studying almost all the bodies around us. It's clear how important this is.

Single crystals are immediately recognizable by their regular shape. Flat faces and straight edges are a characteristic property of the crystal; the correctness of the form is undoubtedly related to the correctness of the internal structure of the crystal. If a crystal is especially elongated in a certain direction, it means that the structure of the crystal in that direction is somehow special.

There is a center of symmetry in a cube of rock salt, in the octahedron of a diamond, and in the star of a snowflake. But in a quartz crystal there is no center of symmetry.

The most accurate symmetry is achieved in the world of crystals, but even here it is not ideal: cracks and scratches invisible to the eye always make equal faces slightly different from each other.

All crystals are symmetrical. This means that in each crystalline polyhedron one can find planes of symmetry, axes of symmetry, a center of symmetry or other symmetry elements so that identical parts of the polyhedron are aligned with each other.

All elements of symmetry repeat the same parts of the figure, all give it symmetrical beauty and completeness, but the center of symmetry is the most interesting. Not only the shape, but also many physical properties of the crystal can depend on whether a crystal has a center of symmetry or not.

Honeycombs are a real design masterpiece. They consist of a number of hexagonal cells. This is the densest packaging, allowing the most advantageous placement of the larva in the cell and, with the maximum possible volume, the most economical use of the building material - wax.

III Conclusion

Symmetry permeates literally everything around, capturing seemingly completely unexpected areas and objects. It, manifesting itself in the most diverse objects of the material world, undoubtedly reflects its most general, most fundamental properties. The principles of symmetry play an important role in physics and mathematics, chemistry and biology, technology and architecture, painting and sculpture, poetry and music.

We see that nature designs any living organism according to a certain geometric pattern, and the laws of the universe have a clear justification. Therefore, the study of the symmetry of various natural objects and the comparison of its results is a convenient and reliable tool for understanding the basic laws of the existence of matter.

The laws of nature that govern the inexhaustible picture of phenomena in their diversity, in turn, are subject to the principles of symmetry. There are many types of symmetry, both in the plant and animal world, but with all the diversity of living organisms, the principle of symmetry always operates, and this fact once again emphasizes the harmony of our world. Symmetry underlies things and phenomena, expressing something common, characteristic of different objects, while asymmetry is associated with the individual embodiment of this common thing in a specific object.

So, on the plane we have four types of movements that transform figure F into an equal figure F1:

1) parallel transfer;

2) axial symmetry (reflection from a straight line);

3) rotation around a point (Partial case - central symmetry);

4) “sliding” reflection.

In space, mirror symmetry is added to the above types of symmetry.

I believe that the goal set in the abstract has been achieved. When writing my essay, the greatest difficulty for me was drawing my own conclusions. I think that my work will help schoolchildren expand their understanding of symmetry. I hope that my essay will be included in the methodological fund of the mathematics classroom.

Study the topic “Symmetry”
Explore the question “Symmetry in the world around us”
Consider different types of symmetry in natural objects

Why does a person need to know about symmetry?

1. Reveal the meaning of the basic concepts of symmetry.
2. Show that nature is a world of symmetry.

study of literature;
comparison of essential features;
analysis, comparison, generalization.

O symmetry!
I sing your anthem!
I recognize you everywhere in the world.
You are in the Eiffel Tower, in a small midge,
Are you in Christmas tree near the forest path.
With you in friendship and tulip and rose,
And the snow swarm is a creation of frost!

The topic of my scientific research work is “Many-faced symmetry”.
I chose this topic because we encounter symmetry everywhere - in nature, architecture, art, science. I would like to become more deeply acquainted with symmetry in mathematics and biology, technology and architecture since the concept of symmetry is widely used by all areas of modern science.

What is it symmetry ?
What deep meaning lies in this concept?
Why does symmetry literally permeate the entire world around us?

Symmetry (from the Greek symmetria - “proportionality”) - a concept meaning persistence, repeatability, “invariance” of any structural features of the object being studied when certain transformations are carried out with it .

Symmetry - this is balance,

orderliness,

beauty,

perfection.

a) symmetry about a point (central symmetry); b) symmetry relative to a straight line (axial symmetry);
c) symmetry relative to the plane (mirror symmetry);
G) Rotation symmetry (turn)
d) Sliding symmetry

OA 1 = OA

Definition

Points A and A 1 are called symmetrical about the point ABOUT, if O is the middle of the segment AA 1.

Definition

The figure is called symmetrical about the center

Symmetry of points relative to a straight line

Definition

Two points A and A 1 are called symmetrical relative to straight line a , if this line passes through the middle of segment AA 1 and is perpendicular to it.

Symmetrical figure relative to a straight line

Definition

The figure is called symmetrical relative to straight , if for each point of a figure the point symmetrical to it also belongs to this figure. Straight l called the axis of symmetry of the figure.

Transformation in which each point A of the figure (body) is rotated by the same angle α around a given center O is called rotation or rotation of the plane. Point O is called the center of rotation, and angle α is called the angle of rotation. Point O is a fixed point of this transformation.

Central symmetry is a rotation of a figure by 180°.

Sliding symmetry is a transformation in which axial symmetry and parallel translation are performed sequentially.

a segment goes into an equal segment;
the angle goes into an angle equal to it;
the circle turns into an equal circle;
any polygon goes into an equal polygon, etc.
parallel lines turn into parallel, perpendicular into perpendicular.

So, on the plane we have four types of movements that translate the figure F into an equal figure F 1 :

parallel transfer;
axial symmetry (reflection from a straight line);
rotation around a point (partial case - central symmetry);
"sliding" reflection.

RADIAL SYMMETRY

(radial symmetry) - symmetry with respect to any planes passing through the longitudinal axis of the animal’s body.

Bilateral symmetry (bilateral symmetry) - mirror reflection symmetry, in which an object has one plane of symmetry, relative to which its two halves are mirror symmetrical.

Symmetry has many faces.

It is associated with orderliness, proportionality and proportionality of parts, beauty and harmony, with expediency and usefulness.

While working on the project, I touched the mysterious mathematical beauty. Mathematics is a language, the language of nature. Without knowing the language, you cannot understand the beauty of the world around you.

But one thing is certain: The world is symmetrical!

1. This amazingly symmetrical world” - L. Tarasov
2. “Explanatory Dictionary” - V. Dalya
“Geometry 7-9 grades” - L. Atanasyan
Malakhov V.V. // Journal. total biology. 1977. T.38.
I.G. Zenkevich “Aesthetics of a mathematics lesson.”
http://900igr.net/fotografii/geometrija/Simmetrija/O-simmetrii.html

Municipal educational institution

"Secondary school in the village of Storozhevka"

Tatishchevsky district, Saratov region

Design and research work

on topic:

Completed by: 11th grade students

"Municipal secondary school in the village of Storozhevka"

Davydova Katerina Olegovna,

Oreshenkova Daria Olegovna.

Head: mathematics teacher

Zhogal Marina Alexandrovna

2011

Content

I. Brief summary……………………………………………………………..3

II. Introduction……………………………………………………4

III. This amazingly symmetrical world………………………......5

1.What is symmetry? The place of symmetry in the surrounding world…..5

2. Types of symmetry……………………………………………………..8

3. Symmetry in physics and technology…………………………………….10

4.Symmetry in nature…………………………………………….14

In the plant world -

In the animal world

5.Symmetry in creativity…………………………………………………………….18

In architecture

In literature

In fine arts

In music and dance

6.Symmetry nearby……………………………………………………22

Symmetry in clothing

Symmetry in everyday life (at home, at school)

Symmetry of the village of Storozhevka and the city of Saratov

IV. Conclusion……………………………………………………….24

V. Literature…………………………………………………………….25

VI.Appendix………………………………………………………..26

Brief summary of the project

This project is designed for students in grades 9-11. It covers the study of educational topics: “Symmetry” in geometry, “Cities and countries”, “Transport”, “Architecture” in geography, “Structural features of plant and animal organisms” in biology, literature, “laws of conservation” in physics. This project creates awareness of the need to live in peace and harmony with nature, develops observation and creative abilities.

When conducting a project, the teacher helps students develop their critical thinking skills, the ability to find and process a large amount of information, develop communication skills, and organize independent research on the educational topic.

Introduction

Mathematics is inexhaustible and multi-valued.

Not a single mathematician, even the very best, is able to study all of mathematics, but chooses only some branch. So today we are choosing a small branch of symmetry.

Mathematicians and biologists, crystallographers and art historians, engineers and philosophers, astronomers and plant breeders, physicists and doctors are trying together to solve the mysteries of symmetry.

In the school mathematics course, the topic “Symmetry” is devoted to only a few hours. In grade 8, students are introduced to axial and central symmetry; in grade 10, the concept of mirror symmetry is introduced. The guys have a question: why is this topic needed and where is it used?

The project “This Amazingly Symmetrical World” is designed to expand students’ knowledge on the topic “Symmetry” in various fields of science, technology, living and inanimate nature, and in the world around us.

Fundamental question:

How does symmetry manifest itself in the world around us?

Goal: studying the concept of symmetry, conducting research work to study the phenomena of symmetry in nature, architecture, technology, in the everyday reality around us, acquiring skills for independent work with a large amount of information.

Tasks:

Deepen and expand knowledge on the topic “Symmetry”;

Learn about the types of symmetry and be able to distinguish one type from another;

Get a visual representation of the manifestation of symmetry in nature, various fields of science and human activity;

Develop teamwork and decision-making skills

III. This amazingly symmetrical world

§1. What is symmetry? The place of symmetry in the surrounding world.

“Symmetry is the idea through which man throughout the ages has tried to comprehend and create order, beauty and perfection.”

G. Weil.

We encounter symmetry everywhere - in nature, technology, art, science, for example, the symmetry of the shapes of a car and an airplane, symmetry in the rhythmic structure of a poem and a musical phrase, the symmetry of ornaments and borders, the symmetry of the atomic structure of molecules and crystals. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human knowledge; it is widely used by all areas of modern science without exception. The principles of symmetry play an important role in physics and mathematics, chemistry and biology, technology and architecture, painting and sculpture, poetry and music.

The laws of nature that govern the inexhaustible picture of phenomena in their diversity, in turn, are subject to the principles of symmetry.

What is symmetry? Why does symmetry literally permeate the entire world around us? What kind of symmetry is there? What types of symmetry do you already know (axial and central, mirror). Symmetry is divided into two groups.

The first group includes symmetry of positions, shapes, structures. This is the symmetry that can be directly seen. She may be called geometric symmetry.

The second group characterizes the symmetry of physical phenomena and laws of nature. This symmetry lies at the very basis of the natural scientific picture of the world: it can be called physical symmetry. Over thousands of years, in the course of social practice and knowledge of the laws of objective reality, humanity has accumulated numerous data indicating the presence of two tendencies in the world around us: on the one hand, towards strict orderliness and harmony, and on the other, towards their violation.

To do this, let's turn to the definition of symmetry. The term “symmetry” in Greek means proportionality, proportionality, uniformity in the arrangement of parts.

According to Weil, an object is called symmetrical if it is possible to perform some operation on it, resulting in its original state. People have long paid attention to the correct shape of crystals, flowers, honeycombs and other natural objects and reproduced this proportionality in works of art, in the objects they created, through the concept of symmetry. “Symmetry,” writes the famous scientist J. Newman, “establishes a funny and surprising relationship between objects, phenomena and theories that outwardly seem to be unrelated to anything: terrestrial magnetism, the female veil, polarized light, natural selection, group theory, invariants and transformations, the working habits of bees in a hive, the structure of space, vase designs, quantum physics, flower petals, the interference pattern of x-rays, sea urchin cell division, equilibrium configurations of crystals, Romanesque cathedrals, snowflakes, music, the theory of relativity..."

The word "symmetry" has two meanings.

In one sense, symmetrical means something very proportionate, balanced; symmetry shows the way many parts are coordinated, with the help of which they are combined into a whole. The second meaning of this word is balance. Aristotle also spoke about symmetry as a state that is characterized by the relationship of extremes. From this statement it follows that Aristotle, perhaps, was closest to the discovery of one of the most fundamental laws of Nature - the law of duality. Pythagoras and his students paid close attention to symmetry. Based on the doctrine of number, the Pythagoreans gave the first mathematical interpretation of harmony, symmetry, which has not lost its meaning to this day.

Science came to the most interesting results precisely when the facts of symmetry violation were established. The consequences arising from the principle of symmetry were intensively developed by physicists in the last century and led to a number of important results. Such consequences of the laws of symmetry are, first of all, the conservation laws of classical physics.

Animals are symmetrical, plants are quite symmetrical, crystals are completely symmetrical, our spherical planet is almost perfectly symmetrical, its trajectory is close to symmetry. After what has been said, perhaps the statement that all laws of nature are determined by the symmetry of the world will not seem so fantastic. (Appendix Fig. 1)

So we live in a fairly symmetrical world. It is not surprising that we ourselves are symmetrical and tend to consider everything symmetrical beautiful.

§2.Types of symmetry

Types of symmetry:

ROTARY SYMMETRY. An object is said to have rotational symmetry if it aligns with itself when rotated through an angle of 2π/n, where n=2,3,4, etc. The axis of symmetry is called the axis of symmetry of the nth order. (Figure 2)

TRANSPORTABLE (TRANSLATIONAL) SYMMETRY. Such symmetry is spoken of when, when moving a figure along a straight line to some distance a, or a distance that is a multiple of this value, it coincides with itself. The straight line along which the transfer occurs is called the transfer axis, the distance a is called the elementary transfer or period.

Associated with this type of symmetry is the concept of periodic structures or lattices, which can be both flat and spatial. (Figure 3)

MIRROR SYMMETRY. An object consisting of two halves that are twins in relation to each other is considered mirror symmetrical. A three-dimensional object transforms into itself when reflected in a mirror plane, which is called the plane of symmetry. (Figure 4)

The shape of all objects that move on or near the surface of the Earth - walking, swimming, flying, rolling - have a plane of symmetry.

Everything that develops or moves only in the vertical direction is characterized by cone symmetry, that is, it has many planes of symmetry intersecting along the vertical axis. Both are explained by the action of gravity.

SYMMETRY OF SIMILARITY are peculiar analogues of previous symmetries with the only difference that they are associated with a simultaneous decrease or increase in similar parts of the figure and the distances between them.

The simplest example of such symmetry are nesting dolls. (Figure 5)

SWITCHING SYMMETRY, which consists in the fact that if identical particles are swapped, then no changes occur.

HEREDITY is also a certain symmetry. (Figure 7)

GAUGE SYMMETRIES involve changes in scale.

The layout is a reduced copy of the original (Fig. 8)

CONFORMAL symmetry (circular) symmetry is a transformation relative to a sphere with a center at point O of radius R, which transforms any point P to a point lying on the extension of the radius passing through point P at a distance from the center = R2/OR. Conformal symmetry has great generality. Mirror reflections, rotations, and parallel shifts are only special cases of conformal symmetry.

(Figure 9a,b)

§3.Symmetry in physics and technology.

In physics.

There is an old parable about Buridan's donkey. One philosopher, named Buridan, had a donkey. Once, while leaving for a long time, the philosopher placed two completely identical armfuls of hay in front of the donkey - one on the left and the other on the right. The donkey could not decide which armful to start with, and died of hunger... Left and right are so much the same that it is impossible to give preference to either one or the other. In other words, in both cases we are dealing with symmetry, manifested in complete equality, balance between left and right.

In fact, if the ball is motionless on the table, then the table is level and the slope on the left is the same as on the right. If the current does not flow through the wire, then there is no potential difference. If a cloud has frozen in the sky, it means the pressure around is the same and the wind has died down. It would be strange if everything happened the other way around. Nature never gives preference in equality.

Symmetry is equality in the broad sense of the word. For example, mirror symmetry means that the right side is exactly equal to the left. This means that if there is symmetry, then something will not happen and, therefore, something will definitely remain unchanged, preserved.

In nature, as in people, there are two types of laws. One type says what should happen under certain circumstances. For example, Ohm's law states that at such and such a voltage and such and such a resistance of a conductor, the strength of the electric current passing through it will be equal to the quotient of the first divided by the second. There is only one answer. The second type of laws are the so-called conservation laws. They describe what should not happen. For example, the law of conservation of matter and energy states that during any process these quantities must be conserved.

In 1915, the German mathematician Amy Noether proved purely mathematically that all conservation laws are related to the symmetries of nature. The law of conservation of momentum rests on the equality of space (homogeneity of space). On the equality of directions (isotropy of space) - the laws of conservation of angular momentum. On the equality of time - the law of conservation of matter and energy. This was an outstanding discovery.

There are a huge number of laws in physics and they are all imbued with several general principles that are contained in each law. Examples of such principles are some properties of symmetry. One of the most important properties of the symmetry of physical laws is constancy in time; the law of universal gravitation formulated by Newton describes the fact of the mutual attraction of bodies that does not change in time. This attraction existed before Newton, and it will exist in subsequent centuries. The ideal gas law is widely used in modern science and technology. If physical laws changed over time, then each physical study would have a “momentary” significance. An important conservation law in physics is the law of conservation of momentum of a closed system.

Everything that is symmetrical in nature is considered to be a reflection of the fundamental qualities of the world, and everything that is asymmetrical is considered to be a game of chance.

Speaking about symmetry in inanimate nature, a point of view arises that symmetry in inanimate nature is by no means a frequent visitor. For example, a pile of stones, an irregular line of hills on the horizon. Of course, a pile of stones is a mess, but every stone is made of crystals. And crystals bring the charm of symmetry to the world of inanimate nature. Crystals of any substance can have very different shapes, but the angles between the faces are always constant. For each given substance there is its own ideal form of its crystal, unique to it. 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)

Remember snowflakes. These are small crystals of frozen water. They have rotational and mirror (axial, central) symmetry. Why are snowflakes hexagonal? Why there are no pentagonal snowflakes; (honeycomb, pomegranate seeds).

Each snowflake is a small crystal of frozen water. The shape of snowflakes can be very diverse, but they all have symmetry (Figure 2)

All solids are made of crystals.

In technology

Symmetry can be observed in technology, in everyday life, and in our surrounding life. Why is symmetry used in technology?

Technical objects - airplanes, cars, rockets, hammers, nuts - almost all of them, from small to large, have some form of symmetry. Is this a coincidence? In technology, the beauty and proportionality of mechanisms is often associated with their reliability and stability in operation (Fig. 10 a, b, c)

Symmetrical shape of an airship, airplane, submarine, car, etc. provides good streamlining by air or water, and therefore minimal resistance to movement.

At the dawn of the development of aviation, our famous scientists N. E. Zhukovsky and S. A. Chaplygin studied the flight of birds in order to draw conclusions regarding the most advantageous shape of the wing and its flight conditions. (Appendix Fig. 11a, b)

Of course, symmetry played a big role in this.

Looking at vehicles, the question arises: What explains the frequent presence of symmetry in technology? Having studied the necessary literature, you understand that symmetry, first of all, is determined by expediency. Nobody needs a crooked car or a plane with wings of different lengths. Plus, symmetrical objects are beautiful.

Types of symmetry in technology:

-Axial

-Central

-Rotatable

-Mirror

§4. Symmetry in nature

Symmetry permeates the entire world around us.

Currently, in natural science, definitions of the categories of symmetry and asymmetry based on the enumeration of certain characteristics prevail. For example, symmetry is defined as a set of properties: order, uniformity, proportionality, harmony. All signs of symmetry in many of its definitions are considered equal, equally significant, and in certain specific cases, when establishing the symmetry of a phenomenon, any of them can be used. So, in some cases symmetry is homogeneity, in others it is proportionality, etc.

The issue of the emergence of life on Earth is closely related to the issues of mirror symmetry - asymmetry - after all, living matter arose at one time from non-living matter. This is due to the violation of the previously existing mirror symmetry, the formation of pure molecules, i.e. mirror symmetrical. Modern science has come to the conclusion that the transition from the mirror world of symmetrical connections to the pure world did not occur in the process of long evolution, but in a leap in the form of a kind of big biological explosion.

So, we owe our life on Earth to the violation of mirror symmetry and the formation of asymmetric molecules.

Symmetry is found everywhere in living nature. (Figure 12)

Symmetry also appears in natural phenomena:

Seasons;

In flowering plants;

In the appearance of snow there is a relative time shift of 12 months,

Symmetry is present in the regularity of day and night;

Thunderclaps repeat at a certain time interval.

In the plant world .

“On Earth, life originated in spherically symmetrical forms, and then began to develop along two main lines: the world of plants with cone symmetry was formed, and the world of animals with bilateral symmetry”

M. Gardner

The term “mirror” is used in geometry and physics, and “bilateral” is used in biology.

Flowers are characterized by rotational symmetry.

The following plants have rotational symmetry: hawthorn twig, St. John's wort flower, acacia twig, and cinquefoil. (Fig. 13 a, b, c)

An acacia branch has mirror and rotational symmetry. (Fig. 14) A hawthorn branch has a sliding axis of symmetry. The cinquefoil has rotational and mirror symmetry.

Taking a closer look at the plants, you can find numerous manifestations of helical symmetry in the arrangement of leaves on the stem, branches on the trunk, and in the structure of the cones. Climbing plants are pronounced screws. (Fig. 15a, b, c)

In the world of flowers, there are rotary axes of symmetry of different orders. The most common rotational symmetry is 5th order.

“The fivefold axis is a kind of instrument in the struggle for existence, insurance against petrification, against crystallization...”

(N.V. Belov)

Rotational symmetry of the 5th order is found in: bell, meadow geranium, forget-me-not, St. John's wort, cherry, pear, rowan, hawthorn, rose hip. (Fig. 16 a, b, c)

The symmetry of a cone is visible in virtually any tree. A tree, with the help of its root system, absorbs moisture and nutrients from the soil, that is, from below, and the remaining vital functions are performed by the crown, that is, from above. (Fig. 17a,b)

Radial symmetry. Look closely and you will see that the petals of many flowers radiate in all directions, like rays from a light source. In mathematics it is symmetry about a point, in biology it is radial symmetry. (Fig. 18a, b)

A person passes on his hereditary characteristics from generation to generation. Also, plants passing from one generation to another, the preservation of certain properties is observed. This is how a new sunflower (sunflower) grows from a seed with the same huge inflorescence-basket, and also regularly turns towards the Sun. This is also symmetry, it is usually called heredity.

In the plant world there are bilateral (mirror), radial, rotational, cone symmetry, axial, central, hereditary symmetry, helical symmetry.

Symmetry in the animal world .

“What could be more like my hand or my ear than their own reflections in the mirror? And the hand that I see in the mirror cannot be put in the place of a real hand...”

I. Kant

If you mentally draw a vertical line dividing a human figure in half, then the left and right sides will also turn into parts of a symmetrical “composition.” (Fig. 19a, b)

The shape of all objects that move on or near the earth's surface - walking, swimming, flying, rolling - usually has one more or less well-defined plane of symmetry.

Another interesting manifestation of the symmetry of life processes is biological rhythms, cyclical fluctuations in biological processes and their characteristics (heart contractions, respiration, fluctuations in the intensity of cell division, metabolism, motor activity, number of plants and animals), often associated with the adaptation of organisms to geophysical cycles.

The question of beauty associated with symmetry is obvious. Looking at proportionate, mutually balanced, naturally repeating parts of a symmetrical object, we feel peace, order, and stability. And as a result, we perceive the object as beautiful. On the contrary, an accidental deviation from symmetry (a collapsing corner of a building, a torn piece of a letter, snow falling unusually early) is perceived negatively, as an unexpected effect that threatens our confidence.

Let's try to imagine a world that is completely symmetrical. Such a world would have to combine with itself at any turn, at any reflection in the mirror. It would be something homogeneous, unchanging. Such a world is impossible. The world exists thanks to the unity of symmetry and asymmetry.

§5.Symmetry in creativity.

A wonderful example of the use of symmetry is human activity, namely creative activity.

In architecture.

Works of architecture demonstrate excellent examples of symmetry.

We can say that as an art, architecture begins precisely when it is possible to find an elegant, harmonious and original relationship between symmetry and asymmetry.

The example of architecture clearly shows the dialectical unity of symmetry and asymmetry.

Many architectural objects in the surrounding world have an axis of symmetry or a center of symmetry.

What symmetry does the Egyptian pyramid have? (rotary, if rotated 90 degrees around a vertical axis passing through the top of the pyramid), mirror (combines with itself when reflected (mentally) in any of the 4 vertical planes passing through the top perpendicular to the base). (fig20)

Most buildings have mirror symmetry. General plans of buildings, facades, ornaments, cornices, columns reveal proportionality and harmony. Old Russian architecture provides many examples of the use of symmetry: bell towers, internal support pillars. All church churches are built on symmetry, which have axes and centers of symmetry.

Examples of symmetry can be seen in the architecture of Saratov:

Temple “quench my sorrows”, circus, Central Department Store, house of books, conservatory, ancient buildings in the city center, etc. (Fig. 21a, b, c, d, Fig. 25a, b)

The proportion that is present in symmetry brings beauty to architecture. This means symmetry is the soul of harmony.

Russian language and literary creativity

Let's discuss the symmetry of the letters A, B, D, E, F, Z, K, L, M, N, P, S, T, F, X, W, E, Y, -

this is an example of mirror symmetry. The letters O, ZH, N, F, X have central (rotational) and mirror symmetry.

In literary works, beauty, associated with symmetry, is contrasted with ugliness due to asymmetry. So, in Pushkin’s “The Tale of Tsar Saltan” this is the beautiful Swan Princess and the twisted villains of the weaver and cook. In literary works, there are a number of funny verbal constructions based on the properties of mirror symmetry. For example, the words “topot”, “Cossack”, “hut” in the literature, this type of words is called palindromes.

All poetry is symmetry. Symmetry in the work of A. A. Fet is represented quite widely, as in the work of any Russian poet. This is a ring composition, and a uniform alternation of stressed and unstressed syllables: size

Quiet starry night...

The moon shines tremblingly

Sweet are the lips of beauty

On a quiet starry night.

Dactyl: stressed and unstressed syllables are repeated absolutely precisely, creating a melodious quality.

The refrains are symmetrical: repetition of lines at a certain interval.

Quietly the evening is burning down,

Mountains of gold;

The sultry air is getting colder

Sleep child

The nightingales have been singing for a long time,

Heralding darkness;

The strings rang timidly -

Sleep, child.

Conclusions:

Symmetry plays a decisive role not only in the process of scientific knowledge of the world, but also in the process of its sensory emotional perception.

Symmetry is a source of aesthetic satisfaction and artistic perception.

Symmetry in fine arts

Many artists paid close attention to the symmetry and proportions of the human body. Leonardo da Vinci discovered that the body fits into a circle and a square. We are all symmetrical! Some artists especially emphasize this symmetry in their works.

RAPHAEL. Sistine Madonna (Fig.22a)

Artists of different eras used a symmetrical construction of the picture. Many ancient mosaics were symmetrical. In a symmetrical composition, people or objects are located almost mirror-like with respect to the central axis of the picture. This construction allows you to achieve the impression of peace, majesty, special solemnity and significance of events.

F. HODLER. Lake Tan (Figure 22b)

Symmetry in art is based on reality. For example, a human figure, a butterfly, a snowflake and much more are arranged symmetrically. Symmetrical compositions are static (stable), the left and right halves are balanced.

V. VASNETSOV. Bogatyrs (Fig. 22c)

Curbs.

“The mathematician, like the artist or the poet, creates patterns.” G. Hardy.

A periodically repeating pattern on a long strip is called a border. This can be a wall painting decorating the walls of buildings, galleries, staircases. This can be cast iron used in park fences, bridge gratings and embankments. These can be plaster bas-reliefs or ceramics. Borders have mirror and figurative symmetry. (Fig.23-25)

Ornaments.

Amazing designs that are often found in decorative art are called ornaments. In them you can find an intricate combination of portable, mirror and rotational symmetry. Depending on what elements the ornament consists of, it is classified as one type or another.) 1geometric ornament (clear alternation of geometric elements). 2) floral ornament.

3) calligraphic (can consist of either individual letters or entire sentences, sayings, proverbs, slogans).

Geometric ornament: clear alternation of geometric elements. Floral ornament: floral motif. Calligraphic ornament: alternation of individual letters, sentences, proverbs. Fantastic ornament: images of mythical creatures. Animal ornament: images of birds and animals. Heraldic ornament: coats of arms, attributes of war, musical and theatrical art. (Figure 26)

Decorations (Fig. 27)

Symmetry exists in music and choreography (dancing). It depends on the alternation of beats. It turns out that many folk songs and dances are built symmetrically. (Fig. 28a, b)

§6. Symmetry is nearby.

In clothes

In clothes, a person also tries to maintain the impression of symmetry: the right sleeve corresponds to the left, the right trouser leg corresponds to the left. The buttons on the jacket and on the shirt sit exactly in the middle, and if they move away from it, then at symmetrical distances.

But against the background of this general symmetry, in small details we deliberately allow asymmetry. For example, placing an asymmetrical pocket on the chest on a suit.

Complete flawless symmetry would look unbearably boring. It is a slight deviation from it that gives characteristic, individual features. And at the same time, sometimes a person tries to emphasize and strengthen the difference between left and right. In the Middle Ages, men at one time sported trousers with legs of different colors. In not so distant days, jeans with bright patches or colored stains were popular. But such fashion is always short-lived. Only tactful, modest deviations from symmetry remain for a long time.

Business clothes are always strictly symmetrical. (Fig. 29-30) A festive outfit can be made asymmetrical to add individuality to the image. But at the same time, the right sleeve (or trouser leg) will not be shorter than the left. The right and left parts of the clothing are cut most often according to the same pattern, placing the pattern of half the product on the material folded in half. (Fig. 31)

Shoes are always strictly symmetrical.

In everyday life.

“The study of archaeological monuments shows that humanity, at the dawn of its culture, already had an idea of symmetry and implemented it in drawings and in everyday objects.

…The use of symmetry in primitive production was determined not only by aesthetic motives, but to a certain extent by human confidence in its greater suitability for the practice of correct forms.”

A.V.Shubnikov

Billiard players are familiar with the action of reflection. Their mirrors are the sides of the playing field, and the role of the beam is played by the trajectories of the balls.

Household appliances and furniture, dishes and cutlery, blankets and carpets, curtains, napkins, vases, etc. are symmetrical (Fig. 40-45)

Symmetry of the villages of Storozhevka and Saratov

You can see many examples of symmetry in the architecture of the city of Saratov and your village. (Figure 21,25, Figure 32-39)

IV. Conclusion.

Considering some aspects of the use of symmetry in physics, art, technology, biology, literature, one can notice an important aspect - this is the philosophical aspect of symmetry, or more precisely, the dialectic of symmetry and asymmetry. It forms the basis of any scientific classification. It is this that determines the degree of beauty contained in a particular work of art or architecture. If symmetry is associated with preservation, the common, the necessary. That asymmetry is associated with change, particular, different, random. The world could not be absolutely symmetrical (nothing would change, there would be no differences, in such a world nothing would be observed - no phenomena, no objects). A completely asymmetrical world could not exist. It would be a world without any laws, where nothing is preserved, where there are no causal connections.

V. Literature used:

Pogorelov Geometry 7-11, Moscow: Education, 1992.

L. Tarasov, This amazingly symmetrical world, Moscow: Enlightenment, 1982

M. Gardner, This Right, Left World.

Weil G. Symmetry. M.: Editorial URSS, 2003.

Zenkevich I.G., Aesthetics of a mathematics lesson: A manual for teachers. – M.: Education, 1981.

Magazine "Around the World"

INTERNET resources:

III scientific and practical conference of schoolchildren

Dovolensky district

Symmetry is all around us

Sobolev Roman Municipal educational institution secondary school No. 2, grade 10, Dovolnoye village, Dovolensky district

Scientific supervisor:

Dobrenkaya Galina Vasilievna,

mathematics teacher of the first qualification category

Contact phone: 22-377

S. Satisfied, 2010

Table of contents:

1. Introduction 3-4

2. The concept of symmetry. Types of symmetry in geometry. 4-8

3. Man is a symmetrical creature 8-9

4. Perfect symmetry is boring 9-10

5. Why the world around us is beautiful. 10-14

6. References 15

1. INTRODUCTION

This abstract is devoted to such a concept of modern natural science as SYMMETRY.

The leitmotif of the entire abstract is the concept of symmetry playing ( there is an opinion) leading, although not always conscious, role in modern science, art, technology and the life around us. Symmetry permeates literally everything around, capturing seemingly completely unexpected areas and objects. It is appropriate here to quote the statement of J. Newman, who especially successfully emphasized the all-encompassing and omnipresent manifestations of symmetry: “Symmetry establishes a funny and surprising affinity between objects, phenomena and theories...”

A truly vast literature is devoted to the problem of symmetry.

In the Concise Oxford Dictionary, symmetry is defined as “beauty due to the proportionality of the parts of the body or any whole, balance, similarity, harmony, consistency” (the term “symmetry” itself in Greek means “proportionality,” which ancient philosophers understood as a special case of harmony - coordination of parts within the whole).

Symmetry is one of the most fundamental and one of the most general patterns of the universe: inanimate, living nature and society.

We are very familiar with the word symmetry. Probably, when we pronounce it, we remember a butterfly or a maple leaf, in which we can mentally draw a straight axis and the parts that will be located on different sides of this straight line will be almost identical. (Slide 3) We encounter symmetry everywhere. The concept of symmetry runs through the entire centuries-old history of human creativity. It is found already at the origins of human knowledge; it is widely used by all areas of modern science without exception.

The principles of symmetry play an important role in physics and mathematics, chemistry and biology, technology and architecture, painting and sculpture, poetry and music. The laws of nature that govern the inexhaustible picture of phenomena in their diversity, in turn, are subject to the principles of symmetry.

2. What is symmetry?

Proportionality, sameness in the arrangement of parts of something on opposite sides of a point, straight line or plane.

in geometry - the property of geometric figures.

proportionality, proportionality, equal (or different) similarity, uniformity, equivalence, correspondence, similarity; sameness, or proportionate similarity in the arrangement of parts of a whole, two halves; understanding, conformity; anti-equality, contrast.

Pythagoras and his students paid close attention to symmetry. Based on the doctrine of number, the Pythagoreans gave the first mathematical interpretation of harmony, symmetry, which has not lost its meaning to this day. The views of Pythagoras and his school were further developed in Plato's doctrine of knowledge. Of particular interest are Plato's views on the structure of the world, which, according to him, consists of regular polygons with perfect symmetry.

Types of symmetry:

The main types of symmetry are: symmetry about a point (central symmetry), symmetry about an axis (axial symmetry), rotation about a given point, parallel translation and mirror symmetry.

It has been noticed that when certain transformations are performed on geometric figures, their parts, having moved to a new position, will again form the original figure. For example, if we draw a straight line through the height of an isosceles triangle to the base, and swap parts of the triangle located on opposite sides of this straight line, we will get the same (in terms of shape and size) isosceles triangle.

Axial symmetry is a mapping of the plane onto itself relative to some straight line, which is the axis of symmetry. Axial symmetry is a movement because it preserves the distance between points. But it doesn't maintain direction. (Slide

Rotation is a movement around a point at an angle α, at which the point remains, and all others rotate around it in a given direction at an angle α. (Slide 5)

A five-pointed star, when rotated at an angle of 72 degrees around the central point (the point of intersection of its rays), will take its original position.

In the plant world, there is also rotational symmetry. Take a chamomile flower in your hand. The combination of different parts of the flower occurs if they are rotated around the stem (Slide 6).

The examples given discuss different types of symmetry. In the first case we are talking about axial symmetry. The parts, which, so to speak, replace each other, are formed by a certain straight line. This straight line is usually called the axis of symmetry. In space, the analogue of the axis of symmetry is the plane of symmetry. If you draw a plane in a cube parallel to the side faces and passing through the point of intersection of the diagonals of the cube, then the side faces will be symmetrical relative to this plane. Or the plane containing the diagonals of the side faces will be a plane of symmetry for parts located on opposite sides of this plane.

Taking into account both cases (plane and space), this type of symmetry is sometimes called mirror symmetry. This name is justified by the fact that both parts of the figure, located on opposite sides of the axis of symmetry or plane of symmetry, are similar to some object and its reflection in the mirror. Note that you may also come across another name for this type of symmetry. For example, in biology, this type of symmetry is called bilateral, and the plane of symmetry is called a bilateral plane.

Another type of symmetry that we haven’t talked about yet is transference symmetry. This type of symmetry consists in the fact that parts of the whole form are organized in such a way that each next one repeats the previous one and is separated from it by a certain interval in a certain direction. This interval is called the symmetry step. (Slide 7)

Portable symmetry is usually used when constructing borders (Slide 8). In works of architectural art it can be seen in the ornaments or grilles that are used to decorate them. Portable symmetry is also used in the interiors of buildings.

Ornament

3. Man is a symmetrical creature

Let’s not figure out for now whether an absolutely symmetrical person actually exists. Everyone, of course, will have a mole, a strand of hair or some other detail that breaks the external symmetry. The left eye is never exactly the same as the right, and the corners of the mouth are at different heights, at least for most people.

Yet these are only minor inconsistencies. No one will doubt that outwardly a person is built symmetrically: the left hand always corresponds to the right and both hands are exactly the same!

BUT! It's worth stopping here. If our hands were really exactly the same, we could change them at any time. It would be possible, say, by transplantation to transfer the left palm to the right hand, or, more simply, the left glove would then fit the right hand, but in fact this is not the case.

Everyone knows that the similarity between our hands, ears, eyes and other parts of the body is the same as between an object and its reflection in a mirror.

Many artists paid close attention to the symmetry and proportions of the human body, at least as long as they were guided by the desire to follow nature as closely as possible in their works. The well-known canons of proportions compiled by Albrecht Durer and Leonardo da Vinci. According to these canons, the human body is not only symmetrical, but also proportional.

The size of the head is proportional not only to the length of the body, but also to the size of other parts of the body. All people are built on this principle, which is why we are generally similar to each other. However, our proportions are only approximately consistent, and therefore people are only similar, but not the same. In any case, we are all symmetrical! In addition, some artists especially emphasize this symmetry in their works.

4. Perfect symmetry is boring.

And in clothing, a person, as a rule, also tries to maintain the impression of symmetry: the right sleeve corresponds to the left, the right trouser leg corresponds to the left.

The buttons on the jacket and on the shirt sit exactly in the middle, and if they move away from it, then at symmetrical distances.

Complete flawless symmetry would look unbearably boring. It is small deviations from it that give characteristic, individual features. To do this, use asymmetry and dissymmetry

But against the background of this general symmetry, in small details we deliberately allow asymmetry - this is a complete lack of symmetry, for example, combing our hair in a side parting - on the left or on the right. Or, say, placing an asymmetrical pocket on the chest on a suit. Or putting a ring on the ring finger of only one hand. Orders and badges are worn on only one side of the chest (usually on the left).

Dissymmetry is a partial lack of symmetry, a disorder of symmetry, expressed in the presence of some symmetrical properties and the absence of others.

And at the same time, sometimes a person tries to emphasize and strengthen the difference between left and right. In the Middle Ages, men at one time wore trousers with legs of different colors (for example, one red and the other black or white). In not so distant days, jeans with bright patches or colored stains were popular. But such fashion is always short-lived. Only tactful, modest deviations from symmetry remain for a long time.

5. Why is the world around us beautiful?

Symmetry is widely used in architecture.

Architectural structures created by man are for the most part symmetrical. They are pleasing to the eye and people consider them beautiful. What is this connected with? Here we can only make assumptions.
Firstly, you and I all live in a symmetrical world, which is determined by the living conditions on planet Earth, primarily by the gravity that exists here. And, most likely, subconsciously a person understands that symmetry is a form of stability, and therefore existence on our planet. Therefore, in man-made things he intuitively strives for symmetry.
Secondly, the people, plants, animals and things around us are symmetrical. However, upon closer examination, it turns out that natural objects (as opposed to man-made ones) are only almost symmetrical. But this is not always perceived by the human eye. The human eye gets used to seeing symmetrical objects. They are perceived as harmonious and perfect.
Symmetry is perceived by a person as a manifestation of regularity, and therefore internal order. Externally, this internal order is perceived as beauty.
Symmetrical objects have a high degree of expediency - after all, symmetrical objects have greater stability and equal functionality in different directions. All this led a person to the idea that for a structure to be beautiful it must be symmetrical. Symmetry was used in the construction of religious and domestic buildings in Ancient Egypt. The decorations of these buildings also represent examples of the use of symmetry. But symmetry is most clearly manifested in the ancient buildings of Ancient Greece (Slide 16-17), luxury items and ornaments that decorated them. From then to the present day, symmetry in the human mind has become an objective sign of beauty.
Maintaining symmetry is the first rule of an architect when designing any structure. One has only to look at the magnificent work of A.N. Voronikhin, Kazan Cathedral in St. Petersburg, to be convinced of this.
If we mentally draw a vertical line through the spire on the dome and the top of the pediment, we will see that on both sides of it there are absolutely identical parts of the structure (colonnades and cathedral buildings). (Slide 18) But it is possible that you do not know what is in the Kazan Cathedral there is one more, so to speak, “failed” symmetry.

The fact is that, according to the canons of the Orthodox Church, the entrance to the cathedral should be from the east, i.e. it should be from the street, which is located to the right of the cathedral and runs perpendicular to Nevsky Prospekt. But, on the other hand, Voronikhin understood that the cathedral should be facing the main highway of the city. And then he made an entrance to the cathedral from the east, but planned another entrance, which he decorated with a beautiful colonnade. To make the building perfect, and therefore symmetrical, the same colonnade had to be located on the other side of the cathedral. Then, if we looked at the cathedral from above, its plan would have not one, but two axes of symmetry. But the architect's plans were not destined to come true.

Kazan Cathedral in St. Petersburg

In addition to symmetry in architecture, one can consider antisymmetry and dissymmetry. Antisymmetry is the opposite of symmetry, its absence. An example of antisymmetry in architecture is St. Basil's Cathedral in Moscow, where symmetry is completely absent in the structure as a whole (Slide 19). However, it is surprising that the individual parts of this cathedral are symmetrical and this creates its harmony. Dissymmetry is a partial lack of symmetry, a disorder of symmetry, expressed in the presence of some symmetrical properties and the absence of others. An example of dissymmetry in an architectural structure is the Catherine Palace in Tsarskoe Selo near St. Petersburg (Slide 20-21). Almost all the properties of symmetry are fully maintained in it, with the exception of one detail. The presence of the Palace Church upsets the symmetry of the building as a whole. If we do not take this church into account, then the Palace becomes symmetrical.

In modern architecture, techniques of both antisymmetry and dissymmetry are increasingly used. These searches often lead to very interesting results. A new aesthetics of urban planning is emerging. Thus, beauty is the unity of symmetry, asymmetry and dissymmetry (Slide 22-25).

6. Conclusion

So we live in a fairly symmetrical world. It is not surprising that we ourselves are symmetrical and tend to consider everything symmetrical beautiful. Sometimes, however, it’s nice to slightly break the ideal symmetry; it gives some liveliness, but not too much, not to the point of chaos. Animals are very symmetrical, plants are quite symmetrical, crystals are completely symmetrical, our spherical planet is almost perfectly symmetrical (Slide 26), its trajectory is close to symmetry. After what has been said, perhaps the statement that all laws of nature are determined by the symmetry of the world will not seem so fantastic.

References:

1. Atanasyan. L.S. “Geometry 7-9 grades” 2003 M. "Enlightenment"

3.Moscow University Publishing House “A manual on geometry for those entering universities” 1974.

4.Kritsman.V.A “Book for reading on geometry” 1975 M. "Enlightenment"

5. Pogorelov A.V. “Geometry 7-9 grades” 2005 M. "Enlightenment"

6. Stanzo.V.V “Encyclopedic Dictionary of Geometry” 1982 M. "Enlightenment"

7.http://yandex.ru

Regional research conference "Junior"

Research work

Symmetry in the world around us

(section of exact sciences)

Completed: Merizanova Anna,

Eliseenko Vera,

8th grade student

Supervisor: Kolesnikova

Lyudmila Alexandrovna,

math teacher

Introduction. . 2

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

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

1.3. Symmetry through the centuries . 7

Chapter 2. Symmetry around us. 8

.. 8

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

Conclusion. 11

Bibliography. 12

Introduction

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

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

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

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

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

We have set ourselves a goal:

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

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

Learn to recognize symmetrical figures among others.

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

Demonstrate the varied applications of mathematics in real life.

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

To write the work, I used various methods:

2) the method of inductive generalization and specification;

3) use of computer equipment.

Chapter 1. First ideas about symmetry

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

1.1. Historical development and understanding of the concept of symmetry

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

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

1.2. Mathematical idea of symmetry

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

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

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

Rosettes and snowflakes are symmetrical and very beautiful figures.

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

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

Every point is straight A

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

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

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

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

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

Research assignments.

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

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

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

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

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

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

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

Discussion. O and X

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

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

Discussion. A, E, O

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

1.3. Symmetry of Old Russian ornament

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

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

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

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

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

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

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

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

1.4. Symmetry through the centuries

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

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

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

Chapter 2. Symmetry around us

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

2.1. The role of symmetry in knowledge of nature

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

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

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

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

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

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

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

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

2.2. Symmetry in architectural structures

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

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

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

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

Conclusion

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

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

As a result of the project:

u expanded knowledge about symmetry;

u learned what phenomena from life and

some sciences are described by symmetry;

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

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

Bibliography

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

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

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

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

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

Construction portal. Honing our skills

"symmetry and the world around us" research work. The world around us is a world of symmetry. Many faces of symmetry in the world around us.