A geometric figure with a center of symmetry. There are figures that do not have a single axis of symmetry

Definition of symmetry;

• Definition of symmetry;

• Central symmetry;

• Axial symmetry;

• Symmetry relative to the plane;

• Rotation symmetry;

• Mirror symmetry;

• Symmetry of similarity;

• Plant symmetry;

• Animal symmetry;

• Symmetry in architecture;

• Is man a symmetrical creature?

• Symmetry of words and numbers;

SYMMETRY

• SYMMETRY- proportionality, sameness in the arrangement of parts of something on opposite sides of a point, straight line or plane.

• (Ozhegov's Explanatory Dictionary)

• So, a geometric object is considered symmetrical if something can be done to it, after which it will remain unchanged.

ABOUT ABOUT ABOUT called center of symmetry of the figure.

• The figure is said to be symmetrical about the point ABOUT, 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.

circle and parallelogram center of the circle ). Schedule odd function

Examples of figures that have central symmetry are circle and parallelogram. The center of symmetry of a circle is center of the circle, and the center of symmetry of the parallelogram is the point of intersection of its diagonals. Any straight line also has central symmetry ( any point on a line is its center of symmetry). Schedule odd function symmetrical about the origin.

• An example of a figure that does not have a center of symmetry is arbitrary triangle.

A A a called axis of symmetry of the figure.

• 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 of the figure.

At an unturned corner one axis of symmetry angle bisector one axis of symmetry three axes of symmetry two axes of symmetry, and the square is four axes of symmetry relative to the y-axis.

At an unturned corner one axis of symmetry- straight line on which it is located angle bisector. An isosceles triangle also has one axis of symmetry, and an equilateral triangle is three axes of symmetry. A rectangle and a rhombus that are not squares have two axes of symmetry, and the square is four axes of symmetry. The circle has an infinite number of them. The graph of an even function is symmetrical when constructed relative to the y-axis.

• There are figures that do not have a single axis of symmetry. Such figures include parallelogram, other than a rectangle, scalene triangle.

Points A And A1 A A AA1 And perpendicular A counts symmetrical to itself

Points A And A1 are called symmetrical relative to the plane A(plane of symmetry), if the plane A passes through the middle of the segment AA1 And perpendicular to this segment. Each point of the plane A counts symmetrical to itself. Two figures are called symmetrical relative to the plane (or mirror-symmetrical relative) if they consist of pairwise symmetrical points. This means that for each point of one figure, a point symmetrical (relatively) to it lies in another figure.

The body (or figure) has rotational symmetry, if when turning an angle 360º/n, where n is an integer fully compatible

• The body (or figure) has rotational symmetry, if when turning an angle 360º/n, where n is an integer, near some straight line AB (axis of symmetry) it fully compatible with its original position.

• Radial symmetry- a form of symmetry that is preserved when an object rotates around a specific point or line. Often this point coincides with the center of gravity of the object, that is, the point at which intersects an infinite number of axes of symmetry. Similar objects can be circle, ball, cylinder or cone.

Mirror symmetry binds anyone

Mirror symmetry binds anyone an object and its reflection in a plane mirror. One figure (or body) is said to be mirror symmetrical to another if together they form a mirror symmetrical figure (or body). Symmetrically mirrored figures, for all their similarities, differ significantly from each other. Two mirror-symmetrical flat figures can always be superimposed on each other. However, to do this it is necessary to remove one of them (or both) from their common plane.

Symmetry of similarity nesting dolls.

• Symmetry of similarity are peculiar analogues of previous symmetries with the only difference being that they are associated with simultaneous reduction or increase in similar parts of the figure and the distances between them. The simplest example of such symmetry is nesting dolls.

• Sometimes figures can have different types of symmetry. For example, some letters have rotational and mirror symmetry: AND, N, M, ABOUT, A.

• There are many other types of symmetries that are abstract in nature. For example:

• Commutation symmetry, which consists in the fact that if identical particles are swapped, then no changes occur;

• Gauge symmetries connected with zoom change. In inanimate nature, symmetry primarily arises in such a natural phenomenon as crystals, from which almost all solids are composed. It is this that determines their properties. The most obvious example of the beauty and perfection of crystals is the well-known snowflake.

We encounter symmetry everywhere: in nature, technology, art, science. The concept of symmetry runs through the entire centuries-old history of human creativity. Symmetry principles play an important role in physics and mathematics, chemistry and biology, technology and architecture, painting and sculpture, poetry and music. The laws of nature are also subject to the principles of symmetry.

axis of symmetry.

• Many flowers have an interesting property: they can be rotated so that each petal takes the position of its neighbor, and the flower aligns with itself. This flower has axis of symmetry.

• Helical symmetry observed in the arrangement of leaves on the stems of most plants. Arranging in a spiral 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.

• Bilateral symmetry Plant organs are also present, for example, the stems of many cacti. Often found in botany radially symmetrically arranged flowers.

dividing line.

• Symmetry in animals means correspondence in size, shape and outline, as well as the relative arrangement of body parts located on opposite sides dividing line.

• The main types of symmetry are radial(radial) – it is possessed by echinoderms, coelenterates, jellyfish, etc.; or bilateral(two-sided) - we can say that every animal (be it an insect, fish or bird) consists of two halves- right and left.

• Spherical symmetry occurs in radiolarians and sunfishes. Any plane drawn through the center divides the animal into equal halves.

• The symmetry of a structure is associated with the organization of its functions. The projection of the plane of symmetry - the axis of the building - usually determines the location of the main entrance and the beginning of the main traffic flows.

• Every detail in a symmetrical system exists like a double to your obligatory couple, located on the other side of the axis, and due to this it can only be considered as part of the whole.

• Most common in architecture mirror symmetry. The buildings of Ancient Egypt and the temples of ancient Greece, amphitheatres, baths, basilicas and triumphal arches of the Romans, palaces and churches of the Renaissance, as well as numerous structures of modern architecture are subordinate to it.

accents

• To better reflect symmetry, buildings are placed accents- particularly significant elements (domes, spiers, tents, main entrances and staircases, balconies and bay windows).

• To design the decoration of architecture, an ornament is used - a rhythmically repeating pattern based on the symmetrical composition of its elements and expressed by line, color or relief. Historically, several types of ornaments have developed based on two sources - natural forms and geometric figures.

• But an architect is first and foremost an artist. And therefore, even the most “classical” styles were more often used dissymmetry– nuanced deviation from pure symmetry or asymmetry- deliberately asymmetrical construction.

• 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 the similarities between our hands, ears, eyes and other parts of the body are the same as between an object and its reflection in a mirror.

right his half rough features characteristic of the male sex. Left half

Numerous measurements of facial parameters in men and women have shown that right his half compared to the left, it has more pronounced transverse dimensions, which gives the face a more rough features characteristic of the male sex. Left half the face has more pronounced longitudinal dimensions, which gives it smooth lines and femininity. This fact explains the predominant desire of females to pose in front of artists with the left side of their faces, and males with the right.

Palindrome

• Palindrome(from the gr. Palindromos - running back) is an object in which the symmetry of its components is specified from beginning to end and from end to beginning. For example, a phrase or text.

• The straight text of a palindrome, read according to the normal reading direction of a given script (usually from left to right), is called upright, reverse – by rover or reverse(from right to left). Some numbers also have symmetry.

Homothety and similarity.Homothety is a transformation in which each point M (plane or space) is assigned to a point M", lying on OM (Fig. 5.16), and the ratio OM":OM= λ the same for all points other than ABOUT. Fixed point ABOUT called the center of homothety. Attitude OM": OM considered positive if M" and M lie on one side of ABOUT, negative - on opposite sides. Number X called the homothety coefficient. At X< 0 homothety is called inverse. Atλ = - 1 homothety turns into a symmetry transformation about a point ABOUT. With homothety, a straight line goes into a straight line, the parallelism of straight lines and planes is preserved, angles (linear and dihedral) are preserved, each figure goes into it similar (Fig. 5.17).

The converse is also true. A homothety can be defined as an affine transformation in which the lines connecting the corresponding points pass through one point - the center of the homothety. Homothety is used to enlarge images (projection lamp, cinema).

Central and mirror symmetries.Symmetry (in a broad sense) is a property of a geometric figure F, characterizing a certain regularity of its shape, its invariability under the action of movements and reflections. A figure Φ has symmetry (symmetrical) if there are non-identical orthogonal transformations that take this figure into itself. The set of all orthogonal transformations that combine the figure Φ with itself is the group of this figure. So, a flat figure (Fig. 5.18) with a point M, transforming-

looking into yourself in the mirror reflection, symmetrical about the straight axis AB. Here the symmetry group consists of two elements - a point M converted to M".

If the figure Φ on the plane is such that rotations relative to any point ABOUT to an angle of 360°/n, where n > 2 is an integer, translate it into itself, then the figure Ф has nth-order symmetry with respect to the point ABOUT - center of symmetry. An example of such figures is regular polygons, for example star-shaped (Fig. 5.19), which has eighth-order symmetry relative to its center. The symmetry group here is the so-called nth order cyclic group. The circle has symmetry of infinite order (since it is compatible with itself by rotating through any angle).

The simplest types of spatial symmetry are central symmetry (inversion). In this case, relative to the point ABOUT the figure Ф is combined with itself after successive reflections from three mutually perpendicular planes, i.e. a point ABOUT - the middle of the segment connecting the symmetrical points F. So, for a cube (Fig. 5.20) the point ABOUT is the center of symmetry. Points M and M" cube

Symmetry is associated with harmony and order. And for good reason. Because the question of what symmetry is, there is an answer in the form of a literal translation from ancient Greek. And it turns out that it means proportionality and immutability. And what could be more orderly than a strict definition of location? And what can be called more harmonious than something that strictly corresponds to size?

What does symmetry mean in different sciences?

Biology. An important component of symmetry in it is that animals and plants have regularly arranged parts. Moreover, there is no strict symmetry in this science. There is always some asymmetry. It admits that the parts of the whole do not coincide with absolute precision.

Physics. A system of bodies and changes in it are described using equations. They contain symmetrical components, which simplifies the entire solution. This is accomplished by searching for conserved quantities.

Mathematics. It is there that basically explains what symmetry is. Moreover, it is given greater importance in geometry. Here, symmetry is the ability to display in figures and bodies. In a narrow sense, it comes down simply to a mirror image.

How do different dictionaries define symmetry?

No matter which of them we look at, the word “proportionality” will appear everywhere. In Dahl one can also see such an interpretation as uniformity and equality. In other words, symmetrical means the same. It also says that it is boring; something that doesn’t have it looks more interesting.

When asked what symmetry is, Ozhegov’s dictionary already talks about the sameness in the position of parts relative to a point, line or plane.

Ushakov’s dictionary also mentions proportionality, as well as the complete correspondence of two parts of the whole to each other.

When do we talk about asymmetry?

The prefix “a” negates the meaning of the main noun. Therefore, asymmetry means that the arrangement of elements does not lend itself to a certain pattern. There is no immutability in it.

This term is used in situations where the two halves of an item are not completely identical. Most often they are not at all similar.

In living nature, asymmetry plays an important role. Moreover, it can be both useful and harmful. For example, the heart is placed in the left half of the chest. Due to this, the left lung is significantly smaller. But it is necessary.

About central and axial symmetry

In mathematics, the following types are distinguished:

• central, that is, made relative to one point;
• axial, which is observed near a straight line;
• specular, it is based on reflections;
• transfer symmetry.

What is an axis and center of symmetry? This is a point or line relative to which any point on the body can find another. Moreover, such that the distance from the original to the resulting one is divided in half by the axis or center of symmetry. As these points move, they describe identical trajectories.

In situations where it is necessary to find the center of symmetry, you need to proceed as follows. If there are two figures, then find their identical points and connect them with a segment. Then divide in half. When there is only one figure, knowledge of its properties can help. Often this center coincides with the point of intersection of diagonals or heights.

What shapes are symmetrical?

Geometric figures can have axial or central symmetry. But this is not a necessary condition; there are many objects that do not possess it at all. For example, a parallelogram has a central one, but it does not have an axial one. But non-isosceles trapezoids and triangles have no symmetry at all.

If central symmetry is considered, there are quite a lot of figures that have it. These are a segment and a circle, a parallelogram and all regular polygons with a number of sides that is divisible by two.

The center of symmetry of a segment (also a circle) is its center, and for a parallelogram it coincides with the intersection of the diagonals. While for regular polygons this point also coincides with the center of the figure.

If a straight line can be drawn in a figure, along which it can be folded, and the two halves coincide, then it (the straight line) will be an axis of symmetry. What's interesting is how many axes of symmetry different shapes have.

For example, an acute or obtuse angle has only one axis, which is its bisector.

If you need to find the axis in an isosceles triangle, then you need to draw the height to its base. The line will be the axis of symmetry. And just one. And in an equilateral one there will be three of them at once. In addition, the triangle also has central symmetry relative to the point of intersection of the heights.

A circle can have an infinite number of axes of symmetry. Any straight line that passes through its center can fulfill this role.

A rectangle and a rhombus have two axes of symmetry. In the first one they pass through the middles of the sides, and in the second they coincide with the diagonals.

The square combines the previous two figures and has 4 axes of symmetry at once. They are the same as those of a rhombus and a rectangle.

“Point of symmetry” - Symmetry in architecture. Examples of symmetry of plane figures. Two points A and A1 are called symmetrical with respect to O if O is the midpoint of the segment AA1. Examples of figures that have central symmetry are the circle and parallelogram. Point C is called the center of symmetry. Symmetry in science and technology.

“Construction of geometric figures” - Educational aspect. Control and correction of assimilation. Study of the theory on which the method is based. In stereometry there are not strict constructions. Stereometric constructions. Algebraic method. Method of transformations (similarity, symmetry, parallel translation, etc.). For example: straight; angle bisector; midperpendicular.

“Human figure” - The shape and movements of the human body are largely determined by the skeleton. Fair with theatrical performance. Do you think there will be work for an artist in the circus? The skeleton plays the role of a frame in the structure of the figure. Main Body (stomach, chest) Didn't pay attention to Head, face, hands. A. Mathis. Proportions. Ancient Greece.

“Symmetry about a straight line” - Symmetry about a straight line is called axial symmetry. Straight line a is the axis of symmetry. Symmetry is relatively straight. Bulavin Pavel, 9B grade. How many axes of symmetry does each figure have? A figure can have one or more axes of symmetry. Central symmetry. Isosceles trapezoid. Rectangle.

“Areas of figures geometry” - Pythagorean Theorem. Areas of various figures. Solve the puzzle. Figures having equal areas are called equal in area. Units of area measurement. Area of ​​a triangle. Rectangle, triangle, parallelogram. Square centimeter. Figures of equal area. Equal figures b). Square millimeter. V). What is the area of ​​the figure made up of figures A and D?

“The limit of a function at a point” - , Then in this case. When striving. Limit of a function at a point. Continuous at a point. Equal to the function value in. But when calculating the limit of the function at. Equal to value. Expression. Aspiration. Or we can say this: in a fairly small neighborhood of the point. Compiled from. Solution. Continuous at intervals. In between.

SYMMETRY OF SPATIAL FIGURES

According to the famous German mathematician G. Weyl (1885-1955), “symmetry is the idea through which man for centuries has tried to comprehend and create order, beauty and perfection.”
Beautiful images of symmetry are demonstrated by works of art: architecture, painting, sculpture, etc.
The concept of symmetry of figures on a plane was discussed in the planimetry course. In particular, the concepts of central and axial symmetry were defined. For spatial figures, the concept of symmetry is defined in a similar way.
Let's look at central symmetry first.
symmetrical about the point O called center of symmetry, if O is the midpoint of the segment AA." Point O is considered symmetrical to itself.
A transformation of space in which each point A is associated with a point A that is symmetrical to it (with respect to a given point O) is called central symmetry. Point O is called center of symmetry.
Two figures Ф and Ф" are called centrally symmetrical, if there is a symmetry transformation that takes one of them to the other.
The figure F is called centrally symmetrical, if it is centrally symmetrical to itself.
For example, a parallelepiped is centrally symmetrical about the point of intersection of its diagonals. The ball and sphere are centrally symmetrical about their centers.
Of the regular polyhedra, the cube, octahedron, icosahedron and dodecahedron are centrally symmetrical. The tetrahedron is not a centrally symmetrical figure.
Let's consider some properties of central symmetry.
Property 1. If O 1 , O 2 are the centers of symmetry of the figure Ф, then the point O 3, symmetrical O 1 relative to O 2 is also the center of symmetry of this figure.
Proof. Let A be a point in space, A 2 – a point symmetrical to it, relative to O 2 , A 1 – point symmetrical to A 2 relative to O 1 and A 3 – symmetrical point A 1 relative to O 2 (Fig. 1).

Then the triangles O 2 O 1 A 1 and O 2 O 3 A 3 , O 2 O 1 A 2 and O 2 O 3 A are equal. Therefore A and A 3 symmetric about O 3 . Thus, symmetry about O 3 is a composition of symmetries with respect to O 2, O 1 and O 2 . Consequently, with this symmetry, the figure F transforms into itself, i.e. O 3 is the center of symmetry of the figure F.

Consequence.Any figure either has no center of symmetry, or has one center of symmetry, or has infinitely many centers of symmetry

Indeed, if O 1 , O 2 are the centers of symmetry of the figure Ф, then the point O 3, symmetrical O 1 relative to O 2 is also the center of symmetry of this figure. Likewise, point O 4 symmetrical O 2 relative to O 3 is also the center of symmetry of the figure Ф, etc. Thus, in this case the figure Ф has infinitely many centers of symmetry.

Let us now consider the concept axial symmetry.
Points A and A" in space are called symmetrical about a straight line a, called axis of symmetry, if straight a passes through the middle of segment AA" and is perpendicular to this segment. Each point of a straight line a is considered symmetrical to itself.
A transformation of space in which each point A is associated with a point A that is symmetrical to it (relative to a given line a), called axial symmetry. Straight a in this case it is called axis of symmetry.
The two figures are called symmetrical about a straight line a, if a symmetry transformation about this line transforms one of them into the other.
The figure F in space is called symmetrical relative to straight a, if it is symmetrical to itself.
For example, a rectangular parallelepiped is symmetrical about a straight line passing through the centers of opposite faces. A right circular cylinder is symmetrical about its axis, a ball and a sphere are symmetrical about any straight lines passing through their centers, etc.
The cube has three axes of symmetry passing through the centers of opposite faces and six axes of symmetry passing through the middles of opposite edges.
The tetrahedron has three axes of symmetry passing through the midpoints of opposite edges.
The octahedron has three axes of symmetry passing through opposite vertices and six axes of symmetry passing through the midpoints of opposite edges.
The icosahedron and dodecahedron each have fifteen axes of symmetry passing through the midpoints of opposite edges.
Property 3. Ifa 1 , a 2 – axes of symmetry of the figure Ф, then the straight linea 3, symmetrical a 1 relative a 2 is also the axis of symmetry of this figure.

The proof is similar to the proof of Property 1.

Property 4.If two intersecting perpendicular lines in space are axes of symmetry of a given figure F, then the straight line passing through the point of intersection and perpendicular to the plane of these lines will also be the axis of symmetry of the figure F.
Proof. Consider the coordinate axes O x, O y, O z. Symmetry about the O axis x x, y, z) to the point of the figure Ф with coordinates ( x, –y, –z). Similarly, symmetry about the O axis y translates a point of the figure Ф with coordinates ( x, –y, –z) to the point of the figure Ф with coordinates (– x, –y, z) . Thus, the composition of these symmetries translates the point of the figure Ф with coordinates ( x, y, z) to the point of the figure Ф with coordinates (– x, –y, z). Therefore, the O axis z is the axis of symmetry of the figure F.

Consequence.Any figure in space cannot have an even (non-zero) number of axes of symmetry.
Indeed, let us fix some axis of symmetry a. If b– axis of symmetry, does not intersect a or does not intersect it at a right angle, then there is another axis of symmetry for it b', symmetrical with respect to a. If the axis of symmetry b crosses a at a right angle, then there is another axis of symmetry for it b', passing through the point of intersection and perpendicular to the plane of lines a And b. Therefore, in addition to the axis of symmetry a either an even or an infinite number of axes of symmetry is possible. Thus, a total even (non-zero) number of axes of symmetry is impossible.
In addition to the symmetry axes defined above, we also consider axis of symmetry n-th order, n 2 .
Straight a called axis of symmetry n-th order figure Ф, if when rotating the figure Ф around a straight line a at an angle, the figure F is combined with itself.

It is clear that the 2nd order axis of symmetry is simply an axis of symmetry.
For example, in the correct n-a carbon pyramid, the straight line passing through the top and the center of the base is the axis of symmetry n-th order.
Let's find out which symmetry axes regular polyhedra have.
The cube has three 4th order axes of symmetry passing through the centers of opposite faces, four 3rd order axes of symmetry passing through opposite vertices, and six 2nd order axes of symmetry passing through the midpoints of opposite edges.
The tetrahedron has three axes of second-order symmetry passing through the midpoints of opposite edges.
The icosahedron has six 5th order axes of symmetry passing through opposite vertices; ten 3rd order symmetry axes passing through the centers of opposite faces and fifteen 2nd order symmetry axes passing through the midpoints of opposite edges.
The dodecahedron has six 5th order axes of symmetry passing through the centers of opposite faces; ten 3rd order symmetry axes passing through opposite vertices and fifteen 2nd order symmetry axes passing through the midpoints of opposite edges.
Let's consider the concept mirror symmetry.
Points A and A" in space are called symmetrical relative to the plane, or, in other words, mirror symmetrical, if this plane passes through the middle of the segment AA" and is perpendicular to it. Each point of the plane is considered symmetrical to itself.
A transformation of space in which each point A is associated with a point A that is symmetrical to it (relative to a given plane) is called mirror symmetry. The plane is called plane of symmetry.
The two figures are called mirror symmetrical relative to the plane if a symmetry transformation relative to this plane transforms one of them into the other.
The figure F in space is called mirror symmetrical, if it is mirror symmetrical to itself.
For example, a rectangular parallelepiped is mirror symmetrical about a plane passing through the axis of symmetry and parallel to one of the pairs of opposite faces. The cylinder is mirror-symmetrical with respect to any plane passing through its axis, etc.
Among regular polyhedra, the cube and the octahedron each have nine planes of symmetry. The tetrahedron has six planes of symmetry. The icosahedron and dodecahedron each have fifteen planes of symmetry passing through pairs of opposite edges.
Property 5. The composition of two mirror symmetries about parallel planes is a parallel translation onto a vector perpendicular to these planes and equal in magnitude to twice the distance between these planes.
Consequence. Parallel transport can be thought of as a composition of two mirror symmetries.
Property 6. The composition of two mirror symmetries about planes intersecting in a straight line is a rotation around this straight line by an angle equal to twice the dihedral angle between these planes. In particular, axial symmetry is the composition of two mirror symmetries about perpendicular planes.
Consequence. A rotation can be thought of as a composition of two mirror symmetries.
Property 7. Central symmetry can be represented as a composition of three mirror symmetries.
Let us prove this property using the coordinate method. Let point A in space has coordinates ( x, y, z). Mirror symmetry with respect to the coordinate plane changes the sign of the corresponding coordinate. For example, mirror symmetry about the O plane xy translates the point with coordinates ( x, y, z) to a point with coordinates ( x, y, –z). The composition of three mirror symmetries with respect to coordinate planes translates a point with coordinates ( x, y, z) to a point with coordinates (– x, –y, –z), which is centrally symmetrical to the original point A.
Movements that transform the figure F into itself form a group relative to the composition. It's called symmetry group F figures
Let's find the order of the symmetry group of the cube.
It is clear that any movement that transfers the cube into itself leaves the center of the cube in place, transfers the centers of faces to the centers of faces, the midpoints of edges to the midpoints of edges, and vertices to vertices.
Thus, to specify the movement of the cube, it is enough to determine where the center of the face goes, the middle of the edge of this face and the vertex of the edge.
Let us consider the division of a cube into tetrahedrons, the vertices of each of which are the center of the cube, the center of the face, the middle of the edge of this face, and the vertex of the edge. There are 48 such tetrahedra. Since the movement is completely determined by which of the tetrahedra a given tetrahedron is translated into, the order of the group of symmetries of the cube will be equal to 48.
The orders of the symmetry groups of the tetrahedron, octahedron, icosahedron and dodecahedron are found in a similar way.
Let us find the symmetry group of the unit circle S 1 . This group is denoted O(2). It is an infinite topological group. Let's imagine the unit circle as a group of complex numbers modulo one. There is a natural epimorphism p:O(2) --> S 1 , which associates an element u of the group O(2) with an element u(1) in S 1 . The kernel of this mapping is the group Z 2 , generated by the symmetry of the unit circle relative to the Ox axis. Therefore O(2)/Z 2S 1 . Moreover, if we do not take into account the group structure, then there is a homeomorphism of O(2) and the direct product S 1 and Z 2.
Similarly, the symmetry group of the two-dimensional sphere S 2 is denoted O(3), and for it there is an isomorphism O(3)/O(2) S 2 .
Symmetry groups of n-dimensional spheres play an important role in modern branches of topology: the theory of manifolds, the theory of fiber spaces, etc.
One of the most striking manifestations of symmetry in nature are crystals. The properties of crystals are determined by the features of their geometric structure, in particular, the symmetrical arrangement of atoms in the crystal lattice. The external shapes of crystals are a consequence of their internal symmetry.
The first, still vague assumptions that atoms in crystals are arranged in a regular, regular, symmetrical arrangement were expressed in the works of various natural scientists already at a time when the very concept of an atom was unclear and there was no experimental evidence of the atomic structure of matter. The symmetrical external shape of the crystals involuntarily suggested the idea that the internal structure of the crystals should be symmetrical and regular. The laws of symmetry of the external form of crystals were fully established in the middle of the 19th century, and by the end of this century the laws of symmetry to which atomic structures in crystals are subject were clearly and accurately deduced.
The founder of the mathematical theory of the structure of crystals is the outstanding Russian mathematician and crystallographer - Evgraf Stepanovich Fedorov (1853-1919). Mathematics, chemistry, geology, mineralogy, petrography, mining - E.S. Fedorov made a significant contribution to each of these areas. In 1890, he strictly mathematically derived all possible geometric laws for the combination of symmetry elements in crystal structures, in other words, the symmetry of the arrangement of particles inside crystals. It turned out that the number of such laws is limited. Fedorov showed that there are 230 space symmetry groups, which were subsequently named Fedorov in honor of the scientist. It was a gigantic effort, undertaken 10 years before the discovery of X-rays, 27 years before they were used to prove the existence of the crystal lattice itself. The existence of 230 Fedorov groups is one of the most important geometric laws of modern structural crystallography. “The gigantic scientific feat of E.S. Fedorov, who managed to bring the entire natural “chaos” of countless crystal formations under a single geometric scheme, still evokes admiration. This discovery is akin to the discovery of the periodic table of D.I. Mendeleev. “The Kingdom of Crystals” is an unshakable monument and the ultimate the pinnacle of classical Fedorov crystallography,” said Academician A.V. Shubnikov.

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