Consider axial and central symmetries as properties of some geometric figures; Consider axial and central symmetries as properties of some geometric figures; Be able to build symmetrical points and be able to recognize figures that are symmetrical about a point or a line; Be able to build symmetrical points and be able to recognize figures that are symmetrical about a point or a line; Improving problem solving skills; Improving problem solving skills; Continue work on the accuracy of recording and performing a geometric drawing; Continue work on the accuracy of recording and performing a geometric drawing;

Oral work "Gentle poll" Oral work "Gentle poll" What point is called the midpoint of the segment? Which triangle is called an isosceles triangle? What property do the diagonals of a rhombus have? Formulate the property of the bisector of an isosceles triangle. Which lines are called perpendicular? What is an equilateral triangle? What property do the diagonals of a square have? What figures are called equal?

What new concepts did you learn in class? What new concepts did you learn in class? What have you learned about geometric shapes? What have you learned about geometric shapes? Give examples of geometric figures with axial symmetry. Give examples of geometric figures with axial symmetry. Give an example of figures with central symmetry. Give an example of figures with central symmetry. Give examples of objects from the surrounding life that have one or two types of symmetry. Give examples of objects from the surrounding life that have one or two types of symmetry.

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

"Construction of geometric shapes" - Educational aspect. Control and correction of assimilation. The study of the theory on which the method is based. In stereometry - not strict constructions. stereometric constructions. algebraic method. Transformation method (similarity, symmetry, parallel translation, etc.). For example: straight; angle bisector; median perpendicular.

"Human Figure" - The shape and movement of the human body is largely determined by the skeleton. Fair with a theatrical performance. Do you think there is a job for an artist in a circus? The skeleton plays the role of a frame in the structure of the figure. Main Body (belly, chest) Didn't pay attention Head, face, hands. A. Mathis. Proportions. Ancient Greece.

"Symmetry about a line" - Symmetry about a line is called axial symmetry. The straight line a is the axis of symmetry. Symmetry about a straight line. Bulavin Pavel, 9B class. How many axes of symmetry does each figure have? A figure can have one or more axes of symmetry. central symmetry. Equosceles trapezoid. Rectangle.

"Squares of figures geometry" - Pythagorean theorem. Areas of various figures. Solve the puzzle. Figures with equal areas are called equal areas. Area units. Area of ​​a triangle. Rectangle, triangle, parallelogram. square centimeter. Figures of equal area. Equal figures b). square millimeter. in). What will be the area of ​​the figure made up of figures A and D.

"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 value of the function in. But when calculating the limit of the function at. Equal to value. Expression. Aspiration. Or you can say this: in a sufficiently small neighborhood of the point. Compiled from. Decision. Continuous on intervals. In between.

Homothety and similarity.Homothety - a transformation in which each point M (plane or space) is assigned a point M", lying on OM (Fig. 5.16), and the ratio OM":OM= λ the same for all points other than O. fixed point O is called the homothety center. Attitude OM": OM considered positive if M" and M lie on one side of O, negative - on opposite sides. Number X is called the homothety coefficient. At X< 0 homothety is called inverse. Atλ = - 1 homothety becomes a symmetry transformation about a point O. With homothety, a straight line passes into a straight line, parallel lines and planes are preserved, angles (linear and dihedral) are preserved, each figure passes 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 symmetry.Symmetry (in broad sense) - a property of a geometric figure Ф, characterizing a certain correctness of its shape, its invariance under the action of movements and reflections. The figure Ф has symmetry (symmetric) if there are non-identical orthogonal transformations that take this figure into itself. The set of all orthogonal transformations that combine the figure Ф with itself is the group of this figure. So, a flat figure (Fig. 5.18) with a dot M, transforming-

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

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

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

SYMMETRY OF SPATIAL FIGURES

According to the famous German mathematician G. Weyl (1885-1955), "symmetry is the idea through which man has been trying for centuries 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 the plane was considered in the course of planimetry. In particular, the concepts of central and axial symmetry were defined. For spatial figures, the concept of symmetry is defined in a similar way.
Consider first the central symmetry.
symmetrical about a point Oh, called center of symmetry, if O is the midpoint of the segment AA". The point O is considered to be symmetrical to itself.
A space transformation in which each point A is associated with a point A symmetric to it (with respect to a given point O) is called central symmetry. The point O is called center of symmetry.
The two figures F and F" are called centrally symmetrical, if there is a symmetry transformation that takes one of them to the other.
Figure F is called centrally symmetrical if it is centrally symmetrical to itself.
For example, a box is centrally symmetrical about the intersection point 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.
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 to O 1 with respect to O 2 is also the center of symmetry of this figure.
Proof. Let A be a point in space, A 2 is a point symmetrical to it with respect to O 2 , A 1 – point symmetrical to A 2 relative to O 1 and A 3 – symmetric 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 are symmetric with respect to O 3 . So the symmetry with respect to O 3 is a composition of symmetries with respect to O 2 , O 1 and O 2 . Consequently, with this symmetry, the figure Ф transforms into itself, i.e. O 3 is the center of symmetry of the F.

Consequence.Any figure either does not have a center of symmetry, or has one center of symmetry, or has an infinite number of centers of symmetry

Indeed, if O 1 , O 2 are the centers of symmetry of the figure Ф, then the point O 3 symmetrical to O 1 with respect to O 2 is also the center of symmetry of this figure. Similarly, point O 4 symmetrical O 2 with respect 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.

Consider now the concept axial symmetry.
Points A and A" of space are called symmetrical about a straight line a called axis of symmetry if straight a passes through the midpoint of the segment AA "and is perpendicular to this segment. Each point of the line a considered symmetrical to itself.
A space transformation in which each point A is associated with a point A symmetric to it (with respect to a given line a), is called axial symmetry. Straight a it is called axis of symmetry.
The two figures are called symmetrical about a straight line a if the symmetry transformation about this line takes one of them to the other.
The figure Ф in space is called symmetrical about a straight line a if it is symmetrical to itself.
For example, a cuboid 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 midpoints of opposite edges.
A 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. If aa 1 , a 2 - the axis of symmetry of the figure Ф, then the straight linea 3, symmetrical a 1 relatively 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 the axes of symmetry of the given figure Ф, then the line passing through the intersection point and perpendicular to the plane of these lines will also be the axis of symmetry of the figure Ф.
Proof. Consider the coordinate axes O x, O y, O z. Symmetry around 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 the point of the figure Ф with coordinates ( x, –y, –z) to a 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 a point of the figure Ф with coordinates (– x, -y, z). Therefore, the O axis z is the axis of symmetry of the F.

Consequence.Any figure in space cannot have an even (non-zero) number of symmetry axes.
Indeed, we fix some axis of symmetry a. If a b- axis of symmetry, does not intersect a or intersects it not at a right angle, then for it there is one more axis of symmetry b', symmetrical with respect to a. If the axis of symmetry b crosses a at a right angle, then for it there is one more axis of symmetry b' passing through the point of intersection and perpendicular to the plane of the 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 symmetry axes is impossible.
In addition to the axes of symmetry defined above, we also consider axes of symmetry n-th order, n 2 .
Straight a called axis of symmetry n-th order figure Ф, if when turning the figure Ф around a straight line a at an angle, the figure Ф 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-angular pyramid, a straight line passing through the top and center of the base is the axis of symmetry n-th order.
Let us find out which axes of symmetry have regular polyhedra.
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 symmetry of the second order passing through the midpoints of opposite edges.
The icosahedron has six 5th-order axes of symmetry passing through opposite vertices; ten axes of symmetry of the 3rd order passing through the centers of opposite faces and fifteen axes of symmetry of the 2nd order passing through the midpoints of opposite edges.
The dodecahedron has six 5th-order axes of symmetry passing through the centers of opposite faces; ten axes of symmetry of the 3rd order passing through opposite vertices and fifteen axes of symmetry of the 2nd order passing through the midpoints of opposite edges.
Consider the concept mirror symmetry.
Points A and A" in space are called symmetrical about the plane, or, in other words, mirror symmetrical, if this plane passes through the midpoint of the segment AA "and is perpendicular to it. Each point of the plane is considered symmetrical to itself.
The transformation of space, in which each point A is associated with a point A symmetric to it (with respect to the given plane), is called mirror symmetry. The plane is called plane of symmetry.
The two figures are called mirror symmetrical with respect to a plane if a symmetry transformation with respect to that plane takes one of them to the other.
The figure Ф in space is called mirror symmetrical if it is mirror symmetric to itself.
For example, a cuboid is mirror-symmetrical with respect to a plane passing through the axis of symmetry and parallel to one of the pairs of opposite faces. The cylinder is mirror-symmetric with respect to any plane passing through its axis, etc.
Among the 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 with respect to parallel planes is a parallel translation by a vector perpendicular to these planes and equal in magnitude to twice the distance between these planes.
Consequence. Parallel transport can be represented as a composition of two mirror symmetries.
Property 6. The composition of two mirror symmetries with respect to 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 with respect to the plane O xy translates a point with coordinates ( x, y, z) to a point with coordinates ( x, y, –z). The composition of three mirror symmetries about the coordinate planes translates the point with coordinates ( x, y, z) to a point with coordinates (– x, -y, -z), which is centrally symmetric to the starting point A.
Movements that translate the figure F into itself form a group with respect to the composition. It is called symmetry group figures F.
Let us find the order of the symmetry group of the cube.
It is clear that any movement that takes the cube into itself leaves the center of the cube in place, moves the centers of the faces to the centers of the faces, the midpoints of the edges to the midpoints of the edges, and the vertices to the vertices.
Thus, to set the movement of the cube, it is enough to determine where the center of the face, the middle of the edge of this face, and the vertex of the edge go.
Consider partitioning a cube into tetrahedra, the vertices of each of which are the center of the cube, the center of the face, the midpoint of the edge of this face, and the vertex of the edge. There are 48 such tetrahedra. Since the motion is completely determined by which of the tetrahedra the given tetrahedron is transferred to, the order of the cube symmetry group will be 48.
Similarly, the orders of the symmetry groups of the tetrahedron, octahedron, icosahedron, and dodecahedron are found.
Find the symmetry group of the unit circle S 1 . This group is denoted O(2). It is an infinite topological group. We represent the unit circle as a group of complex numbers modulo one. There is a natural epimorphism p:O(2) --> S 1 , which assigns to an element u of the group O(2) 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 about the axis Ox. Therefore, O(2)/Z 2S1 . Moreover, if the group structure is not taken into account, then there is a homeomorphism O(2) and the direct product S 1 and Z 2 .
Similarly, the symmetry group of the two-dimensional sphere S 2 is denoted by O(3), and it satisfies the isomorphism O(3)/O(2) S 2 .
The 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, by 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 order 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 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 symmetry laws that govern atomic structures in crystals were clearly and accurately deduced.
The founder of the mathematical theory of the structure of crystals is an 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 deduced 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 later, in honor of the scientist, were named Fedorov's. It was a gigantic work undertaken 10 years before the discovery of X-rays, 27 years before they proved 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 all the natural "chaos" of countless crystal formations under a single geometric scheme, still arouses 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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