A polyhedron is a geometric body bounded on all sides by planes - flat polygons.

Convex polyhedron - if it is located on one side of each of its faces.

A prism is a polyhedron, 2 of whose faces are n-gons lying in a parallel plane, and the remaining n-faces are parallelograms.

Polygons located in parallel planes-bases.

The set of lateral faces forms a lateral surface.

Prisms are divided into:

1) by the number of corners of the base (triangular, quadrangular, etc.)

2) by the slope of the ribs to the base (straight, inclined)

correct prism - base regular polygon.

The height of the prism is the distance between the bases.

The construction of a drawing of a prism is reduced to the construction of its vertices (characteristic points) and the construction of straight lines limited by the projection.

The development of the polyhedron is the figure obtained as a result of combining all its faces with the plane.

Sweeps are depicted as solid main lines. If necessary, apply bending lines. For development, only the natural values ​​​​of the elements are accepted.

A pyramid is a polyhedron, one face is an n-gon, and the rest are triangles that have a common vertex.

If the base of the pyramid is a regular polygon, then it is a regular pyramid. The height will pass through the center of the base. There are other types of polyhedra - prismatoids, tetrahedron, etc.

10. Surfaces. Formation and assignment of surfaces. Surfaces of revolution.

A surface is a common part of two adjacent parts of space, a continuous set of positions of lines moving in space (a trajectory of motion). Surfaces of revolution are such surfaces that are formed by the rotation of some generatrix around a fixed straight line - the axis of rotation.

During rotation, each point of the generatrix describes a circle, the center of rotation of which is located on the axis of rotation. These circles are called parallel.

The parallel of the largest diameter is naz equator.

A cylinder is a geometric body bounded by a cylindrical surface and 2 parallel planes.

If the guide is a circle, then a circular cylinder.

If the generatrix is ​​perpendicular to the line, it is a straight cylinder.

A cone is a geometric body bounded by a conical surface located on one side of the top and a plane at the base that has crossed all generators.

spherical surface. It is obtained by rotating a circle or its part located in the plane of this circle, provided that the center of the circle is on the axis of rotation.

A toric surface is obtained by rotating a circle or part of it around an axis located in the plane of this circle but not passing through its center.

11. Intersection of surfaces by a plane.

When a surface or any geometric figure intersects with a plane, a plane figure is obtained, which is called a section.

The definition of the projections of the section lines should begin with the construction of reference points - points located on the outline generators of the surface (points that determine the boundaries of visibility of the projections of the curve); points remote at extreme (maximum and minimum) distances from the projection planes. After that, arbitrary points of the section line are determined.

Construction of a section of polyhedra.

A polyhedron is a spatial figure bounded by a closed surface, consisting of compartments of planes that have the shape of polygons (in the particular case of triangles).

The sides of the polygons form edges, and the planes of the polygons form the faces of the polyhedron.

The projections of the section of polyhedra, in the general case, are polygons whose vertices belong to the edges, and the sides belong to the faces of the polyhedron*. Therefore, the problem of determining the section of a polyhedron can be reduced to a multiple solution of the problem of determining the meeting point of a straight line (edges of a polyhedron) with a plane or to the problem of finding the line of intersection of two planes (faces of a polyhedron and a cutting plane).

The first solution is called the edge method, the second is called the edge method.

Construction of a section of a surface of revolution.

The shape of the cross-sectional figure of bodies of revolution by a plane depends on the position of the secant plane.

When a circular cylinder is intersected by a plane in section, three figures of the cylinder section can be obtained:

a) a circle, if the cutting plane is perpendicular to the axis of the cylinder;

b) an ellipse, if the cutting plane is inclined to the axis of the cylinder

c) a rectangle if the cutting plane is parallel to the axis of the cylinder

Although stereometry is studied only in high school, every student is familiar with the cube, regular pyramids and other simple polyhedra. The theme "Polyhedrons" has bright applications, including in painting and architecture. In addition, according to the figurative expression of Academician Aleksandrov, it combines "ice and fire", that is, a vivid imagination and strict logic. But in school course In stereometry, little time is devoted to regular polyhedra. But for many, regular polyhedra are of great interest, but there is no way to learn more about them in the lesson. That is why I decided to talk about all the regular polyhedra that have a variety of shapes, and about their interesting properties.

The structure of regular polyhedra is very convenient for studying the many transformations of a polyhedron into itself (rotations, symmetries, etc.). The resulting transformation groups (they are called symmetry groups) turned out to be very interesting from the point of view of the theory of finite groups. The same symmetry made it possible to create a series of puzzles in the form of a regular polyhedron, which began with the "Rubik's Cube" and the "Moldovan Pyramid".

To compile the abstract, the Popular Science Physics and Mathematics Journal "Kvant" was used, from which information was taken about what a regular polyhedron is, about their number, about building all regular polyhedra and describing all the rotations at which the polyhedron is combined with its original position. From the newspaper "Mathematics" I received interesting information about stellated regular polyhedra, their properties, discovery and their application.

Now you have the opportunity to plunge into the world of the correct and magnificent, into the world of the beautiful and extraordinary, which bewitches our eyes.

1. Regular polyhedra

1. 1 Definition of regular polyhedra.

A convex polyhedron is called regular if its faces are equal regular polyhedra and all polyhedral angles are equal.

Let us consider possible regular polyhedra and, first of all, those of them whose faces are regular triangles. The simplest such regular polyhedron is a triangular pyramid whose faces are regular triangles. Three faces converge at each of its vertices. With only four faces, this polyhedron is also called a regular tetrahedron, or simply a tetrahedron, which is translated from Greek means quadrilateral.

A polyhedron whose faces are regular triangles and four faces converge at each vertex, its surface consists of eight regular triangles, therefore it is called an octahedron.

A polyhedron in which five regular triangles converge at each vertex. Its surface consists of twenty regular triangles, which is why it is called an icosahedron.

Note that since more than five regular triangles cannot converge at the vertices of a convex polyhedron, there are no other regular polygons whose faces are regular triangles.

Similarly, since only three squares can converge at the vertices of a convex polyhedron, there are no other regular polyhedra with squares as faces besides the cube. The cube has six sides and is therefore also called a hexahedron.

A polyhedron whose faces are regular pentagons and three faces converge at each vertex. Its surface consists of twelve regular pentagons, which is why it is called a dodecahedron.

It follows from the definition of a regular polyhedron that a regular polyhedron is "perfectly symmetrical": if we mark some face Г and one of its vertices A, then for any other face Г1 and its vertex А1 we can combine the polyhedron with itself by movement in space so that face G will be aligned with G1 and at the same time vertex A falls into point A1.

1. 2. Historical reference.

The five regular polyhedra listed above, often also called "Plato's solids," captured the imagination of ancient mathematicians, mystics, and philosophers over two thousand years ago. The ancient Greeks even established a mystical correspondence between the tetrahedron, cube, octahedron and icosahedron and the four natural principles - fire, earth, air and water. As for the fifth regular polyhedron, the dodecahedron, they considered it as the shape of the universe. These ideas are not only the heritage of the past. And now, after two millennia, many are attracted by the aesthetic principle underlying them.

The first four polyhedra were known long before Plato. Archaeologists have found a dodecahedron made during the Etruscan civilization at least 500 BC. e. But, apparently, in the school of Plato, the dodecahedron was discovered independently. There is a legend about a student of Plato, Hippase, who died at sea because he divulged the secret of "a ball with twelve pentagons."

It has been well known since the time of Plato and Euclid that there are exactly five types of regular polyhedra.

Let's prove this fact. Let all faces of some polyhedron be regular p-gons and k be the number of faces adjoining a vertex (it is the same for all vertices). Consider the vertex A of our polyhedron. Let M1, M2,. , Mk - ends of k edges emerging from it; since the dihedral angles at these edges are equal, AM1M2Mk is a regular pyramid: when rotated through an angle of 360º/k around the altitude AH, the vertex M goes into M, the vertex M1 goes into M2. Mk to M1 .

Let's compare the isosceles triangles AM1M2 and HM1M2 They have a common base, and the side of AM1 is larger than HM1, so M1AM2

Tetrahedron 3 3 4 4 6

Cube 4 3 8 6 12

Octahedron 3 4 6 8 12

Dodecahedron 5 3 20 12 30

Icosahedron 3 5 12 20 30

1. 3. Construction of regular polyhedra.

All the corresponding polyhedra can be constructed using the cube as a basis.

To get a regular tetrahedron, it is enough to take four non-adjacent vertices of the cube and cut off pyramids from it with four planes, each of which passes through three of the taken vertices

Such a tetrahedron can be inscribed in a cube in two ways.

The intersection of two such regular tetrahedra is just a regular octahedron: a polyhedron of eight triangles with vertices located at the centers of the faces of the cube.

2. Properties of regular polyhedra.

2. 1. Sphere and regular polyhedra.

The vertices of any regular polyhedron lie on a sphere (which is hardly surprising, given that the vertices of any regular polygon lie on a circle). In addition to this sphere, called the "circumscribed sphere", there are two other important spheres. One of them, the "middle sphere", passes through the midpoints of all edges, and the other, the "inscribed sphere", touches all faces at their centers. All three spheres have a common center, which is called the center of the polyhedron.

Radius of the circumscribed sphere Name of the polyhedron Radius of the inscribed sphere

Tetrahedron

Dodecahedron

icosahedron

2. 1. Self-matching of polyhedra.

What self-combinations (rotations that translate into themselves) do the cube, tetrahedron and octahedron have? Note that some point - the center of the polyhedron - passes into itself under any self-coincidence, so that all self-coincidences have a common fixed point.

Let's see what rotations with a fixed point A are in general in space. Let's show that such a rotation is necessarily a rotation through some angle around some straight line passing through the point A. It is enough for our motion F (c F (A) \u003d A) to indicate a fixed straight line. You can find it like this: consider three points M1, M2 = F(M1) and M3 = F(M2) that are different from the fixed point A, draw a plane through them and drop the perpendicular AH onto it - this will be the desired line. (If M3 = M1, then our line passes through the midpoint of the segment M1M2, and F is axial symmetry: rotation through an angle of 180°).

So, self-alignment of a polyhedron is necessarily a rotation around an axis passing through the center of the polyhedron. This axis intersects our polyhedron at a vertex or an interior point of an edge or face. Therefore, our self-alignment translates into itself a vertex, edge or face, which means that it translates into itself a vertex, the middle of an edge or the center of a face. Conclusion: the movement of a cube, tetrahedron or octahedron, combining it with itself, is a rotation around the axis of one of three types: the center of the polyhedron is the vertex, the center of the polyhedron is the middle of the edge, the center of the polyhedron is the center of the face.

In general, if a polyhedron coincides with itself when rotated around a straight line through an angle of 360 ° / m, then this straight line is called the m-th order symmetry axis.

2. 2. Movement and symmetry.

The main interest in regular polyhedra is the large number of symmetries they possess.

Considering the self-coincidences of polyhedra, one can include in their number not only rotations, but also any movements that transform the polyhedron into itself. Here motion is any transformation of space that preserves pairwise distances between points.

In the number of movements, in addition to rotations, you need to include mirror movements. Among them are symmetry with respect to a plane (reflection), as well as the composition of reflection with respect to a plane and rotation around a straight line perpendicular to it (this is a general view of a mirror movement that has a fixed point). Of course, such movements cannot be realized by continuous movement of the polyhedron in space.

Let us consider in more detail the symmetries of the tetrahedron. Any line passing through any vertex and center of the tetrahedron passes through the center of the opposite face. A rotation of 120 or 240 degrees around this line is one of the symmetries of the tetrahedron. Since the tetrahedron has 4 vertices (and 4 faces), we get a total of 8 direct symmetries. Any straight line passing through the center and midpoint of an edge of a tetrahedron passes through the midpoint of the opposite edge. A 180 degree turn (half turn) around such a straight line is also a symmetry. Since the tetrahedron has 3 pairs of edges, we get 3 more direct symmetries. Therefore, the total number of direct symmetries, including the identity transformation, goes up to 12. It can be shown that there are no other direct symmetries and that there are 12 inverse symmetries. Thus, the tetrahedron allows a total of 24 symmetries.

Direct symmetries of the remaining regular polyhedra can be calculated by the formula [(q - 1)N0 + N1 + (p - 1)N2]/2 + 1, where p is the number of sides of regular polygons that are faces of the polyhedron, q is the number of faces adjacent to each vertex, N0 is the number of vertices, N1 is the number of edges, and N2 is the number of faces of each polyhedron.

The hexahedron and octahedron each have 24 symmetries, while the icosahedron and dodecahedron each have 60 symmetries.

All regular polyhedra have planes of symmetry (a tetrahedron has 6, a cube and an octahedron have 9 each, an icosahedron and a dodecahedron have 15 each).

2. 3. Star polyhedra.

In addition to regular polyhedra, stellated polyhedra have beautiful shapes. There are only four of them. The first two were discovered by I. Kepler (1571 - 1630), and the other two were built almost 200 years later by L. Poinsot (1777 - 1859). That is why regular stellated polyhedra are called Kepler-Poinsot solids. They are obtained from regular polyhedra by extending their faces or edges. The French geometer Poinsot constructed four regular stellated polyhedra in 1810: the small stellated dodecahedron, the great stellated dodecahedron, the great dodecahedron, and the great icosahedron. These four polyhedra have faces that are intersecting regular polyhedra, and for two of them, each of the faces is a self-intersecting polygon. But Poinsot failed to prove that there are no other regular polyhedra.

A year later (in 1811) this was done by the French mathematician Augustin Louis Cauchy (1789 - 1857). He took advantage of the fact that, according to the definition of a regular polyhedron, it can be superimposed on itself in such a way that its arbitrary face is combined with a pre-selected one. It follows from this that all faces of the star polyhedron are equidistant from some point-center of the sphere inscribed in the polyhedron.

The planes of the faces of the stellated polyhedron, intersecting, also form a regular convex polyhedron, that is, a Platonic solid described around the same sphere. Cauchy called this Platonic solid the core of this stellated polyhedron. Thus, a stellated polyhedron can be obtained by continuing the planes of the faces of one of the Platonic solids.

From the tetrahedron, cube and octahedron, stellated polyhedra cannot be obtained. Consider the dodecahedron. Extending its edges causes each face to be replaced by a stellated regular pentagon, resulting in a small stellated dodecahedron.

On the continuation of the faces of the dodecahedron, the following two cases are possible: 1) if we consider regular pentagons, then we get a large dodecahedron.

2) if stellated pentagons are considered as faces, then a large stellated dodecahedron is obtained.

The icosahedron has one stellation. Extending the face of a regular icosahedron produces a great icosahedron.

Thus, there are four types of regular star polyhedra.

Star-shaped polyhedrons are very decorative, which allows them to be widely used in the jewelry industry in the manufacture of all kinds of jewelry.

Many forms of stellated polyhedra are suggested by nature itself. Snowflakes are stellated polyhedra. Since ancient times, people have tried to describe all possible types of snowflakes, and have compiled special atlases. Several thousand different types of snowflakes are now known.

Conclusion

The following topics are covered in the work: regular polyhedra, construction of regular polyhedra, self-combination, motion and symmetries, stellated polyhedra and their properties. We learned that there are only five regular polyhedra and four star regular polyhedra, which have found wide application in various fields.

The study of the Platonic solids and related figures continues to this day. And although the main motives of modern research are beauty and symmetry, they also have some scientific significance especially in crystallography. Common salt, sodium thioantimonide, and chromic alum crystals occur naturally in the form of a cube, tetrahedron, and octahedron, respectively. The icosahedron and dodecahedron are not found among crystalline forms, but they can be observed among the forms of microscopic marine organisms known as radiolarians.

The ideas of Plato and Kepler about the connection of regular polyhedra with the harmonious structure of the world have found their continuation in our time in an interesting scientific hypothesis, which in the early 80s. expressed by Moscow engineers V. Makarov and V. Morozov. They believe that the core of the Earth has the form and properties of a growing crystal that affects the development of all natural processes walking on the planet. The rays of this crystal, or rather, its force field, determine the icosahedral-dodecahedral structure of the Earth. It manifests itself in earth's crust as if the projections of those inscribed in Earth regular polyhedra: icosahedron and dodecahedron.

Many mineral deposits stretch along the icosahedron-dodecahedron grid; The 62 vertices and midpoints of the edges of polyhedra, called nodes by the authors, have a number of specific properties that make it possible to explain some incomprehensible phenomena. Here are the hearths ancient cultures and civilizations: Peru, Northern Mongolia, Haiti, Ob culture and others. Highs and lows are observed at these points. atmospheric pressure, giant swirls of the oceans. At these nodes are Loch Ness, the Bermuda Triangle. Further studies of the Earth, perhaps, will determine the attitude towards this scientific hypothesis, in which, apparently, regular polyhedra occupy an important place.

The structure of regular polyhedra is very convenient for studying the many transformations of a polyhedron into itself (rotations, symmetries, etc.). The resulting transformation groups (they are called symmetry groups) turned out to be very interesting from the point of view of the theory of finite groups. The same symmetry made it possible to create a series of puzzles in the form of regular polyhedrons, which began with the "Rubik's Cube" and the "Moldovan Pyramid".

Sculptors, architects, and artists also showed great interest in the forms of regular polyhedra. They were all amazed by the perfection, the harmony of polyhedrons. Leonardo da Vinci (1452 - 1519) was fond of the theory of polyhedra and often depicted them on his canvases. Salvador Dali in the painting The Last Supper"Depicted I. Christ with his disciples against the backdrop of a huge transparent dodecahedron.

The purpose of the lesson:

1. Introduce the concept of regular polyhedra.
2. Consider the types of regular polyhedra.
3. Problem solving.
4. To instill interest in the subject, to teach to see beauty in geometric bodies, the development of spatial imagination.
5. Intersubject communications.

Visibility: tables, models.

During the classes

I. Organizational moment. Inform the topic of the lesson, formulate the objectives of the lesson.

II. Learning new material/

Available in school geometry special topics, which you look forward to, anticipating a meeting with incredibly beautiful material. These topics include “Regular polyhedra”. Here, not only the wonderful world of geometric bodies with unique properties opens up, but also interesting scientific hypotheses. And then the geometry lesson becomes a kind of study of unexpected aspects of the usual school subject.

None of the geometric bodies possess such perfection and beauty as regular polyhedra. “Regular polyhedra are defiantly few,” L. Carroll once wrote, “but this detachment, which is very modest in number, managed to get into the very depths of various sciences.”

Definition of a regular polyhedron.

A polyhedron is called regular if:

1. it is convex;
2. all its faces are regular polygons equal to each other;
3. the same number of edges converge at each of its vertices;
4. all its dihedral angles are equal.

Theorem: There are five different (up to similarity) types of regular polyhedra: regular tetrahedron, regular hexahedron (cube), regular octahedron, regular dodecahedron, and regular icosahedron.

Table 1.Some properties of regular polyhedra are given in the following table.

Consider the types of polyhedra:

regular tetrahedron

<Рис. 1>

Regular octahedron

<Рис. 2>

Regular icosahedron

<Рис. 3>

Regular hexahedron (cube)

<Рис. 4>

Regular dodecahedron

<Рис. 5>

Table 2. Formulas for finding volumes of regular polyhedra.

"Platonic solids".

The cube and the octahedron are dual, i.e. are obtained from each other if the centroids of the faces of one are taken as the vertices of the other and vice versa. The dodecahedron and the icosahedron are similarly dual. The tetrahedron is dual to itself. A regular dodecahedron is obtained from a cube by constructing “roofs” on its faces (Euclid's method), the vertices of a tetrahedron are any four vertices of the cube that are not pairwise adjacent along an edge. This is how all other regular polyhedra are obtained from the cube. The very fact of the existence of only five really regular polyhedra is amazing - after all, there are infinitely many regular polygons on the plane!

All regular polyhedra were known back in Ancient Greece, and the final, XII book of the famous beginnings of Euclid is dedicated to them. These polyhedra are often called the same Platonic solids in the idealistic picture of the world given by the great ancient Greek thinker Plato. Four of them personified the four elements: the tetrahedron-fire, the cube-earth, the icosahedron-water and the octahedron-air; the fifth polyhedron, the dodecahedron, symbolized the entire universe. In Latin, they began to call him quinta essentia (“fifth essence”).

Apparently, it was not difficult to come up with the correct tetrahedron, cube, octahedron, especially since these forms have natural crystals, for example: a cube is a monocrystal of sodium chloride (NaCl), an octahedron is a single crystal of potassium alum ((KAlSO 4) 2 l2H 2 O). There is an assumption that the ancient Greeks obtained the shape of the dodecahedron by considering crystals of pyrite (sulphurous pyrite FeS). Having the same dodecahedron, it is not difficult to build an icosahedron: its vertices will be the centers of 12 faces of the dodecahedron.

Where else can you see these amazing bodies?

In a very beautiful book by the German biologist of the beginning of our century, E. Haeckel, “The Beauty of Forms in Nature,” one can read the following lines: “Nature nourishes in its bosom an inexhaustible number of amazing creatures that far surpass all forms created by human art in beauty and diversity.” The creations of nature in this book are beautiful and symmetrical. This is an inseparable property of natural harmony. But here one-celled organisms are visible - feodarii, the shape of which accurately conveys the icosahedron. What caused this natural geometrization? Maybe because of all the polyhedra with the same number of faces, it is the icosahedron that has the largest volume and the smallest surface area. This geometric property helps the marine microorganism overcome the pressure of the water column.

It is also interesting that it was the icosahedron that turned out to be the focus of attention of biologists in their disputes regarding the shape of viruses. The virus cannot be perfectly round, as previously thought. To establish its shape, they took various polyhedrons, directed light at them at the same angles as the flow of atoms to the virus. It turned out that the properties mentioned above make it possible to save genetic information. Regular polyhedra are the most profitable figures. And nature takes advantage of this. Regular polyhedra determine the shape of the crystal lattices of some chemical substances. The next task will illustrate this idea.

Task. The model of the CH 4 methane molecule has the shape of a regular tetrahedron, with hydrogen atoms at four vertices and a carbon atom in the center. Determine the bond angle between two CH bonds.

<Рис. 6>

Solution. Since a regular tetrahedron has six equal edges, it is possible to choose a cube such that the diagonals of its faces are the edges of a regular tetrahedron. The center of the cube is also the center of the tetrahedron, because the four vertices of the tetrahedron are also the vertices of the cube, and the sphere described around them is uniquely determined by four points that do not lie in the same plane.

Triangle AOC is isosceles. Hence, a is the side of the cube, d is the length of the diagonal of the side face or edge of the tetrahedron. So, a = 54.73561 0 and j = 109.47 0

Task. In a cube of one vertex (D), diagonals of faces DA, DB and DC are drawn and their ends are connected by straight lines. Prove that the polytope DABC formed by four planes passing through these lines is a regular tetrahedron.

<Рис. 7>

Task. The edge of the cube is a. Calculate the surface of a regular octahedron inscribed in it. Find its relation to the surface of a regular tetrahedron inscribed in the same cube.

<Рис. 8>

Generalization of the concept of a polyhedron.

A polyhedron is a collection of a finite number of plane polygons such that:

1. each side of any of the polygons is at the same time a side of the other (but only one (called adjacent to the first) along this side);
2. from any of the polygons that make up the polyhedron, one can reach any of them by passing to the one adjacent to it, and from this, in turn, to the one adjacent to it, etc.

These polygons are called faces, their sides are called edges, and their vertices are the vertices of the polyhedron.

The following definition of a polyhedron takes on a different meaning depending on how the polygon is defined:

- if a polygon is understood as flat closed broken lines (even though they intersect themselves), then they come to this definition polyhedron;

- if a polygon is understood as a part of a plane bounded by broken lines, then from this point of view, a polyhedron is understood as a surface composed of polygonal pieces. If this surface does not intersect itself, then it is the full surface of some geometric body, which is also called a polyhedron. From here, a third point of view arises on polyhedra as geometric bodies, and the existence of “holes” in these bodies, limited by a finite number of flat faces, is also allowed.

The simplest examples of polyhedra are prisms and pyramids.

The polyhedron is called n- coal pyramid, if it has one of its faces (base) any n- a square, and the remaining faces are triangles with a common vertex that does not lie in the plane of the base. A triangular pyramid is also called a tetrahedron.

The polyhedron is called n-coal prism, if it has two of its faces (bases) equal n-gons (not lying in the same plane), obtained from each other by parallel translation, and the remaining faces are parallelograms, the opposite sides of which are the corresponding sides of the bases.

For any polytope of genus zero, the Euler characteristic (the number of vertices minus the number of edges plus the number of faces) is equal to two; symbolically: V - P + G = 2 (Euler's theorem). For a polyhedron of the genus p the relation B - R + G \u003d 2 - 2 p.

A convex polyhedron is a polyhedron that lies on one side of the plane of any of its faces. The most important are the following convex polyhedra:

<Рис. 9>

1. regular polyhedra (Plato's solids) - such convex polyhedra, all faces of which are the same regular polygons and all polyhedral angles at the vertices are regular and equal<Рис. 9, № 1-5>;
2. isogons and isohedra - convex polyhedra, all polyhedral angles of which are equal (isogons) or equal to all faces (isohedra); moreover, the group of rotations (with reflections) of an isogon (isohedron) around the center of gravity takes any of its vertices (faces) to any of its other vertices (faces). The polyhedra obtained in this way are called semi-regular polyhedra (Archimedes solids)<Рис. 9, № 10-25>;
3. parallelohedrons (convex) - polyhedra, considered as bodies, by parallel intersection of which it is possible to fill the entire infinite space so that they do not enter into each other and do not leave voids between themselves, i.e. formed a division of space<Рис. 9, № 26-30>;
4. If by a polygon we mean flat closed broken lines (even if they are self-intersecting), then 4 more non-convex (star-shaped) regular polyhedra (Poinsot bodies) can be indicated. In these polyhedra, either the faces intersect each other, or the faces are self-intersecting polygons.<Рис. 9, № 6-9>.

III. Homework assignment.

IV. Solving problems No. 279, No. 281.

V. Summing up.

List of used literature:

1. “Mathematical Encyclopedia”, edited by I. M. Vinogradova, publishing house " Soviet Encyclopedia”, Moscow, 1985. Volume 4, pp. 552–553 Volume 3, pp. 708–711.
2. “Small Mathematical Encyclopedia”, E. Fried, I. Pastor, I. Reiman et al. Publishing House of the Hungarian Academy of Sciences, Budapest, 1976. Pp. 264–267.
3. “Collection of problems in mathematics for applicants to universities” in two books, edited by M.I. Scanavi, book 2 - Geometry, publishing house " graduate School”, Moscow, 1998. Pp. 45–50.
4. “Practical lessons in mathematics: Tutorial for technical schools”, publishing house “Vysshaya Shkola”, Moscow, 1979. Pp. 388–395, pp. 405.
5. “Repeat Mathematics”, edition 2–6, supplementary, Textbook for applicants to universities, publishing house “Vysshaya Shkola”, Moscow, 1974. Pp. 446–447.
6. Encyclopedic Dictionary of a Young Mathematician, A. P. Savin, publishing house "Pedagogy", Moscow, 1989. Pp. 197–199.
7. “Encyclopedia for children. T.P. Mathematics", Chief Editor M. D. Aksenova; method, and resp. editor V. A. Volodin, Avanta+ publishing house, Moscow, 2003. Pp. 338–340.
8. Geometry, 10–11: Textbook for educational institutions/ L.S. Atanasyan, V.F. Butuzov, S.B. Kadomtsev and others - 10th edition - M .: Education, 2001. Pp. 68–71.
9. “Kvant” No. 9, 11 - 1983, No. 12 - 1987, No. 11, 12 - 1988, No. 6, 7, 8 - 1989. Popular scientific and mathematical journal of the Academy of Sciences of the USSR and the Academy pedagogical sciences THE USSR. Publishing house "Science". The main edition of physical and mathematical literature. Page 5-9, 6-12, 7-9, 10, 4-8, 13, 16, 58.
10. Problem solving increased complexity in geometry: 11th grade - M .: ARKTI, 2002. Pp. 9, 19–20.

Municipal Educational Institution

Gymnasium No. 26

Geometry

The main types of polyhedra and their properties

Performed:

9th grade student

Baisakova Lyazzat

Teacher:

Sysoeva Elena Alekseevna

Chelyabinsk

Introduction

Until now, in the course of geometry, we have been engaged in planimetry - we have studied the properties of flat geometric figures, that is, figures that are completely located in a plane. But most of the objects around us are not completely flat, they are located in space. The section of geometry that studies the properties of figures in space is called stereometry ( from other Greek. στερεός, "stereos" - "solid, spatial" and μετρέω - "I measure").

The main figures in space are dot , straight And plane. Along with these simple figures, stereometry considers geometric bodies and their surfaces. When studying geometric bodies, use the images in the drawing.

Figure 1 Figure 2

Figure 1 shows a pyramid, figure 2 - a cube. These geometric bodies are called polyhedra. Consider some types and properties of polyhedra.

multifaceted surface. Polyhedron

A polyhedral surface is a union of a finite number of plane polygons such that each side of any of the polygons is at the same time a side of another (but only one) polygon, called adjacent to the first polygon.

From any of the polygons that make up a polyhedral surface, one can reach any other by moving along adjacent polygons.

The polygons that make up a polyhedral surface are called its faces; the sides of the polygons are called edges, and the vertices are the vertices of the polyhedral surface.

Figure 1 shows unions of polygons that meet the specified requirements and are polyhedral surfaces. Figure 2 shows figures that are not polyhedral surfaces.

The polyhedral surface divides the space into two parts - the inner area of ​​the polyhedral surface and the outer area. Of the two outer regions, there will be one in which it is possible to draw straight lines that entirely belong to the region.

5 The union of a polyhedral surface and its interior is called a polyhedron. In this case, the polyhedral surface and its inner region are called, respectively, the surface and the inner region of the polyhedron. The faces, edges and vertices of the surface of a polyhedron are called, respectively, the faces, edges and vertices of the polyhedron.

Pyramid

A polyhedron, one of whose faces is an arbitrary polyhedron, and the remaining faces are triangles having one common vertex, is called a pyramid.

The polygon is called the base of the pyramid, and the remaining faces (triangles) are called the side faces of the pyramid.

There are triangular, quadrangular, pentagonal, etc. pyramids depending on the type of polygon lying at the base of the pyramid.

A triangular pyramid is also called a tetrahedron. Figure 1 shows a quadrangular pyramid SABCD with base ABCD and side faces SAB, SBC, SCD, SAD.

The sides of the faces of the pyramid are called the edges of the pyramid. The ribs belonging to the base of the pyramid are called base ribs, and all other ribs are called lateral ribs. The common vertex of all triangles (side faces) is called the top of the pyramid (in Fig. 1 point S is the top of the pyramid, the segments SA, SB, SC, SD are the side edges, the segments AB, BC, CD, AD are the edges of the base).

The height of the pyramid is the segment of the perpendicular drawn from the top of the pyramid S to the plane of the base (the ends of this segment are the top of the pyramid and the base of the perpendicular). In Fig.1 SO - the height of the pyramid.

Correct pyramid. A pyramid is called regular if the base of the pyramid is a regular polygon, and orthogonal projection vertex to the plane of the base coincides with the center of the polygon lying at the base of the pyramid.

All side ribs correct pyramid are equal to each other; all side faces are equal isosceles triangles.

The height of the side face of a regular pyramid, drawn from its top, is called the apothem of this pyramid. In Figure 2, SN is an apothem. All apothems of a regular pyramid are equal to each other.

Prism

A polyhedron whose two faces are equal n-gons lying in parallel planes, and the rest n faces - parallelograms, called n- coal prism.

polyhedron pyramid prism parallelepiped

A couple of equals n-gons are called the bases of the prism. The remaining faces of the prism are called its lateral faces, and their union is called the lateral surface of the prism. Figure 1 shows a pentagonal prism.

The sides of the faces of the prism are called ribs, and the ends of the ribs are called the vertices of the prism. Edges that do not belong to the base of the prism are called lateral edges.

A prism whose side edges are perpendicular to the planes of the bases is called a straight prism. Otherwise, the prism is called oblique.

The segment perpendicular to the planes of the bases of the prism, the ends of which belong to these planes, is called the height of the prism.

A right prism whose base is a regular polygon is called a regular prism.

Parallelepiped

A parallelepiped is a hexahedron whose opposite faces are pairwise parallel. Parallelepiped has 8 vertices, 12 edges; its faces are pairwise equal parallelograms.

Parallelepiped is called straight if its side edges are perpendicular to the plane of the base (in this case, 4 side faces are rectangles); rectangular, if parallelepiped a straight line and a rectangle serves as the base (hence, 6 faces are rectangles);

Parallelepiped, all of whose faces are squares, is called a cube.

Volume Parallelepiped is equal to the product of the area of ​​its base and its height.

body volume

Each polyhedron has a volume that can be measured using the chosen volume unit. A cube is taken as a unit of measurement of volumes, the edge of which is equal to the unit of measurement of segments. A cube with an edge of 1 cm is called cubic centimeter. Similarly defined cubic meter And cubic millimeter, etc.

In the process of measuring volumes with the selected unit of measurement, the volume of the body is expressed positive number, which shows how many units of measurement of volumes and its parts fit into this body. The number expressing the volume of the body depends on the choice of unit for measuring volumes. Therefore, the unit of measurement of volumes is indicated after this number.

The main properties of volumes:

1. Equal bodies have equal volumes.

2. If the body is composed of several bodies, then its volume is equal to the sum of the volumes of these bodies.

To find the volumes of bodies, in a number of cases it is convenient to use a theorem called Cavalieri principle .

Cavalieri's principle is as follows: if at the intersection of two bodies by any plane parallel to some given plane, we get sections equal area, then the volumes of the bodies are equal to each other.

Conclusion

So, polyhedra studies a section of geometry called stereometry. Polyhedra are different types(pyramid, prism, etc.) and have different properties. Also, it should be noted that polyhedra, unlike flat figures, have volume and are located in space.

Most of the objects around us are in space, and the study of polyhedra helps us to get an idea of ​​the reality around us in terms of geometry.

Bibliography

1. Geometry. Textbook for grades 7-9.

3. Wikipedia

Polyhedra not only occupy a prominent place in geometry, but also occur in Everyday life each person. Not to mention artificially created household items in the form of various polygons, starting with a matchbox and ending with architectural elements, crystals in the form of a cube (salt), prism (crystal), pyramid (scheelite), octahedron (diamond), etc. d.

The concept of a polyhedron, types of polyhedra in geometry

Geometry as a science contains a section of stereometry that studies the characteristics and properties of three-dimensional bodies, the sides of which in three-dimensional space are formed by limited planes (faces), are called "polyhedra". Types of polyhedra include more than a dozen representatives, differing in the number and shape of faces.

However, all polyhedra have common properties:

1. All of them have 3 integral components: a face (the surface of a polygon), a vertex (the corners formed at the junction of the faces), an edge (the side of the figure or a segment formed at the junction of two faces).
2. Each polygon edge connects two, and only two, faces that are adjacent to each other.
3. Convexity means that the body is completely located only on one side of the plane on which one of the faces lies. The rule applies to all faces of the polyhedron. Such geometric figures in stereometry are called convex polyhedra. The exception is star-shaped polyhedra, which are derivatives of regular polyhedral geometric solids.

Polyhedra can be divided into:

1. Types of convex polyhedra, consisting of the following classes: ordinary or classical (prism, pyramid, parallelepiped), regular (also called Platonic solids), semi-regular (second name - Archimedean solids).
2. Non-convex polyhedra (stellated).

Prism and its properties

Stereometry as a branch of geometry studies the properties of three-dimensional figures, types of polyhedra (a prism is one of them). A prism is a geometric body that necessarily has two absolutely identical faces (they are also called bases) lying in parallel planes, and the n-th number of side faces in the form of parallelograms. In turn, the prism also has several varieties, including such types of polyhedra as:

1. Parallelepiped - formed if the base is a parallelogram - a polygon with 2 pairs of equal opposite angles and 2 pairs of congruent opposite sides.
2. has ribs perpendicular to the base.
3. characterized by the presence of non-right angles (other than 90) between the faces and the base.
4. A regular prism is characterized by bases in the form with equal side faces.

The main properties of a prism:

• Congruent bases.
• All edges of the prism are equal and parallel to each other.
• All side faces are parallelogram-shaped.

Pyramid

A pyramid is a geometric body, which consists of one base and the n-th number of triangular faces, connected at one point - the vertex. It should be noted that if the side faces of the pyramid are necessarily represented by triangles, then at the base there can be either a triangular polygon, or a quadrangle, and a pentagon, and so on ad infinitum. In this case, the name of the pyramid will correspond to the polygon at the base. For example, if there is a triangle at the base of the pyramid - this is a quadrilateral - quadrangular, etc.

Pyramids are cone-like polyhedra. The types of polyhedra of this group, in addition to those listed above, also include the following representatives:

1. has a regular polygon at the base, and its height is projected to the center of a circle inscribed in the base or described around it.
2. A rectangular pyramid is formed when one of the side edges intersects with the base at a right angle. In this case, it is also fair to call this edge the height of the pyramid.

Pyramid properties:

• If all the side edges of the pyramid are congruent (of the same height), then they all intersect with the base at the same angle, and around the base you can draw a circle with a center coinciding with the projection of the top of the pyramid.
• If a regular polygon lies at the base of the pyramid, then all side edges are congruent, and the faces are isosceles triangles.

Regular polyhedron: types and properties of polyhedra

In stereometry, a special place is occupied by geometric bodies with absolutely equal faces, at the vertices of which the same number of edges are connected. These solids are called Platonic solids, or regular polyhedra. Types of polyhedra with such properties have only five figures:

1. Tetrahedron.
2. Hexahedron.
3. Octahedron.
4. Dodecahedron.
5. Icosahedron.

Regular polyhedra owe their name to the ancient Greek philosopher Plato, who described these geometric bodies in his writings and connected them with the natural elements: earth, water, fire, air. The fifth figure was awarded the similarity with the structure of the universe. In his opinion, the atoms of natural elements in shape resemble the types of regular polyhedra. Due to their most fascinating property - symmetry, these geometric bodies were of great interest not only to ancient mathematicians and philosophers, but also to architects, artists and sculptors of all times. The presence of only 5 types of polyhedra with absolute symmetry was considered a fundamental discovery, they were even awarded a connection with the divine principle.

Hexahedron and its properties

In the form of a hexagon, the successors of Plato assumed a similarity with the structure of the atoms of the earth. Of course, at present, this hypothesis has been completely refuted, which, however, does not prevent the figures from attracting the minds of famous figures with their aesthetics in modern times.

In geometry, the hexahedron, also known as a cube, is considered a special case of a parallelepiped, which, in turn, is a kind of prism. Accordingly, the properties of the cube are associated with the only difference is that all the faces and corners of the cube are equal to each other. The following properties follow from this:

1. All edges of a cube are congruent and lie in parallel planes with respect to each other.
2. All faces are congruent squares (there are 6 in total in a cube), any of which can be taken as a base.
3. All interhedral angles are 90.
4. From each peak comes equal amount edges, namely 3.
5. The cube has 9 which all intersect at the intersection point of the diagonals of the hexahedron, called the center of symmetry.

Tetrahedron

A tetrahedron is a tetrahedron with equal faces in the form of triangles, each of the vertices of which is a junction point of three faces.

Properties of a regular tetrahedron:

1. All faces of a tetrahedron - this from which it follows that all faces of a tetrahedron are congruent.
2. Since the base is represented by the correct geometric figure, that is, it has equal sides, then the faces of the tetrahedron converge at the same angle, that is, all angles are equal.
3. The sum of the flat angles at each of the vertices is 180, since all angles are equal, then any angle of a regular tetrahedron is 60.
4. Each of the vertices is projected to the point of intersection of the heights of the opposite (orthocenter) face.

Octahedron and its properties

Describing the types of regular polyhedra, one cannot fail to note such an object as an octahedron, which can be visually represented as two quadrangular regular pyramids glued together at the bases.

Octahedron properties:

1. The very name of a geometric body suggests the number of its faces. The octahedron consists of 8 congruent equilateral triangles, in each of the vertices of which an equal number of faces converge, namely 4.
2. Since all the faces of an octahedron are equal, so are its interface angles, each of which is equal to 60, and the sum of the plane angles of any of the vertices is thus 240.

Dodecahedron

If we imagine that all the faces of a geometric body are regular pentagon, then you get a dodecahedron - a figure of 12 polygons.

Dodecahedron properties:

1. Three faces intersect at each vertex.
2. All faces are equal and have the same edge length and equal area.
3. The dodecahedron has 15 axes and planes of symmetry, and any of them passes through the vertex of the face and the middle of the opposite edge.

icosahedron

No less interesting than the dodecahedron, the icosahedron is a three-dimensional geometric body with 20 equal faces. Among the properties of a regular twenty-hedron, the following can be noted:

1. All faces of the icosahedron are isosceles triangles.
2. Five faces converge at each vertex of the polyhedron, and the sum of the adjacent angles of the vertex is 300.
3. The icosahedron, like the dodecahedron, has 15 axes and planes of symmetry passing through the midpoints of opposite faces.

Semiregular polygons

In addition to the Platonic solids, the group of convex polyhedra also includes the Archimedean solids, which are truncated regular polyhedra. The types of polyhedra of this group have the following properties:

1. Geometric bodies have pairwise equal faces of several types, for example, a truncated tetrahedron has 8 faces, just like a regular tetrahedron, but in the case of an Archimedean solid, 4 faces will be triangular shape and 4 - hexagonal.
2. All angles of one vertex are congruent.

Star polyhedra

Representatives of non-volumetric types of geometric bodies are star-shaped polyhedra, the faces of which intersect with each other. They can be formed by merging two regular three-dimensional bodies or by continuing their faces.

Thus, such stellated polyhedra are known as: stellated forms of the octahedron, dodecahedron, icosahedron, cuboctahedron, icosidodecahedron.