Show the geometric bodies of its face edge vertex. Regular polyhedra. Scope of the Euler formula

Trihedral and polyhedral angles:
A trihedral angle is a shape
formed by three planes bounded by three rays emanating from
one point and not lying in one
planes.
Consider some flat
polygon and a point outside
the plane of this polygon.
Let's draw rays from this point,
passing through the peaks
polygon. We'll get a figure
which is called multifaceted
angle.

A trihedral angle is a part of space
bounded by three flat corners with a common
summit
and
in pairs
common
parties,
not
lying in the same plane. Common top About these
corners
called
summit
trihedral
angle.
The sides of corners are called edges, flat corners
at the vertex of a trihedral angle are called its
faces. Each of the three pairs of faces of a trihedral angle
forms a dihedral angle

Basic properties of a trihedral angle
1. Each plane angle of a trihedral angle is less than the sum
its other two flat corners.
+ > ; + > ; + >
α, β, γ - flat angles,
A, B, C - dihedral angles composed by planes
angles β and γ, α and γ, α and β.
2. The sum of the plane angles of a trihedral angle is less than
360 degrees
3. First cosine theorem
for a trihedral angle
4. The second cosine theorem for a trihedral angle

,
5. Sine theorem
A polyhedral angle whose interior is
located on one side of the plane of each
its faces is called a convex polyhedral
angle. Otherwise, the polyhedral angle
is called nonconvex.

A polyhedron is a body, a surface
which consists of a finite number
flat polygons.

Polyhedron elements
The faces of a polyhedron are
polygons that
form.
The edges of a polyhedron are the sides
polygons.
The vertices of the polyhedron are
polygon vertices.
The diagonal of a polyhedron is
line segment connecting 2 vertices
not belonging to the same face.

Polyhedra
convex
non-convex

The polyhedron is called convex,
if it is on one side
plane of each polygon on its
surfaces.

CONVEX POLYHEDRAL ANGLES

A polyhedral angle is called convex if it is convex
figure, i.e., together with any two of its points, it entirely contains and
the line connecting them.
The figure shows examples
convex
and
non-convex
polyhedral corners.
Theorem. The sum of all plane angles of a convex polyhedral angle is less than 360°.

CONVEX POLYTOPES

An angle polyhedron is called convex if it is a convex figure,
i.e., together with any two of its points, it entirely contains the connecting
their segment.
Cube, parallelepiped, triangular prism and pyramid are convex
polyhedra.
The figure shows examples of a convex and non-convex pyramid.

PROPERTY 1

Property 1. In a convex polyhedron, all faces are
convex polygons.
Indeed, let F be some face of the polyhedron
M, and the points A, B belong to the face F. From the convexity condition
polyhedron M, it follows that the segment AB is entirely contained
in the polyhedron M. Since this segment lies in the plane
polygon F, it will be entirely contained in this
polygon, i.e. F is a convex polygon.

PROPERTY 2

Property 2. Any convex polyhedron can be composed of
pyramids with a common vertex, the bases of which form a surface
polyhedron.
Indeed, let M be a convex polyhedron. Let's take some
an interior point S of the polyhedron M, i.e., a point of it that is not
belongs to no face of the polyhedron M. We connect the point S with
vertices of the polyhedron M as segments. Note that due to the convexity
polyhedron M, all these segments are contained in M. Consider pyramids with
vertex S whose bases are the faces of the polyhedron M. These
pyramids are entirely contained in M, and together they form the polyhedron M.

Regular polyhedra

If the faces of the polyhedron are
regular polygons with one and
the same number of sides and at each vertex
polyhedron converges the same number
edges, then a convex polyhedron
called correct.

Names of polyhedra

came from Ancient Greece,
they indicate the number of faces:
"hedra" face;
"tetra" 4;
"hexa" 6;
"octa" 8;
"ikosa" 20;
dodeca 12.

regular tetrahedron

Rice. one
Made up of four
equilateral
triangles. Each
its top is
top of three
triangles.
Therefore, the sum
flat corners at
each vertex is equal to
180º.

Regular octahedron
Rice. 2
Made up of eight
equilateral
triangles. Each
vertex of the octahedron
is the top
four triangles.
Therefore, the sum
flat corners at
each vertex 240º.

Regular icosahedron
Rice. 3
Made up of twenty
equilateral
triangles. Each
icosahedron vertex
is the top five
triangles.
Therefore, the sum
flat corners at
each vertex is equal to
300º.

Cube (hexahedron)

Rice.
4
Made up of six
squares. Each
the top of the cube is
top of three squares.
Therefore, the sum
flat corners for each
top is 270º.

Regular dodecahedron
Rice. 5
Made up of twelve
correct
pentagons. Each
dodecahedron apex
is the apex of three
correct
pentagons.
Therefore, the sum
flat corners at
each vertex is equal to
324º.

Table No. 1
Right
polyhedron
Number
faces
peaks
ribs
Tetrahedron
4
4
6
Cube
6
8
12
Octahedron
8
6
12
Dodecahedron
12
20
30
icosahedron
20
12
30

Euler formula
The sum of the number of faces and vertices of any
polyhedron
equals the number of edges plus 2.
G+W=R+2
Number of faces plus number of vertices minus number
ribs
in any polyhedron is 2.
H+W R=2

Table number 2
Number
Right
polyhedron
Tetrahedron
faces and
peaks
(G+V)
ribs
(R)
4+4=8
6
"tetra" 4;
Cube
6 + 8 = 14
12
"hexa"
6;
Octahedron
8 + 6 = 14
12
"octa"
Dodecahedron
12 + 20 = 32
30
dodeca"
12.
30
"ikosa"
20
icosahedron
20 + 12 = 32
8

Duality of regular polyhedra

Hexahedron (cube) and octahedron form
dual pair of polyhedra. Number
faces of one polyhedron is equal to the number
vertices of the other and vice versa.

Take any cube and consider a polyhedron with
vertices at the centers of its faces. How easy
make sure we get an octahedron.

The centers of the faces of the octahedron serve as the vertices of the cube.

Polyhedra in nature, chemistry and biology
The crystals of some substances familiar to us are in the form of regular polyhedra.
Crystal
pyrite-
natural
model
dodecahedron.
crystals
cooking
salts pass
cube shape.
Monocrystal
antimony
Crystal
aluminosulfate
(prism)
potassium alum sodium - tetrahedron.
has the form
octahedron.
In a molecule
methane has
form
correct
tetrahedron.
The icosahedron has been at the center of attention of biologists in their disputes over the shape
viruses. The virus cannot be perfectly round, as previously thought. To
to establish its shape, they took various polyhedra, directed light at them
at the same angles as the flow of atoms to the virus. It turned out that only one
the polyhedron gives exactly the same shadow - the icosahedron.
In the process of egg division, a tetrahedron of four cells is first formed, then
the octahedron, the cube, and finally the dodecahedral-icosahedral structure of the gastrula. And finally
perhaps the most important thing - the DNA structure of the genetic code of life - represents
a four-dimensional sweep (along the time axis) of a rotating dodecahedron!

Polyhedra in art
"Portrait of Monna Lisa"
The composition of the drawing is based on golden
triangles that are parts
regular stellated pentagon.
engraving "Melancholy"
In the foreground of the painting
depicted dodecahedron.
"Last Supper"
Christ with his disciples is depicted in
background of a huge transparent dodecahedron.

Polyhedra in architecture
Fruit Museums
The Fruit Museum in Yamanashi was created with the help of
3D modeling.
pyramids
Alexandrian lighthouse
Spasskaya Tower
Kremlin.
Four-tiered Spasskaya Tower with the Church of the Savior
Not made by hands - the main entrance to the Kazan Kremlin.
Erected in the 16th century by Pskov architects Ivan
Shiryayem and Postnik Yakovlev, nicknamed
"Barma". The four tiers of the tower are
cube, polyhedra and pyramid.

Cube, ball, pyramid, cylinder, cone - geometric bodies. Among them are polyhedrons. polyhedron called a geometric body, the surface of which consists of a finite number of polygons. Each of these polygons is called a face of the polyhedron, the sides and vertices of these polygons are called the edges and vertices of the polyhedron, respectively.

Dihedral angles between adjacent faces, i.e. faces that have a common side - an edge of a polyhedron - are also dihedral minds of the polyhedron. The angles of polygons - the faces of a convex polygon - are flat minds of the polyhedron. In addition to flat and dihedral angles, a convex polyhedron also has polyhedral angles. These angles form faces that have a common vertex.

Among the polyhedra, there are prisms and pyramids.

Prism - is a polyhedron whose surface consists of two equal polygons and parallelograms having common sides with each of the bases.

Two equal polygons are called grounds ggrzmg, and parallelograms - her lateral faces. The side faces form side surface prisms. Edges that do not lie in bases are called side ribs prisms.

The prism is called p-coal, if its bases are n-gons. On fig. 24.6 shows a quadrangular prism ABCDA"B"C"D".

The prism is called straight, if its side faces are rectangles (Fig. 24.7).

The prism is called correct , if it is straight and its bases are regular polygons.

A quadrangular prism is called parallelepiped if its bases are parallelograms.

The parallelepiped is called rectangular, if all its faces are rectangles.

Diagonal of the box is a line segment connecting its opposite vertices. A parallelepiped has four diagonals.

Proved that the diagonals of the parallelepiped intersect at one point and bisect that point. The diagonals of a rectangular parallelepiped are equal.

Pyramid- this is a polyhedron, the surface of which consists of a polygon - the base of the pyramid, and triangles that have a common vertex, called the side faces of the pyramid. The common vertex of these triangles is called summit pyramids, edges emerging from the top - side ribs pyramids.

The perpendicular dropped from the top of the pyramid to the base, as well as the length of this perpendicular is called tall pyramids.

The simplest pyramid triangular or a tetrahedron (Fig. 24.8). A feature of a triangular pyramid is that any face can be considered as a base.

The pyramid is called correct, if its base is a regular polygon, and all side edges are equal to each other.

Note that we must distinguish regular tetrahedron(i.e. a tetrahedron in which all edges are equal to each other) and regular triangular pyramid(at its base lies a regular triangle, and the side edges are equal to each other, but their length may differ from the length of the side of the triangle, which is the base of the prism).

Distinguish bulging out and non-convex polyhedra. You can define a convex polyhedron if you use the concept of a convex geometric body: a polyhedron is called convex. if it is a convex figure, i.e. together with any two of its points, it entirely contains the segment connecting them.

One can define a convex polyhedron in another way: the polyhedron is called convex if it lies entirely on one side of each of its bounding polygons.

These definitions are equivalent. We do not provide proof of this fact.

All the polyhedra that have been considered so far have been convex (cube, parallelepiped, prism, pyramid, etc.). The polyhedron shown in Fig. 24.9 is not convex.

Proved that in a convex polyhedron, all faces are convex polygons.

Consider several convex polyhedra (Table 24.1)

From this table it follows that for all the convex polyhedra considered, the equality B - P + G= 2. It turned out that it is also valid for any convex polyhedron. This property was first proved by L. Euler and was called Euler's theorem.

A convex polyhedron is called correct if its faces are equal regular polygons and the same number of faces converge at each vertex.

Using the property of a convex polyhedral angle, one can prove that various kinds There are no more than five regular polyhedra.

Indeed, if the fan and the polyhedron are regular triangles, then 3, 4 and 5 of them can converge at one vertex, since 60 "3< 360°, 60° - 4 < 360°, 60° 5 < 360°, но 60° 6 = 360°.

If three regular triangles converge at each vertex of the polyfane, then we obtain right hand/th tetrahedron, which in translation from Fech means “tetrahedral” (Fig. 24.10, a).

If four regular triangles converge at each vertex of the polyhedron, then we get octahedron(Fig. 24.10, in). Its surface consists of eight regular triangles.

If five regular triangles converge at each vertex of the polyhedron, then we get icosahedron(Fig. 24.10, d). Its surface consists of twenty regular triangles.

If the faces of a polyfane are squares, then only three of them can converge at one vertex, since 90° 3< 360°, но 90° 4 = 360°. Этому условию удовлетворяет только куб. Куб имеет шесть фаней и поэтому называется также hexahedron(Fig. 24.10, b).

If the faces of a polyphane are regular pentagons, then only phi can converge at one vertex of them, since 108° 3< 360°, пятиугольники и в каждой вершине сходится три грани, называется dodecahedron(Fig. 24.10, e). Its surface consists of twelve regular pentagons.

The faces of a polyhedron cannot be hexagonal or more, since even for a hexagon 120° 3 = 360°.

It has been proven in geometry that there are exactly five different kinds of regular polyhedra in three-dimensional Euclidean space.

To make a model of a polyhedron, you need to make it sweep(more precisely, the development of its surface).

A development of a polyhedron is a figure on a plane, which is obtained if the surface of the polyhedron is cut along some edges and unfolded so that all the polygons included in this surface lie in the same plane.

Note that a polyhedron can have several different developments, depending on which edges we have cut. Figure 24.11 shows figures that are different developments of a regular quadrangular pyramid, that is, a pyramid at the base of which lies a square, and all side edges are equal to each other.

For a plane figure to be a development of a convex polyhedron, it must satisfy a number of requirements related to the features of the polyhedron. For example, the figures in Fig. 24.12 are not scans of a regular quadrangular pyramid: in the figure shown in fig. 24.12, a, at the top M four faces converge, which cannot be in the correct quadrangular pyramid; and in the figure shown in Fig. 24.12, b, side ribs A B and Sun not equal.

In general, a development of a polyhedron can be obtained by cutting its surface not only along the edges. An example of such a cube sweep is shown in Fig. 24.13. Therefore, the unfolding of a polyhedron can be more precisely defined as a flat polygon, from which the surface of this polyhedron can be made without overlaps.

Solids of revolution

Body of rotation called the body obtained as a result of the rotation of some figure (usually flat) around a straight line. This line is called axis of rotation.

Cylinder- ego body, which is obtained as a result of the rotation of a rectangle around one of its sides. In this case, the said party is axis of the cylinder. On fig. 24.14 shows a cylinder with an axis OO', resulting from the rotation of a rectangle AA "O" O around a straight line OO". points O and O" are the centers of the bases of the cylinder.

A cylinder, which is obtained by rotating a rectangle around one of its sides, is called direct circular a cylinder, since its bases are two equal circles located in parallel planes so that the segment connecting the centers of the circles is perpendicular to these planes. The lateral surface of the cylinder is formed by segments equal to the side of the rectangle parallel to the axis of the cylinder.

Sweep the side surface of a right circular cylinder, if cut along the generatrix, is a rectangle, one side of which is equal to the length of the generatrix, and the other to the circumference of the base.

Cone- this is the body that is obtained as a result of the rotation of a right triangle around one of the legs.

In this case, the specified leg is motionless and is called cone axis. On fig. 24.15 shows a cone with an axis SO, obtained as a result of the rotation of a right triangle SOA with a right angle O around the leg S0. The point S is called top of the cone, OA is the radius of its base.

The cone that results from the rotation of a right triangle around one of its legs is called straight circular cone since its base is a circle, and the top is projected into the center of this circle. The lateral surface of the cone is formed by segments equal to the hypotenuse of the triangle, during the rotation of which the cone is formed.

If the lateral surface of the cone is cut along the generatrix, then it can be “unfolded” into a plane. Sweep the lateral surface of a right circular cone is a circular sector with radius, equal to the length generatrix.

When a cylinder, cone, or any other body of revolution is intersected by a plane containing the axis of revolution, one obtains axial section. The axial section of the cylinder is a rectangle, the axial section of the cone is an isosceles triangle.

Ball- this is the body that is obtained as a result of the rotation of a semicircle a around its diameter. On fig. 24.16 shows a ball obtained by rotating a semicircle around a diameter AA". point O called the center of the ball and the radius of the circle is the radius of the ball.

The surface of the sphere is called sphere. A sphere cannot be flattened.

Any section of a sphere by a plane is a circle. The radius of the section of the ball will be greatest if the plane passes through the center of the ball. Therefore, the section of a ball by a plane passing through the center of the ball is called big circle ball, and the circle that bounds it - big circle.

IMAGE OF GEOMETRIC BODIES ON THE PLANE

Unlike flat figures, geometric bodies cannot be accurately depicted, for example, on a sheet of paper. However, with the help of drawings on a plane, you can get a fairly clear image of spatial figures. For this, special methods of depicting such figures on a plane are used. One of them is parallel design.

Let a plane a and a line intersecting it be given a. Let us take in space an arbitrary point A" not belonging to the line a, and let's go through X direct a", parallel to a straight line a(Fig. 24.17). Straight a" intersects the plane at some point X", which is called parallel projection of the point X onto the plane a.

If point A lies on a line a, then by parallel projection X" is the point at which the line a crosses the plane a.

If point X belongs to the plane a, then the point X" coincides with the point x.

Thus, if a plane a and a straight line intersecting it are given a. then every point X space can be associated with a single point A" - a parallel projection of the point X on plane a (when designing parallel to a straight line a). plane a called projection plane. About direct a they say that she barks design direction - ggri direct replacement a any other direct result of the design parallel to it will not change. All lines parallel to a line a, set the same direction of design and are called together with a straight line a projecting lines.

projection figures F called set F' projection of all grid points. Mapping to each point X figures F"its parallel projection is a point X" figures F", called parallel design figures F(Fig. 24.18).

A parallel projection of a real object is its shadow falling on a flat surface in sunlight, since the sun's rays can be considered parallel.

Parallel design has a number of properties, the knowledge of which is necessary when depicting geometric bodies on a plane. Let us formulate the main ones without giving their proofs.

Theorem 24.1. In parallel engineering, for straight lines that are not parallel to the design direction, and for the segments lying on them, the following properties are fulfilled:

1) the projection of a straight line is a straight line, and the projection of a segment is a segment;

2) projections of parallel lines are parallel or coincide;

3) the ratio of the lengths of the projections of segments lying on the same straight line or on parallel lines is equal to the ratio of the lengths of the segments themselves.

From this theorem follows consequence: in parallel projection, the middle of the segment is projected into the middle of its projection.

When depicting geometric bodies on a plane, it is necessary to monitor the implementation of these properties. Otherwise, it can be arbitrary. So, the angles and ratios of the lengths of non-parallel segments can change arbitrarily, i.e., for example, a triangle in parallel projection is represented by an arbitrary triangle. But if the triangle is equilateral, then the projections of its medians must connect the vertex of the triangle with the midpoint of the opposite side.

And one more requirement must be observed when depicting spatial bodies on a plane - to contribute to the creation of a correct idea about them.

Let's depict, for example, an inclined prism, the bases of which are squares.

Let's first build the lower base of the prism (you can start from the top). According to the rules of parallel design, oggo will be represented by an arbitrary parallelogram ABCD (Fig. 24.19, a). Since the edges of the prism are parallel, we build parallel lines passing through the vertices of the constructed parallelogram and put equal segments AA", BB ', CC", DD" on them, the length of which is arbitrary. Connecting the points A", B", C", D in series ", we get a quadrilateral A "B" C "D", depicting the upper base of the prism. It is easy to prove that A"B"C"D"- parallelogram equal to parallelogram ABCD and, therefore, we have the image of a prism, the bases of which are equal squares, and the remaining faces are parallelograms.

If you need to depict a straight prism whose bases are squares, then you can show that the side edges of this prism are perpendicular to the base, as is done in Fig. 24.19, b.

In addition, the drawing in Fig. 24.19, b can be considered an image of a regular prism, since its base is a square - a regular quadrilateral, and also a rectangular parallelepiped, since all its faces are rectangles.

Let us now find out how to draw a pyramid on a plane.

To depict a regular pyramid, first draw a regular polygon lying at the base, and its center is a point O. Then a vertical line is drawn OS, representing the height of the pyramid. Note that the verticality of the segment OS provides greater visual clarity. And finally, the point S is connected to all the vertices of the base.

Let us depict, for example, a regular pyramid, the base of which is regular hexagon.

In order to correctly depict a regular hexagon in parallel design, you need to pay attention to the following. Let ABCDEF be a regular hexagon. Then BCEF is a rectangle (Fig. 24.20) and, therefore, with parallel projection, it will be represented by an arbitrary parallelogram B "C" E "F". Since the diagonal AD passes through the point O - the center of the polygon ABCDEF and is parallel to the segments. BC and EF and AO \u003d OD, then with parallel design it will be represented by an arbitrary segment A "D" , passing through a point O" parallel B"C" and E"F" and besides, A "O" \u003d O "D".

Thus, the sequence of constructing the base of the hexagonal pyramid is as follows (Fig. 24.21):

§ depict an arbitrary parallelogram B"C"E"F" and its diagonals; mark the point of their intersection O";

§ through a point O" draw a straight line parallel V'S"(or E "F');

§ choose an arbitrary point on the constructed line BUT" and mark a point D" such that Oh "D" = A "O" and connect the dot BUT" with dots AT" and F", and a point D" - with dots FROM" and E".

To complete the construction of the pyramid, a vertical segment is drawn OS(its length is chosen arbitrarily) and connect the point S with all the vertices of the base.

In parallel projection, the ball is depicted as a circle of the same radius. To make the image of the ball more visual, a projection of some large circle is drawn, the plane of which is not perpendicular to the projection plane. This projection will be an ellipse. The center of the ball will be depicted by the center of this ellipse (Fig. 24.22). Now you can find the corresponding poles N and S provided that the segment connecting them is perpendicular to the plane of the equator. To do this, through the dot O draw a line perpendicular to AB and mark the point C - the intersection of this line with the ellipse; then through point C we draw a tangent to the ellipse representing the equator. It has been proven that the distance CM equal to the distance from the center of the ball to each of the poles. Therefore, putting aside the segments ON and OS, equal CM, get the poles N and S.

Consider one of the methods for constructing an ellipse (it is based on a transformation of the plane, which is called compression): they build a circle with a diameter and draw chords perpendicular to the diameter (Fig. 24.23). Half of each of the chords is divided in half and the resulting points are connected by a smooth curve. This curve is an ellipse whose major axis is the segment AB, and the center is a dot O.

This technique can be used when drawing a straight circular cylinder (Fig. 24.24) and a straight circular cone (Fig. 24.25) on the plane.

A straight circular cone is depicted as follows. First, an ellipse is built - the base, then the center of the base is found - the point O and draw a perpendicular line OS, which represents the height of the cone. From point S, tangents are drawn to the ellipse (this is done “by eye”, applying a ruler) and segments are selected SC and SD these lines from the point S to the points of contact C and D. Note that the segment CD does not match the diameter of the base of the cone.

The purpose of the lesson:

Introduce the concept of regular polyhedra.
Consider the types of regular polyhedra.
Problem solving.
To instill interest in the subject, to teach to see beauty in geometric bodies, the development of spatial imagination.
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/

There are special topics in school geometry that 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:

it is convex;
all its faces are regular polygons equal to each other;
the same number of edges converge at each of its vertices;
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.

Face type	flat corner at the top	View of the polyhedral corner at the vertex	The sum of the flat angles at the vertex	AT	R	G	The name of the polyhedron
right triangle	60º	3-sided	180º	4	6	4	regular tetrahedron
right triangle	60º	4-sided	240º	6	12	8	Regular octahedron
right triangle	60º	5-sided	300º	12	30	20	Regular icosahedron
Square	90º	3-sided	270º	8	12	6	Regular hexahedron (cube)
right triangle	108º	3-sided	324º	20	30	12	Regular dodecahedron

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.

Type of polyhedron	Polyhedron volume
regular tetrahedron
Regular octahedron
Regular icosahedron
Regular hexahedron (cube)
Regular dodecahedron

"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 in ancient Greece, and the final, XII book of the famous principles 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.

A 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

A 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>

A 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:

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);
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>

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>;
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>;
parallelohedrons (convex) - polyhedra, considered as bodies, the parallel intersection of which can 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>;
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:

“Mathematical Encyclopedia”, edited by I. M. Vinogradova, publishing house " Soviet Encyclopedia”, Moscow, 1985. Volume 4, pp. 552–553 Volume 3, pp. 708–711.
“Small Mathematical Encyclopedia”, E. Fried, I. Pastor, I. Reiman et al. Publishing House of the Hungarian Academy of Sciences, Budapest, 1976. Pp. 264–267.
“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.
“Practical lessons in mathematics: Tutorial for technical schools”, publishing house “Vysshaya Shkola”, Moscow, 1979. Pp. 388–395, pp. 405.
“Repeat Mathematics”, edition 2–6, supplementary, Textbook for applicants to universities, publishing house “Vysshaya Shkola”, Moscow, 1974. Pp. 446–447.
Encyclopedic Dictionary of a Young Mathematician, A. P. Savin, publishing house "Pedagogy", Moscow, 1989. Pp. 197–199.
“Encyclopedia for children. T.P. Mathematics”, editor-in-chief M. D. Aksenova; method, and resp. editor V. A. Volodin, Avanta+ publishing house, Moscow, 2003. Pp. 338–340.
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.
“Kvant” No. 9, 11 - 1983, No. 12 - 1987, No. 11, 12 - 1988, No. 6, 7, 8 - 1989. Popular science physics and mathematics journal of the Academy of Sciences of the USSR and the Academy of Pedagogical Sciences of 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.
Solving problems of increased complexity in geometry: 11th grade - M .: ARKTI, 2002. Pp. 9, 19–20.

The content of the article

POLYHEDRON, the portion of space bounded by a collection of a finite number of planar polygons connected in such a way that each side of any polygon is a side of exactly one other polygon (called an adjacent polygon), and there is exactly one cycle of polygons around each vertex. These polygons are called faces, their sides are called edges, and their vertices are called vertices of the polyhedron.

On fig. 1 presents several well-known polyhedra. The first two are examples R-coal pyramids, i.e. polyhedra made up of R-gon, called the base, and R triangles adjacent to the base and having a common vertex (called the top of the pyramid). At R = 3 (cm. rice. one, a) any face of the pyramid can serve as a base. Pyramid, the base of which has the shape of a regular R-gon is called regular R- coal pyramid. So, we can talk about square, regular pentagonal, etc. pyramids. On fig. one, in, 1,G and 1, d examples of a certain class of polyhedra are given, the vertices of which can be divided into two sets of the same number of points; the points of each of these sets are vertices R-gon, and the planes of both p-gons are parallel. If these two R-gons (bases) are congruent and located so that the vertices of the same R R-gon by parallel straight-line segments, then such a polyhedron is called R- coal prism. Two R-angular prisms can serve as a triangular prism ( R= 3) in fig. one, in and a pentagonal prism ( R= 5) in fig. one, G. If the bases are located so that the vertices of one R-gon are connected to the vertices of another R-gon of a zigzag broken line consisting of 2 R straight line segments, as in Fig. one, d, then such a polyhedron is called R-coal antiprism.

Apart from two reasons, R- carbon prism available R faces are parallelograms. If the parallelograms are in the form of rectangles, then the prism is called a straight line, and if, in addition, the bases are regular R-gons, then the prism is called a straight line R- coal prism. R-coal antiprism has (2 p+ 2) faces: 2 R triangular faces and two p- coal bases. If the bases are congruent regular R-gons, and the line connecting their centers is perpendicular to their planes, then the antiprism is called a regular line R-coal antiprism.

In the definition of a polyhedron, the last reservation is made in order to exclude from consideration such anomalies as two pyramids with a common apex. We now introduce an additional restriction on the set of admissible polyhedra by requiring that no two faces intersect, as in Fig. one, e. Any polyhedron that satisfies this requirement divides the space into two parts, one of which is finite and is called "internal". The other, the remaining part, is called the outer.

A polyhedron is called convex if no straight line segment connecting any two of its points contains points belonging to the outer space. Polyhedra in fig. one, a, 1,b, 1,in and 1, d convex, and the pentagonal prism in Fig. one, G not convex, since, for example, the segment PQ contains points lying in the outer space of the prism.

REGULAR POLYTOPES

A convex polyhedron is called regular if it satisfies the following two conditions:

283(i) all its faces are congruent regular polygons;

(ii) each vertex is adjacent to the same number of faces.

If all edges are correct R-gons and q of them are adjacent to each vertex, then such a regular polyhedron is denoted by ( p, q). This notation was proposed by L. Schläfli (1814–1895), a Swiss mathematician who produced many elegant results in geometry and mathematical analysis.

There are non-convex polyhedra whose faces intersect and are called "regular star polyhedra". Since we agreed not to consider such polyhedra, by regular polyhedra we mean exclusively convex regular polyhedra.

Platonic Solids.

On fig. 2 shows regular polyhedra. The simplest of them is a regular tetrahedron, the faces of which are four equilateral triangles and three faces adjoin each of the vertices. The tetrahedron corresponds to the notation (3, 3). It's nothing but special case triangular pyramid. The most famous of the regular polyhedra is the cube (sometimes called the regular hexahedron) - a straight square prism, all six faces of which are squares. Since there are 3 squares adjacent to each vertex, the cube is denoted (4, 3). If two congruent square pyramids with faces in the form of equilateral triangles are combined with bases, then a polyhedron is obtained, called a regular octahedron. It is bounded by eight equilateral triangles, four triangles adjoin each of the vertices, and therefore, it corresponds to the notation (3, 4). A regular octahedron can also be considered as a special case of a right regular triangular antiprism. Consider now a direct regular pentagonal antiprism whose faces are in the form of equilateral triangles, and two regular pentagonal pyramids whose bases are congruent to the base of the antiprism and whose faces are shaped as equilateral triangles. If these pyramids are attached to an antiprism, aligning their bases, then another regular polyhedron will be obtained. Twenty of its faces are in the form of equilateral triangles, with five faces adjoining each vertex. Such a polyhedron is called a regular icosahedron and is denoted (3, 5). In addition to the four regular polyhedra mentioned above, there is one more - a regular dodecahedron, limited by twelve pentagonal faces; three faces adjoin each of its vertices, therefore the dodecahedron is denoted as (5, 3).

The five regular polyhedra listed above, often also called "Plato's solids", captured the imagination of mathematicians, mystics and philosophers of antiquity more than 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 fact that they have not lost their attractiveness to this day is very convincingly evidenced by the picture of the Spanish artist Salvador Dali. The Last Supper.

The ancient Greeks also studied many of the geometric properties of the Platonic solids; the fruits of their research can be found in the 13th book Began Euclid. The study of the Platonic solids and related figures continues to this day. And although the main motives contemporary research beauty and symmetry serve, they also have some scientific value, 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 number of regular polyhedra.

It is natural to ask whether there are other regular polyhedra besides the Platonic solids. As the following simple considerations show, the answer must be no. Let ( p, q) is an arbitrary regular polyhedron. Since its faces are correct R-gons, their interior angles, as it is easy to show, are equal (180 - 360 / R) or 180 (1 – 2/ R) degrees. Since the polyhedron ( p, q) is convex, the sum of all internal angles along the faces adjacent to any of its vertices must be less than 360 degrees. But to each vertex adjoin q faces, so the inequality

It is easy to see that p and q must be greater than 2. Substituting in (1) R= 3, we find that the only valid values q in this case are 3, 4 and 5, i.e. we obtain the polytopes (3, 3), (3, 4) and (3, 5). At R= 4 is the only valid value q is 3, i.e. polyhedron (4, 3), with R= 5 inequality (1) also satisfies only q= 3, i.e. polyhedron (5, 3). At p> 5 allowed values q does not exist. Therefore, there are no other regular polyhedra, except Plato's solids.

All five regular polyhedra are listed in the table below. The last three columns indicate N 0 - number of vertices, N 1 is the number of edges and N 2 is the number of faces of each polyhedron.

Unfortunately, the definition of a regular polyhedron given in many geometry textbooks is incomplete. A common mistake is that the definition only requires condition (i) above to be met, but omits condition (ii). Meanwhile, condition (ii) is absolutely necessary, which is easiest to verify by considering a convex polyhedron that satisfies condition (i) but does not satisfy condition (ii). The simplest example of this kind can be constructed by identifying a face of a regular tetrahedron with a face of another tetrahedron congruent to the first. As a result, we get a convex polyhedron whose six faces are congruent equilateral triangles. However, three faces are adjacent to some vertices, and four to others, which violates condition (ii).

FIVE REGULAR POLYTOPES
Name	Schläfli's entry	N 0 (number of vertices)	N 1 (number of ribs)	N 2 (number of faces)
Tetrahedron
Cube
Octahedron
icosahedron
Dodecahedron

Properties of 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.

Dual polyhedra.

Consider a regular polyhedron ( p, q) and its median sphere S. The midpoint of each edge touches the sphere. Replacing each edge with a segment perpendicular to the line tangent to S at the same point, we get N 1 edges of the polytope dual to the polytope ( p, q). It is easy to show that the faces of the dual polyhedron are regular q-gons and that each vertex is adjacent R faces. Therefore, the polyhedron ( p, q) the regular polyhedron is dual ( q, p). The polytope (3, 3) is dual to another polytope (3, 3) congruent to the original one (therefore (3, 3) is called a self-dual polytope), the polytope (4, 3) is dual to the polytope (3, 4), and the polytope (5, 3) is the polyhedron (3, 5). On fig. 3 polyhedra (4, 3) and (3, 4) are shown in the position of duality to each other. In addition, each vertex, each edge and each face of the polyhedron ( p, q) corresponds to a single face, a single edge, and a single vertex of the dual polytope ( q, p). Therefore, if ( p, q) It has N 0 vertices, N 1 rib and N 2 faces, then ( q, p) It has N 2 tops, N 1 rib and N 0 faces.

Since each of N 2 faces of a regular polyhedron ( p, q) limited R edges and each edge is common to exactly two faces, then there are pN 2/2 ribs, so N 1 = pN 2/2. The dual polyhedron ( q, p) edges also N 1 and N 0 edges, so N 1 = qN 0/2. So the numbers N 0 , N 1 and N 2 for any regular polyhedron ( p, q) are related by the relation

Symmetry.

The main interest in regular polyhedra is big number symmetries they have. By the symmetry (or symmetry transformation) of a polyhedron, we mean its movement as solid body in space (for example, rotation around a certain straight line, reflection about a certain plane, etc.), which leaves the set of vertices, edges and faces of the polyhedron unchanged. In other words, under the action of a symmetry transformation, a vertex, edge, or face either retains its original position or is transferred to the original position of another vertex, another edge, or another face.

There is one symmetry that is common to all polyhedra. It's about about the identical transformation leaving any point in its original position. We encounter a less trivial example of symmetry in the case of a straight line R-coal prism. Let l- a straight line connecting the centers of the bases. turn around l to any integer multiple of the angle 360/ R degrees is symmetry. Let, further, p- a plane passing in the middle between the bases parallel to them. Reflection about a plane p(movement that translates any point P exactly Pў , such that p crosses the segment PP¢ at a right angle and bisects it) is another symmetry. Combining reflection with respect to a plane p with a turn around a straight line l, we get another symmetry.

Any symmetry of a polyhedron can be represented as a product of reflections. The product of several motions of a polyhedron as a rigid body here means the execution of individual motions in a certain predetermined order. For example, the 360 rotation mentioned above R degrees around a straight line l is the product of reflections with respect to any two planes containing l and forming relative to each other an angle of 180 / R degrees. A symmetry that is the product of an even number of reflections is called direct, otherwise it is called inverse. Thus, any rotation around a straight line is a direct symmetry. Any reflection is an inverse symmetry.

Let us consider in more detail the symmetries of the tetrahedron, i.e. regular polyhedron (3, 3). 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. Consequently, total number 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. For clarity, it is useful to build a cardboard model of a regular tetrahedron and make sure that the tetrahedron really has 24 symmetries. Developments that can be cut out of thin cardboard and, having folded, glue five regular polyhedra out of them, are shown in fig. four.

The direct symmetries of the remaining regular polyhedra can be described not individually, but all together. Let us agree to understand by ( p, q) any regular polyhedron, except for (3, 3). The straight line passing through the center ( p, q) and any vertex passes through the opposite vertex, and any rotation by an integer multiple of 360/ q degrees around this line is symmetry. Therefore, for every such line there exist, including the identity transformation, ( q– 1) different symmetries. Each such line connects two of N 0 vertices; therefore, all such straight lines - N 0 /2, which gives ( q – 1) > N 0 /2 symmetries. In addition, the line passing through the center of the polyhedron ( p, q) and the center of any face passes through the center of the opposite face, and any rotation around such a straight line by an integer multiple of 360/ R degrees is symmetry. Since the total number of such lines is N 2 /2, where N 2 is the number of faces of the polyhedron ( p, q), we get ( p – 1) N 2/2 different symmetries, including the identity transformation. Finally, the line passing through the center and midpoint of any edge of the polyhedron ( p, q) passes through the midpoint of the opposite edge, and the symmetry is a half turn around this line. Since there is N 1/2 such lines, where N 1 is the number of edges of the polyhedron ( p, q), we get more N 1/2 symmetry. Taking into account the identical transformation, we obtain

direct symmetries. There are no other direct symmetries, and there are just as many inverse symmetries.

Although formula (3) was not obtained for the polyhedron (3, 3), it is easy to verify that it is also true for it. Thus, the polytope (3, 3) has 12 direct symmetries, the polyhedra (4, 3) and (3, 4) each have 24 symmetries, and the polytopes (5, 3) and (3, 5) each have 60 symmetries.

Readers familiar with abstract algebra will understand that the symmetries of the polyhedron ( p, q) form a group with respect to the "multiplication" defined above. In this group, direct symmetries form a subgroup of index 2, while inverse symmetries do not form a group, since they violate the closure property and do not contain the identity transformation (the identity element of the group). Usually, the group of direct symmetries is referred to as the group of a polyhedron, and the full group of symmetries is called its extended group. From the properties of dual polytopes considered above, it is clear that any regular polytope and its dual polytope have the same group. The tetrahedron group is called the tetrahedral group, the cube and octahedron group is called the octahedral group, and the dodecahedron and icosahedron group is called the icosahedral group. They are isomorphic to the alternating group BUT 4 of four characters, symmetrical group S 4 of four characters and an alternating group BUT 5 out of five characters respectively .

Euler's formula

Looking at the table, one can notice an interesting relationship between the number of vertices N 0 , the number of edges N 1 and the number of faces N 2 of any convex regular polytope ( p, q). It's about the ratio

Substituting the obtained expressions into formulas (3) and (4), we obtain that the number of direct symmetries of the polyhedron ( p, q) equals

This number can also be written in one of the equivalent forms: qN 0 , 2N 1 or pN 2 .

Scope of the Euler formula.

The significance of Euler's formula is enhanced by the fact that it is applicable not only to Platonic solids, but also to any polyhedron homeomorphic to a sphere ( cm. TOPOLOGY). This assertion is proved as follows.

Let P is any polyhedron homeomorphic to a sphere, with N 0 tops, N 1 ribs and N 2 faces; let c = N 0 – N 1 + N 2 - Euler characteristic of the polyhedron P. It is required to prove that c= 2. Since R is homeomorphic to a sphere, we can remove one face and turn the rest into some configuration on the plane (for example, in Fig. 5, a and 5, b you see a prism with its front plane removed). A "planar configuration" is a network of points and straight line segments, called "vertices" and "edges" respectively, with the vertices serving as the ends of the edges. We consider the vertices and edges of the configuration we are considering to be displaced and deformed vertices and edges of the polyhedron. So this configuration is N 0 vertices and N 1 rib. Rest N 2 - 1 faces of the polyhedron are deformed into N 2 – 1 non-overlapping areas on the plane defined by the configuration. Let's call these areas "faces" of the configuration. The vertices, edges and faces of the configuration determine the Euler characteristic, which in this case is equal to c – 1.

Now we will flatten so that if the removed face was R-gon, then all N 2 - 1 configuration faces will fill the inside R-gon. Let BUT- some vertex inside R-gon. Let's assume that in BUT converge r ribs. If you delete BUT and all r edges converging in it, then the number of vertices will decrease by 1, edges - by r, faces - on r – 1 (cm. rice. 5, b and 5, in). The new configuration Nў 0 = N 0 - 1 vertices, Nў 1 = N 1 – r ribs and Nў 2 = N 2 – 1 – (r– 1) faces; Consequently,

Thus, removing one internal vertex and edges converging at it does not change the Euler characteristic of the configuration. Therefore, by removing all internal vertices and edges converging at them, we thereby reduce the configuration to R-gon and its interior (Fig. 5, G). But the Euler characteristic remains equal to c– 1, and since the configuration has R peaks, R edges and 1 face, we get

In this way, c= 2, which was to be proved.

Further, one can prove that if the Euler characteristic of a polytope is 2, then the polytope is homeomorphic to a sphere. In other words, we can generalize the above result by showing that a polytope is homeomorphic to a sphere if and only if its Euler characteristic is 2.

Generalized Euler formula.

The generalized Euler formula is used to classify other polyhedra. If some polyhedron has 16 vertices, 32 edges and 16 faces, then its Euler characteristic is equal to 16 - 32 + 16 = 0. This allows us to assert that this polyhedron belongs to the class of polyhedra homeomorphic to a torus. A distinctive feature of this class is the Euler characteristic equal to zero. More generally, let R- a polyhedron with N 0 tops, N 1 ribs and N 2 edges. A given polyhedron is said to be homeomorphic to a surface of genus n if and only if

Finally, it should be noted that the situation becomes much more complicated if the previous restriction is relaxed, according to which no two faces of a polyhedron should intersect. For example, there appears the possibility of the existence of two non-homeomorphic polyhedra with the same Euler characteristic. They should be distinguished by other topological properties.

The purpose of the lesson:

Introduce the concept of regular polyhedra.
Consider the types of regular polyhedra.
Problem solving.
To instill interest in the subject, to teach to see beauty in geometric bodies, the development of spatial imagination.
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/

Definition of a regular polyhedron.

A polyhedron is called regular if:

it is convex;
all its faces are regular polygons equal to each other;
the same number of edges converge at each of its vertices;
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.

Face type	flat corner at the top	View of the polyhedral corner at the vertex	The sum of the flat angles at the vertex	AT	R	G	The name of the polyhedron
right triangle	60º	3-sided	180º	4	6	4	regular tetrahedron
right triangle	60º	4-sided	240º	6	12	8	Regular octahedron
right triangle	60º	5-sided	300º	12	30	20	Regular icosahedron
Square	90º	3-sided	270º	8	12	6	Regular hexahedron (cube)
right triangle	108º	3-sided	324º	20	30	12	Regular dodecahedron

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.

Type of polyhedron	Polyhedron volume
regular tetrahedron
Regular octahedron
Regular icosahedron
Regular hexahedron (cube)
Regular dodecahedron

"Platonic solids".

Where else can you see these amazing bodies?

<Рис. 6>

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

<Рис. 7>

A 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:

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);
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 of a polyhedron;

The simplest examples of polyhedra are prisms and pyramids.

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>

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>;
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>;
parallelohedrons (convex) - polyhedra, considered as bodies, the parallel intersection of which can 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>;
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:

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“Kvant” No. 9, 11 - 1983, No. 12 - 1987, No. 11, 12 - 1988, No. 6, 7, 8 - 1989. Popular science physics and mathematics journal of the Academy of Sciences of the USSR and the Academy of Pedagogical Sciences of 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.
Solving problems of increased complexity in geometry: 11th grade - M .: ARKTI, 2002. Pp. 9, 19–20.