A geometric body consisting of 6 faces is called. Drawing. I. Organizational moment

3 4 6 12 8 home O 3 5 12 30 20 h O I 4 3 8 12 6 home O 5 3 20 30 12 h O

Hexahedron or cube

The name of each polyhedron comes from the Greek name for the number of its faces and the word "face".

• Combinatorial properties Euler derived a formula connecting the number of vertices (V), faces (G) and edges (P) of any convex polyhedron simple relation

• : B + G = P + 2.

• The ratio of the number of vertices of a regular polyhedron to the number of edges of one of its faces is equal to the ratio of the number of faces of the same polyhedron to the number of edges emerging from one of its vertices. For a tetrahedron this ratio is 4:3, for a hexahedron and octahedron it is 2:1, and for a dodecahedron and icosahedron it is 4:1. A regular polyhedron can be described combinatorially by the Schläfli symbol (, p q A regular polyhedron can be described combinatorially by the Schläfli symbol (), Where: p- number of sides of each face;

- the number of edges converging at each vertex.

Schläfli symbols for regular polyhedra are given in the following table:		Polyhedron	Peaks	Ribs	Edges
Schläfli symbol		4	6	4	{3, 3}
tetrahedron		8	12	6	{4, 3}
cube		6	12	8	{3, 4}
octahedron		20	30	12	{5, 3}
dodecahedron		12	30	20	{3, 5}

icosahedron

From these relations and Euler’s formula we can obtain the following expressions for B, P and G: Geometric properties

Angles

Each regular polyhedron is associated with certain angles that characterize its properties. The dihedral angle between adjacent faces of a regular polyhedron (p, q) is given by the formula:

Sometimes it is more convenient to use an expression in terms of tangent:

where takes the values ​​4, 6, 6, 10 and 10 for the tetrahedron, cube, octahedron, dodecahedron and icosahedron, respectively.

Schläfli symbols for regular polyhedra are given in the following table:	The angular defect at the vertex of a polyhedron is the difference between 2π and the sum of the angles between the edges of each face at this vertex. Defect at any vertex of a regular polyhedron: θ		Dihedral angle	Flat angle between ribs at apex	Angular defect (δ)		Solid angle at vertex (Ω)
Schläfli symbol	Solid angle subtended by a face		70.53°	π			π
tetrahedron	60°	1	60°
cube	90°	√2	109.47°
octahedron	60°, 90°		116.57°
dodecahedron	108°		138.19°

60°, 108°

Radii, areas and volumes

• Associated with each regular polyhedron are three concentric spheres:
• A circumscribed sphere passing through the vertices of a polyhedron;
• A median sphere touching each of its edges in the middle;

An inscribed sphere touching each of its faces at its center.

The radii of the circumscribed () and inscribed () spheres are given by the formulas:

where h is the value described above when determining dihedral angles (h = 4, 6, 6, 10 or 10). The ratio of circumscribed radii to inscribed radii is symmetrical with respect to p and q:

Surface area S of a regular polyhedron (p, q) is calculated as the area of ​​a regular p-gon multiplied by the number of faces Г:

The volume of a regular polyhedron is calculated as the volume multiplied by the number of faces regular pyramid, the base of which is a regular p-gon, and the height is the radius of the inscribed sphere r:

The table below contains a list of various radii, surface areas and volumes of regular polyhedra. The value of edge length a in the table is equal to 2.

Polyhedron (a = 2)	Radius of the inscribed sphere ( r)	Radius of the median sphere (ρ)	Radius of the circumscribed sphere ( R)

The constants φ and ξ are given by the expressions

Among regular polyhedra, both the dodecahedron and the icosahedron represent the best approximation to a sphere. The icosahedron has greatest number faces, the largest dihedral angle and is most closely pressed against its inscribed sphere. On the other hand, the dodecahedron has the smallest angular defect, the largest solid angle at the vertex, and maximally fills its circumscribed sphere.

GEOMETRIC BODIES, THEIR SURFACES AND VOLUMES

GEOMETRIC BODY. POLYHEDRON

Definition: The union of a limited spatial region and its boundary is called a geometric body.

Boundary is the surface of a geometric body.

Spatial region is the internal region of a geometric body.

Definition: A polyhedron is a geometric body whose surface is a finite number of polygons; each side of any polygon is the side of two and only two faces that do not lie in the same plane. Polygons are the faces of a polyhedron.

Vertices and sides of faces are the vertices and edges of a polyhedron.

Polyhedra are classified according to the number of faces: Schläfli symbol(tetrahedron), pentahedron(pentahedron), hexahedron(hexagon), cube(octahedron), octahedron(dodecahedron), dodecahedron(twenty-sided).

Definition: A diagonal of a polyhedron is a segment connecting two vertices that do not belong to the same face.

PRISM. PARALLELEPIPED

Definition: A polyhedron, two of whose faces are polygons belonging to parallel planes, and the remaining faces are parallelograms, is called a prism. Polygons belonging to parallel planes are the bases of a prism. Parallelograms are the lateral faces of a prism.

The sides of parallelograms connecting the corresponding vertices of the bases of the prism are the lateral edges of the prism.

A 1 A 2 ...A p V 1 V 2 ...V p – n-gonal prism;

A 1 A 2 ...A p; V 1 V 2…V p – bases of n-gonal prism;

A 1 B 1 B 2 A 2; ...; A 1 B 1 B p A p – lateral faces of the n-gonal prism;

A 1 B 1; A 2 B 2; ... ; A p B p – lateral edges of the n-gonal prism.

Properties:

The bases of the prism are equal and parallel.

The lateral edges of the prism are equal and parallel.

Definition: A prism is called straight if its side edges are perpendicular to the bases (Fig. 1.), otherwise the prism is called inclined (Fig. 2.).

Fig.1. Rice. 2. Fig.3.

A prism is called triangular, quadrangular, pentagonal, ... depending on which polygon lies at its base.

Definition: Perpendicular drawn from which - or the points of one base to the plane of another base is called the height of the prism (Fig. 3.).

B 1 M^ A 1 A 2 A 3; O 1 O 2^A 1 A 2 A 3;

B 1 M = O 1 O 2 = h – prism height.

Comment: The height of a straight prism is equal to its lateral edge .

Definition: A right prism is called regular if its bases are regular polygons.

Comment: The lateral faces of a regular prism are equal rectangles.

Reference:

1. Regular quadrilateral is a square;

2. Regular triangle - equilateral triangle;

3. Regular hexagon.

Definition: A prism whose base is a parallelogram is called a parallelepiped (Fig. 1.).

Definition: A right parallelepiped is a parallelepiped whose side edges are perpendicular to the bases (Fig. 2.).

Properties:

1. The opposite faces of a parallelepiped are equal and parallel.
2. The diagonals of a parallelepiped intersect and are divided in half by the intersection point.
3. In a rectangular parallelepiped, the square of any diagonal equal to the sum squares of its linear dimensions .d 2 = a 2 + b 2 + c 2
4. The diagonals of a rectangular parallelepiped are equal.

Exercises:

1. Determine the diagonals of a rectangular parallelepiped from its measurements:

a) 8, 9, 12;

B) 12, 16, 21.

Reference: The sum of the squares of the diagonals of a parallelogram is equal to the sum of the squares of all its sides.

1. In a right parallelepiped, the sides of the base are 5 cm and 3 cm, and one of the diagonals is 4 cm. Find the larger diagonal of the parallelepiped, knowing that the smaller diagonal forms an angle of 60° with the plane of the base.
2. In a regular quadrangular prism, the base area is 144 cm 2 and the height is 14 cm. Determine the diagonal of this prism.

PRISM SURFACE

Definition: The total surface area of ​​a prism is the sum of the areas of all its faces.

Definition: The lateral surface area of ​​a prism is the sum of the areas of its lateral faces.

Definition: The perpendicular section of a prism is the polygon obtained by intersecting the prism with a plane perpendicular to its edges.

Theorem: The area of ​​the lateral surface of the prism is equal to the product of the lateral edge and the perimeter of the perpendicular section.

Given:

ABCDA 1 B 1 C 1 D 1 – prism;

A A 1 = l;

l^ KLMNP;

P^= P(KLMNP)

Prove:

Consequence: The lateral surface area of ​​a straight prism is equal to the product of the perimeter of its base and its height.

; ;

Exercises:

Given an inclined triangular prism, two lateral faces of which are mutually perpendicular, their common edge is 9.6 cm and is located at a distance of 4.8 cm and 14 cm from the other two edges. Find the area of ​​the lateral surface of the prism.

6. In a rectangular parallelepiped, its dimensions are in the ratio 1:2:3 (3:7:8). The total surface area of ​​the parallelepiped is 352 cm 2. Find its measurements.

7. Find the total surface area of ​​a right parallelepiped, the sides of which are 8 in. and 12 in. and form an angle of 30°, and the side edge is 6 in.

8. The total surface area of ​​a cube is 36 cm2. Determine its diagonal.

9. Find the edge of a cube if its total surface area is 24 m2.

In a right parallelepiped, the sides of the base are 10 cm and 17 cm, one of the diagonals of the base is 21 cm. The major diagonal of the parallelepiped is 29 cm. Determine the total surface area of ​​the parallelepiped.

15. In a right parallelepiped, the sides of the base are 3 cm and 8 cm, the angle between them is 60°. The area of ​​the lateral surface of the parallelepiped is 220 cm 2. Determine the total surface area of ​​the parallelepiped and the area of ​​the smaller diagonal section.

16. The diagonal of a regular quadrangular prism is 9 cm. The total surface area of ​​the prism is 144 cm 2. Determine the side of the base and the side edge of the prism.

VOLUME OF DIRECT PRISM

BASIC PROPERTIES OF VOLUME

1. Two equal polyhedra have the same volume, regardless of their location in space.
2. The volume of a polyhedron, which is the sum of two adjacent polyhedra, is equal to the sum of the volumes of these polyhedra.
3. If of two polyhedra the first is entirely contained within the second, then the volume of the first polyhedron does not exceed the volume of the second polyhedron.

Definition: Polyhedra that have equal volumes are called equal-sized.

Definition: A unit of volume is the volume of a cube whose edge is equal to a unit of length.

VOLUME OF STRAIGHT PRISM

Theorem: The volume of a rectangular parallelepiped is equal to the product of its linear dimensions.

– linear dimensions(measurements)

Theorem: The volume of a straight prism is equal to the product of the area of ​​the base and the height of the prism.

Given:

ABCA 1 B 1 C 1 – straight prism;

– base of the prism;

; ;

VOLUME OF TILTED PRISM

Theorem: The volume of an inclined prism is equal to the product of the perpendicular cross-sectional area of ​​the prism and its lateral edge.

Given:

- inclined prism;

- side rib;

- perpendicular section;

Prove:

Consequence: The volume of an inclined prism is equal to the product of the area of ​​the base and the height of the prism.

Exercises:

1. In an inclined parallelepiped, the sides of the perpendicular section, equal to 3 cm and 4 cm, form an angle of 30° with each other. The lateral edge of the parallelepiped is 1 dm. Find the volume of the parallelepiped.

2. The base of the prism is a regular triangle with a side of 4 cm. The lateral edge of the prism is 6 cm and makes an angle of 60° with the plane of the base. Find the volume of the prism and the perpendicular cross-sectional area of ​​the prism.

3. The base of a right parallelepiped is a parallelogram, one of the angles of which is 30°. The base area of ​​the parallelepiped is 16 dm2. The areas of the side faces of the parallelepiped are 24 dm 2 and 48 dm 2. Find the volume of the parallelepiped.

4. In a rectangular parallelepiped, the sides of the base are in the ratio 7:24, and the diagonal cross-sectional area is 50 cm 2. Find the area of ​​the lateral surface of the parallelepiped.

5. At the base of a straight prism lies a rhombus with side a and angle 60°. The section drawn through the major diagonal of the base and the vertex of the obtuse angle of the other base is a right triangle. Find the total surface area of ​​the prism.

6. The areas of the lateral faces of a right triangular prism are 425 cm 2, 250 cm 2, 225 cm 2, and the area of ​​the base of the prism is 100 cm 2. Find the volume of the prism.

7. Given an inclined parallelepiped, the base of which is a square with a side of 5 dm. Find the volume of a parallelepiped if one of the side edges forms an angle of 60° with each adjacent side of the base and is equal to 1 m.

The base of a straight prism is an isosceles triangle, the side of which is 1 m, and the base is 1 m 20 cm. The lateral edge of the prism is equal to the height of the base, lowered to its side. Find the total surface area of ​​the prism.

Rice. 1. Fig. 2.

Exercises:

1. The base of the pyramid is a rectangle with sides 12 cm and 16 cm. Each side edge of the pyramid is 26 cm. Find the height of the pyramid.
2. The base of the pyramid is a parallelogram with sides 3 cm and 7 cm and a diagonal of 6 cm. The height of the pyramid is 4 cm and passes through the intersection point of the diagonals of the parallelogram. Find side ribs pyramids.
3. The height of a regular quadrangular pyramid is 7 cm, and the side of the base is 8 cm. Find the side edge of the pyramid.
4. The base of the pyramid is an isosceles triangle, the base of which is 6 cm and the height is 9 cm. The lateral edges of the pyramid are equal to each other and each contains 13 cm. Find the height of the pyramid.
5. The base of the pyramid is an isosceles triangle with a base of 12 cm and a side of 10 cm. The lateral faces of the pyramid form equal dihedral angles of 45° with the base. Find the height of the pyramid.

Point O is equally distant from the vertices of triangle ABC, therefore, it is the center of the circle circumscribed about this triangle. Center of a circle circumscribed about right triangle, there is the middle of the hypotenuse. Point O is the middle of the hypotenuse.

.

; .

; ; ; ; .

; , hence, .

- an equilateral triangle, which means .

; .

on three sides, therefore, .

;

; ;

;

.

Answer: .

Comment: The lateral surface area is incorrect truncated pyramid is calculated by definition as the sum of the areas of its lateral faces.

Exercises:

VOLUME OF THE PYRAMID

Theorem: The volume of a pyramid is equal to one third of the product of the area of ​​the base of the pyramid and its height.

Given:

SABC - pyramid;

S(ABC)= S basic

SO^ ABC; SO = h.

Prove:

9. VOLUME OF A TRUNCATED PYRAMID

Given:

ABCDA 1 B 1 C 1 D 1 - truncated pyramid;

S(ABCD) = S n.o. ; S (A 1 B 1 C 1 D 1) = S v.o.

h - height of the truncated pyramid;

Define: V us.pir. - ?

.

Exercises:

1. Diagonal of a square base regular pyramid is 6 cm, the height of the pyramid is 15 cm. Find its volume.
2. The lateral edge of a regular hexagonal pyramid is 14 dm, the side of its base is 2 dm. Find the volume of the pyramid.
3. The base of the pyramid is a rhombus with a side of 15 cm. The side faces of the pyramid are inclined to the plane of the base at an angle of 45°. The major diagonal of the base is 24 cm. Find the volume of the pyramid.
4. Find the volume of a truncated pyramid if the areas of its bases are 98 cm 2 and 32 cm 2, and the height of the corresponding full pyramid equal to 14 cm.
5. In a pyramid, a plane is drawn through the middle of the height, parallel to its base. Determine the volume of the resulting truncated pyramid if the height of this pyramid is 18 cm and the area of ​​its base is 400 cm 2.
6. Find volume triangular pyramid, the side edges of which are perpendicular in pairs and equal to 10 cm, 15 cm, 9 cm.
7. In a triangular truncated pyramid, the height is 10 cm, the sides of the lower base are 27 m, 29 m, 52 m, and the perimeter of the upper base is 72 m. Find the volume of the truncated pyramid.
8. The sides of the bases of a regular quadrangular truncated pyramid are 40 cm and 10 cm. Its total surface area is 3400 cm 2. Find the volume of a truncated pyramid.

CYLINDER. SURFACE AND VOLUME OF THE CYLINDER.

Definition: The geometric body obtained by rotating a rectangle around one of its sides is called a right circular cylinder.

Definition: A cylinder is called straight if its generators are perpendicular to the planes of the bases.

AB– axis of symmetry, height of the cylinder; AB = H ;

AD– radius of the cylinder base; AD = R .

Definition: The distance between the planes of the bases is the height of a right circular cylinder.

The radius of a cylinder is the radius of its base. The axis of a cylinder is a straight line passing through the centers of the bases. It is parallel to the generators.

The two circles are reasons straight circular cylinder. A segment connecting the points of the circles of the bases and perpendicular to the planes of the bases is called generatrix straight circular cylinder.

Definition: A rectangle, one side of which is equal to the circumference of the base of the cylinder, and the other to its height, is called the development of the lateral surface of the cylinder.

The surface of the cylinder consists of the base and side surface. The lateral surface is composed of generatrices.

In what follows, we will consider only the straight cylinder, calling it simply a cylinder for brevity.

Definition: A cylinder is called equilateral if its height is equal to the diameter of the base.

Sections of a cylinder.

The cross section of a cylinder with a plane parallel to its axis is a rectangle. Its two sides are the generators of the cylinder, and the other two are parallel chords of the bases.

In particular, the rectangle is the axial section. Axial section- section of a cylinder by a plane passing through its axis.

The cross section of a cylinder by a plane parallel to the base is a circle.

The cross section of a cylinder with a plane not parallel to the base and its axis is an oval.

Theorem: The area of ​​the lateral surface of a cylinder is equal to the product of the circumference of its base and its height ( S side = 2πRH, Where R− radius of the cylinder base, N− height of the cylinder).

Definition: The total surface area of ​​a cylinder is the sum of the areas of the lateral surface and the two bases.

S basic = πR 2 S side = 2πRH S full = 2πRH + 2πR 2 .

Let's consider P -gonal straight prism. At p→∞ the perimeter of the polygon lying at the base of the prism will tend to the circumference of the base of the cylinder, the area of ​​the polygon lying at the base of the prism will tend to the area of ​​the circle that is the base of the cylinder. Volume P -a straight carbon prism will tend to the volume of a right circular cylinder.

Definition: A prism is said to be inscribed in a cylinder if its bases are inscribed in the bases of the cylinder.

Definition: A cylinder is said to be inscribed in a prism if its bases are inscribed in the bases of the prism.

Exercises:

1. The diagonal of the axial section of the cylinder is 48 cm. The angle between this diagonal and the generatrix of the cylinder is 60°. Find: height, base radius, base area of ​​the cylinder.

2. The axial cross-sectional area of ​​the cylinder is 10 cm 2, and the base area is 5 cm 2. Find the height of the cylinder.

3. The radius of the base of the cylinder is 4 cm, and the area of ​​its axial section is 72 cm 2. Find the volume of the cylinder.

A square with a side equal to a rotates around an external axis that is parallel to its side. The axis is removed from the square at a distance equal to the side of the square. Find the total surface area and volume of the body of rotation.

11. At the base of a straight prism lies a square with side 2. The lateral edges are equal

12. At the base of a straight prism lies a right triangle with legs 6 and 8. The lateral edges are equal . Find the volume of the cylinder circumscribed around this prism.

13. Find the volume part of the cylinder shown in Figure 1.

14. Find the volume part of the cylinder shown in Figure 2.

Rice. No. 1. Rice. No. 2.

CONE. SURFACE AND VOLUME OF THE CONE.

Cone (from Greek "konos")- Pine cone.

The cone has been known to people since ancient times. In 1906, the book “On the Method”, written by Archimedes (287-212 BC), was discovered; this book gives a solution to the problem of the volume of the common part of intersecting cylinders. Archimedes says that this discovery belongs to the ancient Greek philosopher Democritus (470-380 BC), who, using this principle, obtained formulas for calculating the volume of a pyramid and a cone.

Circular cone called a body that consists of a circle - base of the cone, point not lying in the plane of this circle - vertices of the cone and all segments connecting the top of the cone with the points of the base (Fig. 1) The segments connecting the top of the cone with the points of the base circle are called forming a cone.

The cone is called direct, if the straight line connecting the top of the cone with the center of the base is perpendicular to the plane of the base.

For a straight cone, the base of the height coincides with the center of the base. The axis of a right cone is the straight line containing its height.

Definition: The geometric body obtained by rotating a right triangle around one of its legs is called a right circular cone.

Definition: The altitude of a cone is the perpendicular descended from its top to the plane of the base.

Definition: The development of the lateral surface of a cone is a sector of a circle, the radius of which is equal to the generatrix of the cone, and the length of the arc is the circumference of the base of the cone.

Cone sections.

A plane perpendicular to the axis of the cone intersects the cone in a circle, and lateral surface– in a circle with the center on the axis of the cone.

A plane perpendicular to the axis of the cone cuts off a smaller cone from it. The remaining part is called a truncated cone.

The section of a cone by a plane passing through its vertex is an isosceles triangle, the sides of which form the cone.

Definition: The axial section of a cone is the section passing through the axis of the cone.

Conclusion: The axial section of the cone is an isosceles triangle, the base of which is the diameter of the base of the cone, and the sides are the forming parts of the cone.

The surface of the cone consists of a base and a side surface.

Cone lateral surface area can be found using the formula:

S side = πRL, where R– radius of the base, L– length of the generatrix.

Total surface area of ​​a cone is found by the formula:

S full = πRL + πR 2 , where R– radius of the base, L– length of the generatrix.

The volume of a circular cone is equal to V = 1/3 πR 2 H, Where R– radius of the base, N– height of the cone.

Definition: A pyramid inscribed in a cone is a pyramid whose base is a polygon inscribed in the circle of the base of the cone, and whose apex is the vertex of the cone. The lateral edges of a pyramid inscribed in a cone form the cone.

Definition: A pyramid circumscribed around a cone, is called a pyramid, the base of which is a polygon circumscribed around the base of the cone, and the apex coincides with the top of the cone.

Exercises:

1. An isosceles triangle with an apex angle of 120° and a side of 20 cm rotates around its base. Find the volume of the body of revolution.

2. Find the height of the cone if the area of ​​its lateral surface is 427.2 cm 2 and the generatrix is ​​17 cm.

A right triangle whose legs are 3 cm and 4 cm is rotated about an axis parallel to the hypotenuse and passing through the vertex right angle. Find the total surface area and volume of the body of rotation.

FRUSTUM. SURFACE AND VOLUME OF A TRUNCATED CONE

Definition: A truncated cone is the part of the cone enclosed between its base and a section parallel to the base. Circles lying in parallel planes are called the bases of a truncated cone.

Definition: The geometric body obtained by rotating a rectangular trapezoid around its side perpendicular to the bases is called a right circular truncated cone.

Definition: The generatrix of a truncated cone is a part of the generatrix full cone, enclosed between the bases.

Definition: The height of a truncated cone is the distance between its bases.

Task: Let a truncated cone be given, the radii of the bases and the height of which are known: r = 5, R = 7, H = Ö60. Find the generatrix of the truncated cone.

Definition: The straight line connecting the centers of the bases is called the axis of the truncated cone. The section passing through the axis is called axial. The axial section is an isosceles trapezoid.

Task: Find the axial cross-sectional area if the radius of the upper base, height and generatrix are known: R = 6, H = 4, L = 5.

Lateral surface area of ​​a truncated cone can be found using the formula:

S side = π(R + r)L,

Where R – radius of the lower base, r L – length of the generatrix.

Total surface area of ​​a truncated cone can be found using the formula:

S full = πR 2 + πr 2 + π(R + r)L,

Where R – radius of the lower base, r – radius of the upper base, L – length of the generatrix.

Volume of a truncated cone can be found as follows:

V = 1/3 πH(R 2 + Rr + r 2),

Where R – radius of the lower base, r – radius of the upper base, N – height of the cone.

Exercises:

From the history of its origin.

It is customary to call a body a ball, limited to the sphere, i.e. a ball and a sphere are different geometric bodies. However, both the words ball and sphere come from the same Greek word sphaira - ball. Moreover, the word “ball” was formed from the transition of the consonants sf to sh. In Book XI of the Elements, Euclid defines a ball as a figure described by a semicircle rotating around a fixed diameter. In ancient times, the sphere was held in high esteem. Astronomical observations of the firmament invariably evoked the image of a sphere. The sphere has always been widely used in various fields of science and technology.

Definition: A geometric body obtained by rotating a semicircle around its diameter is called a ball.

Definition: The radius of a sphere (ball) is a segment connecting the center of the sphere (ball) with any point on it.

Definition: A chord of a sphere is a segment connecting any two of its points.

Definition: The diameter of a sphere is the chord passing through its center.

Section of a sphere by a plane.

Any section of a ball by a plane is a circle. The center of this circle is the base of the perpendicular dropped from the center of the ball onto the cutting plane. The section passing through the center of the ball is called the diametrical section (great circle).

Tangent plane to a sphere.

A plane that has only one common point with a sphere is called a tangent plane to the sphere, and their common point is called the point of tangency between the plane and the sphere.

The first geometric concepts arose in prehistoric times. Different shapes Man observed material bodies in nature: the forms of plants and animals, mountains and river meanders, the circle and crescent of the Moon, etc. However, man not only passively observed nature, but practically mastered and used its riches. Practical human activity served as the basis for the discovery of the simplest geometric relationships and relationships.

Polyhedra

In the monuments of Babylonian and ancient Egyptian architecture there are such geometric figures as a cube, parallelepiped, and prism. The most important task Egyptian and Babylonian geometry was the determination of the volume of various spatial figures. This task responded to the need to build houses, palaces, temples and other structures.

The part of geometry in which the properties of the cube, prism, parallelepiped and other geometric bodies and spatial figures are studied has long been called stereometry; The word is Greek origin and is also found in the famous ancient Greek philosopher Aristotle. Stereometry arose later than planimetry. Euclid gives the following definition of a prism: “A prism is a solid figure enclosed between planes, of which two opposite ones are equal and parallel, the rest are parallelograms.” Here, as in many other places, Euclid uses the term “plane” not in the sense of an infinitely extended plane, but in the sense of its limited part, an edge, just as “straight” also means a straight segment.

The term "prism" is of Greek origin and literally means "sawed off." The term “parallelepipedal body” is found for the first time in Euclid and literally means “parallel-plane body.” Greek word"cubos" is used by Euclid in the same sense as our word "cube".

A surface composed of polygons and bounding a certain geometric body will be called a polyhedral surface or polyhedron. Types of polyhedra: parallelepiped, prism, pyramid.

Prism

A polyhedron composed of two equal polygons located in parallel planes and parallelograms is called a prism. The polygons are called the bases, and the parallelograms are called the lateral faces of the prism. The segments are called the lateral edges of the prism.

If the side edges are perpendicular to the bases, then the prism is called a straight one.

If the base of a straight prism contains regular polygons, then the prism is called regular.

Parallelepiped

If a parallelogram lies at the base of a prism, then the prism is called a parallelepiped. Parallelepipeds are inclined, straight and rectangular.

A rectangular parallelepiped has three dimensions: length, height and width. A parallelepiped has 8 vertices, 12 edges, 6 faces. Each face of a parallelepiped is a rectangle. Opposite faces of a parallelepiped are equal. Among all parallelepipeds, the cube plays a special role. A cube is a rectangular parallelepiped with all sides equal. All its faces are squares.

Pyramid

Important and interesting family polyhedra is a pyramid. A pyramid has a base and side faces. The side faces are triangles converging at one vertex, and the base is a polygon opposite this vertex. The base can be a polygon with any number of sides. A pyramid is called by the number of sides of its base: triangular pyramid, quadrangular pyramid, hexagonal pyramid... The simplest pyramid and even the simplest polyhedron is the triangular pyramid. All its faces are triangles, and each of them can be considered a base.

A pyramid is called regular if its base is a regular polygon, and the top of the pyramid is projected into the center of this polygon. All lateral edges of a regular pyramid are equal, and the lateral faces are equal isosceles triangles.

A polyhedron whose faces are polygons located in parallel planes and quadrangles - the lateral faces are called a truncated pyramid.

Regular polyhedra

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

Types of regular polygons

A regular tetrahedron is made up of four equilateral triangles. Each of its vertices is the vertex of three triangles.

A regular octahedron is made up of eight equilateral triangles. Each of its vertices is the vertex of four triangles.

The regular icosahedron is made up of twenty equilateral triangles. Each of its vertices is the vertex of five triangles.

The cube is made up of six squares. Each of its vertices is the vertex of three squares.

The regular dodecahedron is made up of twelve regular pentagons. Each of its vertices is the vertex of three regular pentagons.

Round bodies

Round bodies have a round shape. They can also consist of several circles; round bodies are formed by rotating a square plane. Such figures also have their own characteristics, for example, there are complex round bodies. Examples of round bodies: cylinder, cone, sphere and ball.

Cylinder

A body bounded by a cylindrical surface and two circles with boundaries is called a cylinder. The cylindrical surface is called the lateral surface of the cylinder, and the circles are called the bases of the cylinder. Generators of a cylindrical surface are called generators of the cylinder, straight line 001 is called the axis of the cylinder. All generatrices of the cylinder are parallel and equal to each other as segments of parallel lines enclosed between parallel planes.

The cylindrical surface is called the lateral surface of the cylinder, and the circles are called the bases of the cylinder.

The generatrices of a cylindrical surface are called the generators of the cylinder.

The straight line passing through the centers of the bases is called the axis of the cylinder.

The length of the generatrix is ​​called the height, and the radius of the base is called the radius of the cylinder.

Cone

A body bounded by a conical surface and a circle with a boundary is called a cone. The conical surface is called the lateral surface of the cone, and the circle is called the base of the cone. A cone can be obtained by rotating a right triangle around one of its legs.

Frustum

If you take a cutting plane and draw it along the cone, perpendicular to its axis. This plane intersects with the cone in a circle and splits the cone into two parts. One of the parts is a cone, and the other is called a truncated cone. The base of the original cone and the circle obtained in the section of this cone by a plane are called the bases of the truncated cone, and the segment connecting their centers is called the height of the truncated cone.

The part of the conical surface that bounds the cone is called its lateral surface, and the segments of the generatrices of the conical surface, enclosed between the bases, are called the generators of the truncated cone. All generators of a truncated cone are equal to each other.

Ball and sphere

A sphere is a surface consisting of all points in space located at a given distance from a given point. This point is called the center of the sphere, and the bottom distance is the radius of the sphere.

A body bounded by a sphere is called a ball. The center, radius and diameter of a sphere are also called the center, radius and diameter of a ball.

A plane that has only one common point with a sphere is called a tangent plane to the sphere, and their common point is called the point of tangency between the plane and the sphere.

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

The concept of a polyhedron, types of polyhedra in geometry

Geometry as a science contains the section stereometry, which studies the characteristics and properties of volumetric bodies, the sides of which in three-dimensional space are formed by limited planes (faces), called “polyhedra”. There are dozens of types of polyhedra, differing in the number and shape of faces.

Nevertheless, 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 edge of a polygon connects two, and only two, faces that are adjacent to each other.
3. Convexity means that the body is completely located on only one side of the plane on which one of the faces lies. The rule applies to all faces of the polyhedron. In stereometry, such geometric figures are called convex polyhedra. The exception is stellated polyhedra, which are derivatives of regular polyhedral geometric bodies.

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), semiregular (another name is Archimedean solids).
2. Non-convex polyhedra (stellate).

Prism and its properties

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

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

Basic properties of a prism:

• Congruent bases.
• All edges of the prism are equal and parallel to each other.
• All side faces have the shape of a parallelogram.

Pyramid

A pyramid is a geometric body that consists of one base and the nth number of triangular faces connecting at one point - the apex. It should be noted that if the side faces of the pyramid are necessarily represented by triangles, then at the base there can be a triangular polygon, a quadrangle, 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 at the base of a pyramid there is a triangle - this is a quadrilateral, etc.

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

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

Properties of the pyramid:

• 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 the center coinciding with the projection of the top of the pyramid.
• If a regular polygon lies at the base of the pyramid, then all the 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 bodies are called Platonic solids, or regular polyhedra. There are only five types of polyhedra with these properties:

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 works and associated them with the natural elements: earth, water, fire, air. The fifth figure was awarded similarity to the structure of the Universe. In his opinion, the atoms of natural elements are shaped like regular polyhedra. Thanks 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 find, they were even associated with the divine principle.

Hexahedron and its properties

In the form of a hexagon, Plato's successors 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 in modern times from attracting the minds of famous figures with their aesthetics.

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

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

Tetrahedron

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

Properties of a regular tetrahedron:

1. All faces of a tetrahedron - this means that all faces of a tetrahedron are congruent.
2. Since the base is represented by the correct geometric figure, that is, 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 plane angles at each vertex is 180, since all angles are equal, then any angle of a regular tetrahedron is 60.
4. Each vertex is projected to the point of intersection of the heights of the opposite (orthocenter) face.

Octahedron and its properties

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

Properties of the octahedron:

1. The very name of a geometric body suggests the number of its faces. The octahedron consists of 8 congruent equilateral triangles, at each of the vertices of which an equal number of faces converge, namely 4.
2. Since all the faces of the octahedron are equal, its interface angles are also equal, 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 a regular pentagon, then we get a dodecahedron - a figure of 12 polygons.

Properties of the dodecahedron:

1. Three faces intersect at each vertex.
2. All faces are equal and have the same edge length, as well as 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 edge opposite to it.

Icosahedron

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

1. All faces of the icosahedron are isosceles triangles.
2. Five faces meet 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 in this group have the following properties:

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

Star polyhedra

Representatives of non-volumetric types of geometric bodies are stellate polyhedra, the faces of which intersect with each other. They can be formed by merging two regular three-dimensional bodies or as a result of the extension of their faces.

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

THEORY OF POLYhedra

Faceted geometric bodies

A faceted geometric body or polyhedron is a part 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 another single polygon (called adjacent), and around each vertex there is one cycle of polygons. Simplifying the above definition, we obtain the definition of a polyhedron, familiar from a school textbook.

Schläfli symbols for regular polyhedra are given in the following table:- a geometric body bounded on all sides by flat polygons called faces. The sides of the faces are called the edges of the polyhedron, and the ends of the edges are called the vertices of the polyhedron.

From the history

Greek mathematics, in which the theory of polyhedra first appeared, developed under the great influence of the famous thinker Plato.

Plato(427–347 BC) - great ancient Greek philosopher, founder of the Academy and founder of the tradition of Platonism. One of the essential features of his teaching is the consideration of ideal objects - abstractions. Mathematics, having adopted the ideas of Plato, has been studying abstract, ideal objects since the time of Euclid. However, both Plato himself and many ancient mathematicians put into the term ideal not only an abstract meaning, but also the best meaning. In accordance with the tradition coming from ancient mathematicians, among all polyhedra the best are those that have regular polygons as their faces.

Polyhedra can be classified according to several criteria: for example, by the number of faces, tetrahedrons, pentahedrons, etc. are distinguished.

There are regular and semiregular polyhedra. Regular polyhedra are those in which all faces are regular equal polygons and all angles at the vertices are equal. If the faces of the polyhedron are various regular polygons, then a polyhedron is obtained, which is called semiregular (equiangular semiregular). A semiregular polyhedron is a convex polyhedron whose faces are regular polygons (possibly with a different number of sides), and all polyhedral angles are equal.

In addition to regular and semiregular polyhedra, the so-called regular stellate polyhedra have beautiful shapes. They are obtained from regular polyhedra by the continuation of faces or edges in the same way as regular stellated polygons are obtained by the continuation of the sides of regular polygons.

Of the many polyhedra, we highlight the most famous: prism and pyramid (Fig. 1).

A prism is a polyhedron that has two identical mutually parallel faces - bases, and the rest - side faces - parallelograms.

A pyramid is a polyhedron in which one face - an arbitrary polygon - is taken as the base, and the remaining faces (side) are triangles with a common vertex, called the top of the pyramid.

In Fig. 2 shows several prisms and pyramids. A pyramid whose base is shaped like a triangle is called a triangular pyramid. So, we can talk about square, pentagonal, etc. pyramids fig. 2, A and 2, b. The base of a triangular pyramid can be any face.

In Fig. 2, V, 2, G and 2, 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 the vertices of a p-gon, and the planes of both p-gons are parallel. If these two p-gons (bases) are congruent and arranged so that the vertices of one p-gon are connected to the vertices of another p-gon by parallel straight segments, then such a polyhedron is called a p-gonal prism. Examples of two p-angular prisms are the triangular prism (p = 3) in Fig. 2, V and pentagonal prism (p = 5) in Fig. 2, G. If the bases are located so that the vertices of one p-gon are connected to the vertices of another p-gon by a zigzag broken line, consisting of 2p straight segments, as in Fig. 2, d, then such a polyhedron is called a p-gonal antiprism.

In addition to two bases, a p-gonal prism has p faces - parallelograms. If parallelograms have the shape of rectangles, then the prism is called a straight line. In such a prism, the edges of the side faces are perpendicular to the base. A prism whose bases are not parallel is called truncated.

2. Regular polyhedra. A convex polyhedron is called regular if it satisfies the following two conditions:

All its faces are congruent regular polygons;

Each vertex has the same number of faces adjacent to it.

If all faces of a regular polyhedron are regular polygons, then in regular polyhedra all plane, polyhedral and dihedral angles are equal.

If all faces are regular p-gons and q of them are adjacent to each vertex, then such a regular polyhedron is denoted (p, q). The first number in parentheses indicates how many sides each face has, the second number indicates the number of faces adjacent to each vertex. This notation was proposed by L. Schläfli (1814-1895), a Swiss mathematician who was responsible for many elegant results in geometry and mathematical analysis. There are non-convex polyhedra whose faces intersect and which are called "regular stellated polyhedra". In geometry, by convention, regular polyhedra mean exclusively convex regular polyhedra.

Regular polyhedra are sometimes called Platonic solids, since they occupy a prominent place in the philosophical picture of the world developed by the great thinker Ancient Greece Plato.

There are 5 types of regular polyhedra: tetrahedron, cube, octahedron, dodecahedron, icosahedron.

TETRAHEDRON is a regular polyhedron whose surface consists of four regular triangles.

HEXAHEDRON (CUBE) - a regular polyhedron, the surface of which consists of six regular quadrilaterals (squares)

OCTAHEDRON is a regular polyhedron whose surface consists of eight regular triangles.

DODECAHEDRON is a regular polyhedron whose surface consists of twelve regular pentagons.

ICOSAHEDRON is a regular polyhedron whose surface consists of twenty regular triangles.

The names of these polyhedra come from Ancient Greece, and they indicate the number of faces:

"edra" - edge;

"tetra" - 4;

"hexa" - 6;

"okta" - 8;

“Ikosa” - 20;

"dodeka" - 12.

In Fig. 3 shows regular polyhedra

From the history

Plato believed that the world is built from four “elements” - fire, earth, air and water, and the atoms of these “elements” have shape of four regular polyhedra. The tetrahedron personified fire, since its apex points upward, like a flaring flame; icosahedron - as the most streamlined - water; the cube is the most stable of the figures - the earth, and the octahedron is the air. In our time, this system can be compared to the four states of matter - solid, liquid, gaseous and fiery. The fifth polyhedron, the dodecahedron, symbolized the whole world and was considered the most important. This was one of the first attempts to introduce the idea of ​​systematization into science.

The ancient Greeks considered the dodecahedron as the shape of the Universe. They also studied many geometric properties of Platonic solids; the fruits of their research can be found in the 13th book of Euclid’s Elements.

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

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

Number of regular polyhedra. It is natural to ask whether, besides the Platonic solids, there are other regular polyhedra.

Platonic solids are a three-dimensional analogue of flat regular polygons. However, between the two-dimensional and three-dimensional cases there is important difference: There are infinitely many different regular polygons, but only five different regular polyhedra. The proof of this fact has been known for more than two thousand years; With this proof and the study of the five regular solids, Euclid's Elements are completed

As the following simple considerations show, the answer must be negative. Let (p, q) be an arbitrary regular polyhedron. Since its faces are regular p-gons, their internal angles, as is easy to show, are equal to (180 - 360/p) or 180 (1 - 2/p) 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 each vertex has q faces adjacent to it, so the inequality must hold.

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It is easy to see that p and q must be greater than 2. Substituting p = 3 into (1), we find that the only valid values ​​for q in this case are 3, 4 and 5, i.e. we obtain the polyhedra (3, 3), (3, 4) and (3, 5). For p = 4, the only valid value for q is 3, i.e. the polyhedron (4, 3), with p = 5, also satisfies inequality (1) only q = 3, i.e. polyhedron (5, 3). For p > 5 there are no valid values ​​of q. Consequently, there are no other regular polyhedra except the Platonic solids.

3. Semiregular polyhedra. Above we looked at regular polyhedra, i.e. These are convex polyhedra whose faces are equal regular polygons, and the same number of faces meet at each vertex. If in this definition we allow that the faces of a polyhedron can be different regular polygons, then we obtain polyhedra that are called semiregular (equiangular semiregular).

A semiregular polyhedron is a convex polyhedron whose faces are regular polygons (possibly with a different number of sides), and all polyhedral angles are equal.

Semi-regular polyhedra include regular n-gonal prisms, all of whose edges are equal. For example, the regular pentagonal prism in Figure 4, A has with its faces two regular pentagons - the base of the prism and five squares forming the lateral surface of the prism. Semi-regular polyhedra also include the so-called antiprisms. In Figure 4, b we see a pentagonal antiprism obtained from a pentagonal prism by rotating one of the bases relative to the other at an angle of 36. Each vertex of the upper and lower bases is connected to the two closest vertices of the other base.

a B C

In addition to these two endless series of semi-regular polyhedra, there are 13 more semi-regular polyhedra that were first discovered and described by Archimedes - these are the Archimedean solids.

The simplest of them are obtained from regular polyhedra by the “truncation” operation, which consists of cutting off the corners of the polyhedron with planes. If we cut off the corners of the tetrahedron with planes, each of which cuts off a third of its edges emerging from one vertex, we obtain a truncated tetrahedron with eight faces (Fig. 4, V). Of these, four are regular hexagons and four are regular triangles. Three faces meet at each vertex of this polyhedron.

If we cut off the vertices of the octahedron and icosahedron in this manner, we obtain a truncated octahedron (Fig. 5, a) and a truncated icosahedron (Fig. 5, b), respectively. Please note that the surface of the soccer ball is made in the shape of the surface of a truncated icosahedron. From a cube and a dodecahedron you can also get a truncated cube (Fig. 5, c) and a truncated dodecahedron (Fig. 5, d).

a B C D

We examined 4 of the 13 semiregular polyhedra described by Archimedes. The remaining ones are polyhedra of a more complex type.

From the history

Kepler's cosmological hypothesis is very original, in which he tried to connect some properties solar system with properties of regular polyhedra. Kepler suggested that the distances between the six then known planets were expressed in terms of the sizes of five regular convex polyhedra (Platonic solids). Between each pair of celestial spheres along which, according to this hypothesis, the planets rotate, Kepler inscribed one of the Platonic solids. An octahedron is described around the sphere of Mercury, the planet closest to the Sun. This octahedron is inscribed in the sphere of Venus, around which the icosahedron is described. The sphere of the Earth is described around the icosahedron, and the dodecahedron is described around this sphere.

A serious step in the science of polyhedra was made in the 18th century by Leonhard Euler (1707-1783), who, without exaggeration, “believed harmony in algebra.” Euler's theorem on the relationship between the number of vertices, edges and faces of a convex polyhedron, the proof of which Euler published in 1758 in the Proceedings of the St. Petersburg Academy of Sciences, finally brought mathematical order to the diverse world of polyhedra.

Vertices + Faces - Edges = 2.

Elements of symmetry of regular polyhedra

Some of the regular and semi-regular bodies are found in nature in the form of crystals, others - in the form of viruses, simple microorganisms

Star polyhedra

Stellate polyhedra are obtained from regular polyhedra by extending faces or edges in the same way as regular stellate polygons are obtained by extending the sides of regular polygons.

The first two regular stellate polyhedra were discovered by J. Kepler (1571-1630), and the other two were built almost 200 years later by the French mathematician and mechanic L. Poinsot (1777-1859). That is why regular stellated polyhedra are called Kepler-Poinsot bodies.

In his work “On polygons and polyhedra” (1810), Poinsot described four regular stellate polyhedra, but the question of the existence of other such polyhedra remained open. The answer was given a year later, in 1811, by the French mathematician O. Cauchy (1789-1857). In his work "A Study on Polyhedra" he proved that there are no other regular stellated polyhedra.

Let us consider the question of which regular polyhedra can be used to obtain regular stellated polyhedra. Regular stellated polyhedra cannot be obtained from a tetrahedron, cube, or octahedron. Let's take the dodecahedron. The continuation of its edges leads to the replacement of each face with a stellated regular pentagon (Fig. 30, a), and as a result a polyhedron arises, which is called the small stellated dodecahedron (Fig. 30, b).

When extending the faces of the dodecahedron, two possibilities arise. Firstly, if we consider regular pentagons, then you get the so-called great dodecahedron (Fig. 31). If, secondly, we consider stellated pentagons as faces, then we get a large stellated dodecahedron (Fig. 32).

The icosahedron has a single star shape. When the faces of a regular icosahedron are extended, a large icosahedron is obtained (Fig. 33).

Thus, there are 4 types of regular stellated polyhedra.

Star-shaped polyhedra 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 stellate polyhedra are suggested by nature itself. Snowflakes are star-shaped polyhedra (Figure 34). Since ancient times, people have tried to describe all possible types of snowflakes and compiled special atlases. Several thousand are now known various types snowflakes.

Related information.