Flat polygons that make up the surface of a polyhedron. A polyhedron is a body whose surface consists of a finite number of flat polygons

Cube, ball, pyramid, cylinder, cone - geometric bodies. Among them are polyhedra. Polyhedron is a geometric body whose surface 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, respectively, the edges and vertices of the polyhedron.

Dihedral angles between adjacent faces, i.e. faces that have a common side - the edge of the 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 that have common sides with each of the bases.

Two equal polygons are called reasons ggrizmg, and parallelograms are her lateral edges. The side faces form lateral surface prisms. Edges that do not lie at the base are called lateral ribs prisms.

The prism is called p-coal, if its bases are i-gons. In 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 a parallelepiped is a segment connecting its opposite vertices. A parallelepiped has four diagonals.

It has been proven that The diagonals of a parallelepiped intersect at one point and are bisected by this point. The diagonals of a rectangular parallelepiped are equal.

Pyramid 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 lateral faces of the pyramid. The common vertex of these triangles is called top pyramids, ribs extending from the top, - lateral ribs pyramids.

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

The simplest pyramid - triangular or tetrahedron (Fig. 24.8). The peculiarity 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 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 also entirely contains the segment connecting them.

A convex polyhedron can be defined differently: a polyhedron is called convex, if it lies entirely on one side of each of the polygons bounding it.

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

All 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.

It has been proven that in a convex polyhedron, all faces are convex polygons.

Let's consider several convex polyhedra (Table 24.1)

From this table it follows that for all considered convex polyhedra the equality B - P + G= 2. It turned out that this is also true for any convex polyhedron. This property was first proven 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 There are no more than five different types of regular polyhedra.

Indeed, if fan and polyhedron are regular triangles, then 3, 4 and 5 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 a polyfan, then we get right-handed tetrahedron, which translated from Phetic means “tetrahedron” (Fig. 24.10, A).

If four regular triangles meet at each vertex of a polyhedron, then we get octahedron(Fig. 24.10, V). Its surface consists of eight regular triangles.

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

If the faces of a polyfan 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 edges of a polyfan are regular pentagons, then only phi can converge at one vertex, since 108° 3< 360°, пятиугольники и в каждой вершине сходится три грани, называется dodecahedron(Fig. 24.10, d). 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°.

In geometry, it has been proven that in three-dimensional Euclidean space there are exactly five different types of regular polyhedra.

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

The development of a polyhedron is a figure on a plane that is obtained if the surface of the polyhedron is cut along certain 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 cut. Figure 24.11 shows figures that are various developments of a regular quadrangular pyramid, i.e. a pyramid with a square at its base and all side edges equal to each other.

For a figure on a plane 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 developments of a regular quadrangular pyramid: in the figure shown in Fig. 24.12, A, at the top M four faces converge, which cannot happen in a regular quadrangular pyramid; and in the figure shown in Fig. 24.12, b, lateral ribs A B And Sun not equal.

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

Bodies of revolution

Body of rotation called a 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 rotation of a rectangle around one of its sides. In this case, the specified party is axis of the cylinder. In Fig. 24.14 shows a cylinder with an axis OO', obtained by rotating a rectangle AA"O"O around a straight line OO". Points ABOUT And ABOUT"- centers of the cylinder bases.

A cylinder that results from rotating a rectangle around one of its sides is called straight 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 cylinder axis.

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

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

In this case, the indicated leg is motionless and is called the axis of the cone. In Fig. Figure 24.15 shows a cone with an axis SO, obtained by rotating a right triangle SOA with a right angle O around leg S0. Point S is called apex of the cone, OA- 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 its 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, upon rotation of which a cone is formed.

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

When a cylinder, cone or any other body of rotation intersects a plane containing the axis of rotation, it turns out axial section. The axial section of the cylinder is a rectangle, the axial section of the cone is an isosceles triangle.

Ball- this is a body that is obtained as a result of rotation of a semicircle around its diameter. In Fig. 24.16 shows a ball obtained by rotating a semicircle around the diameter AA". Full stop ABOUT called the center of the ball, and the radius of the circle is the radius of the ball.

The surface of the ball is called sphere. The sphere cannot be turned onto a plane.

Any section of a ball by a plane is a circle. The cross-sectional radius 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 large circle of the ball, and the circle that bounds it is large 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. To do this, special methods are used to depict such figures on a plane. One of them is parallel design.

Let a plane and a straight line intersecting a be given A. Let us take an arbitrary point A in space that does not belong to the line A, and we'll guide you through X direct A", parallel to the line A(Fig. 24.17). Straight A" intersects the plane at some point X", which is called parallel projection of point X onto plane a.

If point A lies on a straight line A, then with parallel projection X" is the point at which the line A intersects the plane A.

If the 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 each point X space can be associated with a single point A" - a parallel projection of the point X onto the plane a (when designing parallel to the straight line A). Plane A called projection plane. About the line A they say she will bark design direction - ggri replacement direct A any other direct design result parallel to it will not change. All lines parallel to a line A, specify the same design direction and are called along with the straight line A projecting straight lines.

Projection figures F call a set F' projection of all the points. Mapping 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, knowledge of which is necessary when depicting geometric bodies on a plane. Let us formulate the main ones without providing their proof.

Theorem 24.1. During parallel design, the following properties are satisfied for straight lines not parallel to the design direction and for segments lying on them:

1) the projection of a line is a 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 line or on parallel lines is equal to the ratio of the lengths of the segments themselves.

From this theorem it follows consequence: with 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 ensure that the specified properties are met. Otherwise it can be arbitrary. Thus, the angles and ratios of the lengths of non-parallel segments can change arbitrarily, i.e., for example, a triangle in parallel design is depicted as an arbitrary triangle. But if the triangle is equilateral, then the projection of its median must connect the vertex of the triangle with the middle of the opposite side.

And one more requirement must be observed when depicting spatial bodies on a plane - to help create a correct idea of ​​them.

Let us depict, for example, an inclined prism whose bases 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 depicted as an arbitrary parallelogram ABCD (Fig. 24.19, a). Since the edges of the prism are parallel, we build parallel straight lines passing through the vertices of the constructed parallelogram and lay on them equal segments AA", BB', CC", DD", the length of which is arbitrary. By connecting points A", B", C", D in series ", we obtain a quadrilateral A" B "C" D", depicting the upper base of the prism. It is not difficult to prove that A"B"C"D"- parallelogram equal to parallelogram ABCD and, consequently, 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, the bases of which 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 depict 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 ABOUT. Then draw a vertical segment OS depicting the height of the pyramid. Note that the verticality of the segment OS provides greater clarity of the drawing. Finally, point S is connected to all the vertices of the base.

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

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

Thus, the sequence of constructing the base of a 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 ABOUT" draw a straight line parallel V'S"(or E"F');

§ choose an arbitrary point on the constructed line A" and mark the point D" such that O"D" = A"O" and connect the dot A" with dots IN" And F", and point D" - with dots WITH" And E".

To complete the construction of the pyramid, draw a vertical segment OS(its length is chosen arbitrarily) and connect point S to all 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, draw a projection of some large circle, the plane of which is not perpendicular to the projection plane. This projection will be an ellipse. The center of the ball will be represented by the center of this ellipse (Fig. 24.22). Now we can find the corresponding poles N and S, provided that the segment connecting them is perpendicular to the equatorial plane. To do this, through the point ABOUT draw a straight line perpendicular AB and mark 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, we get the poles N and S.

Let's consider one of the techniques for constructing an ellipse (it is based on a transformation of the plane, which is called compression): construct a circle with a diameter and draw chords perpendicular to the diameter (Fig. 24.23). Half of each chord 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 point ABOUT.

This technique can be used to depict a straight circular cylinder (Fig. 24.24) and a straight circular cone (Fig. 24.25) on a plane.

A straight circular cone is depicted like this. First, they build an ellipse - the base, then find the center of the base - the point ABOUT and draw a line segment perpendicularly 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 straight lines from point S to points of tangency C and D. Note that the segment CD does not coincide with the diameter of the base of the cone.

“Types of polyhedra” - Regular stellate polyhedra. Dodecahedron. Small stellated dodecahedron. Polyhedra. Hexahedron. Plato's solids. Prismatoid. Pyramid. Icosahedron. Octahedron. A body limited by a finite number of planes. Star octahedron. Two faces. Law of reciprocity. Mathematician. Tetrahedron.

“Geometric body polyhedron” - Polyhedra. Prisms. The existence of incommensurable quantities. Poincare. Edge. Volume measurement. Faces of a parallelepiped. Rectangular parallelepiped. We often see a pyramid on the street. Polyhedron. Interesting Facts. Alexandrian lighthouse. Geometric shapes. Distance between planes. Memphis.

“Cascades of polyhedra” - Edge of a cube. Octahedron edge. Cube and dodecahedron. Unit tetrahedron. Dodecahedron and icosahedron. Dodecahedron and tetrahedron. Octahedron and icosahedron. Polyhedron. Regular polyhedron. Octahedron and dodecahedron. Icosahedron and octahedron. Unit icosahedron. Tetrahedron and icosahedron. Unit dodecahedron. Octahedron and tetrahedron. Cube and tetrahedron.

““Polyhedra” stereometry” - Polyhedra in architecture. Section of polyhedra. Give the polyhedron a name. Great Pyramid of Giza. Platonic solids. Correct the logical chain. Polyhedron. Historical reference. The finest hour of polyhedra. Problem solving. Lesson objectives. "Playing with the Spectators" Do the geometric shapes and their names correspond?

“Stellar forms of polyhedra” - Great stellated dodecahedron. The polyhedron shown in the figure. Star polyhedra. Side ribs. Stellar cuboctahedra. Stellated truncated icosahedron. A polyhedron obtained by truncating a stellated truncated icosahedron. Vertices of the great stellated dodecahedron. Stellated icosahedrons. Great dodecahedron.

“Section of a polyhedron by a plane” - Section of polyhedra. Polygons. The cuts formed a pentagon. Trace of the cutting plane. Section. Let's find the point of intersection of the lines. Plane. Construct a cross section of a cube. Construct a cross section of the prism. We find the point. Prism. Methods for constructing sections. The resulting hexagon. Section of a cube. Axiomatic method.

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Geometric bodies

Introduction

In stereometry, figures in space are studied, which are called geometric bodies.

The objects around us give us an idea of ​​geometric bodies. Unlike real objects, geometric bodies are imaginary objects. Clearly geometric body one must imagine it as a part of space occupied by matter (clay, wood, metal, ...) and limited by a surface.

All geometric bodies are divided into polyhedra And round bodies.

Polyhedra

Polyhedron is a geometric body whose surface consists of a finite number of flat polygons.

Edges polyhedron, the polygons that make up its surface are called.

Ribs of a polyhedron, the sides of the faces of the polyhedron are called.

Peaks of a polyhedron are called the vertices of the faces of the polyhedron.

Polyhedra are divided into convex And non-convex.

The polyhedron is called convex, if it lies entirely on one side of any of its faces.

Exercise. Specify edges, ribs And peaks cube shown in the figure.

Convex polyhedra are divided into prisms And pyramids.

Prism

Prism is a polyhedron with two equal and parallel faces
n-gons, and the rest n faces are parallelograms.

Two n-gons are called prism bases, parallelograms – side faces. The sides of the side faces and bases are called prism ribs, the ends of the edges are called the vertices of the prism. Side edges are edges that do not belong to the bases.

Polygons A 1 A 2 ...A n and B 1 B 2 ...B n are the bases of the prism.

Parallelograms A 1 A 2 B 2 B 1, ... - side faces.

Prism properties:

· The bases of the prism are equal and parallel.

· The lateral edges of the prism are equal and parallel.

Prism diagonal called a segment connecting two vertices that do not belong to the same face.

Prism height is called a perpendicular dropped from a point of the upper base to the plane of the lower base.

A prism is called 3-gonal, 4-gonal, ..., n-coal, if its base
3-gons, 4-gons, ..., n-gons.

Direct prism called a prism whose side ribs are perpendicular to the bases. The lateral faces of a straight prism are rectangles.

Inclined prism called a prism that is not straight. The lateral faces of an inclined prism are parallelograms.

With the right prism called straight a prism with regular polygons at its base.

Area full surface prisms is called the sum of the areas of all its faces.

Area lateral surface prisms is called the sum of the areas of its lateral faces.

S full = S side + 2 S basic

Polyhedron

• Polyhedron- this is a body whose surface consists of a finite number of flat polygons.

The polyhedron is called convex

• The polyhedron is called convex ,if it is located on one side of each flat polygon on its surface.

• Euclid (presumably 330-277 BC) - mathematician of the Alexandrian school of Ancient Greece, author of the first treatise on mathematics that has come down to us, “Elements” (in 15 books)

side faces.

• A prism is a polyhedron, which consists of two flat polygons lying in different planes and combined by parallel translation, and all the segments connecting the corresponding points of these polygons. Polygons Ф and Ф1 lying in parallel planes are called prism bases, and the remaining faces are called side faces.

• The surface of the prism thus consists of two equal polygons (bases) and parallelograms (side faces). There are triangular, quadrangular, pentagonal, etc. prisms. depending on the number of vertices of the base.

• If the lateral edge of a prism is perpendicular to the plane of its base, then such a prism is called straight ; if the lateral edge of the prism is not perpendicular to the plane of its base, then such a prism is called inclined . A straight prism has rectangular side faces.

The bases of the prism are equal.

• The bases of the prism are equal.

• The bases of a prism lie in parallel planes.

• The side edges of a prism are parallel and equal.

• The height of a prism is the distance between the planes of its bases.

• It turns out that a prism can be not only a geometric body, but also an artistic masterpiece. It was the prism that became the basis for the paintings of Picasso, Braque, Griss, etc.

• It turns out that a snowflake can take the shape of a hexagonal prism, but this will depend on the air temperature.

• In the 3rd century BC. e. a lighthouse was built so that ships could safely pass the reefs on their way to Alexandria Bay. At night they were helped in this by the reflection of flames, and during the day by a column of smoke. It was the world's first lighthouse, and it stood for 1,500 years.

• The lighthouse was built on the small island of Pharos in the Mediterranean Sea, off the coast of Alexandria. It took 20 years to build and was completed around 280 BC.

• In the 14th century, the lighthouse was destroyed by an earthquake. Its debris was used in the construction of a military fort. The fort has been rebuilt several times and still stands on the site of the world's first lighthouse.

Mausolus was the ruler of Caria. The capital of the region was Halicarnassus. Mausolus married his sister Artemisia. He decided to build a tomb for himself and his queen. Mavsol dreamed of a majestic monument that would remind the world of his wealth and power. He died before work on the tomb was completed. Artemisia continued to lead the construction. The tomb was built in 350 BC. e. It was named Mausoleum after the king.

The ashes of the royal couple were kept in golden urns in a tomb at the base of the building. A row of stone lions guarded this room. The structure itself resembled a Greek temple, surrounded by columns and statues. At the top of the building was a step pyramid. At a height of 43 m above the ground, it was crowned with a sculpture of a chariot drawn by horses. There were probably statues of the king and queen on it.

• Eighteen centuries later, an earthquake destroyed the Mausoleum to the ground. Another three hundred years passed before archaeologists began excavations. In 1857, all the finds were transported to the British Museum in London. Now, in the place where the Mausoleum once was, only a handful of stones remain.

crystals.

There are not only geometric shapes created by human hands. There are many of them in nature itself. The impact on the appearance of the earth’s surface of such natural factors as wind, water, sunlight is very spontaneous and chaotic. However, sand dunes, pebbles on the seashore, The crater of an extinct volcano, as a rule, has geometrically regular shapes. Sometimes stones are found in the ground of such a shape, as if someone had carefully cut them out, ground them, and polished them. This is - crystals.

parallelepiped.

• If the base of the prism is a parallelogram, then it is called parallelepiped.

• The models of a rectangular parallelepiped are:

• cool room

• It turns out that calcite crystals, no matter how much they are crushed into smaller parts, always break up into fragments shaped like a parallelepiped.

• City buildings most often have the shape of polyhedra. As a rule, these are ordinary parallelepipeds. And only unexpected architectural solutions decorate cities.

• 1. Is a prism regular if its edges are equal?

• a) yes; c) no. Justify your answer.

• 2. The height of a regular triangular prism is 6 cm. The side of the base is 4 cm. Find the total surface area of ​​this prism.

• 3. The areas of the two lateral faces of an inclined triangular prism are 40 and 30 cm2. The angle between these faces is straight. Find the lateral surface area of ​​the prism.

• 4. In the parallelepiped ABCDA1B1C1D1, sections A1BC and CB1D1 are drawn. In what ratio do these planes divide diagonal AC1?

• 1) a tetrahedron with 4 faces, 4 vertices, 6 edges;

• 2) cube - 6 faces, 8 vertices, 12 edges;

• 3) octahedron - 8 faces, 6 vertices, 12 edges;

• 4) dodecahedron - 12 faces, 20 vertices, 30 edges;

• 5) icosahedron - 20 faces, 12 vertices, 30 edges.

Thales of Miletus, founder Ionian Pythagoras of Samos

Scientists and philosophers of Ancient Greece adopted and reworked the achievements of culture and science of the Ancient East. Thales, Pythagoras, Democritus, Eudoxus and others traveled to Egypt and Babylon to study music, mathematics and astronomy. It is no coincidence that the beginnings of Greek geometric science are associated with the name Thales of Miletus, founder Ionian schools. The Ionians, who inhabited the territory that bordered the eastern countries, were the first to borrow the knowledge of the East and began to develop it. Scientists of the Ionian school were the first to subject to logical processing and systematize mathematical information borrowed from the ancient Eastern peoples, especially from the Babylonians. Proclus and other historians attribute many geometric discoveries to Thales, the head of this school. About attitude Pythagoras of Samos to geometry, Proclus writes the following in his commentary to Euclid’s Elements: “He studied this science (i.e., geometry), starting from its first foundations, and tried to obtain theorems using purely logical thinking.” Proclus attributes to Pythagoras, in addition to the well-known theorem on the square of the hypotenuse, the construction of five regular polyhedra:

Plato's solids

Plato's solids are convex polyhedra, all of whose faces are regular polygons. All polyhedral angles of a regular polyhedron are congruent. As follows from calculating the sum of plane angles at a vertex, there are no more than five convex regular polyhedra. Using the method indicated below, one can prove that there are exactly five regular polyhedra (this was proven by Euclid). They are regular tetrahedron, cube, octahedron, dodecahedron and icosahedron.

Octahedron (Fig. 3).

• Octahedron -octahedron; a body bounded by eight triangles; a regular octahedron is bounded by eight equilateral triangles; one of the five regular polyhedra. (Fig. 3).

• Dodecahedron -dodecahedron, a body bounded by twelve polygons; regular pentagon; one of the five regular polyhedra . (Fig. 4).

• Icosahedron -twenty-hedron, a body bounded by twenty polygons; the regular icosahedron is limited by twenty equilateral triangles; one of the five regular polyhedra. (Fig. 5).

The faces of the dodecahedron are regular pentagons. The diagonals of a regular pentagon form the so-called star pentagon - a figure that served as an emblem, an identification mark for the students of Pythagoras. It is known that the Pythagorean League was at the same time a philosophical school, a political party and a religious brotherhood. According to legend, one Pythagorean fell ill in a foreign land and could not pay the owner of the house who cared for him before his death. The latter painted a star-shaped pentagon on the wall of his house. Seeing this sign a few years later, another wandering Pythagorean inquired about what had happened from the owner and generously rewarded him.

• Reliable information about the life and scientific activities of Pythagoras has not been preserved. He is credited with creating the doctrine of the similarity of figures. He was probably among the first scientists to view geometry not as a practical and applied discipline, but as an abstract logical science.

The school of Pythagoras discovered the existence of incommensurable quantities, that is, those whose relationship cannot be expressed by any integer or fractional number. An example is the ratio of the length of the diagonal of a square to the length of its side, equal to C2. This number is not rational (i.e., an integer or a ratio of two integers) and is called irrational, i.e. irrational (from the Latin ratio - attitude).

Tetrahedron (Fig. 1).

• Tetrahedron -tetrahedron, all faces of which are triangles, i.e. triangular pyramid; a regular tetrahedron is bounded by four equilateral triangles; one of the five regular polygons. (Fig. 1).

• Cube or regular hexahedron (Fig. 2).

Tetrahedron -tetrahedron, all faces of which are triangles, i.e. triangular pyramid; a regular tetrahedron is bounded by four equilateral triangles; one of the five regular polygons. (Fig. 1).

• Tetrahedron -tetrahedron, all faces of which are triangles, i.e. triangular pyramid; a regular tetrahedron is bounded by four equilateral triangles; one of the five regular polygons. (Fig. 1).

• Cube or regular hexahedron - a regular quadrangular prism with equal edges, limited by six squares. (Fig. 2).

Pyramid

• Pyramid- a polyhedron, which consists of a flat polygon - the base of the pyramid, points that do not lie in the plane of the base-top of the pyramid and all segments connecting the top of the pyramid with the points of the base

The picture shows a pentagonal pyramid SABCDE and its development. Triangles that have a common vertex are called side faces pyramids; common vertex of the side faces - top pyramids; a polygon to which this vertex does not belong is basis pyramids; the edges of the pyramid converging at its apex - lateral ribs pyramids. Height pyramid is a perpendicular segment drawn through its top to the base plane, with ends at the top and on the base plane of the pyramid. In the figure there is a segment SO- height of the pyramid.

• Definition . A pyramid whose base is a regular polygon and whose vertex is projected into its center is called regular.

• The figure shows a regular hexagonal pyramid.

The volumes of grain barns and other structures in the form of cubes, prisms and cylinders were calculated by the Egyptians and Babylonians, the Chinese and Indians by multiplying the base area by the height. However, the ancient East knew mainly only certain rules, found experimentally, which were used to find volumes for the areas of figures. At a later time, when geometry was formed as a science, a general approach to calculating the volumes of polyhedra was found.

• Among the remarkable Greek scientists of the V - IV centuries. BC, who developed the theory of volumes were Democritus of Abdera and Eudoxus of Cnidus.

• Euclid does not use the term "volume". For him, the term “cube,” for example, also means the volume of a cube. In Book XI of the "Principles" the following theorems are presented, among others.

• 1. Parallelepipeds with equal heights and equal bases are equal in size.

• 2. The ratio of the volumes of two parallelepipeds with equal heights is equal to the ratio of the areas of their bases.

• 3. In parallelepipeds of equal area, the areas of the bases are inversely proportional to the heights.

• Euclid's theorems relate only to the comparison of volumes, since Euclid probably considered the direct calculation of the volumes of bodies to be a matter of practical manuals in geometry. In the applied works of Heron of Alexandria, there are rules for calculating the volume of a cube, prism, parallelepiped and other spatial figures.

• A prism whose base is a parallelogram is called a parallelepiped.

• According to the definition a parallelepiped is a quadrangular prism, all of whose faces are parallelograms. Parallelepipeds, like prisms, can be straight And inclined. Figure 1 shows an inclined parallelepiped, and Figure 2 shows a straight parallelepiped.

• A right parallelepiped whose base is a rectangle is called rectangular parallelepiped. All faces of a rectangular parallelepiped are rectangles. Models of a rectangular parallelepiped are a classroom, a brick, and a matchbox.

• The lengths of three edges of a rectangular parallelepiped having a common end are called measurements. For example, there are matchboxes with dimensions of 15, 35, 50 mm. A cube is a rectangular parallelepiped with equal dimensions. All six faces of the cube are equal squares.

• Let's consider some properties of a parallelepiped.

• Theorem. The parallelepiped is symmetrical about the middle of its diagonal.

• It follows directly from the theorem important properties of a parallelepiped:

• 1. Any segment with ends belonging to the surface of the parallelepiped and passing through the middle of its diagonal is divided in half by it; in particular, all diagonals of a parallelepiped intersect at one point and are bisected by it. 2. Opposite faces of a parallelepiped are parallel and equal

Introduction

A surface composed of polygons and bounding some geometric body is called a polyhedral surface or polyhedron.

A polyhedron is a bounded body whose surface consists of a finite number of polygons. The polygons that bound a polyhedron are called faces, and the lines of intersection of the faces are called edges.

Polyhedra can have a varied and very complex structure. Various structures, such as houses being built using bricks and concrete blocks, are examples of polyhedra. Other examples can be found among furniture, such as a table. In chemistry, the shape of hydrocarbon molecules is a tetrahedron, a regular twenty-hedron, a cube. In physics, crystals serve as examples of polyhedra.

Since ancient times, ideas about beauty have been associated with symmetry. This probably explains people's interest in polyhedra - amazing symbols of symmetry that attracted the attention of outstanding thinkers who were amazed by the beauty, perfection, and harmony of these figures.

The first mentions of polyhedra are known three thousand years BC in Egypt and Babylon. Suffice it to recall the famous Egyptian pyramids and the most famous of them, the Pyramid of Cheops. This is a regular pyramid, at the base of which is a square with a side of 233 m and the height of which reaches 146.5 m. It is no coincidence that they say that the Pyramid of Cheops is a silent treatise on geometry.

The history of regular polyhedra goes back to ancient times. Starting from the 7th century BC, philosophical schools were created in Ancient Greece, in which there was a gradual transition from practical to philosophical geometry. Reasoning with the help of which it was possible to obtain new geometric properties acquired great importance in these schools.

One of the first and most famous schools was the Pythagorean school, named after its founder Pythagoras. The distinctive sign of the Pythagoreans was the pentagram, in the language of mathematics it is a regular non-convex or star-shaped pentagon. The pentagram was assigned the ability to protect a person from evil spirits.

The Pythagoreans believed that matter consisted of four basic elements: fire, earth, air and water. They attributed the existence of five regular polyhedra to the structure of matter and the Universe. According to this opinion, the atoms of the main elements must have the form of different bodies:

§ The Universe is a dodecahedron

§ Earth - cube

§ Fire - tetrahedron

§ Water - icosahedron

§ Air - octahedron

Later, the teaching of the Pythagoreans about regular polyhedra was outlined in his works by another ancient Greek scientist, the idealist philosopher Plato. Since then, regular polyhedra have become known as Platonic solids.

Platonic solids are regular homogeneous convex polyhedra, that is, convex polyhedra, all of whose faces and angles are equal, and the faces are regular polygons. The same number of edges converge to each vertex of a regular polyhedron. All dihedral angles at the edges and all polyhedral angles at the vertices of a regular polygon are equal. Platonic solids are a three-dimensional analogue of flat regular polygons.

The theory of polyhedra is a modern branch of mathematics. It is closely related to topology, graph theory, and is of great importance both for theoretical research in geometry and for practical applications in other branches of mathematics, for example, algebra, number theory, applied mathematics - linear programming, optimal control theory. Thus, this topic is relevant, and knowledge on this issue is important for modern society.

Main part

A polyhedron is a bounded body whose surface consists of a finite number of polygons.

Let us give a definition of a polyhedron that is equivalent to the first definition of a polyhedron.

Polyhedron – This is a figure that is the union of a finite number of tetrahedra for which the following conditions are met:

1) every two tetrahedra do not have common points, or have a common vertex, or only a common edge, or an entire common face;

2) from each tetrahedron to another you can go along a chain of tetrahedrons, in which each subsequent one is adjacent to the previous one along an entire face.

Polyhedron elements

The face of a polyhedron is a certain polygon (a polygon is a limited closed area whose boundary consists of a finite number of segments).

The sides of the faces are called the edges of the polyhedron, and the vertices of the faces are called the vertices of the polyhedron. The elements of a polyhedron, in addition to its vertices, edges and faces, also include the flat angles of its faces and the dihedral angles at its edges. The dihedral angle at an edge of a polyhedron is determined by its faces approaching this edge.

Classification of polyhedra

Convex polyhedron - is a polyhedron, any two points of which can be connected by a segment. Convex polyhedra have many remarkable properties.

Euler's theorem. For any convex polyhedron V-R+G=2,

Where IN – the number of its vertices, R - the number of its ribs, G - the number of its faces.

Cauchy's theorem. Two closed convex polyhedra, identically composed of respectively equal faces, are equal.

A convex polyhedron is considered regular if all its faces are equal regular polygons and the same number of edges converge at each of its vertices.

Regular polyhedron

A polyhedron is called regular if, firstly, it is convex, secondly, all its faces are equal regular polygons, thirdly, the same number of faces meet at each of its vertices, and, fourthly, all its dihedral angles are equal.

There are five convex regular polyhedra - the tetrahedron, octahedron and icosahedron with triangular faces, the cube (hexahedron) with square faces and the dodecahedron with pentagonal faces. The proof of this fact has been known for more than two thousand years; with this proof and the study of the five regular bodies, the Elements of Euclid (the ancient Greek mathematician, the author of the first theoretical treatises on mathematics that have come down to us) are completed. Why did regular polyhedra get such names? This is due to the number of their faces. A tetrahedron has 4 faces, translated from the Greek “tetra” - four, “hedron” - face. A hexahedron (cube) has 6 faces, a “hexa” has six; octahedron - octahedron, "octo" - eight; dodecahedron - dodecahedron, "dodeca" - twelve; The icosahedron has 20 faces, and the icosi has twenty.

2.3. Types of regular polyhedra:

1) Regular tetrahedron(composed of four equilateral triangles. Each of its vertices is the vertex of three triangles. Therefore, the sum of the plane angles at each vertex is 180 0);

2)Cube- a parallelepiped, all of whose faces are squares. The cube is made up of six squares. Each vertex of the cube is the vertex of three squares. Therefore, the sum of the plane angles at each vertex is 270 0.

3) Regular octahedron or simply octahedron– a polyhedron with eight regular triangular faces and four faces meeting at each vertex. The octahedron is made up of eight equilateral triangles. Each vertex of the octahedron is the vertex of four triangles. Therefore, the sum of plane angles at each vertex is 240 0. It can be built by folding the bases of two pyramids, the bases of which are squares, and the side faces are regular triangles. The edges of an octahedron can be obtained by connecting the centers of adjacent faces of a cube, but if we connect the centers of adjacent faces of a regular octahedron, we obtain the edges of a cube. They say that the cube and the octahedron are dual to each other.

4)Icosahedron- composed of twenty equilateral triangles. Each vertex of the icosahedron is the vertex of five triangles. Therefore, the sum of the plane angles at each vertex is equal to 300 0.

5) Dodecahedron- a polyhedron made up of twelve regular pentagons. Each vertex of the dodecahedron is the vertex of three regular pentagons. Therefore, the sum of the plane angles at each vertex is 324 0.

The dodecahedron and icosahedron are also dual to each other in the sense that by connecting the centers of adjacent faces of the icosahedron with segments, we get a dodecahedron, and vice versa.

A regular tetrahedron is dual to itself.

Moreover, there is no regular polyhedron whose faces are regular hexagons, heptagons, and n-gons in general for n ≥ 6.

A regular polyhedron is a polyhedron in which all faces are regular equal polygons and all dihedral angles are equal. But there are also polyhedra in which all polyhedral angles are equal, and the faces are regular, but opposite regular polygons. Polyhedra of this type are called equiangular semiregular polyhedra. Polyhedra of this type were first discovered by Archimedes. He described in detail 13 polyhedra, which were later named the bodies of Archimedes in honor of the great scientist. These are truncated tetrahedron, truncated oxahedron, truncated icosahedron, truncated cube, truncated dodecahedron, cuboctahedron, icosidodecahedron, truncated cuboctahedron, truncated icosidodecahedron, rhombicuboctahedron, rhombicosidodecahedron, "snub" (snub) cube, "snub" (kur nose) dodecahedron.

2.4. Semiregular polyhedra or Archimedean solids are convex polyhedra with two properties:

1. All faces are regular polygons of two or more types (if all faces are regular polygons of the same type, it is a regular polyhedron).

2. For any pair of vertices, there is a symmetry of the polyhedron (that is, a movement that transforms the polyhedron into itself) transferring one vertex to the other. In particular, all polyhedral vertex angles are congruent.

In addition to semiregular polyhedra, from regular polyhedra - Platonic solids - you can obtain so-called regular stellate polyhedra. There are only four of them, they are also called Kepler-Poinsot bodies. Kepler discovered a small dodecahedron, which he called the prickly or hedgehog, and a large dodecahedron. Poinsot discovered two other regular stellated polyhedra, respectively dual to the first two: the great stellated dodecahedron and the great icosahedron.

Two tetrahedrons passing through one another form an octahedron. Johannes Kepler gave this figure the name “stella octangula” - “octagonal star”. It is also found in nature: this is the so-called double crystal.

In the definition of a regular polyhedron, the word “convex” was deliberately not emphasized - counting on apparent obviousness. And it means an additional requirement: “and all the faces of which lie on one side of the plane passing through any of them.” If we abandon such a restriction, then to the Platonic solids, in addition to the “extended octahedron,” we will have to add four more polyhedra (they are called Kepler-Poinsot solids), each of which will be “almost regular.” All of them are obtained by Platonov’s “starring” body, that is, by extending its edges until they intersect with each other, and therefore are called stellate. The cube and tetrahedron do not generate new figures - their faces, no matter how much you continue, do not intersect.

If you extend all the faces of the octahedron until they intersect with each other, you will get a figure that appears when two tetrahedra interpenetrate - the “stella octangula,” which is called “extended octahedron."

The icosahedron and dodecahedron give the world four “almost regular polyhedra” at once. One of them is the small stellated dodecahedron, first obtained by Johannes Kepler.

For centuries, mathematicians did not recognize the right of all kinds of stars to be called polygons due to the fact that their sides intersect. Ludwig Schläfli did not expel a geometric body from the family of polyhedra simply because its faces intersected themselves; however, he remained adamant as soon as the conversation turned to the small stellated dodecahedron. His argument was simple and weighty: this Keplerian animal does not obey Euler’s formula! Its spines are formed twelve faces, thirty edges and twelve vertices, and, therefore, B+G-R does not equal two at all.

Schläfli was both right and wrong. Of course, the geometric hedgehog is not so prickly as to rebel against the infallible formula. You just need to not consider that it is formed by twelve intersecting star-shaped faces, but look at it as a simple, honest geometric body made up of 60 triangles, having 90 edges and 32 vertices.

Then B+G-R=32+60-90 is equal, as expected, to 2. But then the word “correct” does not apply to this polyhedron - after all, its faces are now not equilateral, but just isosceles triangles. Kepler didn't realized that the figure he received had a double.

The polyhedron, which is called the “great dodecahedron,” was built by the French geometer Louis Poinsot two hundred years after Kepler’s star figures.

The great icosahedron was first described by Louis Poinsot in 1809. And again Kepler, having seen a large stellated dodecahedron, left the honor of discovering the second figure to Louis Poinsot. These figures also half obey Euler's formula.

Practical use

Polyhedra in nature

Regular polyhedra are the most advantageous shapes, which is why they are widespread in nature. This is confirmed by the shape of some crystals. For example, table salt crystals are cube-shaped. In the production of aluminum, aluminum-potassium quartz is used, the single crystal of which has the shape of a regular octahedron. The production of sulfuric acid, iron, and special types of cement cannot be done without sulfurous pyrites. The crystals of this chemical are dodecahedron shaped. Antimony sodium sulfate, a substance synthesized by scientists, is used in various chemical reactions. The crystal of sodium antimony sulfate has the shape of a tetrahedron. The last regular polyhedron, the icosahedron, conveys the shape of boron crystals.

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. They are also used in architecture. Many forms of stellate polyhedra are suggested by nature itself. Snowflakes are star-shaped polyhedra. Since ancient times, people have tried to describe all possible types of snowflakes and compiled special atlases. Several thousand different types of snowflakes are now known.

Regular polyhedra are also found in living nature. For example, the skeleton of the single-celled organism Feodaria (Circjgjnia icosahtdra) is shaped like an icosahedron. Most feodaria live in the depths of the sea and serve as prey for coral fish. But the simplest animal protects itself with twelve spines emerging from the 12 peaks of the skeleton. It looks more like a star polyhedron.

We can also observe polyhedra in the form of flowers. A striking example is cacti.

Related information.