Determination of a regular polyhedron types of regular polyhedra. Why regular polyhedra got such names

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Polyhedron called a body bounded by flat polygons. The elements of the polyhedron are tops , ribs and facets ... The polyhedron is called convex if all of it lies on one side of the plane of any of its faces. Correct is called a polyhedron whose faces are a regular polygon. In total, there are five regular convex polyhedra, which were first investigated and described by Plato, who lived in the 5th-4th centuries BC. Therefore, these polyhedra are also called “ Platonic solids ».

1. Tetrahedron (tetrahedron - regular triangular pyramid) - 4 vertices, 4 faces - triangles.

2. Hexahedron (hexagon - cube) - 8 vertices, 6 faces - squares.

3. Octahedron (octahedron) - 6 vertices, 8 faces - triangles.

4. Icosahedron (twenty-sided) - 12 vertices, 20 faces - triangles.

5. Dodecahedron (dodecahedron) - 20 vertices, 12 faces - pentagons.

Euler's formula for a regular polyhedron:

B + G - P \u003d 2

where AT -the number of vertices of the polyhedron,

G -the number of faces of the polyhedron,

R -the number of edges of the polyhedron.

Of the whole variety of convex polyhedra, the following are of greatest practical interest:

1) prisms - polyhedra, in which the lateral edges are parallel to each other, and the lateral faces are parallelograms;

2) pyramids - polyhedra, whose lateral edges intersect at one point - the vertex;

3) prismatoid - polyhedrons bounded by any two polygons located in parallel planes and called bases, and triangles or trapezoids, the vertices of which are the vertices of the bases (Figure 8.1).

Regular polyhedron is called a convex polyhedron, whose faces are equal regular polygons, and the dihedral angles at all vertices are equal to each other. It is proved that the same number of faces and the same number of edges converge at each of the vertices of a regular polytope.

There are five regular polyhedra in nature. Compared to the number of regular polygons, this is very small: for each integer n\u003e 2, there is one regular n-gon, i.e. there are infinitely many regular polygons. Regular polyhedrons are named according to the number of faces: tetrahedron (4 faces): hexahedron (6 faces), octahedron (8 faces), dodecahedron (12 faces), and icosahedron (20 faces). In Greek, "hedron" means a face, "tetra", "hexa", etc. - the indicated number of faces. It is not hard to guess that the hexahedron is nothing more than the familiar cube. The faces of the tetrahedron, octahedron and icosahedron are regular triangles, the cube are squares, and the dodecahedron are regular pentagons.

Polyhedron called convexif it all lies on one side of the plane of any of its faces; then its faces are convex too. The convex polyhedron cuts the space into two parts - an external and an internal one. Its inner part is a convex body. Conversely, if the surface of a convex body is polyhedral, then the corresponding polyhedron is convex.

Not a single geometric body possesses such perfection and beauty as regular polyhedra. “There are defiantly few regular polyhedra,” L. Carroll once wrote, “but this very modest detachment managed to get into the very depths of various sciences.

What is this defiantly small number and why there are just so many of them. How much? It turns out exactly five - no more, no less. This can be confirmed by developing a convex polyhedral angle. Indeed, in order to obtain any regular polyhedron according to its definition, the same number of faces must converge at each vertex, each of which is a regular polygon. The sum of the flat angles of a polyhedral angle must be less than 360, otherwise no polyhedral surface will work. Sorting out possible integer solutions to inequalities: 60k< 360, 90к < 360 и 108к < 360, можно доказать, что правильных многогранников ровно пять (к - число плоских углов, сходящихся в одной вершине многогранника).

The names of regular polyhedra come from Greece. In literal translation from the Greek "tetrahedron", "octahedron", "hexahedron", "dodecahedron", "icosahedron" mean: "tetrahedron", "octahedron", "hexahedron", "dodecahedron", "dyadiahedron". The 13th book of Euclid's Principles is dedicated to these beautiful bodies. They are also called Plato's bodies, tk. they occupied an important place in Plato's philosophical concept of the structure of the universe. Four polyhedrons personified in her four essences or "elements". The tetrahedron symbolized fire, because its top is directed upward; icosahedron - water, because he is the most "streamlined"; cube - the earth, as the most "stable"; octahedron - air, as the most "airy". The fifth polyhedron, the dodecahedron, embodied "everything that exists", symbolized the entire universe, and was considered the main one.

If you put on the globe the centers of the largest and most remarkable cultures and civilizations of the Ancient World, you will notice a pattern in their location relative to the geographic poles and the planet's equator. Many mineral deposits stretch along the icosahedral-dodecahedron grid. Even more amazing things happen at the intersection of these ribs: here are the centers of the most ancient cultures and civilizations: Peru, Northern Mongolia, Haiti, the Ob culture and others. At these points, there are highs and lows of atmospheric pressure, giant eddies of the World Ocean, here is the Scottish Loch Ness, the Bermuda Triangle. Further studies of the Earth, perhaps, will determine the attitude to this beautiful scientific hypothesis, in which, as you can see, regular polyhedra take an important place.

So, it was found that there are exactly five regular polyhedra. And how to determine the number of edges, faces, vertices in them? This is not difficult to do for polytopes with a small number of edges, but how, for example, can one obtain such information for an icosahedron? The famous mathematician L. Euler obtained the formula В + Г-Р \u003d 2, which connects the number of vertices / В /, faces / Г / and edges / Р / of any polyhedron. The simplicity of this formula is that it is not related to distance or angles. In order to determine the number of edges, vertices and faces of a regular polyhedron, we first find the number k \u003d 2y - xy + 2x, where x is the number of edges belonging to one face, y is the number of faces converging at one vertex.

So, regular polyhedra revealed to us the attempts of scientists to approach the secret of world harmony and showed the irresistible attractiveness of geometry.

List of regular polyhedra

There are only five regular polyhedra:

Our world is full of symmetry. Since ancient times, our ideas of beauty have been associated with it. Perhaps this explains the enduring interest of man in polyhedra - amazing symbols of symmetry, which attracted the attention of many prominent thinkers, from Plato and Euclid to Euler and Cauchy.

However, polyhedra are by no means only an object of scientific research. Their forms are complete and whimsical and are widely used in decorative arts. Typically, polyhedron models are constructed from reamers. But there is another way.

Mathematicians have long since proven the possibility of constructing three-dimensional objects from tape. In fig. 1 shows how to obtain a tetrahedron by bending a paper tape along the sides of the equilateral triangles drawn on it.

Figure: 1

In a similar way, you can collapse the cube (Fig. 2). Its edges also line up in a chain, and in order to change the direction of the tape to complete the shaping, it is enough to bend it along the diagonal of the square.

Figure: 2

So, at first glance, an unremarkable paper tape, when a pattern is applied to its surface, turns into a blank for constructing a wide variety of polyhedra. All regular polyhedra can be created from various patterns, except for the dodecahedron. This is due to the fact that flat patterns do not have symmetry axes of the 5th, 7th and higher orders - in other words, it is impossible to construct a continuous pattern of pentagons.

Fig. 3

The construction of an octahedron and an icosahedron is carried out on the basis of a pattern of regular triangles (Fig. 3 and Fig. 4). Having folded a ring of six for an octahedron, and for an icosahedron - of ten triangles, bend the tape in the opposite direction and continue to fold the same rings.

Fig. 4

The patterns of our ribbons are a special case of Shubnikov-Laves symmetry networks (see Fig. 5). Triangular cells are obtained by superimposing two pairs of mirror hexagonal lattices rotated relative to each other by 90 °, and square ones - by combining square lattices at an angle of 45 ° to each other. From these positions, the process of formation of polyhedra from a focus turns into a theoretically substantiated and natural phenomenon.

Figure: five

In fact, when the ring of the future polyhedron is folded, then, in the literal sense, the unit cell of the lattice is transferred to a certain step, that is, portable symmetry is performed. Changing the direction of shaping due to the bending of the tape in the opposite direction, we make a mental rotation of the cell around the lattice node, that is, the rotational symmetry is already manifested. Consequently, the strip blank provides rotary-transfer symmetry. In our constructions, such rotational-transferable symmetry can be realized with the angles of the turns; 30 ° 45 °, 60 °, 90 °, 120 °, 150 °, 180 °. This is the whole secret of the method of forming volumetric bodies from a flat strip.

Thus, it is clear that there can only be two types of tapes with split angles in multiples of 30 ° and 45 °. From them four regular polyhedra are obtained: a cube, an octahedron, a tetrahedron, an icosahedron - and a whole family of homogeneous polyhedra (see Fig. 6). In Johannes Kepler's excellent work "On Hexagonal Snowflakes" there is a very apt remark: "Among regular bodies, the first is considered to be the cube, the primordial figure, the father of all other bodies, the Octahedron, which has as many vertices as the cube has faces, is, as it were, his wife ... "Indeed, all the elements of the complex shapes formed from our ribbon are elements of a cube or octahedron, or both.

Fig. 6

polyhedron tetrahedron cube octahedron dodecahedron icosahedron

The construction of simple polytopes is not particularly difficult. But in order to fold complex star-shaped shapes from the tape, you will need special devices to hold the rings that are not yet connected to each other - clips, clamps, and the like. The creation of polyhedrons, original in their form, is extremely entertaining by the very process of shaping.

Students, graduate students, young scientists using the knowledge base in their studies and work will be very grateful to you.

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Polyhedroncalled convexif it all lies on one side of the plane of any of its faces; then its faces are convex too. The convex polyhedron cuts the space into two parts - an external and an internal one. Its inner part is a convex body. Conversely, if the surface of a convex body is polyhedral, then the corresponding polyhedron is convex.

So, regular polyhedra revealed to us the attempts of scientists to approach the secret of world harmony and showed the irresistible attractiveness of geometry.

List of regular polyhedra

There are only five regular polyhedra:

Picture	Regular polyhedron type	The number of sides at the edge	The number of edges adjacent to the vertex	Total number of vertices	Total number of edges	Total number of faces
	Tetrahedron


	Dodecahedron
	Icosahedron

Figure: 1

Figure: 2

Fig. 3

Fig. 4

Figure: five

Fig. 6

polyhedron tetrahedron cube octahedron dodecahedron icosahedron

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Regular polyhedron type

The number of sides at the edge

The number of edges adjacent to the vertex

Total number of vertices

Total number of edges

Total number of faces

Tetrahedron

Dodecahedron

Icosahedron

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

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

Among polyhedra, there are prisms and pyramids.

Prism - it is a polyhedron, the surface of which consists of two equal polygons and parallelograms that have common sides with each of the bases.

Two equal polygons are called grounds ggrismg, and parallelograms - its lateral faces. The side faces form lateral surface prisms. Ribs that do not lie in the bases are called lateral ribs prisms.

The prism is called n-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.

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

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

Diagonal of a parallelepiped is a line segment connecting its opposite vertices. The parallelepiped has four diagonals.

It has been proven thatthe diagonals of the parallelepiped meet at one point and are halved 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 side faces of the pyramid. The common vertex of these triangles is called top pyramids, edges 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 a tetrahedron (Figure 24.8). The peculiarity of the triangular pyramid is that any face can be considered as a base.

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

Note that one should distinguish regular tetrahedron (i.e., a tetrahedron in which all edges are equal) 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 vomit 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 entirely contains the segment connecting them.

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

These definitions are equivalent. The proof of this fact is omitted.

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 thatin a convex polytope, all faces are convex polygons.

Consider several convex polyhedra (table 24.1)

From this table it follows that for all the considered convex polytopes the equality B - P + D\u003d 2. It turned out that it is also valid for any convex polytope. This property was first proved by L. Euler and was called Euler's theorem.

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

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

Indeed, if the fan and the 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 the polyphane, then we obtain right-handed tetrahedron, which in translation from the fecal means "tetrahedron" (Fig. 24.10, and).

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

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

If the faces of the polyphane 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 grains of the polyphane 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, e). Its surface consists of twelve regular pentagons.

The faces of a polyhedron cannot be hexagonal or more, since even for a hexagon 120 ° 3 \u003d 360 °.

It is proved in geometry that in three-dimensional Euclidean space there are exactly five different kinds of regular polyhedra '.

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

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

Note that a polyhedron can have several different sweeps, depending on which edges we cut. Figure 24.11 shows figs "urs, which are different sweeps of a regular quadrangular pyramid, that is, a pyramid at the base of which is a square, and all side edges are 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 singularities of the polyhedron. For example, the figures in Fig. 24.12 are not sweeps of a regular quadrangular pyramid: in the figure shown in Fig. 24.12, and, at the top M four faces converge, which cannot be in a regular quadrangular pyramid; and in the figure shown in Fig. 24.12, b, side ribs A B and Sun not equal.

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

Rotation bodies

Body of rotation is called a body resulting from the rotation of a figure (usually flat) around a straight line. This line is called axis of rotation.

Cylinder - ego body, which is obtained by rotating a rectangle around one of its sides. Moreover, the specified party is axis of the cylinder. In fig. 24.14 depicts a cylinder with an axis OO ', rotated rectangle AA "O" Oaround straight OO ". Points ABOUT and ABOUT" - the centers of the bases of the cylinder.

The cylinder, which is obtained by 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 axis of the cylinder.

Sweep the lateral surface of a straight 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 circumference of the base.

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

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

The cone, which is obtained by rotating a right-angled triangle around one of its legs, is called straight circular cone, Since its base is a circle, and the top is projected into the center of this circle. The lateral surface of the cone is formed by segments equal to the hypotenuse of the triangle, the rotation of which forms a cone.

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

When a cylinder, cone, or any other solid of revolution intersects by a plane containing the axis of revolution, we get 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 by rotating a semicircle a around its diameter. In fig. 24.16 shows a ball obtained by rotating a semicircle around a diameter AA ". Point ABOUTcalled 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 sphere by a plane is a circle. The radius of the sphere will be greatest if the plane passes through the center of the sphere. Therefore, the section of the ball by a plane passing through the center of the ball is called a large circle of the ball, and the circle that bounds it - a large circle.

IMAGE OF GEOMETRIC BODIES ON A 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 visual representation of spatial figures. For 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 it be given and. Take an arbitrary point A "in space, which does not belong to the straight line and, and lead through X straight and", parallel line and(fig. 24.17). Straight and" intersects the plane at some point X ", which is called parallel projection of the point X onto the plane a.

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

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

Thus, if the plane a and the line intersecting it are given and. then each point X space can be associated with a single point A "- a parallel projection of the point Xon plane a (when designing parallel to a straight line and). Plane and called plane of projections.About straight and they say she will bark design direction - when replacing the straight line and any other direct design result parallel to it will not change. All lines parallel to a line and, one and the same design direction and are called together with a straight line and projecting straight lines.

Projection figures F call a lot F projection of all points. Display mapping to each point X figures F"its parallel projection is a point X " figures F ", called parallel design figures F(fig.24.18).

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

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

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

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

2) projections of parallel lines are parallel or coincide;

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

This theorem implies consequence: in parallel design, the midpoint of a segment is projected into the middle of its projection.

When depicting geometric bodies on a plane, it is necessary to monitor the fulfillment of these properties. Otherwise, it can be arbitrary. So, the angles and ratios of the lengths of non-parallel segments can change arbitrarily, i.e., for example, a triangle in parallel projection is represented by an arbitrary triangle. But if the triangle is equilateral, then the projection of its median should connect the apex 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 \u200b\u200bthem.

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

Let's build the lower base of the prism first (you can start from the upper one). According to the rules of parallel design, oggo will be represented by an arbitrary parallelogram ABCD (Fig. 24.19, a). Since the edges of the prism are parallel, we construct parallel 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. Connecting successively points A", B ", C", D ", we obtain a quadrangle A" B "C" D ", representing the upper base of the prism. It is easy to prove that A "B" C "D" - parallelogram equal to parallelogram ABCD and, therefore, we have an image of a prism, the bases of which are equal squares, and the rest of the faces are parallelograms.

If you need to depict a straight prism, the bases of which are squares, then you can show that the lateral 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 quadrangle, and also a rectangular parallelepiped, since all its faces are rectangles.

Let us now figure 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 a vertical segment is taken OS, depicting the height of the pyramid. Note that the verticality of the segment OSprovides greater clarity of the drawing. Finally, point S is connected to all vertices of the base.

Let's draw, for example, a regular pyramid, the base of which is a regular hexagon.

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

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

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

§ through point ABOUT" conduct a straight, parallel B's " (or E "F");

§ an arbitrary point is chosen on the constructed line AND" and mark the point D " such that About "D" = A "Oh" and connect the point AND"with dots AT" and F"and point D "- with dots FROM" and E ".

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

In parallel design, the ball is drawn as a circle of the same radius. To make the image of the ball more visual, a projection of some large circle is drawn, the plane of which is not perpendicular to the projection plane. This projection will be an ellipse. The center of the ball will be represented by the center of this ellipse (Fig. 24.22). Now the corresponding poles can be found 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 the point C - the intersection of this line with the ellipse; then through point C we draw a tangent line to the ellipse representing the equator. It is proven that the distance CM is equal to the distance from the center of the ball to each of the poles. Therefore, postponing the segments ON and OS, equal CM, get poles N and S.

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

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

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

Tetrahedron translated from the ancient Greek tetrahedron. This is the simplest polyhedron whose faces are.

A tetrahedron has 4 faces, 4 vertices and 6 edges. The faces are equilateral triangles. Three corners converge at each vertex. Amount of these angles at each vertex is 180º.

Octahedron

Translated from Greek οκτάεδρον (οκτώ - " eight"And έδρα -" base») Is a polyhedron with eight faces. Faces of a regular octahedron -. The octahedron has 6 vertices and 12 edges. 4 triangles converge at each vertex, so sum of angles at each vertex of the octahedron is 240 °.

Cube in translation from ancient Greek κύβος2 or regular hexahedron(« correct hexagon"From the ancient Greek ἑξάς-" six"And ἕδρα -" seat, base») Is a regular polyhedron, each face of which is.

The number of sides at the edge - 4; the total number of faces is 6; the number of edges adjacent to the vertex - 3; the total number of vertices is 8; the total number of edges is 12. Sum of angles at each peak 90º + 90º + 90º \u003d 270º

Dodecahedron from the ancient Greek δώδεκα - " twelve"And εδρον -" edge". The dodecahedron is composed of twelve regular pentagons, which are its faces.

Each vertex of the dodecahedron is a vertex. Thus, a dodecahedron has 12 faces (pentagonal), 30 edges and 20 vertices (3 edges converge in each). Sum of angles at each peak 108º + 108º + 108º \u003d 324º

Icosahedronfrom the ancient Greek εἴκοσι " twenty»; ἕδρον « sitting», « base»Is a regular convex polyhedron, a 20-sided polyhedron. Each of the 20 faces is an equilateral triangle.

The number of edges is 30, the number of vertices is 12. The icosahedron has 59 stellate shapes. Leonard Euler in 1750 was the first to derive a formula connecting the number of vertices (B), faces (G) and edges (P) of any convex polyhedron by a simple relation: B + G \u003d P + 2.

Table 1

Regular polyhedra since ancient times they have attracted the attention of scientists, architects and artists. They were amazed by the beauty, perfection, harmony of these polyhedra.

Leonardo da Vinci was fond of the theory of polyhedra and often depicted them in his canvases. He illustrated the book of the monk Luca Pacioli “ About divine proportion».

Another famous artist who was also fond of geometry was Albrecht Durer. In his engraving “ MelancholyHe gave a perspective view of the dodecahedron.

The German astronomer and mathematician Johannes Kepler in his work, using regular polyhedrons, deduced the principle to which the shapes and sizes of the planets of the solar system obey. This model received the model “ Space cup»Kepler.

The famous painting by Salvador Dali " Last supper»Contains a perspective view of a regular dodecahedron.