A polyhedron that consists of two flat polygons. Polyhedra and their types

Introduction

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

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

Polyhedra can have a varied and very complex structure. Various structures, such as brick and concrete block houses under construction, 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 20-sided, cube. In physics, crystals are an example of polyhedra.

Since ancient times, the concept of beauty has been associated with symmetry. Probably, this explains the human interest in polyhedrons - amazing symbols of symmetry that attracted the attention of prominent thinkers who were struck by the beauty, perfection, harmony of these figures.

The first mentions of polyhedra are known as early as 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 a height of 146.5 m. It is not by chance that they say that the pyramid of Cheops is a mute 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 a gradual transition from practical to philosophical geometry took place. Of great importance in these schools are the reasoning with the help of which it was possible to obtain new geometric properties.

One of the first and most famous schools was Pythagorean, named after its founder, Pythagoras. A 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 is made up 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 should have the form of various bodies:

§ The universe is a dodecahedron

§ Earth - cube

§ Fire is a tetrahedron

§ Water is an icosahedron

§ Air is an octahedron

Later, the teaching of the Pythagoreans about regular polyhedra was set forth in his writings by another ancient Greek scientist, philosopher - the idealist Plato. Since then, regular polyhedra have been called Platonic solids.

Platonic solids are called regular homogeneous convex polyhedra, that is, convex polyhedra, all faces and angles of which are equal, and the faces are regular polygons. The same number of edges converges to each vertex of a regular polytope. 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.

Polyhedron theory is a modern branch of mathematics. It is closely related to topology, graph theory, is of great importance both for theoretical research in geometry and for practical applications in other branches of mathematics, for example, in 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, which is equivalent to the first definition of a polyhedron.

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

1) every two tetrahedra have no common points, or have a common vertex, or only a common edge, or a whole common face;

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

Polyhedron elements

The face of a polyhedron is some polygon (a bounded closed region is called a polygon, the boundary of which 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 dihedral angles at its edges. The dihedral angle at an edge of a polyhedron is determined by its faces that fit this edge.

Classification of polyhedra

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

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

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

Cauchy's theorem. Two closed convex polyhedra, equally composed of correspondingly 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 to each other regular polygons, thirdly, the same number of faces converge at each of its vertices, and, fourthly, all its dihedral angles are equal.

There are five convex regular polyhedrons - a tetrahedron, octahedron, and icosahedron with triangular faces, a cube (hexahedron) with square faces, and a dodecahedron with pentagonal faces. The proof of this fact has been known for over two thousand years; with this proof and the study of five regular bodies, the "Beginnings" 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. The tetrahedron has 4 faces, in translation from the Greek "tetra" - four, "edron" - a face. Hexahedron (cube) has 6 faces, "hexa" - six; octahedron - octahedron, "octo" - eight; dodecahedron - dodecahedron, "dodeca" - twelve; the icosahedron has 20 faces, the ikosi 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 flat angles at each vertex is 180 0);

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

3) Regular octahedron or simply octahedron– a polyhedron with eight regular triangular faces and four faces converging at each vertex. The octahedron is composed of eight equilateral triangles. Each vertex of the octahedron is the vertex of four triangles. Therefore, the sum of the flat angles at each vertex is 240 0. It can be built by adding two pyramids at the bases, at the base of which are squares, and the side faces are regular triangles. The edges of the octahedron can be obtained by connecting the centers of the adjacent faces of the cube, but if we connect the centers of the adjacent faces of the regular octahedron, we will get the edges of the cube. The cube and octahedron are said to be dual to each other.

4)Icosahedron- made up of twenty equilateral triangles. Each vertex of the icosahedron is the vertex of five triangles. Therefore, the sum of the flat angles at each vertex is 300 0.

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

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

A regular tetrahedron is dual to itself.

Moreover, there is no regular polytope whose faces are regular hexagons, heptagons, and generally n-gons 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 such polyhedra in which all polyhedral angles are equal, and the faces are regular, but opposite regular polygons. Polyhedra of this type are called equi-semiregular polytopes. For the first time, polyhedra of this type were discovered by Archimedes. He described in detail 13 polyhedrons, which were later named the bodies of Archimedes in honor of the great scientist. This is a truncated tetrahedron, a truncated oxahedron, a truncated icosahedron, a truncated cube, a truncated dodecahedron, a cuboctahedron, an icosidodecahedron, a truncated cuboctahedron, a truncated icosidodecahedron, a rhombocubooctahedron, a "currenoid"

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, this is a regular polyhedron).

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

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

Two tetrahedra, passing one through the other, form an octahedron. Johannes Kepler gave this figure the name "stella octangula" - "octagonal star". It is also found in nature: it is the so-called double crystal.

In the definition of the correct polyhedron, the word "convex" was deliberately - based on apparent obviousness - not emphasized. And it means an additional requirement: "and all the faces, which lie on one side of the plane passing through any of them." If we refuse 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 bodies), each of which will be "almost regular". All of them are obtained by "staring" Platonov bodies, that is, the extension of its edges until they intersect with each other, and therefore are called star-shaped. The cube and the tetrahedron do not generate new shapes - their faces, no matter how much you go on, do not intersect.

If we extend all the faces of the octahedron until they intersect with each other, then we get a figure that arises when two tetrahedrons interpenetrate - "stele octangula", which is called "continued octahedron ".

Icosahedron and dodecahedron give the world four "almost regular polyhedrons" at once. One of them is the small stellated dodecahedron, first obtained by Johannes Kepler.

For centuries, mathematicians have not recognized the right to be called polygons for all kinds of stars due to the fact that their sides intersect. Ludwig Schläfli did not expel a geometric body from the family of polyhedrons just because its faces are self-intersecting; nevertheless, he remained adamant as soon as the small stellated dodecahedron was discussed. His argument was simple and weighty: this Keplerian animal does not obey Euler's formula! Its thorns are formed twelve faces, thirty edges and twelve vertices, and, therefore, B + G-R is not at all equal to two.

Schläfli was both right and wrong. Of course, the geometric hedgehog is not so prickly as to rebel against an infallible formula. It is only necessary not to assume that it is formed by twelve intersecting star-shaped faces, but to look at it as a simple, honest geometric body, composed of 60 triangles, with 90 edges and 32 vertices.

Then В + Г-Р = 32 + 60-90 is equal, as it should be, 2. But then the word "correct" is not applicable to this polyhedron - after all, its faces are now not equilateral, but only isosceles triangles. Kepler not thought that the figure he received has a double.

The polyhedron called the "great dodecahedron" was built by the French geometer Louis Poinseau two hundred years after the Keplerian star-shaped figures.

The large icosahedron was first described by Louis Poinseau in 1809. And again Kepler, seeing a large stellated dodecahedron, left Louis Poinseau with the honor of discovering the second figure. These figures also obey half of 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 grades of cement is not complete without pyrite. The crystals of this chemical are in the shape of a dodecahedron. Antimony sodium sulfate, a substance synthesized by scientists, is used in various chemical reactions. The crystal of antimony sodium sulfate has the shape of a tetrahedron. The last regular polyhedron, the icosahedron, gives the shape of boron crystals.

Star polyhedrons 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 star polyhedra are suggested by nature itself. Snowflakes are star polyhedra. Since ancient times, people have tried to describe all possible types of snowflakes, they have made special atlases. Several thousand different types of snowflakes are now known.

Regular polyhedra are also found in nature. For example, the skeleton of the unicellular organism of Feodaria (Circjgjnia icosahtdra) resembles an icosahedron in shape. Most feudariums live in the deep sea and serve as prey for coral fish. But the simplest animal defends itself with twelve needles extending from the 12 vertices of the skeleton. It looks more like a stellated polyhedron.

We can also observe polyhedrons in the form of flowers. Cacti are a prime example.

Similar information.

"Types of polyhedra" - Regular star polyhedra. Dodecahedron. Small stellated dodecahedron. Polyhedra. Hexahedron. Plato's bodies. Prismatoid. Pyramid. Icosahedron. Octahedron. A body bounded by a finite number of planes. Stellated octahedron. Two faces. The law of reciprocity. Mathematician. Tetrahedron.

"Geometric body polyhedron" - Polyhedra. Prisms. The existence of incommensurable quantities. Poincaré. Edge. Measurement of volumes. The faces of the 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" - Cube edge. 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. Single icosahedron. Tetrahedron and icosahedron. Unit dodecahedron. Octahedron and tetrahedron. Cube and tetrahedron.

"Polyhedrons" stereometry "- Polyhedrons in architecture. Section of polyhedra. Give a name to the polyhedron. Great Pyramid of Giza. Platonic solids. Correct the logical chain. Polyhedron. Historical reference. The finest hour of polyhedrons. Solving problems. Lesson objectives. "Playing with the audience". Whether the geometric shapes and their names match.

"Stellated forms of polyhedrons" - Large stellated dodecahedron. The polyhedron shown in the figure. Star polyhedra. Side ribs. Stellated cuboctahedra. Stellated truncated icosahedron. A polyhedron obtained by truncating a stellated truncated icosahedron. The 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. Cut plane trail. Section. Let's find the point of intersection of the lines. Plane. Construct a section of the cube. Construct a section of the prism. We find the point. Prism. Sectioning methods. The resulting hexagon. Section of a cube. Axiomatic method.

There are 29 presentations in total

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 that have 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 ygrismg, and parallelograms are 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 depicts a quadrangular prism ABCDA "B" C "D".

The prism is called straight, if its lateral 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 is proved that the diagonals of the parallelepiped intersect 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 pinnacle 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 a regular polygon lies 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, it entirely contains the segment connecting them.

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

These definitions are equivalent. We omit the 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 is proved that in 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= 2. It turned out that it is also valid for any convex polytope. This property was first proved by L. Euler and is 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 a fan and a 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 fey means "tetrahedron" (Fig. 24.10, but).

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

If five regular triangles converge at each vertex of the polyhedron, then we 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 = 360 °.

In geometry it is proved 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 surface of a polyhedron is cut but some edges and unfolded so that all the polygons included in this surface lie in the same plane.

Note that a polyhedron can have several different 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, but, 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 sweep 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. In this case, the specified party is axis of the cylinder. In fig. 24.14 depicts a cylinder with an axle OO ', rotated rectangle AA "O" O around the 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 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- this 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 apex 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 length of the generatrix.

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 is a body that is obtained by rotating a semicircle a around its diameter. In fig. 24.16 depicts a ball obtained by rotating a semicircle around a diameter AA ". Point ABOUT are 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 flattened.

Any section of a sphere by a plane is a circle. The radius of the ball will be greatest if the plane passes through the center of the ball. 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 a and a straight line intersecting it be given but. Let us take an arbitrary point A "in space, which does not belong to the straight line but, and lead through X straight but", parallel line but(fig. 24.17). Straight but" 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 but, then with a parallel projection X " is the point at which the line but crosses the plane but.

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 but. then each point X space can be associated with a single point A "- a parallel projection of the point X on plane a (when designing parallel to a straight line but). Plane but called plane of projections. About straight but they say she will bark design direction - when replacing the straight line but any other direct design result parallel to it will not change. All straight lines parallel to a straight line but, one and the same direction of design and are called together with a straight line but 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 under sunlight, since the sun's rays can be considered parallel.

Parallel design has a number of properties, the knowledge of which is necessary when depicting geometric bodies on a plane. Let us formulate the main ones without presenting 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 midpoint of its projection.

When depicting geometric bodies on a plane, it is necessary to monitor the fulfillment of the specified 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 with 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 contribute to the creation of a correct representation of them.

Let's depict, 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 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. 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 OS provides 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.

To correctly depict a regular hexagon in parallel design, you need to pay attention to the following. Let ABCDEF be a regular hexagon. Then BCEF is a rectangle (Fig. 24.20) and, therefore, in 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 = 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" = O "D".

Thus, the sequence for 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 point ABOUT" conduct a straight, parallel B's "(or E "F");

§ an arbitrary point is chosen on the constructed line BUT" and mark the point D " such that About "D" = A "O" and connect the point BUT" with dots IN" 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 plane of the projection. This projection will be an ellipse. The center of the ball will be represented by the center of this ellipse (Fig. 24.22). The corresponding poles can now 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 straight line with the ellipse; then through point C we draw a tangent to the ellipse representing the equator. It is proved 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 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 on the plane a straight circular cylinder (Fig. 24.24) and a straight circular cone (Fig. 24.25).

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

Geometric solids

Introduction

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

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

All geometric bodies are divided into polyhedra and round bodies.

Polyhedra

Polyhedron Is a geometric body, the surface of which consists of a finite number of flat polygons.

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

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

Peaks 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 all lies on one side of any of its edges.

The task... Please indicate facets, ribs and tops the 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 - 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 ribs are called tops of the prism... Side ribs are ribs 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 А 1 А 2 B 2 B 1,… - side faces.

Prism properties:

· The bases of the prism are equal and parallel.

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

Diagonal prism a segment connecting two vertices that do not belong to the same face is called.

The height of the prism is called the perpendicular dropped from the point of the upper base to the plane of the lower base.

The prism is called 3-sided, 4-sided, ..., n-gonal if its bases
3-gons, 4-gons, ..., n-gons.

Straight prism called a prism in which the lateral edges are perpendicular to the bases. The side faces of a straight prism are rectangles.

Oblique prism called a prism that is not straight. The side faces of the inclined prism are parallelograms.

Correct prism called straight a prism with regular polygons at its bases.

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

Square lateral surface prisms called the sum of the areas of its lateral faces.

S full = S side + 2 S main

Polyhedron

• Polyhedron 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, the author of the first surviving treatise on mathematics "Beginning" (in 15 books)

side faces.

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

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

• If the lateral edge of the 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 oblique ... A straight prism has side faces - rectangles.

The bases of the prism are equal.

• The bases of the prism are equal.

• At the prism, the bases lie in parallel planes.

• The side edges of the 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 of paintings by 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 III century BC. e. a lighthouse was built so that ships could safely pass the reefs on their way to the bay of Alexandria. At night they were helped in this by the reflection of the flames, and during the day - by a column of smoke. It was the world's first lighthouse, and it stood for 1500 years.

• The lighthouse was built on the small island of Pharos in the Mediterranean, 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 fragments were used in the construction of a military fort. The fort was rebuilt more than once and still stands on the site of the world's first lighthouse.

Mausol was the ruler of Kariy. The capital of the region was Halicarnassus. Mavsol 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 the completion of work on the tomb. Artemisia continued to supervise the construction. The tomb was built in 350 BC. e. It was named the Mausoleum after the king.

The ashes of the royal couple were kept in golden urns in the burial vault 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 stepped pyramid. At a height of 43 m above the ground, it was crowned with a sculptural image 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 its foundations. Another three hundred years passed before archaeologists began excavations. In 1857, all 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 craters of an extinct volcano have, as a rule, geometrically regular shapes. In the ground, stones are sometimes found in such a shape as if someone had carefully cut them out, grinded, polished. crystals.

parallelepiped.

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

• Models of a rectangular parallelepiped are:

• cool room

• It turns out that calcite crystals, no matter how much they are fractions into smaller parts, always disintegrate into parallelepiped-shaped fragments.

• City buildings are most often polyhedron-shaped, as a rule, they are ordinary parallelepipeds, and only unexpected architectural solutions decorate cities.

• 1. Is a prism correct 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 the inclined triangular prism are 40 and 30 cm2. The angle between these faces is straight. Find the area of ​​the lateral surface of the prism.

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

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

• 2) a 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 revised 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 on the eastern countries, were the first to borrow the knowledge of the East and began to develop it. Scientists of the Ionian school for the first time subjected to logical processing and systematized mathematical information borrowed from ancient Eastern peoples, especially from the Babylonians. Thales, the head of this school, Proclus and other historians attribute many geometric discoveries. About attitude Pythagoras of Samos to geometry, Proclus writes in his commentary to Euclid's "Principles" the following: "He studied this science (ie, geometry), proceeding from its first foundations, and tried to obtain theorems using purely logical thinking." Proclus ascribes to Pythagoras, in addition to the well-known theorem on the square of the hypotenuse, the construction of five regular polyhedra:

Plato's bodies

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

Octahedron (fig. 3).

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

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

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

The faces of the dodecahedron are regular pentagons. The diagonals of the regular pentagon form the so-called star-shaped pentagon - a figure that served as an emblem, an identification mark for the students of Pythagoras. It is known that the Pythagorean Union 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 off the owner of the house who looked after 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 survived. He is credited with creating the doctrine of the similarity of figures. He was probably among the first scientists to consider geometry not as a practical and applied discipline, but as an abstract logical science.

In the school of Pythagoras, the existence of incommensurable quantities was discovered, that is, such, the relationship between which cannot be expressed by any whole 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 Ts2. This number is not rational (i.e., an integer or the 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 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 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 five regular polygons. (fig. 1).

• Cube or regular hexahedron - a regular quadrangular prism with equal edges, bounded 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 figure shows a pentagonal pyramid SABCDE and its sweep. Triangles with a common vertex are called side faces pyramids; the common vertex of the side faces - pinnacle pyramids; a polygon to which this vertex does not belong - basis pyramids; the edges of the pyramid, converging at its top, - lateral ribs pyramids. Height a pyramid is a segment of a perpendicular drawn through its apex to the plane of the base, with ends at the apex and on the plane of the base of the pyramid. The figure shows the segment SO- the height of the pyramid.

• Definition . A pyramid, the base of which is a regular polygon and the 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, Chinese and Indians by multiplying the area of ​​the base by the height. However, the ancient East was mainly aware of only individual rules found empirically, which were used to find volumes for the areas of figures. In 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 5th - 4th 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, means the volume of a cube. In the XI book "Beginnings", among others, the theorems of the following content are set forth.

• 1. Parallelepipeds with the same heights and equal bases of the same 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 equal-sized parallelepipeds, the areas of the bases are inversely proportional to the heights.

• Euclid's theorems refer only to the comparison of volumes, since the direct calculation of the volumes of bodies was probably considered by Euclid to be a matter of practical manuals on geometry. In works of the applied nature 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 faces of which are parallelograms... Parallelepipeds, like prisms, can be straight and oblique... Figure 1 shows an oblique parallelepiped, and Figure 2 shows a straight parallelepiped.

• A straight parallelepiped, the base of which 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, a matchbox.

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

• The theorem immediately implies 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 halved by it; in particular, all the diagonals of the parallelepiped meet at one point and are bisected by it. 2. Opposing faces of the parallelepiped are parallel and equal