In this lesson, we will describe the types of symmetry in space, get acquainted with the concept of a regular polyhedron.

As in planimetry, in space we will consider symmetry with respect to a point and with respect to a line, but in addition, symmetry with respect to a plane will appear.

Definition.

Points A and are called symmetrical about the point O (center of symmetry), if O is the midpoint of the segment. Point O is symmetrical to itself.

In order to obtain a point symmetrical to it with respect to point O for a given point A, you need to draw a straight line through points A and O, set aside a segment equal to OA from point O, and get the desired point (Figure 1).

Rice. 1. Symmetry about a point

Similarly, points B and are symmetrical about the point O, since O is the midpoint of the segment.

So, a law is given according to which each point of the plane goes to another point of the plane, and we said that any distances are preserved, that is, .

Consider symmetry with respect to a line in space.

To get a symmetrical point for a given point A with respect to some line a, you need to lower the perpendicular from point A to the line and set an equal segment on it (Figure 2).

Rice. 2. Symmetry with respect to a straight line in space

Definition.

Points A and are called symmetrical with respect to the line a (axis of symmetry) if the line a passes through the middle of the segment and is perpendicular to it. Each point of the line is symmetrical to itself.

Definition.

Points A and are called symmetrical with respect to the plane (plane of symmetry) if the plane passes through the middle of the segment and is perpendicular to it. Each point of the plane is symmetrical to itself (Figure 3).

Rice. 3. Symmetry with respect to the plane

Some geometric figures may have a center of symmetry, an axis of symmetry, a plane of symmetry.

Definition.

Point O is called the center of symmetry of a figure if each point of the figure is symmetrical with respect to it to some point of the same figure.

For example, in a parallelogram and a parallelepiped, the intersection point of all diagonals is the center of symmetry. Let's illustrate for a parallelepiped.

Rice. 4. Center of symmetry of the parallelepiped

So, with symmetry about the point O in the parallelepiped point A goes to point , point B goes to point, etc., thus, the box goes into itself.

Definition.

A straight line is called an axis of symmetry of a figure if each point of the figure is symmetrical about it to some point of the same figure.

For example, each diagonal of a rhombus is an axis of symmetry for it, a rhombus transforms into itself when it is symmetric about any of the diagonals.

Consider an example in space - a rectangular parallelepiped (lateral edges are perpendicular to the bases, equal rectangles at the bases). Such a parallelepiped has axes of symmetry. One of them passes through the center of symmetry of the parallelepiped (the intersection point of the diagonals) and the centers of the upper and lower bases.

Definition.

A plane is called the plane of symmetry of a figure if each point of the figure is symmetrical with respect to it to some point of the same figure.

For example, a cuboid has planes of symmetry. One of them passes through the middle of the opposite edges of the upper and lower bases (Figure 5).

Rice. 5. Plane of symmetry of a rectangular parallelepiped

Elements of symmetry are inherent in regular polyhedra.

Definition.

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

Theorem.

There is no regular polyhedron whose faces are regular n-gons for .

Proof:

Consider the case when is a regular hexagon. All its interior angles are equal:

Then at the internal angles will be larger.

At each vertex of the polyhedron, at least three edges converge, which means that each vertex contains at least three flat angles. Their total sum (assuming that each is greater than or equal to ) is greater than or equal to . This contradicts the statement: in a convex polyhedron, the sum of all plane angles at each vertex is less than .

The theorem has been proven.

Cube (Figure 6):

Rice. 6. Cube

The cube is made up of six squares; a square is a regular polygon;

Each vertex is a vertex of three squares, for example, vertex A is common to the square faces ABCD, ;

The sum of all plane angles at each vertex is , since it consists of three right angles. This is less than , which satisfies the notion of a regular polyhedron;

The cube has a center of symmetry - the point of intersection of the diagonals;

The cube has axes of symmetry, for example, straight lines a and b (Figure 6), where straight line a passes through the midpoints of opposite faces, and b through the midpoints of opposite edges;

A cube has planes of symmetry, such as a plane that passes through lines a and b.

2. Regular tetrahedron (regular triangular pyramid, all edges of which are equal to each other):

Rice. 7. Regular tetrahedron

A regular tetrahedron is made up of four equilateral triangles;

The sum of all plane angles at each vertex is , since a regular tetrahedron consists of three plane angles in . This is less than , which satisfies the notion of a regular polyhedron;

A regular tetrahedron has axes of symmetry; they pass through the midpoints of opposite edges, for example, straight line MN. In addition, MN is the distance between the crossing lines AB and CD, MN is perpendicular to the edges AB and CD;

A regular tetrahedron has planes of symmetry, each passing through an edge and the midpoint of the opposite edge (Figure 7);

A regular tetrahedron has no center of symmetry.

3. Regular octahedron:

Consists of eight equilateral triangles;

Four edges converge at each vertex;

The sum of all plane angles at each vertex is , since a regular octahedron consists of four plane angles along . This is less than , which satisfies the concept of a regular polyhedron.

4. Regular icosahedron:

Consists of twenty equilateral triangles;

Five edges converge at each vertex;

The sum of all plane angles at each vertex is , since a regular icosahedron consists of five plane angles along . This is less than , which satisfies the concept of a regular polyhedron.

5. Regular dodecahedron:

Consists of twelve regular pentagons;

Three edges converge at each vertex;

The sum of all plane angles at each vertex is . This is less than , which satisfies the concept of a regular polyhedron.

So, we considered the types of symmetry in space and gave strict definitions. We also defined the concept of a regular polyhedron, considered examples of such polyhedra and their properties.

Bibliography

1. I. M. Smirnova, V. A. Smirnov. Geometry. Grade 10-11: a textbook for students of educational institutions (basic and profile levels) / I. M. Smirnova, V. A. Smirnov. - 5th ed., Rev. and additional - M.: Mnemosyne, 2008. - 288 p.: ill.
2. Sharygin I. F. Geometry. Grade 10-11: A textbook for general educational institutions / Sharygin I. F. - M .: Bustard, 1999. - 208 p.: ill.
3. E. V. Potoskuev, L. I. Zvalich. Geometry. Grade 10: Textbook for general educational institutions with in-depth and profile study of mathematics / E. V. Potoskuev, L. I. Zvalich. - 6th ed., stereotype. - M.: Bustard, 2008. - 233 p.: ill.

1. Matemonline.com().
2. Fmclass.ru ().
3. 5class.net().

Homework

1. Specify the number of axes of symmetry of the cuboid;
2. indicate the number of symmetry axes of a regular pentagonal prism;
3. indicate the number of planes of symmetry of the octahedron;
4. build a pyramid that has all the elements of symmetry.

- (definition) a geometric body bounded on all sides by flat polygons - faces.

Examples of polyhedra:

The sides of the faces are called edges, and the ends of the edges are called vertices. According to the number of faces, 4-hedrons, 5-hedrons, etc. are distinguished. The polyhedron is called convex, if it is all located on one side of the plane of each of its faces. The polyhedron is called correct, if its faces are regular polygons (that is, those in which all sides and angles are equal) and all polyhedral angles at the vertices are equal. There are five types of regular polyhedra: tetrahedron, cube, octahedron, dodecahedron, icosahedron.

Polyhedron in three-dimensional space (the concept of a polyhedron) - a collection of a finite number of flat polygons such that

1) each side of one is at the same time a side of the other (but only one), called adjacent to the first (on this side);

2) from any of the polygons that make up the polyhedron, you can get to any of them by going to the one adjacent to it, and from this, in turn, to the one adjacent to it, etc.

These polygons are called faces, their sides ribs, and their vertices are peaks polyhedron.

Vertices of the polyhedron

Polyhedron edges

Facets of a polyhedron

A polyhedron is called convex if it lies on one side of the plane of any of its faces.

It follows from this definition that all faces of a convex polyhedron are flat convex polygons. The surface of a convex polyhedron consists of faces that lie in different planes. In this case, the edges of the polyhedron are the sides of the polygons, the vertices of the polyhedron are the vertices of the faces, the flat corners of the polyhedron are the corners of the polygons - faces.

A convex polyhedron all of whose vertices lie in two parallel planes is called prismatoid. A prism, a pyramid, and a truncated pyramid are special cases of a prismatoid. All side faces of a prismatoid are triangles or quadrilaterals, and the quadrangular faces are trapezoids or parallelograms.

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

The concept of a polyhedron, types of polyhedra in geometry

Geometry as a science contains a section of stereometry that studies the characteristics and properties of three-dimensional bodies, the sides of which in three-dimensional space are formed by limited planes (faces), are called "polyhedra". Types of polyhedra include more than a dozen representatives, differing in the number and shape of faces.

However, all polyhedra have common properties:

1. All of them have 3 integral components: a face (the surface of a polygon), a vertex (the corners formed at the junction of the faces), an edge (the side of the figure or a segment formed at the junction of two faces).
2. Each polygon edge connects two, and only two, faces that are adjacent to each other.
3. Convexity means that the body is completely located only on one side of the plane on which one of the faces lies. The rule applies to all faces of the polyhedron. Such geometric figures in stereometry are called convex polyhedra. The exception is star-shaped polyhedra, which are derivatives of regular polyhedral geometric solids.

Polyhedra can be divided into:

1. Types of convex polyhedra, consisting of the following classes: ordinary or classical (prism, pyramid, parallelepiped), regular (also called Platonic solids), semi-regular (second name - Archimedean solids).
2. Non-convex polyhedra (stellated).

Prism and its properties

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

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

The main properties of a prism:

• Congruent bases.
• All edges of the prism are equal and parallel to each other.
• All side faces are parallelogram-shaped.

Pyramid

A pyramid is a geometric body, which consists of one base and the n-th number of triangular faces, connected at one point - the vertex. It should be noted that if the side faces of the pyramid are necessarily represented by triangles, then the base can be either a triangular polygon, or a quadrangle, or a pentagon, and so on ad infinitum. In this case, the name of the pyramid will correspond to the polygon at the base. For example, if there is a triangle at the base of the pyramid - this is a quadrilateral - quadrangular, etc.

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

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

Pyramid properties:

• If all side edges of the pyramid are congruent (of the same height), then they all intersect with the base at the same angle, and around the base you can draw a circle with a center coinciding with the projection of the top of the pyramid.
• If a regular polygon lies at the base of the pyramid, then all side edges are congruent, and the faces are isosceles triangles.

Regular polyhedron: types and properties of polyhedra

In stereometry, a special place is occupied by geometric bodies with absolutely equal faces, at the vertices of which the same number of edges are connected. These solids are called Platonic solids, or regular polyhedra. Types of polyhedra with such properties have only five figures:

1. Tetrahedron.
2. Hexahedron.
3. Octahedron.
4. Dodecahedron.
5. Icosahedron.

Regular polyhedra owe their name to the ancient Greek philosopher Plato, who described these geometric bodies in his writings and connected them with the natural elements: earth, water, fire, air. The fifth figure was awarded the similarity with the structure of the universe. In his opinion, the atoms of natural elements in shape resemble the types of regular polyhedra. Due to their most fascinating property - symmetry, these geometric bodies were of great interest not only to ancient mathematicians and philosophers, but also to architects, artists and sculptors of all times. The presence of only 5 types of polyhedra with absolute symmetry was considered a fundamental discovery, they were even awarded a connection with the divine principle.

Hexahedron and its properties

In the form of a hexagon, the successors of Plato assumed a similarity with the structure of the atoms of the earth. Of course, at present, this hypothesis has been completely refuted, which, however, does not prevent the figures from attracting the minds of famous figures with their aesthetics in modern times.

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

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

Tetrahedron

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

Properties of a regular tetrahedron:

1. All faces of a tetrahedron - this from which it follows that all faces of a tetrahedron are congruent.
2. Since the base is represented by a regular geometric figure, that is, it has equal sides, then the faces of the tetrahedron converge at the same angle, that is, all angles are equal.
3. The sum of the flat angles at each of the vertices is 180, since all angles are equal, then any angle of a regular tetrahedron is 60.
4. Each of the vertices is projected to the point of intersection of the heights of the opposite (orthocenter) face.

Octahedron and its properties

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

Octahedron properties:

1. The very name of a geometric body suggests the number of its faces. The octahedron consists of 8 congruent equilateral triangles, in each of the vertices of which an equal number of faces converge, namely 4.
2. Since all the faces of an octahedron are equal, so are its interface angles, each of which is equal to 60, and the sum of the plane angles of any of the vertices is thus 240.

Dodecahedron

If we imagine that all the faces of a geometric body are a regular pentagon, then we get a dodecahedron - a figure of 12 polygons.

Dodecahedron properties:

1. Three faces intersect at each vertex.
2. All faces are equal and have the same edge length and equal area.
3. The dodecahedron has 15 axes and planes of symmetry, and any of them passes through the vertex of the face and the middle of the opposite edge.

icosahedron

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

1. All faces of the icosahedron are isosceles triangles.
2. Five faces converge at each vertex of the polyhedron, and the sum of the adjacent angles of the vertex is 300.
3. The icosahedron, like the dodecahedron, has 15 axes and planes of symmetry passing through the midpoints of opposite faces.

Semiregular polygons

In addition to the Platonic solids, the group of convex polyhedra also includes the Archimedean solids, which are truncated regular polyhedra. The types of polyhedra of this group have the following properties:

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

Star polyhedra

Representatives of non-volumetric types of geometric bodies are star-shaped polyhedra, the faces of which intersect with each other. They can be formed by merging two regular three-dimensional bodies or by continuing their faces.

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

There are special topics in school geometry that you look forward to, anticipating a meeting with incredibly beautiful material. These topics include "Regular polyhedra".Here, not only the wonderful world of geometric bodies with unique properties opens up, but also interesting scientific hypotheses. And then the geometry lesson becomes a kind of study of unexpected aspects of the usual school subject.

None of the geometric bodies possess such perfection and beauty as regular polyhedra. "Regular polyhedra are defiantly few," L. Carroll once wrote, "but this detachment, which is very modest in number, managed to get into the very depths of various sciences."

What is this defiantly small number and why there are so many of them. And how much? It turns out that exactly five - no more, no less. This can be confirmed by unfolding 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 plane angles of a polyhedral angle must be less than 360 o, otherwise no polyhedral surface will be obtained. Going through possible integer solutions of inequalities: 60k< 360, 90к < 360 и 108к < 360, можно доказать, что правильных многогранников ровно пять (к - число плоских углов, сходящихся в одной вершине многогранника), рис.1.

The names of regular polyhedra come from Greece. In literal translation from Greek "tetrahedron", "octahedron", "hexahedron", "dodecahedron", "icosahedron" mean: "tetrahedron", "octahedron", "hexahedron". dodecahedron, dodecahedron. The 13th book of Euclid's Elements is dedicated to these beautiful bodies. They are also called the bodies of Plato, because. they occupied an important place in Plato's philosophical concept of the structure of the universe. Four polyhedrons personified in it four essences or "elements". The tetrahedron symbolized fire, because. its top is directed upwards; icosahedron - water, because he is the most "streamlined"; cube - earth, as the most "steady"; 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.

The ancient Greeks considered harmonious relationships to be the basis of the universe, therefore, their four elements were connected by such a proportion: earth/water=air/fire. The atoms of the "elements" were tuned by Plato in perfect consonances, like the four strings of a lyre. Let me remind you that a pleasant consonance is called consonance. It must be said that the peculiar musical relationships in the Platonic solids are purely speculative and have no geometric basis. Neither the number of vertices of the Platonic solids, nor the volumes of regular polyhedra, nor the number of edges or faces are connected by these relations.

In connection with these bodies, it would be appropriate to say that the first system of elements, which included four elements - earth, water, air and fire - was canonized by Aristotle. These elements remained the four cornerstones of the universe for many centuries. It is quite possible to identify them with the four states of matter known to us - solid, liquid, gaseous and plasma.

An important place was occupied by regular polyhedra in the system of the harmonious structure of the world by I. Kepler. All the same faith in harmony, beauty and the mathematically regular structure of the universe led I. Kepler to the idea that since there are five regular polyhedra, only six planets correspond to them. In his opinion, the spheres of the planets are interconnected by the Platonic solids inscribed in them. Since for each regular polyhedron the centers of the inscribed and circumscribed spheres coincide, the whole model will have a single center, in which the Sun will be located.

Having done a huge computational work, in 1596 I. Kepler published the results of his discovery in the book "The Secret of the Universe". He inscribes a cube into the sphere of Saturn's orbit, into a cube - the sphere of Jupiter, into the sphere of Jupiter - a tetrahedron, and so on successively fit into each other the sphere of Mars - a dodecahedron, the sphere of the Earth - an icosahedron, the sphere of Venus - an octahedron, the sphere of Mercury. The secret of the universe seems open.

Today it is safe to say that the distances between the planets are not related to any polyhedra. However, it is possible that without the "Secrets of the Universe", "Harmony of the World" by I. Kepler, regular polyhedra there would not have been three famous laws of I. Kepler, which play an important role in describing the motion of the planets.

Where else can you see these amazing bodies? In a very beautiful book by the German biologist of the beginning of our century, E. Haeckel, "The Beauty of Forms in Nature," one can read the following lines: "Nature nourishes in its bosom an inexhaustible number of amazing creatures that far surpass all forms created by human art in beauty and diversity." The creations of nature in this book are beautiful and symmetrical. This is an inseparable property of natural harmony. But here you can also see unicellular organisms - feodarii, the shape of which accurately conveys the icosahedron. What caused such a natural geometrization? Maybe because of all the polyhedra with the same number of faces, it is the icosahedron that has the largest volume and the smallest surface area. This geometric property helps the marine microorganism overcome the pressure of the water column.

It is also interesting that it was the icosahedron that turned out to be the focus of attention of biologists in their disputes regarding the shape of viruses. The virus cannot be perfectly round, as previously thought. To establish its shape, they took various polyhedrons, directed light at them at the same angles as the flow of atoms to the virus. It turned out that only one polyhedron gives exactly the same shadow - the icosahedron. Its geometric properties, mentioned above, allow saving genetic information. Regular polyhedra are the most advantageous figures. And nature takes advantage of this. The crystals of some substances familiar to us are in the form of regular polyhedra. So, the cube conveys the shape of sodium chloride crystals NaCl, the single crystal of aluminum-potassium alum (KAlSO4) 2 12H2O has the shape of an octahedron, the crystal of pyrite sulfide FeS has the shape of a dodecahedron, antimony sodium sulfate is a tetrahedron, boron is an icosahedron. Regular polyhedra determine the shape of the crystal lattices of some chemicals. I will illustrate this idea with the following problem.

Task. The model of the CH4 methane molecule has the shape of a regular tetrahedron, with hydrogen atoms at four vertices and a carbon atom in the center. Determine the bond angle between two CH bonds.

Decision. Since a regular tetrahedron has six equal edges, it is possible to choose such a cube so that the diagonals of its faces are the edges of a regular tetrahedron (Fig. 2). The center of the cube is also the center of the tetrahedron, because the four vertices of the tetrahedron are also the vertices of the cube, and the sphere described around them is uniquely determined by four points that do not lie in the same plane. The desired angle j between two CH bonds is equal to the angle AOC. Triangle AOC is isosceles. Hence, where a is the side of the cube, d is the length of the diagonal of the side face or edge of the tetrahedron. So, from where \u003d 54.73561 O and j \u003d 109.47 O

The ideas of Pythagoras, Plato, I. Kepler about the connection of regular polyhedra with the harmonious structure of the world have already found their continuation in our time in an interesting scientific hypothesis, the authors of which (in the early 80s) were Moscow engineers V. Makarov and V. Morozov. They believe that the core of the Earth has the shape and properties of a growing crystal that affects the development of all natural processes taking place on the planet. The rays of this crystal, or rather, its force field, determine the icosahedron-dodecahedral structure of the Earth (Fig. 3), which manifests itself in the fact that projections of regular polyhedra inscribed in the globe appear in the earth's crust: icosahedron and dodecahedron. Their 62 vertices and midpoints of the edges, called nodes by the authors, have a number of specific properties that make it possible to explain some incomprehensible phenomena.

If you put on the globe the centers of the largest and most remarkable cultures and civilizations of the Ancient World, you can notice a pattern in their location relative to the geographic poles and equator of the planet. Many mineral deposits stretch along an icosahedral-dodecahedral 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 maxima and minima of atmospheric pressure, giant eddies of the World Ocean, here the Scottish Loch Ness, the Bermuda Triangle. Further studies of the Earth, perhaps, will determine the attitude towards this beautiful scientific hypothesis, in which, apparently, regular polyhedra occupy an important place.

So, it was found out 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 polyhedra with a small number of edges, but how, for example, to obtain such information for an icosahedron? The famous mathematician L. Euler obtained the formula В+Г-Р=2, which relates the number of vertices /В/, faces /Г/ and edges /Р/ of any polyhedron. The simplicity of this formula is that it has nothing to do with 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. To find the number of faces, vertices and edges of a regular polyhedron, we use formulas. After that, it is easy to fill out a table that provides information about the elements of regular polyhedra:

polyhedron H W R

tetrahedron 4-4-6

hexahedron 6-8-12

octahedron 8-6-12

dodecahedron 12-20-30

icosahedron 20-12-30

And one more question arises in connection with regular polyhedra: is it possible to fill the space with them so that there are no gaps between them? It arises by analogy with regular polygons, some of which can fill the plane. It turns out that you can fill the space only with the help of one regular polyhedron-cube. Space can also be filled with rhombic dodecahedrons. To understand this, you need to solve the problem.

Task. With the help of seven cubes forming a spatial "cross", build a rhombic dodecahedron and show that they can fill space.

Decision. Cubes can fill space. Consider a part of the cubic lattice shown in Fig.4. We leave the middle cube untouched, and in each of the "bounding" cubes we draw planes through all six pairs of opposite edges. In this case, the "surrounding" cubes will be divided into six equal pyramids with square bases and side edges equal to half the diagonal of the cube. The pyramids adjacent to the untouched cube form together with the latter a rhombic dodecahedron. From this it is clear that the whole space can be filled with rhombic dodecahedrons. As a consequence, we obtain that the volume of a rhombic dodecahedron is equal to twice the volume of a cube whose edge coincides with the smaller diagonal of the dodecahedron face.

Solving the last problem, we came to rhombic dodecahedrons. Interestingly, the bee cells, which also fill the space without gaps, are also ideally geometric shapes. The upper part of the bee cell is a part of the rhombic dodecahedron.

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

Home > Abstract

MINISTRY OF EDUCATION

SECONDARY EDUCATIONAL SCHOOL №3

ESSAY

in geometry

Subject:

"Polyhedra".

Performed: student of 11-"b" class MOU secondary school No. 3 Alyabyeva Yulia. Checked: teacher of mathematics Sergeeva Lyubov Alekseevna.

Zheleznovodsk

Plan

I. Introduction. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 II. Theoretical part

Dihedral angle. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 Trihedral and polyhedral angles. . . . . . . . . . . . . . . . 4 Polyhedron. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 Prism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 The image of a prism and the construction of its sections. . . . . 7 direct prism. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nine Parallelepiped. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . nine Central symmetry of a parallelepiped. . . . . . . . ten Rectangular parallelepiped. . . . . . . . . . . . . . . . . . eleven

10. Symmetry of a rectangular parallelepiped. . . . 12 11. Pyramid. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . thirteen 12. Construction of a pyramid and its plane sections. . . . . . thirteen 13. Truncated pyramid. . . . . . . . . . . . . . . . . . . . . . . . . . . fifteen 14. Correct pyramid. . . . . . . . . . . . . . . . . . . . . . . . . fifteen 15. Regular polyhedra. . . . . . . . . . . . . . . . . . . . sixteen III. Practical part. . . . . . . . . . . . . . . . . . . . . . . . . . . 17 IV. Conclusion. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .nineteen V. Literature. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

I Introduction

There are special topics in school geometry that you look forward to, anticipating a meeting with incredibly beautiful material. Such topics include "Polyhedra". Here, not only the wonderful world of geometric bodies with unique properties opens up, but also interesting scientific hypotheses. And then the geometry lesson becomes a kind of study of unexpected aspects of the usual school subject. None of the geometric bodies possess such perfection and beauty as polyhedra. "There are defiantly few polyhedrons," L. Carroll once wrote, "but this detachment, which is very modest in number, managed to get into the very depths of various sciences."

II. Theoretical part.

1. Dihedral angle dihedral angle called a figure formed by two "half-planes with a common straight line bounding them (Fig. 1). Half-planes are called faces, and the line that bounds them edge dihedral angle. A plane perpendicular to an edge of a dihedral angle intersects its faces along two half-lines. The angle formed by these half-lines is called linear. corner dihedral angle. The measure of a dihedral angle is taken as the measure of the corresponding linear angle. All linear angles of a dihedral angle are combined by parallel translation, which means they are equal. Therefore, the measure of a dihedral angle does not depend on the choice of a linear angle. 2. Trihedral and polyhedral angles Consider three beams a, b, c, emanating from the same point and not lying in the same plane. trihedral angle (abc) called a figure composed of "three flat angles (ab),(bc) and (ac) (Fig. 2). These angles are called faces trihedral angle, and their sides - ribs common vertex of flat corners is called summit triangular angle. The dihedral angles formed by the faces of a trihedral angle are called dihedral angles of a trihedral angle. The concept of a polyhedral angle is defined similarly (Fig. 3).

3. Polyhedron

In stereometry, figures in space, called bodies, are studied. Visually, a (geometric) body must be imagined as a part of space occupied by a physical body and bounded by a surface. A polyhedron is a body whose surface consists of a finite number of flat polygons (Fig. 4). A polyhedron is called convex if it lies on one side of the plane of every flat polygon on its surface. The common part of such a plane and the surface of a convex polyhedron is called a face. The faces of a convex polyhedron are flat convex polygons. The sides of the faces are called the edges of the polyhedron, and the vertices are called the vertices of the polyhedron. Let us explain what was said on the example of a familiar cube (Fig. 5). The cube is a convex polyhedron. Its surface consists of six squares: ABCD, BEFC, .... They are its faces. The edges of the cube are the sides of these squares: AB, BC, BE,.... The vertices of the cube are the vertices of the squares: A, B, C, D, E, .... The cube has six faces, twelve edges and eight vertices. The simplest polyhedra - prisms and pyramids, which will be the main object of our study - we will give such definitions , which essentially do not use the concept of a body. They will be defined as geometric figures with indication of all points of space belonging to them. The concept of a geometric body and its surface in the general case will be given later.

4. Prism

A prism is a polyhedron, which consists of two flat polygons lying in different planes and combined by parallel translation, and all segments connecting the corresponding points of these polygons (Fig. 6). The polygons are called the bases of the prism, and the segments connecting the corresponding vertices are called the lateral edges of the prism. Since parallel translation is movement, the bases of the prism are equal. Since, during parallel transfer, the plane passes into a parallel plane (or into itself), then the bases of the prism lie in parallel planes. edges are parallel and equal. The surface of a prism consists of bases and a side surface. The lateral surface consists of parallelograms. For each of these parallelograms, two sides are the corresponding sides of the bases, and the other two are adjacent side edges. The height of a prism is the distance between the planes of its bases. A segment connecting two vertices of a prism that do not belong to the same face is called the diagonal of the prism. A prism is called n-gonal if its bases are n-gons. In the future, we will consider only prisms whose bases are convex polygons. Such prisms are convex polyhedra. Figure 6 shows a pentagonal prism. Its bases are pentagons. BUT 1 BUT 2 ...BUT 5 , BUT 1 ’ BUT" 2 ...BUT" 5 . XX"- a line segment connecting the corresponding points of the bases. Lateral edges of the prism-segments BUT 1 BUT" 2 , BUT 1 BUT" 2 , ..., BUT 5 BUT" 5 . Side faces of the prism - parallelograms BUT 1 BUT 2 BUT" 2 BUT 1 , BUT 2 BUT 3 BUT ’ 3 BUT" 2 , ... .

5. The image of a prism and the construction of its sections

In accordance with the rules of parallel projection, the image of a prism is constructed as follows. First, one of the bases is built R(Fig. 7). It will be some flat polygon. Then from the vertices of the polygon R the lateral ribs of the prism are drawn in the form of parallel segments of equal length. The ends of these segments are connected, and another base of the prism is obtained. Invisible edges are drawn with dashed lines. Sections of the prism by planes parallel to the side edges are parallelograms. In particular, diagonal sections are parallelograms. These are sections by planes passing through two side edges that do not belong to the same face (Fig. 8). In practice, in particular, when solving problems, it is often necessary to construct a section of a prism by a plane passing through a given straight line g on the plane of one of the bases of the prism. Such a line is called next cutting plane on the plane of the base. To construct a section of a prism, it is sufficient to construct segments of the intersection of the secant plane with the faces of the prism. Let us show how such a section is constructed if any point is known BUT on the surface of the prism belonging to the section (Fig. 9). If this point BUT belongs to another base of the prism, then its intersection with the cutting plane is a segment sun, parallel to the wake g and containing the given point BUT(Fig. 9, a). If this point BUT belongs to the side face, then the intersection of this face with the cutting plane is constructed, as shown in Figure 9, b. Namely: first a point is built D, in which the plane of the face intersects the given trace g. Then a line is drawn through the points BUT and D. Line segment sun straight AD on the considered face is the intersection of this face with the cutting plane. If the face containing the point BUT, parallel to the trace g, then the cutting plane intersects this face along the segment sun, passing through a point BUT and parallel to the line g.

Line ends sun belong to neighboring faces. Therefore, in the described way, it is possible to construct the intersection of these faces with our cutting plane. And so on. Figure 10 shows the construction of a section of a quadrangular prism by a plane passing through a straight line a in the plane of the lower base of the prism and a point BUT on one of the side ribs. 6. Straight prism A prism is called straight if its side edges are perpendicular to the bases. Otherwise, the prism is called oblique. For a straight prism, the side faces are rectangles. When depicting a straight prism in the figure, the side ribs are usually drawn vertically (Fig. 11). A right prism is called regular if its bases are regular polygons. The lateral surface of the prism (more precisely, the area of ​​the lateral surface) is the sum of the areas of the lateral faces. The total surface of the prism is equal to the sum of the lateral surface and the areas of the bases. Theorem 19.1. The lateral surface of a straight prism is equal to the product of the perimeter of the base and the height of the prism, i.e., the length of the lateral edge. Proof. The side faces of a straight prism are rectangles. The bases of these rectangles are the sides of the polygon lying at the base of the prism, and the heights are equal to the length of the side edges. It follows that the lateral surface of the prism is equal to

S=a 1 l+a 1 l+...+a n l=pl,

where a 1 ,..., a n- the length of the edges of the base, R - the perimeter of the base of the prism, and 1 - side rib length. The theorem has been proven. 7. Parallelepiped If the base of a prism is a parallelogram, then it is called a parallelepiped. All the faces of a parallelepiped are parallelograms. In Figure 12, a, an inclined parallelepiped is shown, and in Figure 12, b - a straight parallelepiped. Faces of a parallelepiped that do not have common vertices are called opposite faces. THEOREM 19.2. A parallelepiped has opposite faces that are parallel and equal. Proof. Consider some two opposite faces of the parallelepiped, for example A1A2A"2A"1 and A3A4A"4A"3. (Fig. 13). Since all the faces of the parallelepiped are parallelograms, the line A1A2 is parallel to the line A4A3, and the line A1A"1 is parallel to the line A4A4". It follows from this that the planes of the considered faces are parallel. From the fact that the faces of the parallelepiped are parallelograms, it follows that the segments A1A4, A1 "A4", A "2A" 3 and A2A3 are parallel and equal. Hence we conclude that the face A1A2A"2A"1 is combined by a parallel translation along the edge A1A4. with a face A3A4A "4A" 3. So these edges are equal. Parallelism and equality of any other opposite faces of the parallelepiped are proved similarly. The theorem has been proven.
8. Central symmetry of the parallelepiped Theorem 19.3. The diagonals of the parallelepiped intersect at one point and the intersection point is divided in half. Proof. Consider some two diagonals of the parallelepiped, for example, A 1 A "3 and A 4 A" 2 (Fig. 14). Since the quadrangles A 1 A 2 A 3 A 4 and A 2 A "2 A" 3 A 3 are parallelograms with a common side A 2 A 3, then their sides A 1 A 4 and A "2 A" 3 are parallel to each other, which means they lie in the same plane. This plane intersects the planes of opposite faces of the parallelepiped along parallel lines A 1 A" 2 and A 4 A" 3 . Therefore, the quadrilateral A 4 A 1 A "2 A" 3 is a parallelogram. The diagonals of the parallelepiped A 1 A "3 and A 4 A" 2 are the diagonals of this parallelogram. Therefore, they intersect and the intersection point O is divided in half. Similarly, it is proved that the diagonals A1A"3 and A2A"4, as well as the diagonals A1A"3 and A3A"1 intersect and are bisected by the intersection point. Hence we conclude that all four diagonals of the parallelepiped intersect at one point and the intersection point is divided in half. The theorem has been proven. Theorem 19.3 implies that the point of intersection of the diagonals of the parallelepiped is its center of symmetry. 9. Rectangular box A right parallelepiped whose base is a rectangle is called a rectangular parallelepiped. All faces of a cuboid are rectangles. A rectangular parallelepiped in which all edges are equal is called a cube. The lengths of non-parallel edges of a rectangular parallelepiped are called its linear dimensions (measurements). A cuboid has three dimensions. Theorem 19.4. In a cuboid, the square of any diagonal is equal to the sum of the squares of its three dimensions. Proof. Consider a rectangular parallelepiped ABCDA"B"C"D" (Fig. 15). From the right triangle AC "C, according to the Pythagorean theorem, we get:

AC" 2 = AC 2 + CC" 2 .

From the right triangle ASV, by the Pythagorean theorem, we obtain

AC 2 \u003d AB 2 + BC 2.

Hence AC" 2 \u003d CC" 2 + AB 2 + BC 2.

The edges AB, BC and CC" are not parallel, and, therefore, their lengths are the linear dimensions of the parallelepiped. The theorem is proved. 10. Symmetry of a rectangular parallelepiped A rectangular parallelepiped, like any parallelepiped, has a center of symmetry - the point of intersection of its diagonals. It also has three planes of symmetry passing through the center of symmetry parallel to the faces. Figure 16 shows one of these planes. It passes through the midpoints of four parallel edges of the parallelepiped. The ends of the edges are symmetrical points. If a parallelepiped has all linear dimensions different, then it has no other planes of symmetry other than those named. If the parallelepiped has two linear dimensions equal, then it has two more planes of symmetry. These are the planes of diagonal sections shown in Figure 17. If a parallelepiped has all the linear dimensions equal, that is, it is a cube, then its plane of any diagonal section is a plane of symmetry. Thus, the cube has nine planes of symmetry. 11. Pyramid Pyramid called a polyhedron, which consists of a flat polygon - pyramid bases, point not lying in the plane of the base, - tops of the pyramid and all segments connecting the top of the pyramid with the points of the base (Fig. 18). The segments connecting the top of the pyramid with the tops of the base are called side ribs. The surface of the pyramid consists of a base and side faces. Each side face is a triangle. One of its vertices is the top of the pyramid, and the opposite side is the side of the base of the pyramid. pyramid height, called the perpendicular dropped from the top of the pyramid to the plane of the base. A pyramid is called n-gonal if its base is an n-gon. The triangular pyramid is also called tetrahedron. The pyramid shown in Figure 18 has a base - a polygon A 1 A 2 ... A n, the top of the pyramid - S, side edges - SA 1, S A 2, ..., S A n, side faces - SA 1 A 2,  SA 2 A 3 , ... . In what follows, we will consider only pyramids with a convex polygon at the base. Such pyramids are convex polyhedra. 12. Construction of a pyramid and its plane sections In accordance with the rules of parallel projection, the image of the pyramid is constructed as follows. First, the foundation is built. It will be some flat polygon. Then the top of the pyramid is marked, which is connected by lateral ribs to the tops of the base. Figure 18 shows an image of a pentagonal pyramid. Sections of the pyramid by planes passing through its top are triangles (Fig. 19). In particular, diagonal sections are triangles. These are sections by planes passing through two non-adjacent side edges of the pyramid (Fig. 20). The section of a pyramid by a plane with a given trace g on the plane of the base is constructed in the same way as the section of a prism. To construct a section of a pyramid by a plane, it is sufficient to construct the intersections of its side faces with the cutting plane. If on a face not parallel to the trace g, some point A is known that belongs to the section, then the intersection of the trace g of the cutting plane with the plane of this face is first constructed - point D in Figure 21. Point D is connected to point A by a straight line. Then the segment of this line belonging to the face is the intersection of this face with the cutting plane. If the point A lies on a face parallel to the trace g, then the secant plane intersects this face along a segment parallel to the line g. Going to the adjacent side face, they build its intersection with the cutting plane, etc. As a result, the required section of the pyramid is obtained.
Figure 22 shows a section of a quadrangular pyramid by a plane passing through the side of the base and point A on one of its side edges.

13. Truncated pyramid Theorem 19.5. A plane intersecting a pyramid and parallel to its base cuts off a similar pyramid. Proof. Let S be the vertex of the pyramid, A the vertex of the base, and A "- the point of intersection of the secant plane with the lateral edge SA (Fig. 23). We subject the pyramid to a homothety transformation with respect to the vertex S with the homothety coefficient

With this homothety, the plane of the base passes into a parallel plane passing through the point A ", i.e., into the cutting plane, and, consequently, the entire pyramid into the part cut off by this plane. Since the homothety is a similarity transformation, the cut-off part of the pyramid is a pyramid, similar to this one, the theorem is proved.

By Theorem 19.5, a plane parallel to the base plane of a pyramid and intersecting its side edges cuts off a similar pyramid from it. The other part is a polyhedron, which is called a truncated pyramid (Fig. 24). The faces of a truncated pyramid lying in parallel planes are called bases; the rest of the faces are called side edges. The bases of the truncated pyramid are similar (moreover, homothetic) polygons, the side faces are trapezoids. 14. Correct pyramid A pyramid is called regular if its base is a regular polygon, and the base of the height coincides with the center of this polygon. The axis of a regular pyramid is a straight line containing its height. Obviously, the side edges of a regular pyramid are equal; therefore, the side faces are equal isosceles triangles. The height of the side face of a regular pyramid, drawn from its top, is called apothem. The lateral surface of a pyramid is the sum of the areas of its lateral faces. THEOREM 19.6. The lateral surface of a regular pyramid is equal to the product of the semi-perimeter of the base and the apothem. Proof. If the base side a, number of sides P, then the lateral surface of the pyramid is equal to:

(a1/2)ap \u003d a1p / 2 \u003d p1/2 "

Where I- apothem, a p- perimeter of the base of the pyramid. The theorem has been proven. A truncated pyramid, which is obtained from a regular pyramid, is also called correct. The lateral faces of a regular truncated pyramid are equal isosceles trapezoids; their heights are called apothems. 15. Regular polyhedra A convex polyhedron is called regular if its faces are regular polygons with the same number of sides and the same number of edges converge at each vertex of the polyhedron.) There are five types of regular convex polyhedra (Fig. 25): regular tetrahedron (1), cube (2), octahedron (3), dodecahedron (4); icosahedron (5). A regular tetrahedron has faces that are regular triangles; three edges converge at each vertex. A tetrahedron is a triangular pyramid with all edges equal. In a cube, all faces are squares; three edges converge at each vertex. The cube is a rectangular parallelepiped with equal edges. The octahedron faces are regular triangles, but unlike the tetrahedron, four edges converge at each of its vertices. The faces of the dodecahedron are regular pentagons. Three edges converge at each vertex. The icosahedron faces are regular triangles, but unlike the tetrahedron and octahedron, five edges converge at each vertex.

III. Practical part.

Task 1. From the points A and B lying on the faces of the dihedral angle, the perpendiculars AA\ and BB\ are dropped onto the edge of the angle. Find the length of the segment AB if AA 1 \u003d a, BB 1 \u003d b, A 1 B 1 \u003d c and the dihedral angle is a (Fig. 26). Decision. Draw lines A 1 C||BB 1 and BC||A 1 B 1 . The quadrilateral A 1 B 1 BC is a parallelogram, which means AA 1 \u003d\u003d BB 1 \u003d b. The line A 1 B 1 is perpendicular to the plane of the triangle AA 1 C, since it is perpendicular to two lines in this plane AA 1 and CA 1. Therefore, the line BC parallel to it is also perpendicular to this plane. This means that the triangle ABC is right-angled with a right angle C. According to the cosine theorem, AC 2 \u003d AA 1 2 + A 1 C 2 -2AA 1 A 1 C cos  \u003d a 2 + b 2 -2abcos . According to the Pythagorean theorem, AB \u003d AC 2 + BC 2 \u003d a 2 + b 2 - 2ab cos  + c 2. Task 2. A trihedral angle (abc) has a dihedral angle at an edge with a straight line, a dihedral angle at an edge b is equal to , and a flat angle (bс) is equal to  (, </2). Найдите два других плоских угла: =  (ab), = (ac). Decision. Let us drop from an arbitrary point A the edge a, the perpendicular AB to the edge b and the perpendicular AC to the edge c (Fig. 27). According to the three perpendiculars theorem, CB is the perpendicular to the edge b. From right-angled triangles OAB, OSV, AOC and ABC we get: BC/sin )=tg  sin  Task 3. In an inclined prism, a section is drawn that is perpendicular to the side ribs and intersects all the side ribs. Find the side surface of the prism if the perimeter of the section is p and the side edges are l. Decision. The plane of the section drawn divides the prism into two parts (Fig. 28). Let's subject one of them to a parallel translation that combines the bases of the prism. In this case, we obtain a straight prism, in which the section of the original prism serves as the base, and the side edges are equal to l. This prism has the same side surface as the original one. Thus, the side surface of the original prism is equal to pl. Task 4. The lateral edge of the pyramid is divided into four equal parts and planes parallel to the base are drawn through the division points. The base area is 400 cm2. Find the area of ​​the sections. Decision. The sections are like the base of a pyramid with similarity coefficients of ¼, 2/4, and ¾. The areas of similar figures are related as squares of linear dimensions. Therefore, the ratio of the cross-sectional areas to the area of ​​​​the base of the pyramid is (¼) 2, (2/4) 2, and (¾) 2. Therefore, the cross-sectional areas are 400 (¼) 2 \u003d 25 (cm 2), 400 (2/4) 2 \u003d 100 (cm 2), 400 (¾) 2 \u003d 225 (cm 2). Task 5. Prove that the lateral surface of a regular truncated pyramid is equal to the product of half the sum of the perimeters of the bases and the apothem. Decision. The side faces of a truncated pyramid are trapeziums with the same upper base a, lower b and height (apothem) l. Therefore, the area of ​​one face is equal to ½ (a + b)l. The area of ​​all faces, i.e., the side surface, is equal to ½ (an + bn)l, where n is the number of vertices at the base of the pyramid, an and bn are the perimeters of the bases of the pyramid.

IV. Conclusion

Thanks to this work, I summarized and systematized the knowledge gained during the course of study in the 11th grade, got acquainted with the rules for performing creative work, gained new knowledge and put it into practice. I would like to highlight 3 of my favorite books: A.V. Pogorelov "Geometry", G. Yakusheva "Mathematics - a schoolchild's reference book", L.F. Pichurin "Behind the pages of a geometry textbook". These books have helped me more than others. I would like to use my newly acquired knowledge in practice more often.

V. Literature

1. A.V. Pogorelov Geometry. - M .: Education, 1992 2. G. Yakusheva "Mathematics - a schoolchild's guide." M.: Slovo, 1995 3. L.D. Kudryavtsev "Course of Mathematical Analysis" v.1, Moscow 1981 4. L.F. Pichurin "Behind the pages of a geometry textbook". - M .: Education, 1990 5. I.N. Bashmakov "Geometry".