Polyhedra not only occupy a prominent place in thegeometry, but also occur in the everyday life of each person. Not to mention artificially created objects of everyday use in the form of various polygons, starting with the matchbox and ending with architectural elements, in nature there are also crystals in the form of a cube (salt), prisms (crystal), pyramids (scheelite), octahedra (diamond) and t e.
Geometry as a science contains a section of stereometry,studying the characteristics and properties of three-dimensional figures. Geometric bodies, whose sides in the three-dimensional space are formed by bounded planes (faces), are called "polyhedra." Types of polyhedra number more than one dozen representatives, differing in the number and shape of faces.
Nevertheless, all polyhedra have common properties:
Polyhedra can be conditionally divided into:
Stereometry as a section of geometry studiesproperties of three-dimensional figures, types of polyhedra (prism in their number). A prism is a geometric body that necessarily has two completely identical faces (also called bases) lying in parallel planes, and the n-th number of lateral faces in the form of parallelograms. In turn, the prism also has several varieties, including such types of polyhedra as:
The basic properties of the prism:
A pyramid is a geometric body thatconsists of one base and the n-th number of triangular faces joining at one point-the vertex. It should be noted that if the side faces of the pyramid are represented by triangles, then in the base there can be both a triangular polygon, a quadrilateral, and a pentagon, and so on ad infinitum. The name of the pyramid will correspond to the polygon at the bottom. For example, if there is a triangle at the bottom of the pyramid, it is a triangular pyramid, a quadrilateral is a quadrangular pyramid, and so on.
Pyramids are cone-like polyhedra. Types of polyhedra of this group, besides the above, also include the following representatives:
Properties of the pyramid:
В стереометрии особое место занимают geometric bodies with absolutely equal sides, at the vertices of which the same number of edges are connected. These bodies are called Platonic bodies, or regular polyhedra. Types of polyhedra with these properties have only five figures:
By its name, regular polyhedra are requiredthe ancient Greek philosopher Plato, who described these geometric bodies in his works and connected them with the natural elements: earth, water, fire, air. The fifth figure was awarded similarity to the structure of the universe. In his opinion, the atoms of natural elements resemble in shape the kinds of regular polyhedra. Due to its most fascinating property - symmetry, these geometric bodies were of great interest not only for ancient mathematicians and philosophers, but also for architects, artists and sculptors of all time. The presence of only 5 types of polyhedra with absolute symmetry was considered a fundamental find, they were even awarded a connection with the divine beginning.
In the shape of a hexagon, Plato's successorsassumed a similarity with the structure of the atoms of the earth. Of course, at the present time this hypothesis is completely disproved, which, however, does not prevent the figures and in modern times attract the minds of famous figures with their aesthetics.
In geometry, the hexahedron, the same cube, is considereda particular case of a parallelepiped, which, in turn, is a kind of prism. Accordingly, the properties of the cube are related to the properties of the prism with the only difference that all faces and angles of the cube are equal to each other. This implies the following properties:
The tetrahedron is a tetrahedron with equal faces in the form of triangles, each of whose vertices is a point of connection of three faces.
Properties of a regular tetrahedron:
Describing the types of regular polyhedra, one can not fail to note an object such as an octahedron that can be visualized in the form of two quadrilateral regular pyramids stuck together by bases.
Properties of an octahedron:
If we imagine that all the faces of the geometric body are a regular pentagon, we get a dodecahedron - a figure of 12 polygons.
Properties of the dodecahedron:
No less interesting than a dodecahedron, the icosahedron is a voluminous geometric body with 20 equal faces. Among the properties of a regular twenty-sided can be noted the following:
In addition to the Platonic solids, the group of convexpolyhedra also includes Archimedean bodies, which are truncated regular polyhedra. The types of polyhedra of this group have the following properties:
Representatives of non-existent types of geometric bodies- star-shaped polyhedra whose faces intersect each other. They can be formed by the fusion of two regular three-dimensional bodies or as a result of the continuation of their faces.
Thus, such star polyhedra as are known: star-shaped octahedron, dodecahedron, icosahedron, cuboctahedron, icosododecahedron.