The polyhedra are geometric elements that have flat faces and that house a volume That is not infinite. The etymological roots of the term, found in the Greek language, refer to "Many faces" .
A polyhedron can be understood as a body solid and three-dimensional. When all its faces and angles are equal to each other, it is qualified as a regular polyhedron . Otherwise, it will be a irregular polyhedron .
Another possible classification is linked to the quantity of faces it presents. A six-sided polyhedron is called hexahedron , a five-sided polyhedron is known as pentahedron and so on, always forming the denomination with the corresponding Greek prefix (hexa, penta, tetra, etc.).
On the other hand, you can differentiate between concave polyhedra and convex polyhedra . The concave polyhedra are those that, by joining two points located inside the body, the segment corresponding leaves the surface. Instead, in the convex polyhedra , the segments that link two points of the interior space never leave the geometric body.
An example of polyhedron is the Cube , a regular polyhedron with four equal faces, whose interior angles are congruent to each other. This means that the dice constructed in this way are polyhedra. Boxes whose faces are square also enter the group of polyhedra.
Another example of polyhedron are the prisms : in this case, these are irregular polyhedra. It is important to note that classifications are not always exclusive. The prism is an irregular polyhedron but, in turn, is a convex polyhedron.
Polyhedra are classified into various families, two of which are listed below:
* platonic solids : these are those that have equal faces and angles and that are convex . There are only five polyhedra of this family, which are the cube, the dodecahedron, the tetrahedron, the octahedron and the icosahedron. This family is essential, since others derive from it, such as archimedean solids ;
* Archimedean solids : they are convex, their vertices are uniform and their faces regular (but not uniform). There are only eleven, and some of them are achieved by truncating the platonists, that is by cutting their vertices or its edges. Some of the archimedean solids are the truncated cube, the rombicuboctahedron, the rhombicosidodecahedron and the truncated icosidodecahedron;
It is known by the name of dual polyhedron to the one whose vertices are corresponding to the center of the faces of a second polyhedron. Let's see some data curious: the dual polyhedron of a dual one resembles the original; the dual of one with equivalent vertices also has equivalent faces; that of a polyhedron that has equivalent edges, will also have equivalents. Kepler-Poinsot and Platonic solids are associated with this classification, among other regular polyhedra.
Although you can recognize several kinds of duality from which to relate two figures, among the most used are the polar reciprocity and the topological duality . Let's see below the definition of these concepts:
* polar reciprocity : in general, to define duality by talking about its reciprocity polar a concentric sphere is taken as a reference, so that each pole (or vertex) is associated with a face and its plane (called polar), so that the imaginary line that passes through the vertex and center is perpendicular to said plane and the square of the radius can be obtained if the product of the distances from each side to the center is made;
* topological duality : when a dual polyhedron is distorted so that it can no longer be obtained by reciprocity, it can be said that the original and the current are topologically dual, but not polar reciprocals.