Polyhedron

A ( three-dimensional ) polyhedron [ polye ː dər ] (also polyhedron, polyhedron or Ebenflächner, from gr πολύς polys " much " and ἕδρα hedra, " seat ( area) " ) is strictly speaking a subset of the three-dimensional space, which exclusively of straight surfaces ( planes) is limited, such as a cube or an octant of the three-dimensional coordinate system.

The term is often used ( especially in the topology ) in a wider sense and generalized to higher dimensions. One speaks in the context of geometric problems also from the concept of the polytope. In the broad sense is called a subset of a polyhedron, if it is triangulated, so if they can be formed as the union of simplices of a simplicial complex. The homeomorphic image of such a general polyhedron is called crooked polyhedra and images of the simplices involved as crooked simplexes.

Terms relating to the three-dimensional polyhedron

Examples of polyhedra from everyday life are ( in their usual construction ) cupboards, pyramids, houses, crystals or dice. No polyhedra are spheres, cones, bottles, pieces of cake, because they have curved edge surfaces. The main polyhedra in geometric applications are cuboids, prisms, pyramids and Spate ( parallelepipeds ).

A polyhedron is called bounded if there is a sphere in which the polyhedron is completely contained. Unlimited polyhedron with only one corner are called Polyederkegel.

A polyhedron is convex if the link between these two points is on 2 points of the polyhedron completely in the polyhedron.

For convex and bounded polyhedra the Euler Polyedersatz applies:

Here, the number of corners, the number of areas and the number of edges.

Counter-example: The points of the three-dimensional space with the ( rectangular Cartesian ) coordinates (x, y, z), wherein the absolute value of x, y and z are each less than or equal to 2, to form a cube of edge length 4 If we therefrom Remove points, the coordinates of all of the amount < 1, creates a nonconvex polyhedron, namely a cube, from the interior of a smaller die is drilled, with corners 16, 24 and edge surfaces 12, in which the Euler Polyedersatz not apply.

For related polyhedra ( to which the above example is not a part ) is generally

With the Euler characteristic. For a torus, for example. The polyhedron pictured to the right is an example of this. It has 24 vertices, 72 edges and 48 faces.

Regular polyhedra

Also known polyhedra, which are characterized by a high regularity, as the Platonic Solids (or regular polyhedra ) - the only five convex polyhedra, which are composed only of congruent ( congruent ) polygons and whose vertices are all identical. In contrast, when the congruence of the lateral surfaces are not met and there are several surface types present, the body is either a prism, antiprism or one of the 13 Archimedean body. The convex polyhedra, which are bounded by regular polygons and do not fall into one of the previous categories, the 92 - Johnson body.

Another group of regular convex polyhedra are the 13 Catalan body whose non- regular surfaces are all congruent and appear equally in the body.

Important special polyhedron

Pyramids

Pyramid ( geometry)

Orthogonal polyhedra

The faces of an orthogonal polyhedron meet at right angles. Its edges are parallel to the axes of a Cartesian coordinate system. With the exception of the cuboid orthogonal polyhedra are not convex. Expand the two-dimensional orthogonal polygons into the third dimension. Orthogonal polyhedra come in computational geometry are used. There has limited their structural advantages in coping with otherwise unsolved problems (any polyhedron ). An example is the deployment of the polyhedron into a polygonal mesh.