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Contents [ show ] Basis for definition Any polyhedron can be built up from different kinds of element or entity, each associated with a different number of dimensions: 3 dimensions: The body is bounded by the faces, and is usually the volume enclosed by them. These polygonal faces together make up the polyhedral surface.
The edges together make up the polyhedral skeleton. More generally in mathematics and other disciplines, "polyhedron" is used to refer to a variety of related constructs, some geometric and others purely algebraic or abstract.
A defining characteristic of almost all kinds of polyhedra is that just two faces join along any common edge. This ensures that the polyhedral surface is continuously connected and does not end abruptly or split off in different directions.
A polyhedron is a 3-dimensional example of the more general polytope in any number of dimensions.
Characteristics Naming polyhedra Polyhedra are often named according to the number of faces. The naming system is again based on Classical Greek, for example tetrahedron 4 , pentahedron 5 , hexahedron 6 , heptahedron 7 , triacontahedron 30 , and so on.
Often this is qualified by a description of the kinds of faces present, for example the Rhombic dodecahedron vs. Other common names indicate that some operation has been performed on a simpler polyhedron, for example the truncated cube looks like a cube with its corners cut off, and has 14 faces so it is also an example of a tetrakaidecahedron. Some special polyhedra have grown their own names over the years, such as Miller's monster or the Szilassi polyhedron. Edges Edges have two important characteristics unless the polyhedron is complex : An edge joins just two vertices.
An edge joins just two faces. These two characteristics are dual to each other. For a detailed discussion, see Proofs and Refutations by Imre Lakatos. Duality For every polyhedron there is a dual polyhedron having faces in place of the original's vertices and vice versa. In most cases the dual can be obtained by the process of spherical reciprocation. Vertex figure For every vertex one can define a vertex figure consisting of the vertices joined to it.
The vertex is said to be regular if this is a regular polygon and symmetrical with respect to the whole polyhedron.
Traditional polyhedra A dodecahedron In geometry , a polyhedron is traditionally a three-dimensional shape that is made up of a finite number of polygonal faces which are parts of planes ; the faces meet in pairs along edges which are straight-line segments, and the edges meet in points called vertices. Cubes , prisms and pyramids are examples of polyhedra.
The polyhedron surrounds a bounded volume in three-dimensional space; sometimes this interior volume is considered to be part of the polyhedron, sometimes only the surface is considered, and occasionally only the skeleton of edges. A polyhedron is said to be convex if its surface comprising its faces, edges and vertices does not intersect itself and the line segment joining any two points of the polyhedron is contained in the interior and surface.
Symmetrical polyhedra Many of the most studied polyhedra are highly symmetrical. Of course it is easy to distort such polyhedra so they are no longer symmetrical.
But where a polyhedral name is given, such as icosidodecahedron , the most symmetrical geometry is almost always implied, unless otherwise stated. Some of the most common names in particular are often used with "regular" in front or implied because for each there are different types which have little in common except for having the same number of faces.
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