Polyhedron

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A Polyhedron is a shape, made up of faces. 'Poly-' is from the Greek word for 'many' and '-hedron' is a Greek combining form meaning 'base', 'seat', or 'face'.

Roughly speaking, a polyhedron is a higher dimension version of a polygon - for instance prisms and pyramids are polyhedra. The terminology here is unfortunately inconsistent. Sometimes the word polyhedron is used to apply to figures of any dimension, and sometimes the word polytope is used to apply to those of dimensionality higher than three; other times the words polyhedron and polytope are used to describe things that are bounded and unbounded, respectively. It is often unclear whether polyhedron are supposed to represent surfaces or interiors, and to make matters worse the word polygon is sometimes used for all these things. Here, simply to be as concise as possible, I'm going to use the term polytope to mean the general object and polyhedron to mean a surface in three-dimensions; see also polytope.

There are only five convex polyhedra that are vertex-, edge-, and face-uniform (meaning that all these elements are the same, or to be precise, for any two there is a symmetry of the polyhedron mapping them onto each other). These have been known since ancient times, and are called the Platonic solids:

             Vertices  Edges  Faces  Edges/Face  Edges/Vertex  Symmetry group
Tetrahedron     4        6      4        3           3          Td
Cube            8       12      6        4           3          Oh
Octahedron      6       12      8        3           4          Oh
Dodecahedron   20       32     12        5           3          Ih
Icosahedron    12       32     20        3           5          Ih

Note how these come in natural pairs: the dodecahedron with the icosahedron, the cube with the octahedron, and the tetrahedron with itself (ok, so that's not a pair). These are called duals, and can be obtained by connecting the midpoints of each other's faces, among other interesting things. If you allow the polyhedra to be non-convex, there are four more, called the Kepler-poinsot solids. These nine are all called regular polyhedra. There are also five regular polyhedral compounds.

Polyhedra which are vertex- and edge-uniform, but not necessarily face-uniform, are called quasi-regular and include two more convex forms (the cuboctahedron and icosidodecahedron, as well as a few non-convex forms. The duals of these are the edge- and face-uniform polyhedra: the rhombic dodecahedron, rhombic triacontahedron, plus whatever the non-convex ones are. No other convex edge-uniform polyhedra exist.

Any polyhedra which is vertex-uniform can be deformed slightly to form a vertex-uniform polyhedron with regular polygons as faces. These are called semi-regular polyhedra. Convex forms include two infinite series, one of prisms and one of antiprisms, as well as the thirteen Archimedean solids. The duals of these are of course the face-uniform polyhedra, with the two infinite convex series becoming the bipyramids and trapezohedrons (sp?). These don't have regular faces, but do have regular vertices.

Another thing to consider is what kind of polyhedra, of any symmetry, can be made of regular polygons. There are an infinite amount of non-convex forms, but surprisingly only a finite amount of convex shapes other than the prisms and antiprisms. These include the Platonic solids, Archimedean solids, and 92 extra shapes called Johnson solids.


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