A polyhedron is
- convex if the line segment joining any two points of the polyhedron is contained in the polyhedron's interior
- vertex-uniform if all vertices are the same, in the sense that for any two vertices there exists a symmetry of the polyhedron mapping the first onto the second
- edge-uniform if all edges are the same, in the sense that for any two edges there exists a symmetry of the polyhedron mapping the first onto the second
- face-uniform if all faces are the same, in the sense that for any two faces there exists a symmetry of the polyhedron mapping the first onto the second
- regular if it is vertex-uniform, edge-uniform and face-uniform
|Cube or hexahedron||8||12||6||4||3||Oh|
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. There are also five regular polyhedral compounds.
If you allow the polyhedra to be non-convex, there are four more, called the Kepler-Poinsot solids.
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 polyhedron 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 prismss 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 trapezohedra. 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 number of non-convex forms, but surprisingly only a finite number of convex shapes other than the prisms and antiprisms. These include the Platonic solids, Archimedean solids, and 92 extra shapes called Johnson solids.
Given two polyhedra of equal volume, one may ask whether it is then always possible to cut the first into polyhedral pieces which can be reassembled to yield the second polyhedron. This is a version of Hilbert's third problem; the answer is "no", as was shown by Dehn in 1902.