There are only five convex polyhedra that are vertex, edge, and faceuniform (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:
Name  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 nonconvex, there are four more, called the KeplerPoinsot solids[?]. These nine are all called regular polyhedra. There are also five regular polyhedral compounds.
Polyhedra which are vertex and edgeuniform, but not necessarily faceuniform, are called quasiregular and include two more convex forms (the cuboctahedron and icosidodecahedron, as well as a few nonconvex forms. The duals of these are the edge and faceuniform polyhedra: the rhombic dodecahedron[?], rhombic triacontahedron[?], plus whatever the nonconvex ones are. No other convex edgeuniform polyhedra exist.
Any polyhedron which is vertexuniform can be deformed slightly to form a vertexuniform polyhedron with regular polygons as faces. These are called semiregular 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 faceuniform polyhedra, with the two infinite convex series becoming the bipyramids and trapezohedra[?] (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 nonconvex 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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