Convex polyhedra whose
faces are all
regular polygons and which have the same faces in the same order at each
vertex (i.e., their
solid angles are equal), but not all the faces are equal. Aside from
infinite sets of
prisms and
antiprisms, there are only 13 of these:
In the table below, "8@3 6@4" in the FACES column means the shape has eight triangular faces and 6 square faces.
NAME FACES VERTICES EDGES
cuboctahedron 8@3 6@4 12 24
great rhombicosidodecahedron 30@4 20@6 12@10 120 180
great rhombicuboctahedron 12@4 8@6 6@8 48 72
icosidodecahedron 20@3 12@5 30 60
small rhombicosidodecahedron 20@3 30@4 12@5 60 120
small rhombicuboctahedron 8@3 18@4 24 48
snub cube 32@3 6@4 24 60
snub dodecahedron 80@3 12@5 60 150
truncated cube 8@3 6@8 24 36
truncated dodecahedron 20@3 12@10 60 90
truncated icosahedron 12@5 20@6 60 90
truncated octahedron 6@4 8@6 24 36
truncated tetrahedron 4@3 4@6 12 18
Note that the Archimedean solids all satisfy the equation (2pi - s)V = 4pi, where s is the sum of face angles at a vertex, and V is the number of vertices, as well as Euler's formula F + V = E + 2, where F is the total number of faces and E is the number of edges.
The last five of these shapes are formed by a minor truncation of the platonic solids -- that is, the corners are cut off so that the remaining parts of the original faces are now regular polygons with twice as many edges as before.
The cuboctahedron and icosidodecahedron are major truncations of platonic solids -- that is, truncations such that all of the edges of the original figure are removed; the new faces formed from the truncations just meet at vertices, while the old faces that used to be adjacent now just meet at vertices. There are only two of these, because both the cube and the octahedron have the cuboctahedron as their major truncation, and likewise with the icosahedron and dodecahedron for the other. The tetrahedron major-truncates to form a smaller copy of itself.