A polyhedron is said to be uniform if it has regular faces and admits symmetries that will transform a given vertex into every other vertex in turn. The Platonic, Kepler-Poinsot solids are uniform, and so are the right regular prisms and antiprisms of suitable height; that is, those whose lateral faces are squares and equilaterals, respectively. In Solid Geometry, L. Lines proves that, apart from these, there are only 13 finite, convex uniform polyhedra. These are called the Archimedean solids. Plato is said to have known at least one, the cuboctahedron, and Archimedes wrote about the entire set, though his book on them is lost. Durer gives the nets for some Archimedean solids in his Underweysung, but they were first treated systematically by Kepler.
The Archimedean solids can be broken down into various subsets:
First are the five derived by the process of truncation from each vertex along with the vertex itself. This can be done to the Platonic solids in such a way that the new faces are again regular polygons. For example, on cutting off the corners of a cube, by planes parallel to the faces of the reciprocal octahedron, we obtain small triangles--the square faces of the cube have become octagons. For suitable positions of the cutting planes, these octagons will be regular, and we have an Archimedean solid, namely the truncated cube t{4,3}.
Another subset, containing only two members, is that known as the quasi-regular polyhedra.
When two regular polyhedra, {p,q} and {q,p}, are reciprocal with respect to their common mid-sphere, the solid region interior to both polyhedra forms another polyhedron, say , which has vertices, namely the mid-edge points of either {p,q} or {q,p}. Its faces consist of {q}'s and {p}'s, which are the vertex figures of {p,q} and {q,p}, respectively.
When p = q = 3, we have the octahedron; therefore, = {3,4}.
When {p}'s and {q}'s are different, we have , which is the cuboctahedron, and , which is the icosidodecahedron.
Note that .
Next are the two called the small rhombicuboctahedron and the small rhombiicosidodecahedron.
The faces of the icosidodecahedron consist of 20 triangles and 12 pentagons (corresponding to the faces of the two parent regulars). Its 60 edges are perpendicularly bisected by those of the reciprocal triacontahedron. The 60 points where these pairs of edges cross one another are the vertices of a polyhedron whose faces consist of 20 triangles, 12 pentagons, and 30 rectangles. By slightly displacing the points towards the mid-points of the edges of the triacontahedron, the rectangles can be distorted into squares, and we have the small rhombiicosidodecahedron.
An analogous construction leads to the rhombicuboctahedron, whose faces consist of 8 triangles and 6+12 squares.
By applying the truncation method to the cuboctahedron and to the icosidodecahedron, in addition to a distortion to convert rectangles into squares, we obtain the great rhombicuboctahedron and the great rhombiicosidodecahedron.
All the Archimedean solids so far discussed can be reflexed (by reflection in the plane that perpendicularly bisects the edge). The remaining two; however, cannot be reflexed: the snub cube and the snub dodecahedron. Each of them occurs in two forms, and the two forms of each are related to one another like a left-hand and a right-hand glove: they are enantiomorphic. See Line's Solid Geometry (pp.175 - 184) for discussions about constructions of these two snub polyhedra.
