69 Truncated cube

70 Truncated tetrahedron

71 Truncated cuboctahedron

72 Icosidodecahedron

73 Rhombicuboctahedron

74 Truncated octahedron

75-79 The Platonic solids

The models in this group show the five Platonic solids and some of the thirteen Archimedean solids (which must have regular faces and congruent vertices, but need not have all faces the same).

The Platonic solids. The five regular convex polyhedra, or Platonic solids, are the tetrahedron, cube, octahedron, dodecahedron, icosahedron (75 - 79), with 4, 6, 8, 12, and 20 faces, respectively. These are distinguished by the property that they have equal and regular faces, with the same number of faces meeting at each vertex. From any regular polyhedron we can construct another one, called its dual, by joining the centers of its faces with line segments. For example, if we join the centers of the faces of a cube, we get an octahedron sitting inside the cube, so the dual of a cube is an octahedron. If we repeat the process for the octahedron, we obtain a cube sitting inside the octahedron, so the dual of an octahedron is a cube. Similarly, the dodecahedron and the icosahedron are duals of each other, and the tetrahedron is its own dual. A polyhedron and its dual have the same number of edges (12 for a cube and an octahedron, but the numbers of vertices and faces are interchanged).

The Archimedean solids. If we allow the faces to be unequal, but still insist that they be regular, and that each vertex have the same arrangement of faces around it, we obtain the thirteen quasiregular convex polyhedra, or Archimedean solids. Some are obtained by cutting off, or truncating, the corners of a regular polyhedron. Thus we obtain the truncated cube (69), the truncated tetrahedron (70), the truncated octahedron (74). The cuboctahedron and the icosidodecahedron (72) are obtained by taking the volume simultaneously enclosed by a regular polyhedron and its dual of the same radius (performing the same process with a tetrahedron yields an octahedron). Truncating a cuboctahedron and adjusting the resulting faces to make them squares gives the truncated cuboctahedron (71). The final model in this group is the rhombicuboctahedron (73) (bounded by a cube, and octahedron, and rhombic dodecahedron). The seven Archimedean solids not shown here are the truncated dodecahedron, truncated icosahedron, cuboctahedron, rhombicosidodecahedron, truncated icosidodecahedron, snub cube, and snub dodecahedron. For pictures of these and many other polyhedra, visit the website http://www.li.net/~george/virtual-polyhedra/vp.html.