Each polyhedron has a so-called 'dual', in which the roles ofverticies and faces and some sense swap over. As the name suggests, the dual of the dual of a polyhedron is the original polyhedron itself.
Technically the dual can be defined by inversion of a polyhedron with respect to its inter-sphere, but its easier to think of it being acheieved by replacing each face with a vertex, each vertex with a face, and rotating all of the edges through 90°. Consequently, the number of edges of the dual matches that of the original polyhedron, while the numbers of faces and vertices are exchanged.
Regular polyhedra have duals that are also regular. The dual of a polyhedron with vertex symbol pq has symbol qp. Semi-regular polyhedra have duals in the set of vertically regular solids and vice versa.
External Links: Entry on MathWorld.