The edges and vertices of a polyhedron constitute a special case of a graph in which a set of points, or nodes, is joined in pairs by segments or branches. Therefore, the essential property of a polyhedron is that its faces together form a single unbounded surface. The edges are merely curves drawn on the surface that come together in sets of three or more at the vertices.
In other words, a polyhedron with faces, edges, and vertices may be regarded as a map, that is, as the partition of an unbounded surface into polygonal regions by means of simple curves joining pairs of points.
From a given map, we may derive a second, called the dual map, on the same surface. This second map has vertices, one in the interior of each face of the given map; edges, one crossing each edge of the given map; and faces, one surrounding each vertex of the given map. Corresponding to a p-gonal face of the given map, the dual map will have a vertex where p edges (and p faces) come together.
Duality is a symmetric relation: A map is the dual of its dual.
A map is said to be regular, of type {p,q}, when there are p vertices and p edges for each face, and q edges and q faces at each vertex, that are arranged symmetrically in a sense that can be made precise. Therefore, a regular polyhedron is a special case of a regular map. For each map of type {p,q} is a dual map of type {q,p}.
Consider the regular polyhedron {p,q}, with its vertices, edges, and faces. If we replace each edge by a perpendicular line touching the mid-sphere at the same point, we obtain the edges of the reciprocal polyhedron {q,p}, which has vertices and faces. This process is in fact reciprocation with respect to the mid-sphere: the vertices and face-planes of {p,q} are the poles and the polars, respectively, of the face-planes and vertices of {q,p}.
Reciprocation with respect to another concentric sphere would yield a larger or smaller {q,p}.
This process of reciprocation can evidently be applied to any figure that has a recognizable "center". It agrees with the topological duality that we defined for maps. The 13 Archimedean solids are therefore included in this case; that is, for each Archimedean solid there exists a reciprocal polyhedron with respect to a concentric sphere.
