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.

In Maple, one can define a duality of a regular polyhedron or of an Archimedean solid by using the command duality(dualp,p,s) where dualp is the name of the reciprocal polyhedron of the given polyhedron p with respect to the sphere s which is concentric with p (that is, s and p have the same center).

The following series of Maple commands show how to define and display the reciprocal polyhedron of a given regular polyhedron.

A given regular polyhedron is closed under duality--the duality of a regular polyhedron is also a regular polyhedron.

This is not the case for the Archimedean solids, though. The following table shows the polyhedron type of the duals of the Archimedean solids:

Archimedean Solids             Maple's Schlafli    Reciprocal Polyhedron          Maple's Schlafli
TruncatedTetrahedron           _t([3,3])           TriakisTetrahedron             dual(_t([3,3]))
TruncatedOctahedron            _t([3,4])           TetrakisHexahedron             dual(_t([3,4]))
 TruncatedHexahedron           _t([4,3])           TriakisOctahedron              dual(_t([4,3]))
TruncatedIcosahedron           _t([3,5])           PentakisDodecahedron           dual(_t([3,5]))
 TruncatedDodecahedron         _t([5,3])           TriakisIcosahedron             dual(_t([5,3]))
cuboctahedron                  [[3],[4]]           RhombicDodecahedron            dual([[3],[4]])
icosidodecahedron              [[3],[5]]           RhombicTriacontahedron         dual([[3],[5]])
SmallRhombicuboctahedron       _r([[3],[4]])       TrapezoidaIcositetrahedron     dual(_r([[3],[4]]))
SmallRhombiicosidodecahedron   _r([[3],[5]])       TrapezoidalHexecontahedron     dual(_r([[3],[5]]))
GreatRhombicuboctahedron       _t([[3],[4]])       HexakisOctahedron              dual(_t([[3],[4]]))
GreatRhombiicosidodecahedron   _t([[3],[5]])       HexakisIcosahedron             dual(_t([[3],[5]]))
SnubCube                       _s([[3],[4]])       PentagonalIcositetrahedron     dual(_s([[3],[4]]))
SnubDodecahedron               _s([[3],[5]])       PentagonalHexacontahedron      dual(_s([[3],[5]]))

To access information relating to the reciprocal of an Archimedean solid gon, use the following function calls:

center(gon);    returns the center of the mid-sphere of gon.
faces(gon);     returns the faces of gon, with each face is represented as a list of coordinates of its vertices.
form(gon);      returns the form of gon (TriakisTetrahedron3d, ...).
radius(gon);    returns the mid-radius of gon.
schlafli(gon);  returns the Schlafli symbol of gon.
vertices(gon);  returns the coordinates of vertices of gon.

For example:

The following figure shows the 13 reciprocals of the Archimedean solids: