Dual polytope



For any polytope, one can associate a dual polytope, so that the vertices of one correspond to the facets of the other, the edges of one correspond to the ridges of the other, and so on. If a polytope with central symmetry has no facets passing through its center, a simple construction can build a unique dual polytope with the same symmetry as the original, which is often regarded as the dual polytope. Thus, the dual of any isogonal polytope without any hemi facets is an isotopic polytope, and viceversa.

The cells of a dual polytope are given by the duals of the vertex figures of the base polytope.

If the dual of a polytope is topologically equivalent to the base polytope, then the base polytope is called self-dual. Among the regular polytopes, the regular polygons and the simplexes are self-dual, while the hypercube and the cross polytope form dual pairs. The three- and four-dimensional convex pentagonal polytopes also come in dual pairs. The icositetrachoron is a further self-dual polychoron.

Abstractly, taking a dual amounts to inverting the incidences of the polytope, so that if A was a subelement of B in the original polytope, B becomes a subelement of A on the dual polytope.

Spherical reciprocation
Virtually any polytope can be dualized by means of the spherical reciprocation. For that purpose select some hypersphere of the according dimensionality, i.e. its center and its radius. Take the chosen radius for unit of measure. Now each ray from that center to a vertex of the given polytope also defines 2 quantities: the direction and the distance. Take the reciprocal of that distance (wrt. the radius of the sphere) and errect the affine hyperplane wrt. to the ray at that reciprocal distance. Continue with all former vertices. These become the facets of the dual. Next consider the margins of that dual. These will be constructed by the intersections of those hyperplanes, which correspond to any former pair of vertices, which was connected by an edge. - Note that the choice of the center within a single original facet will lead to a vertex of the dual, which is located at infinity. Similarily, if the center would be taken at a vertex of the former polytope, then clearly the corresponding new facet of the dual would be contained within the hyperplane of infinity. Therefore best results are obtained when that center would be chosen in "general" position.