Dual polyhedron

In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. Starting with any given polyhedron, the dual of its dual is the original polyhedron. The dual of an isogonal polyhedron, having equivalent vertices, is one which is isohedral, having equivalent faces, and of one which is isotoxal, having equivalent edges, is also isotoxal. The regular polyhedra — the Platonic solids and Kepler-Poinsot polyhedra — form dual pairs, with the exception of the regular tetrahedron which is self-dual.

Duality is closely related to reciprocity or polarity.

Kinds of duality

There are many kinds of duality. The kinds most relevant to elementary polyhedra are:

Polar reciprocation

The duality of polyhedra is most commonly defined in terms of polar reciprocation about a concentric sphere. Here, each vertex (pole) is associated with a face plane (polar plane or just polar) so that the ray from the center to the vertex is perpendicular to the plane, and the product of the distances from the center to each is equal to the square of the radius. In coordinates, for reciprocation about the sphere