Dual polyhedron

In geometry, polyhedra are associated into pairs called duals, where the 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; the dual of an isotoxal polyhedron (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.

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

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

the vertex

is associated with the plane

The vertices of the dual are the poles reciprocal to the face planes of the original, and the faces of the dual lie in the polars reciprocal to the vertices of the original. Also, any two adjacent vertices define an edge, and these will reciprocate to two adjacent faces which intersect to define an edge of the dual. This dual pair of edges are always orthogonal (at right angles) to each other.

If $r_{0}$ $r_{0}$ is the radius of the sphere, and $r_{1}$ $r_{1}$ and $r_{2}$ $r_{2}$ respectively the distances from its centre to the pole and its polar, then:

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