Dual polyhedra

In elementary geometry, any polyhedron is associated with a second dual figure, where the vertices of one correspond to the faces of the other and the edges between pairs of vertices of one correspond to the edges between pairs of faces of the other. Such dual figures remain combinatorial or abstract polyhedra, but not all are also geometric polyhedra. Starting with any given polyhedron, the dual of its dual is the original polyhedron.

Duality preserves the symmetries of a polyhedron. Therefore, for many classes of polyhedra defined by their symmetries, the duals also belong to a symmetric class. Thus, the regular polyhedra – the (convex) Platonic solids and (star) Kepler-Poinsot polyhedra – form dual pairs, where the regular tetrahedron is self-dual. 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.

Duality is closely related to reciprocity or polarity, a geometric transformation that, when applied to a convex polyhedron, realizes the dual polyhedron as another convex polyhedron.

There are many kinds of duality. The kinds most relevant to elementary polyhedra are polar reciprocity and topological or abstract duality.

The duality of polyhedra is often 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.

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