Dual polyhedron

Polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the others. The dual of the dual is the original polyhedron. The dual of a polyhedron with equivalent vertices is one with equivalent faces, and of one with equivalent edges is another with equivalent edges. So the regular polyhedra - the Platonic solids and Kepler-Poinsot polyhedra - are arranged into dual pairs.

Duality is defined in terms of polar reciprocation about a given sphere. Here, each vertex is associated with a face plane 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 x² + y² + z² = r², the vertex (x₀, y₀, z₀) is associated with the plane x₀x + y₀y + z₀z = r².

The vertices of the dual, then, are the reciprocals of the face planes of the original, and the faces of the dual lie in the reciprocals of 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 can be generalized to n-dimensional space, so we can talk about dual polytopes. Then the vertices of one correspond to the (n-1)-dimensional elements of the other, and the j points that define a (j-1)-dimensional element will correspond to j hyperplanes that intersect to give a (n-j)-dimensional element. The dual of a honeycomb can be defined similarly.

Notice that the dual of a polyhedron will depend on what sphere we reciprocate with respect to, the resulting forms being distortions of one another. The center of the sphere is sufficient to define the dual up to similarity. If multiple symmetry axes are present, they will necessarily intersect at a single point, and this is usually taken to be the center. Failing that a circumscribed sphere, inscribed sphere, or midsphere (one with all edges as tangents) can be used. It can be shown that all polyhedra can be distorted into a canonical form where a midsphere exists and in fact the points where the edges touch it average out to give the center of the circle, and this form is unique up to congruencies.

If a polyhedron has an element passing through the center of the sphere, it will have an infinite dual. It is worth noting that the vertices and edges of a convex polyhedron can be projected to form a graph on the sphere, and the corresponding graph formed by the dual of this polyhedron is its dual graph.

The concept of duality here is also related to that in projective geometry, where lines and edges are interchanged, and is in fact a particular version of the same.