In geometry, polyhedra are associated into pairs called duals, where the vertices of one correspond to the faces of the other. 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. Geometry (from the Greek words Ge = earth and metro = measure) is the branch of mathematics first introduced by Theaetetus dealing with spatial relationships. ... In mathematics, there are three related meanings of the term polyhedron: in the traditional meaning it is a 3-dimensional polytope, and in a newer meaning that exists alongside the older one it is a bounded or unbounded generalization of a polytope of any dimension. ... A Platonic solid is a convex polyhedron whose faces all use the same regular polygon and such that the same number of faces meet at all its vertices. ... A Kepler solid (also called Kepler-Poinsot solid) is a regular non-convex polyhedron, all the faces of which are identical regular polygons and which has the same number of faces meeting at all its vertices (compare to Platonic solids). ...

Duality is usually defined in terms of polar reciprocation about a concentric 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 polytope correspond to the (n − 1)-dimensional elements, or facets, 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. In geometry polytope means, first, the generalization to any dimension of polygon in two dimensions, and polyhedron in three dimensions. ... Honeycomb on a Langstroth frame A honeycomb is a mass of hexagonal wax cells built by honeybees in their nests to contain their larvae and stores of honey and pollen. ...

Notice that the exact form of the dual will depend on what sphere we reciprocate with respect to, as we move the sphere around the dual form distorts. 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 convex polyhedra can be distorted into a canonical form where a midsphere exists such that the points where the edges touch it average out to give the center of the circle, and this form is unique up to congruencies.

We can distort a dual polyhedron such that it can no longer be obtained by reciprocating the original ln any sphere - in this case we can say that the two poyhedra are still topologically dual.

It is worth noting that the vertices and edges of a convex polyhedron can be projected to form a graph on the sphere or on a flat plane, and the corresponding graph formed by the dual of this polyhedron is its dual graph. In graph theory, a planar graph is a graph that can be embedded in a plane so that no edges intersect. ...

The concept of duality here is also related to the duality in projective geometry, where lines and edges are interchanged; and is in fact a particular version of the same. Duality in the projective plane refers to the interchangeability between points and lines which preserves incidence properties. ... In a historical perspective on mathematics, the field of geometry that developed in the first half of the nineteenth century under the name projective geometry was a stepping stone from analytic geometry to algebraic geometry. ...

If a polyhedron has an element passing through the center of the sphere, it will have an infinite dual. This is because one or more elements of the dual lie in the plane at infinity. Interestingly, traditional infinite "Euclidean" space never reaches infinity. the projective equivalent, called extended Euclidean space, is formed by adding the required plane at infinity.

External links

• Software for displaying duals (http://www.software3d.com/Stella.html)
• The Uniform Polyhedra (http://www.mathconsult.ch/showroom/unipoly/)
• Virtual Reality Polyhedra (http://www.georgehart.com/virtual-polyhedra/vp.html) The Encyclopedia of Polyhedra