Dual polyhedron

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The dual of a cube as an octahedron with vertices at the cube face centers.

Truncation sequence from a cube to its dual octahedron. A polyhedral dual is called a face-rectification or a birectification.

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, with the exception of the regular tetrahedron which is self-dual.

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

x 2 + y 2 + z 2 = r 2,

the vertex

(x 0, y 0, z 0)

is associated with the plane

x 0 x + y 0 y + z 0 z = r 2

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.
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 choice of 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 in which a midsphere exists such that the points where the edges touch it average out to give the center of the sphere, and this form is unique up to congruences.

We can distort a dual polyhedron such that it can no longer be obtained by reciprocating the original in any sphere; in this case we can say that the two polyhedra 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. The concept of duality here is also related to the duality in projective geometry, where lines and edges are interchanged; in fact it is a particular version of the same.

If a polyhedron has an element passing through the center of the sphere, the corresponding element of its dual will pass through or be at infinity. Since traditional infinite "Euclidean" space never reaches infinity, the projective equivalent, called extended Euclidean space, must be formed by adding the required plane at infinity.

[edit] Dual Polytopes

Duality can be generalized to n-dimensional space and dual polytopes.

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.