Dual polyhedron

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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 also sometimes called reciprocity or polarity.


Kinds of duality

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

  • Polar reciprocity
  • Topological duality
  • Abstract duality

Polar reciprocation

Duality 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, then, 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.

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.

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