Dual Polyhedron

By the Duality Principle, for every Polyhedron, there exists another Polyhedron in which faces andVertices occupy complementary locations. This Polyhedron is known as the dual, orReciprocal. The dual polyhedron of a Platonic Solid or Archimedean Solid canbe drawn by constructing Edges tangent to the Reciprocating Sphere (a.k.a. Midsphere and Intersphere) which are Perpendicular to the original Edges.

The dual of a general solid can be computed by connecting the midpoints of the sides surrounding each Vertex, and constructing the corresponding tangent Polygon. (The tangent polygon is the polygon which istangent to the Circumcircle of the Polygon produced by connecting the Midpoint on the sides surroundingthe given Vertex.) The process is illustrated below for the PlatonicSolids. The Polyhedron Compounds consisting of a Polyhedron and itsdual are generally very attractive, and are also illustrated below for the Platonic Solids.

The Archimedean Solids and their duals are illustrated below.

The following table gives a list of the duals of the Platonic Solids and Kepler-PoinsotSolids together with the names of the Polyhedron-dual Compounds.

References

Weisstein, E. W. ``Polyhedron Duals.'' Mathematica notebook Duals.m.

Wenninger, M. Dual Models. Cambridge, England: Cambridge University Press, 1983.

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