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Geometric ideas and concepts

Dual Polyhedra

from Curious and Interesting Geometry by David Wells

The dual of any of the Platonic polyhedra is formed by joining the centers of adjacent faces. In the resulting dual solid, each vertex corresponds to a face of the original, each face of the new solid to an original vertex, and the edges match, one for one.

As it happens, the dual of each Platonic solid is also Platonic. The regular tetrahedron is its own dual, the cube and the regular octahedron are duals of each other, and so are the regular dodecahedron and icosahedron.

The same simple process will not work for the semi-regular or Archimedean polyhedra, because the centers of the faces round a vertex will not lie in a plane. It is necessary instead to inscribe the semi-regular polyhedron in a sphere and construct the tangent plane at each vertex.

The resulting duals of the semi-regular polyhedra are not themselves semi-regular. However, their faces are all congruent and every vertex is regular, though not all faces are necessarily identical.

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