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Duality

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. Because the stellations of the dodecahedron are also Kepler-Poinsot polyhedra, they have existing duals.

Small stellated dodecahedron

Great dodecahedron

Great stellated dodecahedron

Great icosahedron

 Faces Edges Vertices Small stellated dodecahedron 12 30 12 Great dodecahedron 12 30 12 Great stellated dodecahedron 12 30 20 Great icosahedron 20 30 12

Notice that the vertices of the small stellated dodecahedron correspond to faces of the great dodecahedron and vice versa. The great stellated dodecahedron and great icosahedron have the same correspondence. So, the Kepler-Poinsot polyhedra exist in dual pairs:

• Small stellated dodecahedron and great dodecahedron.
• Great stellated dodecahedron and great icosahedron.