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.

 

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