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 KeplerPoinsot polyhedra, are arranged into dual pairs. Because the
stellations of the dodecahedron are also KeplerPoinsot polyhedra, they have
existing duals.

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 KeplerPoinsot
polyhedra exist in dual pairs: