Inscribing 5 Tetrahedra in a Dodecahedron

This model illustrates a correspondence between the group of symmetries of a dodecahedron and a certain permutation group on 5 symbols. The rigid geometric motions that take the dodecahedron to itself serve to permute the 5 inscribed tetrahedra. A careful analysis will show that only even permutations of the 5 tetrahedra can be achieved in this manner.