The Dimensions of the Symmetry Types of Polyhedra with Reflection Groups

Let P and Q be convex polyhedra in E3 with face lattices F(P) and F(Q) and symmetry groups G(P) and G(Q), respectively. Then, P and Q are called face equivalent if there is a lattice isomorphism between F(P) and F(Q); P and Q are called symmetry equivalent if the action of G(P) on F(P) is equivalent to the action of G(Q) on F(Q). It is well known that the set [P] of all polyhedra which are face equivalent to P has the structure of a manifold of dimension {e-1}, up to similarities, where e=e(P) is the number of edges of P. This is a consequence of the Steinitz's classical Theorem. We give a new proof of this fact. The symmetry type of P denoted by <P> is the set of all polyhedron Q symmetry equivalent to P. We show that for polyhedra with symmetry group G(P) a reflection group the dimension of this manifold is {O-1} where O is the number of edge orbits of P under and the action of G(P) on F(P).