On regular polyhedra with hidden symmetries (Q1096176)

The title refers to the class of polyhedra in \(E^ 3\) (with or without selfintersections) which are combinatorically regular. The geometric symmetry group is a subgroup of the automorphism group of a certain index. By previous work of the author there are exactly five such polyhedra of index 2, each with icosahedral symmetry group. There are also examples of index 4 and octahedral group, and index 8 and tetrahedral group. In this paper it is shown that index 2 or 3 is impossible in the octahedral case and that index 2,3,4,5 is impossible in the tetrahedral case.