Subgroups ofSO(3) Associated with Tilings

We give a thorough analysis of those subgroups ofSO(3) generated by rotations about perpendicular axes by 2π/pand 2π/q. A corollary is that such a group is the free product of the cyclic groups of rotations about the separate axes if and only ifp, q ≥ 3 and are both odd. These groups are naturally associated with a family of hierarchical tilings of Euclidean 3-space.