Quaquaversal Tilings And Rotations

This paper is concerned with certain patterns in 3 dimensional Euclidean space, and their symmetries. One step in analyzing these symmetries involves determining the group G(m;n) generated by a pair of rotations about orthogonal axes, one by 2ß=m and the other by 2ß=n, 2 m n. Specifically, we determine presentations for the groups G(3; 3) and G(3; 4), which are perhaps the simplest cases aside from the finite groups, G(4; 4) and G(2; n). (There were previous results for some cases [Swi] where m = n = 1.