ROTATIONS OF THE THREE-SPHERE AND SYMMETRY OF THE CLIFFORD TORUS

JOHN MCCUAN AND LAFE SPIETZ

Abstract. We describe decomposition formulas for rotations of ℝ^{3} and ℝ^{4} that have special properties with respect to stereographic projection. We use the lower dimensional decomposition to analyze stereographic projections of great circles in 𝕊^{2} ⊂ ℝ^{3}. This analysis provides a pattern for our analysis of stereographic projections of the Clifford torus C ⊂ 𝕊^{3} ⊂ ℝ^{4}. We use the higher dimensional decomposition to prove a symmetry assertion for stereographic projections of C which we believe we are the first to observe and which can be used to characterize the Clifford torus among embedded minimal tori in 𝕊^{3}—though this last assertion goes beyond the scope of this paper. An effort is made to intuitively motivate all necessary concepts including rotation, stereographic projection, and symmetry.