Rotations of the three-sphere and symmetry of the Clifford torus
John McCuan and Lafe Spietz
We describe decomposition formulas for rotations of ${\mathbb R}^3$ and ${\mathbb R}^4$ that have special properties with respect to stereographic projection. We use the lower dimensional decomposition to analyze stereographic projections of great circles in ${\mathbb S}^2 \subset {\mathbb R}^3$. This analysis provides a pattern for our analysis of stereographic projections of the Clifford torus ${\mathcal C}\subset {\mathbb S}^3 \subset{\mathbb R}^4$. We use the higher dimensional decomposition to prove a symmetry assertion for stereographic projections of ${\mathcal C}$ which we believe we are the first to observe and which can be used to characterize the Clifford torus among embedded minimal tori in ${\mathbb S}^3$–-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.