Three - Sphere Rotations and The

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 Cliiord 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 rst to observe and which can be used to characterize the Cliiord torus among embedded minimal tori in S 3 |though this last assertion goes beyond the scope of this paper. An eeort is made to intuitively motivate all necessary concepts including rotation, stereographic projection, and symmetry.