The Symmetries of Hypersphere

This shows four loops in the Special Orthogonal group of dimension four. The Lie group SO(4) can be realized by multiplying a quaternionic number on both sides (left and right) with two other quaternionic numbers. There are three vector fields on the 3-sphere: K1, K2 and K3. The natural connection on the Hopf bundle regards K1 and K2 as horizontal directions and K3 as the vertical. But, we get a six-dimensional group of rotations by choosing to act on the 3-sphere either on the left or on the right. The first thirty seconds of this animation shows a loop in the direction of K1 by multiplication on the right. The next thirty seconds shows the rotation of the 3-sphere in the direction of K2. The next thirty seconds in the K3 direction. Finally, the last thirty seconds shows rescaling the coordinates of the map of the Earth that is pulled back by the Hopf map, which takes points from the 3-sphere into points in the 2-sphere. Any closed loop in the 3-sphere is shrinkable to a single point, and that’s the purpose of the last quarter of the animation. That makes the 3-sphere a simply-connected space.
Read the documentation: iamazadi.githu...