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Thinking about rotations of a 3D sphere- We first choose an axis. Then the whole sphere can be thought of as an infinite number of circles, perpendicular to the axis, stacked on top of each other. When we rotate the sphere about the axis, the points on these stacked infinite circles start rotating in the plane of the circles (like normal 2D rotations).

Now coming to rotations of 4D hypersphere : We again first choose an axis. This time, there should be an infinite number of spheres stacked on top of each other, perpendicular to the axis. When we start rotating about the axis, the points on these spheres should start going in circles in some weird way. In case of 3D, the points on the circles stayed on the circles after rotation. So in the 4D case, the points should stay on their spheres after rotation.

I mentioned : When we start rotating about the axis, the points on these spheres should start going in circles in some weird way.

What is this weird way? I can't visualize it.

Say there's a 4D sphere with center $(x,y,z,w)=(0,0,0,0)$. If we start rotating this about the w-axis, then the w-coordinate of all the points on the sphere should stay fixed. But what happens to the x,y,z co-ordinates of the points on the hypersphere?

Ryder Rude
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    Rotations in even-dimensional spaces don’t have a fixed axis. – amd Feb 15 '20 at 06:42
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    Chiming in with amd. Depending on how you define things, a rotation either is just an orientation and distance preserving linear transformation, or possibly (more limited point of view) one such that moves points of a plane (=a 2-dimensional subspace), and leaves everything in the orthogonal complement (=an $(n-2)$-dimensional subspace) fixed pointwise. In the latter sense the axis has dimension $n-2$ (a point in 2D, a line in 3D). In the former sense there may or may not be an axis at all (but a one-dimensional subspace of fixed points always exists if $n$ is odd). – Jyrki Lahtonen Feb 15 '20 at 08:28
  • More discussion. – Jyrki Lahtonen Feb 15 '20 at 08:32
  • @JyrkiLahtonen I like the later sense more. So do quaternions rotate things in 4D space about a planer 2D axis? – Ryder Rude Feb 15 '20 at 11:44
  • When people talk about quaternions and rotation, they usually mean rotations in a three-dimensional space. – David K Feb 15 '20 at 15:21
  • For 4D rotations see https://math.stackexchange.com/questions/2543122/axes-of-rotation-in-4d – David K Feb 15 '20 at 15:22

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