A 3-sphere has two planes of rotation. The two planes are always at right angles to one another, and can rotate with completely different periods. I want to make an animation that shows this. It's been done, but this as far as I know is a different way.

The idea is analogous to animating a 2D map of the surface of a 2-sphere like the Earth. The image would move from west to east. As portions of the image move over the eastern boundary, they reappear in the west. So first we make a 3D map of the surface of the 3-sphere, then animate it to show the rotations.

The 3-sphere can be parameterized with polar coordinates as sin(x)e^it, cos(x)e^iu, with 0<=x
If I ever get around to making a computer graphic, I suppose I'll have the vertices of 4D Platonic polyhedral on the surface of the 4-sphere, with "straight" lines between the vertices. Rotate and display the results on the 3D map.

Could any recommend a graphics package that is capable of this? If I go to all this effort then I'll want to distribute it. For that it would be necessary to render a video.