Take the surface of a 4D ball — the 3-sphere. Hopf discovered it can be perfectly combed into circles: one circle through every point, no two touching, every pair linked exactly once. Each circle corresponds to one point of an ordinary sphere (its color here).
Stereographic projection of those fibers into 3D — the same trick as flattening the Earth onto a map, one dimension up. Circles over one latitude of the base sphere form a donut; the latitudes nest all the donuts inside each other, filling all of space.
Raise latitudes and watch the nested tori appear. The isoclinic spin drives the white beads along their circles — the circles themselves never move, because the flow carries each one onto itself: the whole 3-sphere flowing in place. Alt-drag to tip the projection pole and melt the picture into a new one.