Plasmath wrote:[...]
These are really amazing animations! There's just something about them that's really mesmerizing to look at. I have a feeling that these polygonal distortions would no longer exist if we looked at Villarceau circles on a Clifford torus instead of a normal 3D one, although I don't know how to confirm this.
You're absolutely right. The Clifford torus is a member of a set of decompositions of the 3-sphere into concentric toroids. Villarceau circles in 3D map to great circles on the 3-sphere that lie on the Clifford torus. Great circles that are equally spaced around the Clifford torus remain equidistant at all points; so yes, they absolutely form regular polygon cross-sections. One may thus view the distortions in 3D as an artifact of projecting the 4th dimension into one of the first 3: in 4D, there is no distortion because the curvature of the torus lies entirely in the plane orthogonal to its cross-section (in 4D, the perpendicular of a 2D cross-section is a plane, not just a line). In 3D, however, this curvature is mapped to the plane of the major axis of the torus. Because this now involves one of the axes where the polygons lie, distortion is inevitable.
Look again at the projections of spidrox. The 12 rings of alternating prisms, for example, divide into 1 + 5 + 5 + 1, corresponding to the faces of a regular dodecahedron. Within each group of 5, the rings are always equidistant to its neighbours, and form a spiralling regular pentagonal configuration around each other. Replace the rings with great circles through the middle of the alternating prisms, and you have 5 great circles in pentagonal formation on a Clifford torus. Project this to 3D, and you get a 3D torus with 5 Villarceau circles. In 4D, the pentagon connecting the 5 great circles is regular and has no distortion; when projected into 3D, though, it exhibits the same distortion we see in the above animation.
Also on the topic of Clifford torii, I think that a Hopf fibration-like mapping from one of the torii to a Clifford torus is probably likely, considering the Clifford torii's fundamental domain is a square.
Not only it's likely, it's actually so.
quickfur wrote:I doubt a single-twist strip is possible, because that would imply that a strip of the torus's surface is non-orientable, both sides of which are on the outside, which is a contradiction.
I meant connecting Mobius strips together instead of double-twisted Mobius strips to try to create a surface (although looking back I wasn't exactly very clear that I was talking about that). I think it would create a Klein bottle but I'm not sure.
Ahh I gotcha. You're right, if you glue Möbius strips together side-by-side until it closes up, you'd get a Klein bottle. You can't actually do this physically in 3D, though, because in 3D the resulting surface necessarily intersects itself. In 4D, however, this construction can be carried out without any self-intersection, and you get the Klein bottle, which is a knotted 2-sphere.
Note, however, that the "bottle" part of the name is actually a misnomer: in 4D, the Klein bottle actually cannot hold any liquid, because a 2D manifold in 4D is insufficient to hold liquid in place. You need a 3D manifold in order to contain a liquid in 4D, so the Klein bottle actually isn't a bottle; it's just a 4D equivalent of a twisted hyper-string. I.e., a 4D Mobius strip.