Hyperbolic Hopf Fibrations The Hopf Fibration of S^3 is amazing and beautiful. Rather than describe it here, I'll point you to a lovely online reference with pictures and videos by Niles Johnson. nilesjohnson.net/hopf.html To understand it better (and fibrations in general), I recommend this talk by Niles too. www.youtube.com/watch?v=QXDQsmL-8Us It turns out there is an analogue of the Hopf fibration for H^3. In fact, there is not just one "fiberwise homogenous" fibration in the hyperbolic case. There is a 2-parameter family of them, plus one additional fibration that does not fit the family. As with S^3, fibers in the H^3 cases are geodesics. They are ultraparallel in fibrations from the family, and parallel in the exceptional fibration. I found the following dissertation by Haggai Nuchi a good intro and resource to help think about all this. www.math.upenn.edu/grad/dissertations/NuchiThesis.pdf I'm posting three early pictures of H^3 fibrations below, labeled
The sad thing about spherical models of H3 is that the infinitely dense horizon obscures the large-scale centre! I like that you cut the {4,4,4} in half, presumably for this reason. I wonder about half-space models.
ReplyDeleteYep, that was one of the main reasons. The other nice thing about cutting in half is saving on printing costs. We should do half-ball models of these as well.
ReplyDeleteWe have played some with half-space models too, but haven't printed anything yet. Culling out the large parts in a nice way will be a challenge. I'll post some of the experimental pictures in a bit.