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
A more accurate rendering of the {6,3,3}. This shows the geodesic edges more properly, as arcs orthogonal to the "plane at infinity" of hyperbolic space (vs. straight lines).
Originally shared by Andrea Hawksley Chess secretly makes more sense on a {5,4} hyperbolic chess board. http://blog.andreahawksley.com/non-euclidean-chess-part-2/ http://blog.andreahawksley.com/non-euclidean-chess-part-2
If you stare at this for a bit, then look elsewhere, everything appears to be shrinking.
ReplyDeleteAn animated stereogram version of this would be fun to watch too I think. I mean things like this;
ReplyDeletehttps://www.youtube.com/watch?v=0Vtyoqzt-P8