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

I found it rather hard to predict how the topology of the board affects play. Hexagonal boards in the form of an 10*10 rhombus were interesting, some peices proved more or less useful, and needed some work to make them function.

ReplyDeleteWhat about hyperbolic go?

ReplyDeleteYou'd probably have to play go in the style of reversi, where you can only build onto an existing structure.

ReplyDeleteEduard Baumann, check out Jenn3D for Go played on spherical polytopes! See the very bottom of this page:

ReplyDeletehttp://www.math.cmu.edu/~fho/jenn/

wendy krieger, I believe you that it's difficult, based on my experience with my "global chess" board (spherical chess).

ReplyDeletehttp://www.pa-network.com/global-chess/indexf.html

That's quite hard to play, and I'm sure some of the hyperbolic topologies have to be even more mind-bending.