Today was awesome. Thanks Mark Sheffield and Woods Cycle Country!
Today was awesome. Thanks Mark Sheffield and Woods Cycle Country!
Hyperbolic Hopf Fibrations
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
NIce Motorcycle Roice I really do like it and that is awesome
ReplyDeleteWow. What a beautiful machine!
ReplyDeleteYes! And in this case there is definitely a form-function correlation. Riding has been fantastic :)
ReplyDeleteCan I borrow it for an afternoon? :)
ReplyDeleteYep, just means you have to come down to hang out in Austin sometime :D
ReplyDeleteRoice Nelson yes that is true I would have to come down to Austin Texas and hangout sometime and that would be fun :)
ReplyDeleteRoice Nelson OK, I'll see if I can borrow a plane. :)
ReplyDelete