Plane-filling fractals projected to spheres...pretty!
Plane-filling fractals projected to spheres...pretty!
http://blog.hvidtfeldts.net/index.php/2012/03/spherical-worlds/
http://blog.hvidtfeldts.net/index.php/2012/03/spherical-worlds/
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
Check out this vid, Wau: The Most Amazing, Ancient, and Singular Number gotta love fractals, even when it is just a number
ReplyDeleteGotta love the number 1 ;) I've been meaning to send you a link to another video series by that same author. It's all about Fibonacci numbers, and I think you would like it. There are three in the series, and the first is: Doodling in Math: Spirals, Fibonacci, and Being a Plant [1 of 3]
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