Math .. i get dizzy watching it
John Baez+Vladimir Bulatov does it again! At each moment, this movie shows you a tiling of the hyperbolic plane by pentagons, four meeting at each corner, mapped onto a disc with four slits cut out. This mapping is conformal, meaning that it preserves angles. As time passes, the hyperbolic plane rotates and we see this crazy movie.
For a more detailed explanation, with tons of great pictures, go here: http://bulatov.org/math/1001/
Here's the short version: there's a way to measure distances on a disk that makes it into a model of the hyperbolic plane. There are actually a number of ways, but Bulatov – and Escher – use the Poincare disk model, because in this model straight lines look like portions of circles: very pretty. Then, according to the Riemann mapping theorem you can map this disk in a conformal way onto a disk with 4 slits cut out. The hard part is finding a formula for how to do it, and then implementing it on a computer.
For more details, try these picture-packed pages:
http://en.wikipedia.org/wiki/Conformal_map
http://en.wikipedia.org/wiki/Poincar%C3%A9_disk_modeland this more advanced one:
bulatov_disk_with_cuts.gif
