+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:


and this more advanced one:


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