+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_maphttp://en.wikipedia.org/wiki/Poincar%C3%A9_disk_modeland this more advanced one:

http://en.wikipedia.org/wiki/Riemann_mapping_theorem#sciencesunday