Public

+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

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

For more details, try these picture-packed pages:

http://en.wikipedia.org/wiki/Conformal_map

http://en.wikipedia.org/wiki/Poincar%C3%A9_disk_model

and this more advanced one:

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

#sciencesunday

**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_model

and this more advanced one:

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

#sciencesunday

View 19 previous comments

- SempiternalNov 7, 2012
- I'm having some trouble trying to figure out what this is. For example, what is happening at each frame, I don't know what it means to 'tile"..Nov 7, 2012
- +Misha Kandel - like the sphere, the hyperbolic plane is a non-Euclidean variant of the usual plane. Just as you can cover (or technically, 'tile') the plane with squares, 4 meeting at each corner, you can tile the hyperbolic plane with regular pentagons, 4 meeting at each corner. See:

http://bulatov.org/math/1001/#%282%29

and subsequent slides. Bulatov starts with this, and then applies an angle-preserving transformation that maps the hyperbolic plane to a disc with 4 slits cut out. For more on tilings of the hyperbolic plane see:

http://en.wikipedia.org/wiki/Uniform_tilings_in_hyperbolic_plane

Even if the math is stressful the pictures are enjoyable!Nov 8, 2012 - Oooooh!Nov 14, 2012
- amazingDec 10, 2012
- YES ! amazing to the amazing power ~Feb 22, 2015