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+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:

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:

John Baez's profile photoAaron Fenyes's profile photoVineet George's profile photoArthur Lawrence's profile photo
Hmm, the rotation is not centered at the center of the picture, but a bit to the right.  That's why the curves are moving faster on the left side of the picture!
It was (apparently) my 'Mercator' designs that prompted him to explore mapping the Poincaré disc to other shapes.  Whee, I indirectly made a mark on the world!
I like slide 75:  "I have no idea how I've got this."

Make it a puzzle!
If it's pretty, maybe I will.  I only vaguely remember that one.
If I follow one line all the way round the circle, it reminds me of a Tim Burton film.
+Anton Sherwood - nice!!!  Mind if I post one or two of those someday, citing you of course?
Go to town!  Let me know if you want a variation or a hi-res version for some purpose.
Bulatov's page was the first to mention to me that open subsets of the plane have to be hyperbolic. Gave me an even better feeling on how huge the hyperbolic plane really is. Precious moments. Thanks.
+Refurio Anachro - If we lived in hyperbolic 3-space we could expand our civilization out into space, and after colonizing enough galaxies we'd start noticing that the total volume of the colonized region was growing closer and closer to exponentially.   But the best measurements today seem to show space is flat rather than hyperbolic.  If so, exponential growth is impossible in the long term: the volume of our civilization can only grow as the cube of time, given that we can't go faster than light.  Bummer!
+Anton Sherwood - Thanks!  Since these Google+ photos are fairly small (unless you click on them, but few people do), I probably won't be wanting higher-res versions. 
+sansi soni - I explained it in my article.  Click "expand this post" for more.  It's math!
Thank you  John Baez. idea was superb.
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"..
+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:

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:

Even if the math is stressful the pictures are enjoyable!
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