Fractal madness with heptagons

This picture drawn by +Danny Calegari shows the '{7,3,3} honeycomb' in hyperbolic space.  Hyperbolic space is an infinite space where triangles have angles that add up to less than 180 degrees.  But the mathematician Poincaré figured out how compress it down to a ball so we can see it from the outside, and that's what we see here.

The {7,3,3} honeycomb is built of regular 7-sided shapes - heptagons - in hyperbolic space.  But what does {7,3,3} mean?

The heptagons lie on infinite sheets, which show up as holes here.  If you look at these holes,  you'll see they have three heptagons meeting at each corner.  So, each one is a copy of the {7,3} tiling of the hyperbolic plane.  7 stands for heptagon, 3 stands for three meeting at each corner.

It's impossible to see, but 3 of these {7,3} tilings meet along each edge in the picture.  That's why the whole thing is called the {7,3,3} honeycomb.

It sounds wacky and fun, and it is, but it's also part of a deep theory, which I explain on my blog:

http://blogs.ams.org/visualinsight/2014/08/01/733-honeycomb/

For a nice picture of a {7,3} tiling, try this earlier blog article:

http://blogs.ams.org/visualinsight/2014/07/15/73-tiling/

The topology of the {7,3,3} honeycomb is interesting. It is simply connected, since all the holes extend all the way to the edge of the Poincaré ball.  And its ‘boundary’ is a highly distorted copy of the Sierpinski carpet. 

That's why I had a contest to create a nice picture of the Sierpinski carpet a while back!  To see the winning entry and learn more about what makes that fractal special, go here:

http://blogs.ams.org/visualinsight/2014/07/01/sierpinski-carpet/

#fractals #geometry