John Baez gives us a closer look at the {6,3,3} honeycomb.

John Baez gives us a closer look at the {6,3,3} honeycomb.

Originally shared by John Baez

This is the {6,3,3} honeycomb, as drawn by Roice Nelson.

A 3-dimensional honeycomb is a way of filling 3d space with polyhedra. It’s the 3d analogue of a tiling of the plane.  But besides honeycombs in ordinary Euclidean space, we can also have them in hyperbolic space.  This is a curved 3d space.  The {6,3,3} honeycomb lives in hyperbolic space.  That’s why it looks weirdly distorted.  Actually all the hexagons are the same size… but we have to warp hyperbolic space to draw it in ordinary space.

You can learn more about all these concepts by going to my new blog, Visual Insight:

http://blogs.ams.org/visualinsight/2013/09/15/633-honeycomb-in-upper-half-space/

But let me just answer one obvious question: why is it called the {6,3,3} honeycomb?  

{6,3,3} is a Schläfli symbol.   The symbol for the hexagon is {6}. The symbol for the hexagonal tiling of the plane is {6,3} because 3 hexagons meet at each vertex. Similarly, the symbol for the hexagonal tiling honeycomb is {6,3,3} because 3 hexagonal tilings meet along each edge.

3 hexagonal tilings meeting at each edge!   That’s a bit hard to visualize.  But if you stare carefully at this picture, and look at one of the big fat edges near the top, you can see 2 hexagonal tilings meeting at that edge – one in front that’s easy to see, and one in back.  The third, not shown, goes upward.

#4d