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76 Unique Honeycombs

76 Unique Honeycombs

Last weekend, Tom Ruen and I hit the milestone of uploading to wikipedia at least one image for 9 families of compact, Wythoffian, uniform H3 honeycombs, a total of 76 unique honeycombs.

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

You can easily browse all the images on my wiki user page:
http://commons.wikimedia.org/wiki/User:Roice3

- Compact means the cells are finite in extent. 
- Wythoffian means we can generate them using a kaleidoscopic construction, that is by reflecting in mirrors.   
- Uniform means they are vertex transitive and have uniform polyhedral cells.

There may even be more honeycombs that meet all these criteria, I don't know. (update: see Tom's comment below!)  I do know there are hundreds more which don't meet one or more of these criteria, many undiscovered.  In fact, there are infinitely more because there are some infinite families of honeycombs.  wendy krieger continues to discover and enumerate more of them.  

h…

Chess secretly makes more sense on a {5,4} hyperbolic chess board.

Originally shared by Andrea Hawksley

Chess secretly makes more sense on a {5,4} hyperbolic chess board. http://blog.andreahawksley.com/non-euclidean-chess-part-2/
http://blog.andreahawksley.com/non-euclidean-chess-part-2

A honeycomb of bucky-balls

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A honeycomb of bucky-balls

Here are two lovely honeycombs, the bitruncated {5,3,5} and bitruncated {3,5,3}.  The first is a honeycomb of bucky-balls!  (a.k.a. truncated icosahedra, which look like soccer balls.)

Both are cell transitive, meaning every cell is exactly the same as every other.  They are also vertex transitive, so every vertex is identical as well.  Since they are vertex transitive, this also means they are examples of uniform honeycombs.

I think they are edge transitive too!  A truncated icosahedron is not edge transitive, so how can a honeycomb built of them suddenly be edge transitive?  On a truncated icosahedron, some edges join two hexagons, and some join 1 hexagon and 1 pentagon, so you can't move one edge type to another as a symmetry.  But on the honeycomb, all edges are connected to 2 hexagons and 1 pentagon.

These two honeycombs are so close to being regular, but do fail in one respect.  They are not face transitive because each has two kinds of faces.  Still, …

Cool thought Hamish Todd.

Cool thought Hamish Todd.  I hadn't considered that before.  At first blush it seems like it should be possible to have an analogous chiral geodesic saddle.  

This stackoverflow question looks to have some good information and links about the various possibilities:

http://stackoverflow.com/questions/3031875/math-for-a-geodesic-sphere

Sunrise in Galveston, TX

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Sunrise in Galveston, TX

What a beautiful weekend!  I'm not a morning person, so it is pretty rare for me to see a sunrise, but I got up on Saturday and snapped these pictures with my phone.  The calmness sure made me want to be a morning person!  I like the gradient across the sky in the panoramas.

If you find yourself in Galveston and are looking for a cute place to stay, Sarah and I have been working on a little raised cottage there this past year and would love to host you.

https://www.airbnb.com/rooms/1364851