A honeycomb of bucky-balls
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, they are quite special and unique among the full class of uniform honeycombs.
What other interesting properties do you notice?
Relevant links
Bitruncation
https://en.wikipedia.org/wiki/Bitruncation
Bucky-balls
https://en.wikipedia.org/wiki/Buckminsterfullerene
Uniform Honeycombs
https://en.wikipedia.org/wiki/Uniform_honeycombs_in_hyperbolic_space
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, they are quite special and unique among the full class of uniform honeycombs.
What other interesting properties do you notice?
Relevant links
Bitruncation
https://en.wikipedia.org/wiki/Bitruncation
Bucky-balls
https://en.wikipedia.org/wiki/Buckminsterfullerene
Uniform Honeycombs
https://en.wikipedia.org/wiki/Uniform_honeycombs_in_hyperbolic_space
I suppose it's one of eight tilings of this property. There's one for each regular figure. The second paragraphs, though correct, bring to light some glaring deficiencies in the naming. Still, this looks like o5x3x5o ought. Well done Roice.
ReplyDeleteThe first is gosset's polytope in 4d, the second is octagonny, the third is the tiling of truncated octahedra. Thr fourth is the tiling of truncated dodecahedra, this is ths fifth, the sixth is the paracompact tiling of truncated quadlats, the seventh is the tiling of truncated hexlats, the eighth is indeed regular; being 6,3,3 itself.
Well done, indeed, Roice. I think the first one is the nicest image yet.
ReplyDeleteIf ye look at the cells, no more then three or four faces are pointing at you. Most of the faces are on the other side. A trunc icosa has some 32 faces, but you can only see three or four. Just shows some propetry of hyperbolic space ordinary maths can't.
ReplyDeleteAlso, all the cells in the picture are no more than four steps from the point of view, most three or less. All the fine detal is the back half of the cell. The bit you see looking through the cell.
There's another one hiding in the back, a o3x5x3o, by name.
I too like the honeycomb of buckyballs and am tempted to put it on Visual Insight!
ReplyDeleteBy the way, Scott Vorthmann, I'm looking around for great visual images with serious mathematical content, to put on my Visual Insight blog. If you have some, or know some, please tell me.
Sure, John Baez , I have several possibilities. I'll send you one now, and more later.
ReplyDeleteWhat do you use to generate the honeycombs? I'm trying to write some software that deals with the order-4 dodecahedral honeycomb but I'm stuck somewhere between the "calculate distances and angles between random points" and "construct polygons and polyhedra" stages.
ReplyDeleteRoice Nelson Hi, and thanks for the paper. So far the functions I've written use the hyperboloid model, but I already know how to convert between models if using the Poincare Ball model is necessary.
ReplyDeleteI have 2 and a half specific questions I'd like to ask.
1: Is there an operation that moves points a specific distance along lines perpendicular to a specific plane? If so, what is it called?
2: Regarding isometries: I keep coming across them when I'm trying to find the answer to question 1 but that doesn't seem right because they are supposed to preserve the size and shape of transformed objects. In the hyperboloid model, isometries are represented as linear transformations. How is it possible for linear transformations to preserve a non-linear hyperboloid?
Jeremy List, to give a couple more resources in the direction you are going, check out this page by my friend Nan Ma. He gives details on {5,3,4} coordinates in the hyperboloid model, and also mentions the Lorentz transforms John is talking about (the operation you are looking for). He has a mathematica notebook you can download too.
ReplyDeletehttp://nan.ma/hyperbolic/
Nan references the paper Coordinates for vertices of regular honeycombs in hyperbolic space.
http://www.jstor.org/discover/10.2307/2415373
If you sign up on Jstor, you can view a couple papers for free. I used that to read this paper in the past.
Let me know if you remain stuck finding the right Lorentz transforms you need. I don't know the particular answer, but would be up for digging into the specifics of your problem.
Only other thing I thought I'd mention regarding your #2 question. The size and shape of elements get plenty warped in the models under application of isometries (even though they continue to live on the hyperboloid). It is only in hyperbolic space itself where size and shape are truly preserved.
Hope this is helpful.
Roice Nelson I have been making progress. As it happens: the isometries for pushing points along one of the axes are sufficient for plotting out {5,3,4} because the angles between edges at each vertex are all Ο and Ο/2.
ReplyDeleteIs there a hi-rez (4K) 360 version of this? I'd love to see it in VR.
ReplyDeleteKING SOBIESKI, there is now!
ReplyDeletephotos.google.com - New photo by Roice Nelson
Roice Nelson Lovely! Much appreciated. π
ReplyDeleteBut is it 4K? Or at least exceeding Full HD (1080)?
Honestly, I'm not sure what is considered 4K for spherical images. This is 4096x2048 pixels in equirectangular format. I could render at double that if you wanted 4K pixels in the vertical direction.
ReplyDeleteUh and i did the coordinates for the first 10 rings of x3o3o3o5o. The complete coordinates, if you will.
ReplyDeleteRoice Nelson Oh no, don't worry about it. What I have now is satisfactory. π Even the video of it is good.
ReplyDeleteI certainly do enjoy the VR approach to exploring these weird and wonderful geometric spaces.
BTW, how do you make these things? Are you using a 3D drawing package like Blender? Can hyperbolic spaces be drawn and rendered in something like Blender? (I'm still learning to use Blender. So much to learn! π)
Thanks for the tips π
ReplyDelete