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.
http://hddb.teamikaria.com/forum/viewtopic.php?f=3&t=1971
For those we've rendered, some are very pretty and I find it impossible to pick a favorite. Here are but a few that jumped out at me along the way, but check out the wiki pages to see which ones speak to you.
- Tetrahedral-icosahedral: This is composed of tetrahedra, octahedra, and icosahedra, so every platonic solid with triangular faces! Question: Can you have a honeycomb with every platonic solid in it? Why?
- Omnitruncated order-5 dodecahedral: Look at the amazing way all the decagonal prisms fit together.
- Runcinated order-5 cubic: Honeycombs with octahedral vertex figures remind me of the Euclidean honeycomb of cubes because they have 3 perpendicular axes meeting at each vertex. This one has an especially "cubical" feel to it, but look closer! For another honeycomb with an octahedral vertex figure, check out the cyclotruncated icosahedral-dodecahedral.
- Dodecahedral-icosahedral: Beautiful!
And don't forget the wonderfully symmetric bitruncations I shared here recently. Like I said... impossible to pick! The sheer variety overloads the mind.
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.
http://hddb.teamikaria.com/forum/viewtopic.php?f=3&t=1971
For those we've rendered, some are very pretty and I find it impossible to pick a favorite. Here are but a few that jumped out at me along the way, but check out the wiki pages to see which ones speak to you.
- Tetrahedral-icosahedral: This is composed of tetrahedra, octahedra, and icosahedra, so every platonic solid with triangular faces! Question: Can you have a honeycomb with every platonic solid in it? Why?
- Omnitruncated order-5 dodecahedral: Look at the amazing way all the decagonal prisms fit together.
- Runcinated order-5 cubic: Honeycombs with octahedral vertex figures remind me of the Euclidean honeycomb of cubes because they have 3 perpendicular axes meeting at each vertex. This one has an especially "cubical" feel to it, but look closer! For another honeycomb with an octahedral vertex figure, check out the cyclotruncated icosahedral-dodecahedral.
- Dodecahedral-icosahedral: Beautiful!
And don't forget the wonderfully symmetric bitruncations I shared here recently. Like I said... impossible to pick! The sheer variety overloads the mind.
Your pictures are ever beautiful and inspiring. Thank you very much Roice Nelson Tom Ruen I sit there and look at them. Maybe i could send you annotated pictures :)
ReplyDeleteThis is quite an achievement, Roice Nelson . Congratulations on "finishing" it!
ReplyDeleteThere are other compact wythoffian groups, but they're not convex. x5o5/2o5o3z is a tiling of icosahedra, icosahedra, and great dodecahedra, for example. The groups are not common, but equate to the kepler stars.
ReplyDeleteBeautiful. Thank you!
ReplyDeleteThat's a full winter's worth of staring, Roice Nelson! Incredibly cool!
ReplyDeleteYour criteria for the 76 cases look good, maybe just add "simplectic" Wythoffian honeycombs, that is there are more (countable, but uncounted) reflective ones with fundamental domains that are not tetrahedra.
ReplyDeleteThanks Tom Ruen! I added a note above directing readers to your comment :)
ReplyDeleteGorgeous!
ReplyDeleteWythoff devised a mirror-construction emulation of Stott's expand operators. It's a construction, not an outcome. Not all uniform figures are wythoffian, and not all call down to a coxeter-dynkin graph. The biggest non-wythoffian class is the laminates.
ReplyDeleteAt teamikaria, we generally restrict "wythoff" to mean something that answers to the spreadsheets constructed around the stott-vectors: that is, a simplex-symmetry where the nodes take numeric values. There may be many thousands of uniform figures, but Wythoff's constructions don't lead to them. One has to use a new set of rules, or being ever ready when the fell runs freely. John Conway has some fancy tools he shared with me. I have a few more i shared with him. Marek Ctrnak has some good ideas too. But in the end it's just plain luck on what is found.
In the end, some of us are bashing away at some fine tools, like Melinda Green's tyler java-app. Marek descovered an interesting tiling of pentagons and triangles, three at a corner, which is essentially non-uniform (kind of like penrose-tiles).
wendy krieger If you or Marek send me a snap of that tiling, I'll add it to the Tyler art gallery. I would love to see it.
ReplyDeleteMelinda Green I put one in a post. I suppose i could upload some others for ye to fetch from my website.
ReplyDeletep.s. Roice, a nonsimplectic uniform honeycomb has nonintersecting mirrors, like the Euclidean cubic honeycomb exists as {4,3,4} Schläfli symbol, but also in a "radial subgroup", index 6, [4,3*,4] = [oo]x[oo]x[oo], which is a cubic fundamental domains and {oo}x{oo}x{oo} Schläfli symbol. The infinity (oo) represent nonintersecting or parallel mirrors. So this construction still a "cubic honeycomb", but can be seen as different uniform colorings on the cells created by this subgroup.
ReplyDeleteFor the compact hyperbolic groups, the only example I know is [(4,3,4,3*)], removing the mirrors around the second order-3 branch, and leaves a triagonal trapezoidal fundamental domain, with order-3 dihedral angles in the zig-zagging mirrors along the equator, and order-2 dihedral angles around the two sets of polar axis mirrors. So for this group I draw the Coxeter diagram as a hexagon, with 3 diagonal dotted lines for the ultraparallel/divergent mirrors. The legal permutations for uniform honeycombs have 3,4,5,6 rings, while direct subgroups of [(4,3,4,3)] only generate 3 or 6 ring cases. It would be interesting if you could try to construct these as examples (4 or 5 active mirrors or rings) sometime.
https://en.wikipedia.org/wiki/Uniform_honeycombs_in_hyperbolic_space#Compact_uniform_honeycomb_families
Excellent work absolutely beautiful images. The symmetry is fascinating.
ReplyDeleteWonderful that you created and uploaded these images to Wikipedia. Do you also intend to create a page that discusses these objects?
ReplyDeleteTom Ruen seems to have this in hand. Most of them are wythoffian in the Polygloss sense, but there is one that is not.
ReplyDeleteMelinda Green, the page discussing them is here: http://en.wikipedia.org/wiki/Uniform_honeycombs_in_hyperbolic_space
ReplyDeleteThe link was was a bit mashed with the link to my user page in the post, so I edited it slightly just now, to hopefully avoid the first link getting lost.
Like Wendy said, Tom Ruen has been doing a ton of work on all the honeycomb pages. He's also very good at exposing me to the needed concepts incrementally, so that I can make steady progress on images without getting too overwhelmed!
Roice Nelson
ReplyDeleteOh ho! I didn't realize you were editing that page. The addition of your images is fantastic! What a wonderful contribution you guys are making.
BTW, is there a logic to their background colors?
Tom and Roice ask me from time to time to look over what they are doing, and for assorted advice. It is their achievement though.
ReplyDeleteMelinda Green, it's evolving that direction. I was assigning colors manually at the beginning, and generating images one by one. This was getting tedious, so I starting picking one base hue for each family and batching things. The slight differences in color within a family is calculated based on which nodes are ringed in the Coxeter-Dynkin diagram (John Baez suggested this to me) . Each of these honeycombs has 4 nodes, any of which can be active or not, so I used this as a binary encoding of 16 states. Overall the coloring has been inconsistent though, and there could be a nicer scheme to calculate the hues completely.
ReplyDeletewendy krieger, I feel it's all of us! The pd{3,5,3} is a clear case that wouldn't exist there without you, but I love that this has been a collaborative thing with multiple people chipping in, all crucially. My first wiki contribution experience has been a really positive one.
Generally, because i do 'original research' a lot of my wiki stuff is on the talk pages. It's ok for other people to quote me though. Tom Ruen would know since he has fancy badges from the wiki thing.
ReplyDelete