A more accurate rendering of the {6,3,3}.
A more accurate rendering of the {6,3,3}. This shows the geodesic edges more properly, as arcs orthogonal to the "plane at infinity" of hyperbolic space (vs. straight lines).
Hyperbolic Hopf Fibrations
Hyperbolic Hopf Fibrations The Hopf Fibration of S^3 is amazing and beautiful. Rather than describe it here, I'll point you to a lovely online reference with pictures and videos by Niles Johnson. nilesjohnson.net/hopf.html To understand it better (and fibrations in general), I recommend this talk by Niles too. www.youtube.com/watch?v=QXDQsmL-8Us It turns out there is an analogue of the Hopf fibration for H^3. In fact, there is not just one "fiberwise homogenous" fibration in the hyperbolic case. There is a 2-parameter family of them, plus one additional fibration that does not fit the family. As with S^3, fibers in the H^3 cases are geodesics. They are ultraparallel in fibrations from the family, and parallel in the exceptional fibration. I found the following dissertation by Haggai Nuchi a good intro and resource to help think about all this. www.math.upenn.edu/grad/dissertations/NuchiThesis.pdf I'm posting three early pictures of H^3 fibrations below, labeled
That's awesome.. what software are you using there?
ReplyDeleteThanks! The software to generate these is custom. It outputs STL (stereolithography) files, and various programs can consume that format. In this case, I loaded the output into a program called MeshLab and took a screenshot.
ReplyDelete(The reason my software outputs STL is that this coding has been for 3D printing. See https://www.shapeways.com/shops/roice3)
Wow Cool shapes!! and yes, I'm familiar with the STL format. I'm very interested in the custom software.
ReplyDeleteThanks Lee Brenton . The software has no UI and as is, wouldn't be nearly as friendly as I would like. I would like to share it with those having interest in hyperbolic geometry though, so I could make a goal to clean it up. (No promises on a timeframe though.)
ReplyDeleteIt's in C#. If I got it sharable, would you be comfortable compiling source in that language?
Wow cool, yes hyperbolic geometry fascinates me. My background is with 3D and visual digital art so yes i would love to have a go! Programming is not my forte but tinkering and compiling is fine :)
ReplyDeleteRoice, is this a "Poincare ball" model? It seems to scale the right way, but it looks like the limit surface is a sphere that we are outside of, rather than inside of. Do I have a misunderstanding? Or is the Poincare ball invertible?
ReplyDeleteThis one is the upper half space model, and the limit surface is the z=0 plane. A lot of material is culled out though, so it's not very recognizable. In addition to culling out small hexagons near the limit surface, every edge not living within some radius was culled out as well (so there may be an artificial sphere perceivable from that).
ReplyDeletePatterns in the UHS model can look a lot like the ball model in any case. I sometimes think of UHS as the ball model where the radius has been increased to infinity, or alternatively what you'd see if you zoomed in really close on the boundary of the ball.
(This was experimenting for a possible shapeways print, but I haven't come up with a UHS model I've tried to print yet.)
Roice, I saw this image on the AMS website and featured it in a blog post, crediting you and linking back to your blog. This is utterly wonderful. Thank you for creating it! To see what i wrote, go to:
ReplyDeletehttp://jan-gephardt.blogspot.com/2013/11/and-i-used-to-think-math-was-boring.html
Thank you Jan Gephardt for sharing your post with me. I enjoyed reading it, and I'm really glad you like the image!
ReplyDeleteMy pleasure! It's a really stunning image, and i've gotten a lot of good feedback on the post. Thanks for creating it!
ReplyDeleteHi Roice. Great graphics. I helped get a {6,3,3} article on wikipedia last August, with a single perspective image by another guy, Claudio Rocchini. He's been busy to help with more, but I'd be glad if you have some images you'd be willing to share on Wikipedia!
ReplyDeletehttps://en.wikipedia.org/wiki/Hexagonal_tiling_honeycomb
And enumeration pages, based on Coxeter diagrams.
Complete compact list:
https://en.wikipedia.org/wiki/Uniform_honeycombs_in_hyperbolic_space
Incomplete paracompact lists:
https://en.wikipedia.org/wiki/Paracompact_uniform_honeycomb
Hey Tom, thanks for writing. I ran across your pages recently and was glad to see these objects getting attention on wikipedia :)
ReplyDeleteI'm happy to share images like this (put them in the public domain or whatever is required for Wikipedia). I could try to make some images on demand as well, time permitting.
What do you consider the lowest hanging fruit of images you'd like to see?
Also, do you think images of the physical models we've been making would be appropriate for the pages you are nurturing? For example, images from here:
http://www.shapeways.com/shops/roice3
(Some of these images were done by Henry Segerman, so would require his permission.)
Your request provides further motivation to get the code into a sharable state like Lee requested above.
Hi Roice, thanks for your reply!
ReplyDeleteI love the 3D printed models (of Henry's, which I saw first) as possible photo uploads, but more interested in the renderd images with further limits towards the projection circumference.
All the regular hyperbolic honeycombs are a great start. Here's my summary I gave to Rocchini last summer, so "good' means we have something, but really if you wanted to try all of them in a consistent way, that's even better. Currently we have a mixed of Klein and Poincare projections, perspective views from outside, or interior, so all that can confuse things.
Compact:
Good: {5,3,5}
Needed: {3,5,3}, {4,3,5}, {5,3,4}
Finite Vertex figure paracompact:
Good: {6,3,3}, {6,3,4}
Needed: {4,4,3}, {6,3,5}
Ideal vertex paracompact:
{3,3,6}, {3,4,4}, {3,6,3}, {4,3,6}, {4,4,4}, {5,3,6}, {6,3,6}.
Anyway, no hurry, but a good challenge I hope when you're ready!
If you want to try the {6,3,3} like above again, with a bit deeper recursion towards the boundary, I'd love to start with that one above.
Another fun question is the {6,3,3} has 3 subsymmetries generated from related Coxeter groups and ring permutations in the Coxeter diagrams explained on the article. So you could render them potentially by doubled-sided fill-colorings of the hexagons, although I can't tell you exactly what the colorings would look like, unless I thought about it more!
Tom
p.s. Incidentally Wikipedia has licensing options which generally allow unlimited reuse with attribution, although in practice, even when I've uploaded images as "public domain", various publishers for books who have asked to reuse images WILL ask for explicit permission, so if you have a stable email, or contact process, I'm sure you'll be called for permission for whatever you share.
Excellent, thanks for the direction. The regular hyperbolic honeycombs are the ones I'm able to generate at the moment, so your list gives us the best chance for successfully producing something. Currently, I can do Poincare Ball or Upper Half Space (what this pic is), but it would not be much effort to support Klein or In-Space views.
ReplyDeleteProbably the work here is finding a nice way to generate quality renderings (say by supporting POV-Ray exports), rather than exporting large STL meshes and taking screenshots in programs that can read those. Since this was 3D printing focused, renderings are a bit cumbersome at the moment.
Aside: Of the "ideal vertex paracompact", the {3,6,3}, {4,4,4}, and {6,3,6} were the most exotic and required some special, separate programming routines. They might deserve their own category! (to distinguish from those having cells with a finite number of facets)
Anyway, let me play around with getting a rendering setup, and let's keep in touch.
Cheers,
Roice
Sounds great! Consistent views from the Poincare ball model is useful to see local topology in perspective if projection centered on cells. Otherwise the limited models being printed are good too, with limited iterations to the ideal boundaries.
ReplyDelete@Roice, I would be interested in seeing your C# source. My goal would be to put a nice UI on it using Mathematica (my tool of choice). I too use STL to produce hyper dimensional geometry projections for additive mfg, laser etched optical crystal, and virtual world objects. At work I have access to an immersive CAVE (think a room with video walls in stereo 3D). These objects would be fabulous to get inside and fly around in. jgmoxess@theoryofeverything.org
ReplyDeleteOk, thanks for your interest J Gregory Moxness. We have progressed nicely with wiki images (which I will post here soon), but I haven't yet spent time on getting the code shareable. I may just end up needing to share it in a more raw state. I will keep you posted of any progress.
ReplyDeleteI would love to see these in a CAVE too! I got to experience one of those some years ago in Houston, and it was awesome!
Raw code is fine - I've been doing IT / code slinging 40+ years, so source to me is better than documentation (butI understand wanting it to look good ;-).
ReplyDeleteHere is a public release article on where I work:
http://www.raytheon.com/newsroom/feature/rtn13_virtual/
Lee Brenton and J Gregory Moxness, finally opensourced my honeycomb code, only 2 years later. Sorry for the huge delay, but at least it has evolved since then...
ReplyDeletehttps://github.com/roice3/Honeycombs
Great news, thanks! I'll check it out asap
ReplyDelete