Hyperbolic Honeycomb Photospheres

Hyperbolic Honeycomb Photospheres

Encouraged by Tom Ruen again, I fired up the honeycomb rendering code this week. The result is approximately 70 new honeycomb images for Wikipedia and the album below with spherical images of some of my favorites. Have a look around... literally!

This batch focused on uniform paracompact honeycombs. Uniform means they are vertex-transitive and paracompact means some cells have ideal vertices or centers. See if you can pick out some of the cell types.

There are many more paracompact honeycombs to render, but this was a nice chunk, covering all those with Coxeter diagrams that are linear graphs.

Relevant Links

Uniform Paracompact Honeycombs
The new Wikipedia images can be found here:
https://en.wikipedia.org/wiki/Paracompact_uniform_honeycombs

Visualizing Hyperbolic Honeycombs
This paper by Henry Segerman and myself gives an introduction to exotic honeycombs like these. We focus on regular honeycombs, but much of the content is relevant.
https://arxiv.org/abs/1511.02851

Honeycomb Rendering Code
https://github.com/roice3/Honeycombs

POV-Ray
http://www.povray.org

Gory Implementation Details

I made some changes that improved this batch of images from the last and thought I'd share.

First, Tom noted that the recursion cutoff was jarringly visible for ideal cells. Increasing the number of edges, even by an order-of-magnitude, would not avoid this. However, fading out the edge color near the limit of recursion turned out to be quite effective!

I also made a level-of-detail change that allowed rendering two meeellion edges in each of the images, almost double than before.

Finally, I played with color a little. The darker background colors feel more effective because of increased contrast with the edges.
https://goo.gl/photos/Stg9R7HxQgMwfWBG6

Comments

  1. John Baez and Scott Vorthmann, I think you will like these photospheres. And thanks again Scott for the ExifTool information!

    ReplyDelete
  2. This is beautiful. The hi-res version of the photospheres take a while to load, so I advise viewers with a slow connection to be patient.

    ReplyDelete
  3. Thanks Arnaud Chéritat. Yeah, most of the photosphere files are in the 10-15 MB range in size.

    ReplyDelete
  4. Roice Nelson I had forgotten that discussion! These are really compelling and accessible as photospheres! Well done!

    ReplyDelete
  5. These are great!   I'll do some articles on Visual Insight linking to these.  But as usual, they make me want even more - I want to be able not merely to look around but also move forwards and travel through this honeycomb.

    ReplyDelete
  6. I'll take whatever I can get!  :-)

    The omnitruncated one is certainly optimal if one wants a 'maximally fancy' honeycomb.  I find the struts a bit fat in your rendering: there are so many of them in this particular honeycomb, it's hard to see what's going on: ones field of view is largely filled by shade of light gray.  Perhaps making them thinner would be good... but being able to fly though would also help.

    ReplyDelete
  7. I find a sparser look, like this, visually preferable:

    https://en.wikipedia.org/wiki/Paracompact_uniform_honeycombs#/media/File:H3_444-0110.png

    It's not so much about the honeycomb as about the amount of my visual field that's full of light gray.

    Have you tried thin struts that are darker than the background?  I think that might look nice.

    ReplyDelete
  8. John Baez, here's a spherical render with thinner struts that are darker than the background. Do you like it better?

    https://goo.gl/photos/iBJJBfbdsJS6Bs9P8

    I really like the bitruncated honeycombs in general (those with the 2nd and 3rd simplex mirrors active like the one you linked to), so I'll happily try my first flythrough animation with that one.

    One thought I had was to fade in and out faces of the different cell types during the flythrough. That makes omnitruncated honeycombs a nice choice, but that is going to take some additional setup work anyway and I wasn't going to include that just yet.
    photos.google.com - New photo by Roice Nelson

    ReplyDelete
  9. Roice Nelson - yes, l like that one better!  I can see more fine detail, and it's the potentially infinite amount of beautifully symmetrical fine detail that makes this kind of scene so appealing.

    It's also nice if the color scheme matches some sort of imaginable real-world structure.  Your black on white makes me think of metal scaffolding in a huge domed cathedral.  This by Jos Leys:

    http://blogs.ams.org/visualinsight/2014/02/01/535-honeycomb/

    reminds me of a thorny forest in an evening sky - I love how the background goes from blue to purple as you move across.  You could try that color scheme sometime!

    ReplyDelete
  10. I found a new land to explore ... I'll follow you Roice Nelson

    ReplyDelete

Post a Comment

Popular posts from this blog

Hyperbolic Hopf Fibrations