Hyperbolic Honeycomb Photospheres
Originally shared by Roice Nelson
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
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
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
Roice Nelson Fantastic album! Amazing spherical effect. This enables the detailed views of each hyperbolic honeycomb, as in the virtual (hyperbolic) reality.
ReplyDelete