### What polyhedron is this?

What polyhedron is this?

What polyhedron is this?

*Hints:*

* The dihedral angle is 120 degrees.

* It's perfectly regular! Surprise, there are more than 5 platonic solids :D

Bonus question. What kind of rotation is pictured?

Thank you Naomi Weinberger for suggesting this animation, and thank you Sean Walker for sharing information about how you create.

I love this! Can I ask why the camera looks so close to the figure?

ReplyDeleteI tried to show more of the polyhedron, unsuccessfully. I pushed it away from the camera and decreased the camera viewing angle, but it hardly helped avoid the "wide angle" appearance. It's a characteristic of hyperbolic space that you can't see more than 3 or 4 faces, and all the others get occluded.

ReplyDeleteThe following post shows something similar with hyperbolic truncated icosahedra. Each cell has 32 faces, but just 3 or 4 point at the viewer.

plus.google.com - A honeycomb of bucky-balls Here are two lovely honeycombs, the bitruncated…

Very cool! What did you use to make it? I like the very wide angle lens effect. BTW, why are you thanking me?

ReplyDeleteis this a 3D project from 4D? Initially I thought the distortion was due to a wider angle perspective - but on closer inspection I'm realizing something is quite irregular about this 'regular' polyhedron

Hmm, a closed surface tiled by (6,3) should be a turus. You could film that from close by so you don't see the hole.

ReplyDeleteSean Walker, for the rendering I used my ray tracer of choice (Pov-Ray), but this was my first ever animated gif and that is my reason for thanking you. I went to your stream to see if I could find info about making gifs, and that's where I learned about ImageMagick :)

ReplyDeleteTo answer your question, I'd say this is a genuine 3D object, not a projection from 4D (though to confuse the answer, it probably could be argued there is a sense in which this is higher dimensional).

Gerard Westendorp, on the right track, but there is no hidden hole in this case.

ReplyDeleteHorosphere tiled by (6,3)

ReplyDeleteYep David Eppstein! It's an in-space view of this:

ReplyDeletehttps://commons.m.wikimedia.org/wiki/File:633_honeycomb_one_cell_horosphere.png

The rotation shown is a limit rotation, aka parabolic. commons.m.wikimedia.org - File:633 honeycomb one cell horosphere.png - Wikimedia Commons

Now, what about an animation with transparent faces?

ReplyDeleteWell, the Octahemioctahedron also has Euler characteristic 0. Instead of having a hole, it selfinterscts It has 4 hexagons and 8 triangles, you might be able to get rid of the triangles by reverse truncating. Also, the 120 degrees dihedral angle could be around there somehow, the rhomic dodecahedron has 120 degrees, and that is the dual of the cubocahedron.

ReplyDeleteA star polyhedron with 4 hexagons would be pretty cool, but I don't see how to make it yet.

Intrinsically speaking, is this in any way different from a plane honeycomb? It seems to me that it's not a "polyhedron" at all, but just an embedding of a flat plane in a hyperbolic space.

ReplyDeleteMatt McIrvin, yes, it is different than a plane honeycomb. While all the vertices live on a flat plane embedded in hyperbolic space (a horosphere), the edges and faces are geodesic and curve off of that plane. This is why this shape has a dihedral angle. If all elements lived on the horosphere, the dihedral angle would be 180 degrees.

ReplyDeleteSo this is subtly different than a horosphere tiled by {6,3}.

There's some nice discussion about this in the comments of one of John Baez's posts.

plus.google.com - This is the order 4 hexagonal tiling honeycomb in 3-dimensional space, drawn by…

Btw, here's another regular {6,3} polyhedron, but with a larger dihedral angle (also suggested by Naomi Weinberger). This one can't act as a cell of regular honeycomb because the dihedral angle is too big to fit 3 of these around an edge, but we can see more faces since the hexagons are smaller.

ReplyDeletehttps://lh3.googleusercontent.com/cPdDVkuwVqfuwBKnxhp-R165Z8cCwHDgVQaeSlUpZ3pOOxAb0LNmV67WGsfazP6uRMlahO4dir4OSG9yqj2IZydafRBYKwqAKMUd=s0

I think some of the "uniform paracompact" honeycombs have {6,3} cells with dihedral angles > 120 degrees, e.g.

ReplyDeleteen.wikipedia.org - Hexagonal tiling honeycomb - Wikipedia

It's a cell of {6,3,3}. It's hardly compact. Horris, perchance, but not compact.

ReplyDeleteRoice Nelson I see! So the geodesics on the horosphere are

ReplyDeletenotgeodesics of the embedding hyperbolic space (which makes sense).You could draw an "inside out" version, or basically no solid faces.

ReplyDeleteCan a similar figure be constructed from heptagons? Or with a 6 4 tiling?

ReplyDeleteHallucigeniaIV, yes!

ReplyDeleteYou can use hyperbolic tilings as "polyhedra" in hyperbolic space just as well, and you can adjust their dihedral angles by increasing or decreasing the size of the heptagons, hexagons, etc.

Roice Nelson cool! What software do you used to make those animations?

ReplyDeleteHallucigeniaIV, for these, I used the following software, though it isn't particularly easy to dig into. I have been wanting to do a cleanup pass for some time.

ReplyDeletehttps://github.com/roice3/Honeycombs

That can generate files that POV-Ray can render.

povray.org

And ffmpeg is the utility I use for collating images into video.

ffmpeg.org

I meant to mention earlier the paper

Visualizing Hyperbolic Honeycombs, because it describes drawing pictures of polyhedral "cells" that are hyperbolic tilings.https://arxiv.org/abs/1511.02851

Roice Nelson I'll try to dig into it! I'm not particularily code.versed but I've managed to trace hyperbolic tilings on Geogebra :D

ReplyDeletelt looks like an x3x8o, a truncated {3,8}. The rotation is actually along a equidistant.

ReplyDelete