Rectified {4,4,4}
Rectified {4,4,4}
The regular {4,4,4} has square tiling cells. The vertex figure is a square tiling too.
https://en.wikipedia.org/wiki/Order-4_square_tiling_honeycomb
Therefore, the rectified {4,4,4} contains rectified square tiling cells and square tiling cells. We saw earlier that a rectified square tiling is just another square tiling, so the rectified {4,4,4} has square tiling cells and square tiling cells. Wait a minute, that's just one kind of cell...and hmmmm...that cell is regular. This doesn't sound like a uniform honeycomb.
And indeed it isn't. The rectified {4,4,4} is the regular {4,4,3}! This came as a surprise to me when I saw the rendering. It's the only honeycomb to do this after rectification, so I thought it would be a good one to end on.
While there are 15 regular honeycombs in hyperbolic space, rectification produces just 14 new honeycombs.
I hope you enjoyed having these pictures trickle out one-a-day. I'll likely jump back into less frequent batch posting again as I experiment with truncations and other variants.
The regular {4,4,4} has square tiling cells. The vertex figure is a square tiling too.
https://en.wikipedia.org/wiki/Order-4_square_tiling_honeycomb
Therefore, the rectified {4,4,4} contains rectified square tiling cells and square tiling cells. We saw earlier that a rectified square tiling is just another square tiling, so the rectified {4,4,4} has square tiling cells and square tiling cells. Wait a minute, that's just one kind of cell...and hmmmm...that cell is regular. This doesn't sound like a uniform honeycomb.
And indeed it isn't. The rectified {4,4,4} is the regular {4,4,3}! This came as a surprise to me when I saw the rendering. It's the only honeycomb to do this after rectification, so I thought it would be a good one to end on.
While there are 15 regular honeycombs in hyperbolic space, rectification produces just 14 new honeycombs.
I hope you enjoyed having these pictures trickle out one-a-day. I'll likely jump back into less frequent batch posting again as I experiment with truncations and other variants.
The laws of symmetry say that the rectified p,p,4 is the same as the p,4,3. Maybe i could use my gadget to write all this stuff up.
ReplyDeleteYou will find others. The truncated 3,6,3 and the bitruncated 6,3,6 are the same as 6,3,3. The o3o6x3x6z is identical to the 6,3,4.
Accidental stereo. Neat!
ReplyDeleteThat's correct Adam Majewski. A C# program generates a POV-Ray formatted export file for rendering.
ReplyDeleteThe C# code is not available yet, but I would like to make it public and will certainly post on Google+ when I do that.
The algorithm is roughly:
(1) Calculate the faces of the fundamental tetrahedron.
(2) Calculate the endpoints of initial edge(s).
(3) Using the faces of the fundamental tetrahedron as mirrors, recursively reflect around initial edges(s).
All this can be done with just edge endpoints, and thickness along the edge for rendering can be calculated at the end.
The code works in models of hyperbolic space where edges are arcs, faces are spheres, and reflections are sphere inversions. (Ball and Upper Half Space models).
I tune things to max out my current hardware. I shoot for about 750k to 1M edges, and the POV-Ray files and can be as large as a few gigabytes (bloated text files).
What did you mean by TIA?