The three types of eight-fold way path on the Klein Quartic.﻿<br><br>Source c...
- 2015-12-11 21:38:30+0000 - Updated: 2015-12-11 21:56:07+0000
The three types of eight-fold way path on the Klein Quartic.﻿

Source code and mesh files here: https://github.com/timhutton/klein-quartic﻿
Shared with: Public, Kris Ove, Niles Johnson
- 2015-12-11 21:45:33+0000
It's interesting that the red and green are almost planar. Are these Petrie polygons, and should we expect them to be planar?
- 2015-12-12 02:50:12+0000
Very nice!  The transparency is especially effective.

Yes, the eight-fold paths are along Petrie polygons.  Abstractly (say, in a higher dimensional realization), I think of them as behaving exactly like they do on the dodecahedron (which has ten-fold paths).  That is, they are skew polygons with half the vertices in one plane and half in another.

And the red, green, and blue polygons are all perfectly identical in the abstract, the "three types" here being just an artifact of this embedding.

This actually makes me wonder if a method to get a nice embedding could be to use a relaxation algorithm on these paths, something that tries to make them as close to regular skew polygons as possible.
- 2016-01-14 11:04:54+0000
Great work! But two questions:
1. Is there any method to the coloring of the heptagons or is it only for better illustration and completly arbitrary? Why do some neighboring heptagons have the same color?

2. How can I see the seven rotational symmetries in this tiling? I only see three: I can rotate heptagon  21 to 23, 23 to 22 and 22 to 21.
In this tiling (http://math.ucr.edu/home/baez/Klein168.gif) the seven rotation symmetries are obvious, but where are they in this tiling?

thanks!
- 2016-01-14 13:01:40+0000
In this video the colors are random. In the GitHub repo there are a few different coloring available, including an 8-color one by where no neighbors share a color.

The 7-fold rotational symmetry applies around the center of every heptagon, of course, but by putting a vertex at the center (for symmetry with the above 3D embedding) it is not obvious. I hope to be able to produce a video soon that shows it. For now imagine that when heptagons drop off one side of the finite region they reappear on another side.
- 2016-01-15 18:52:24+0000
Thanks! Looking forward to your next video!

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