A realization of the Klein Quartic, made from 24 heptagons. The heptagons com...
- 2015-12-02 00:24:05+0000 - Updated: 2015-12-02 00:24:05+0000
A realization of the Klein Quartic, made from 24 heptagons. The heptagons come in two different shapes: the 12 on the outside are different to the 12 on the inside.

(Fixed from a previous version where I got the connections wrong.)

Source code and mesh files here: https://github.com/timhutton/klein-quartic﻿

Made from 24 heptagons

Shared with: Public, Roice Nelson
- 2015-12-03 17:24:08+0000
Very cool! I didn't know the quartic could be naturally partitioned into two sets of 12 heptagons.  I wanted to see what the two sets look like on the universal cover, so I used MagicTile to make a picture.

https://goo.gl/photos/nu2CfxmG4XxB41vq5

The sets are identical there.  And they are chiral, both with the same handedness.

I would like to see another embedding like you've done, but with symmetry that takes "affinity groups" to each other.  For what that means, see this excellent short article by Thurston.

http://library.msri.org/books/Book35/files/thurston.pdf
- 2015-12-03 21:46:09+0000
That image you made is useful! How did you work out how to color it? I'm hoping to be able to show the shape folding up from the plane. I haven't got very far yet: https://goo.gl/photos/wwuPdD4JRQBbU2r1A

As guidance for making the mesh I've been using Fig. 10b in http://www.cs.berkeley.edu/~sequin/PAPERS/Bridges06_PatternsOnTetrus.pdf

I haven't understood what you mean by the affine groups in that Thurston paper but I will think about it. Do you know what was on the Plate 2 that is referenced?
- 2015-12-04 17:20:33+0000 - Updated: 2015-12-04 17:26:18+0000
I used your video to work out the coloring!  I could see that three heptagons were meeting at each of the tetroid vertices, so I started with one of those triples, and just followed the edges to see how they connected to the other triples.  MagicTile lets you configure the 24 colors individually, so I set the colors one-by-one, and it colored the rest of the plane from there.

Regarding the plate, it's a bummer the online version of that book doesn't include the images.  Fortuitously, it is cover image of Notices of the AMS this month, because they have a piece remembering Bill Thurston!

http://www.ams.org/notices/201511/rnoti-CVR1.pdf

The three affinity groups are sets of 8 tiles.  The white tiles at the center are one group.  The red tiles (and the white tile with a person they surround) are a second set, and similarly for the green tiles.  Here's the page about the image in the Notices.

http://www.ams.org/notices/201511/rnoti-p1388.pdf
- 2015-12-04 18:05:46+0000
Brilliant Roice, thanks! I'm thinking now that the gray area in my new image is wrong.

For the affinity groups idea the embedding would have to be based on a triangle? Maybe if we flatten the tetrahedron?
- 2015-12-04 21:04:19+0000
Ah, I didn't noticed that before, but I think you are right.  Even though I count 24 heptagons there, I can see some that should be identified, so are shown twice.  That means some other heptagons are missing too.

MagicTile allows showing only fundamental tiles, so here is a vertex centered picture of a working fundamental region (other choices are possible):

https://goo.gl/photos/p6PjBaE34UWTsWmz8

I think you are right on the affinity groups too.  It does seem like the embedding would be a flattened tetrahedron with triangular symmetry, perhaps looking like the pretzel halfway through Egan's animated image:

http://www.math.ucr.edu/home/baez/mathematical/KleinDualInsideOut.gif
- 2015-12-04 21:48:09+0000
I've found a symmetric fundamental region: https://goo.gl/photos/qC1Wn1aa8dwUYdaX9
and I've worked out the matchings between it and the 3D version: https://goo.gl/photos/HnXHqQWjkSFwgQ6s8

So now it would be nice to do two things: animate the folding, and make a net so we can assemble it out of card.
- 2015-12-04 21:51:49+0000
Nice!
- 2015-12-04 23:17:32+0000
I made a version of your inner/outer coloring: https://goo.gl/photos/84r1tFKpmnwWTLYE8

And a version colored by the 4 corners of the tetrahedron: https://goo.gl/photos/ovkf8SNnPpCAhieb9

I'm just looking now for a nice 3-colored version. There are lots that are rotationally symmetric but I'm expecting one where a heptagon is surrounded by others of its own color, as in the 3 affinity groups in the paper.
- 2015-12-05 00:14:13+0000
Here's a coloring of the affinity groups: https://goo.gl/photos/NxAqHy2b68JLGZd87

There are 3 (outer) heptagons of different colors, each surrounded by their own color. I'm not sure a different embedding would help to show this - the shape already has a rotational symmetry around the vertical axis.
- 2015-12-07 17:45:07+0000
These are all awesome colorings!  Now I want to see each extended to the full plane :)

Yeah, maybe a different embedding (placing the 3 central heptagons of affinity groups symmetrically on the surface) wouldn't really help much with respect to seeing the groups, although it would still be nice to see.

On this topic, it is interesting that when you rotate the surface about one of the central heptagons, the other two both rotate in place as well, but by different amounts.  This would be a neat animation to see on the embedding, because I can't picture it.

And rotating the surface about a vertex or edge is different.  Instead of two "opposites", these elements have one and three opposites, respectively.  Meaning for example, that if you rotate the surface about an edge, it leaves three other edges in place.  Weird, and I'd love to see that animated on an embedding too.  Some images that demonstrate this in a roundabout way are at: www.gravitation3d.com/magictile/checkerboards/
- 2015-12-08 22:26:37+0000
Here are the 2, 3 and 4 colorings copied into MagicTile:
https://goo.gl/photos/pqxi2JJsGHFGeaBU7
https://goo.gl/photos/L1WVtJLRqi4vc9V67
https://goo.gl/photos/Fu8fi2dzwZ48s9X19
I can share the xml files if you want.

MagicTile is really great for exploring these. I still can't really wrap my head around hyperbolic tiling though! Like, in the 4-colored one, where each color is a symmetric fractal branching tree, and yet somehow they tessellate the plane perfectly happily.

I haven't understood the other things you said but I will think about it.

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