# A realization of the Klein Quartic, mad

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

(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

+1'd by: Whitt Whitton, Matthew Plymale, David Demma, Torolf Sauermann, Henry Segerman, Luis Guzman, Roice Nelson

Reshared by: Whitt Whitton, William Rutiser

*This post was originally on Google+*

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

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?

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 AMSthis month, because they have a piece remembering Bill Thurston!http://www.ams.org/notices/201511/rnoti-CVR1.pdf

The three

affinity groupsare 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

For the affinity groups idea the embedding would have to be based on a triangle? Maybe if we flatten the tetrahedron?

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

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.

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.

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.

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/

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.