(2 3 7) Triangles and Klein's Quartic



(2 3 7) Triangles and Klein's Quartic


My new favorite t-shirt arrived in the mail today, designed by Henry Segerman.

This design and many others with mathematical themes are available at Henry's website.
http://www.segerman.org/tshirts.html

I used the "create" version of the shop to slightly alter the size and color.
http://math-art-create.spreadshirt.com/us/US/Shop/

So what is this design?

In short, it is a tiling of (2 3 7) triangles.  The (2 3 7) designates a Schwarz triangle.  It means the 3 angles of each triangle are π/2, π/3, and π/7.  Schwarz triangles can tile the sphere, Euclidean plane, or hyperbolic plane depending on the choice of the 3 angles.  (2 3 7) triangles tile the hyperbolic plane, and Henry has drawn these triangles in the Poincare disk model, cutting off the model at some radius from the origin. 

My G+ banner is a similar representation of (2 3 7) triangles tiling the disk, but with a different choice of cutoff and as a 3D printed model.  I previously posted about the model here.
https://plus.google.com/u/0/+RoiceNelson/posts/jUrUZD2EXH8

You can order your very own copy too.
http://shpws.me/nftj

I especially like this t-shirt because the tiling connects to a very special object linked to loads of mathematics, called Klein's Quartic.  Klein's Quartic surface can be constructed from 336 of these triangles.  14 of the triangles form a heptagon, so you can think of the surface as made of 24 heptagons.  The heptagons fit together in a perfectly regular way in higher dimensional space, analogous to how pentagons fit together perfectly to make a dodecahedron.  It is a secret platonic solid!  I've been fascinated with it for a few years, probably having first seen it discussed by John Baez.
http://math.ucr.edu/home/baez/klein.html

Klein's Quartic is special to mathematicians, so there are lots of resources out there to learn more about it.  There is even an entire book freely available online titled The Eightfold Way.  I especially recommend the accessible first chapter by William Thurston.
http://library.msri.org/books/Book35/contents.html

I used puzzling as a way to learn about Klein's Quartic by coding up a working Rubik's Cube analogue played on the surface.  Experimenting with the puzzle is a great way to gain familiarity with Klein's Quartic (even if just moving the puzzle around and noticing the pattern of heptagonal faces).  Here's a static picture.
http://www.gravitation3d.com/magictile/pics/73.png

You can download the program at:
http://www.gravitation3d.com/magictile/

Links for further study

Here are some additional links I've enjoyed and used to learn about Klein's Quartic surface the past few years.

Patterns on the Genus-3 Klein Quartic, paper by Carlo Séquin
http://www.cs.berkeley.edu/~sequin/PAPERS/Bridges06_PatternsOnTetrus.pdf

Magnetic Klein Quartic, blog post by Edmund Harriss 
http://maxwelldemon.com/2011/10/02/magnetic-klein-quartic/

Klein's Quartic Equation by Greg Egan
http://gregegan.customer.netspace.net.au/SCIENCE/KleinQuartic/KleinQuarticEq.html

Wiki pages:
http://en.wikipedia.org/wiki/Klein_quartic
http://en.wikipedia.org/wiki/Schwarz_triangle

...and surely others I'm forgetting.  What resources do you recommend?

Comments

  1. The circle is the size of a cell of a {7,4}.  Start at the black or white point opposite the vertical line, and ye can follow a black or white cell around the perimeter.

    ReplyDelete
  2. Hey, I want a shirt like that!  But a bit lighter in color, since I spend most of my time in hot places (Riverside and Singapore).

    ReplyDelete
  3. Nice shirt.  Also it does not spell "Nerd" all over it, since it is not obvious there is math behind the pattern :)

    ReplyDelete
  4. You mean like my T-shirt with digits of pi on it?

    ReplyDelete
  5. I recall drawing a {5,3,3} on some notepad when i was coming home on the bus, and some girl was trying to implement an escher-style art (of which {5,3,3} isn't), so i had to explain her what really she needed.

    ReplyDelete
  6. The thing on your g+ page is actually a cell of {14,7}, divided into 24*14 symmetries of * 2 3 7.  It's one of the things that attracted me into hyperbolic geometry, but i found the trig to hard, so i rewrote the geometry.

    ReplyDelete
  7. wendy krieger, yep!  I used a {14,7} to help generate the boundary.

    ReplyDelete
  8. Roice Nelson wrote: "the 337 is the one we did, and Danny's picture looks like the 733."

    Sorry, you're right - I wrote the opposite of what I meant.  I have a very nice yellow picture of the {3,3,7} honeycomb from you, I believe you sent a permissions form to Beth Ayer okaying me to put it on the Visual Insight blog, and I'll be doing that after articles about the {7,3} and Danny's version of the {7,3,3}.  I just thought it would be nice to slip a "{7,3,3} meets the plane at infinity" in this series of blog articles before running your "{3,3,7} meets the plane at infinity".

    ReplyDelete
  9. John Baez, I should have mentioned that I have done some renderings of {7,3,3}, because Henry and I worked on a shapeways print for it.  But they are in the Poincare ball, not in the "boundary image" style

    I'll send you a couple examples.

    ReplyDelete
  10. John Baez, cool, ok.  That does sound like a nice progression towards the {3,3,7}.  I'll try a "{7,3,3} meets the plane at infinity" in the next couple days, and will let you know how it goes.

    In case it can be useful, one place I have seen an image of this (but on the ball, not UHS) is here:

    by Curtis McMullen (in the Kleinian groups section)
    http://www.math.harvard.edu/~ctm/gallery/index.html

    ReplyDelete
  11. That's a nice image!  If you feel overworked I can ask him to sign a form to let the AMS use it.  But one in your style (as you did with {3,3,7}) would be nice.  It'll be almost a month before I "need" it.

    ReplyDelete

Post a Comment

Popular posts from this blog

Hyperbolic Hopf Fibrations