(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?
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?
