Folding a Klein Quartic
Jos Leys is making animations of polygons turning into hyperbolic surfaces:https://plus.google.com/108557640546882398221/posts/hdsvak2ZkYt
He agrees it would be cool to do the Klein Quartic, but it is difficult...
I have a lot of old heptagon structures in my home, so I took one apart and converted it to a 14-gon, but instead of drawn on the Poincare disk, it is actually made of regular heptagons.
The colors are the same as in the drawing on the right, which was by Tony Smith, and used on John Baez’s Klein Quartic page.
On each vertex where 3 heptagons meet, the surface is not flat, the total angle is 360 degrees*(15/14). But if you fold the paper through the vertex, the curvature gets tidy, you can make a kind of flower. You could continue folding like that through the entire hyperbolic plane, I tried (perhaps succeeded, but I can’t prove it) to make an isometric C1 embedding of the hyperbolic plane:http://westy31.home.xs4all.nl/Geometry/Geometry.html#Embed
OK, so next we can try to connect the 14 sides of the ‘fundamental polygon’. I will try it tomorrow, but things might get horribly crumpled, to I thought I post this first, while it still looks relatively nice.