How to Glue the Hyperbolic Plane to Form Klein's Quartic?

[Lapidus]

Can someone tell me how I have to glue together the hyperbolic plane
so that I get the surface (Klein's quartic) shown to the left. I found this picture on the net, but without a desription how to glue.
thanks!

 

[mathwonk]

this is a fun question. i don't know the answer but it is fun staring at the pictures. since the klein quartic is a surface of genus 3, one must mnake a minimum of 6 loop cuts to render it topologically a disc as in th picture. but there are 14 edges to the disc in the picture. this has something to do with th symmetries of the klein quartic which has symmetries of orders 2,3, 7, so the decomposition is done in a way that respects these symmetries. e.g. rotation of order 7 about the center gives one of the automorphisms of the quartic, but i don't quite see how to glue it back together.

 

[jim mcnamara]

Oh brother. Beyond my pay grade - so John Baez on understanding part of this problem
http://www.tony5m17h.net/JBGEKQ.html

Neat movie.

 

[mathwonk]

this is a nice link with a description of how to glue, but it puzzles me since again making more than 6 loops cuts would separate the disc, so somehow those cuts are not all loop cuts?

 

[zinq]

Since the genus of a g-holed torus is g, the Klein quartic has genus 3, and hence Euler characteristic 2 - 2g = -4.

After identifying the edges in pairs, there will be 1 2-cell, 7 edges, and say V vertices.

The alternating sum 1 - 7 + V is the Euler characteristic, so we have

1 - 7 + V = -4​

and so

V = 2.​

The reason that the number of cuts seemed too many is that there will be 2 vertices after identification, not just 1.

 

[mathwonk]

equivalently the last cut along the edge joining the two distinct vertices is not a loop cut.

 

[zinq]

In fact, all 7 "cuts" are between the two vertices. Though it is better to call them edges.