Crude Dissection of Elusive Order

archival inkjet print

2015

The Klein Quartic is a genus 3 surface whose orientation-preserving automorphism group is isomorphic to PSL(2,7), the projective special linear group of 2x2 matrices over the field of 7 elements. This is the second smallest non-Abelian simple group, and appears in many interesting contexts. In this image, the Klein Quartic is given a hyperbolic metric and projected into the Poincaré disk as a hyperbolic 14-gon with edges identified to give a surface of genus 3. An isohedral tiling of the surface is shown such that each of the 168 tiles can be put into one-to-one correspondence with the elements of PSL(2,7), with an arbitrary choice of an "identity" tile. The tiles are colored according to the conjugacy class of the corresponding element.