On Klein's Riemann Surface

In autumn 1993, in front of the MSRI in Berkeley, a marble sculpture by Helaman Ferguson called The Eightfold Way was revealed. This sculpture shows a compact Riemann surface of genus 3 with tetrahedral symmetry and with a tessellation by 24 distorted heptagons. The base of the sculpture is a disc which is tessellated by hyperbolic 120◦-heptagons thus suggesting that one should imagine that the surface is “really” tessellated by these regular hyperbolic polygons. In the celebration speech Bill Thurston explained how to see the surface as a hyperbolic analogue of the Platonic solids: Its symmetry group is so large that any symmetry of each of the 24 regular heptagons extends to a symmetry of the whole surface — a fact that can be checked “by hand” in front of the model: Extend any symmetry to the neighboring heptagons, continue along arbitrary paths and find that the continuation is independent of the chosen path. The hyperbolic description was already given by Felix Klein after whom the surface is named. The large number of symmetries — we just mentioned a group of order 24 · 7 = 168 — later turned out to be maximal: Hurwitz showed that a compact Riemann surface of genus g ≥ 2 has at most 84(g − 1) automorphisms (and the same number of antiautomorphisms). The sculpture introduced Klein’s surface to many non-experts. Of course the question came up how the hyperbolic definition of the surface (as illustrated by the sculpture) could be related to the rather different algebraic descriptions. For example the equation W 7 = Z(Z − 1)