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Published online by Cambridge University Press: 25 June 2025
Starting from the hyperbolic definition of Klein's surface we prove platonicity, derive the two classical equations W7 = Z(Z - 1)2 between meromorphic functions and x 3 y+ y3z + z3x = 0 between holomorphic forms, describe a pair of pants decomposition in terms of which the automorphisms can be seen, find a basis for the holomorphic forms for which all periods (hence the Jacobian) can be computed, and from which the lattice of rhombic tori can be determined which are covered by Klein's surface.
1. Introduction
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 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.
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