The geometry behind Galois’ final theorem

In Galois’ last letter he found the values of the primes p for which the group PSL ( 2 , p ) acts transitively on less than p + 1 points. (It always acts transitively on the p + 1 points of the projective line.) He found that these values of p are 2,3,5,7,11. The cases p = 7 , p = 11 have much geometric interest. PSL(2,7) is the automorphism group of the simplest projective plane, the Fano plane on seven points. The simplest biplane is on eleven points, and PSL(2,11) is its automorphism group. The Fano plane can be embedded in Klein’s Riemann surface of genus 3. We find an interesting surface of genus 70, in which we can embed the biplane on eleven points. This surface also contains truncated icosahedra or buckyballs and so is called the buckyball curve.