In a paper published in 1878/79 Klein produced his famous 14-sided polygon representing the Klein quartic, his Riemann surface of genus 3 which has PSL(2,7) as its automorphism group. The construction and method of side pairings are fairly complicated. By considering the Farey map modulo 7 we show how to obtain a fundamental polygon for Klein’s surface using arithmetic. Now the side pairings are immediate and essentially the same as in Klein’s paper. We also extend his work from 7 to 11 as Klein also did in a follow-up paper of 1879.
Ivrissimtzis, I., Singerman, D., & Strudwick, J. (2021). From Farey fractions to the Klein quartic and beyond. Ars mathematica contemporanea (Spletna izd.), 20(1), 37-50. https://doi.org/10.26493/1855-3974.2046.cb6