From Farey fractions to the Klein quartic and beyond - Ivrissimtzis, Ioannis (Ars mathematica contemporanea, 2021)
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.
Keywords: Farey tessellation; Fareyjevo tlakovanje; glavne kongruenčne podgrupe; Klein quartic; Kleinov kvartik; modular group; modularna grupa; principal congruence subgroups; regular maps; regularni zemljevidi; Riemann surfaces; Riemannove ploskve