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From Farey fractions to the Klein quartic and beyond

Ivrissimtzis, Ioannis and Singerman, David and Strudwick, James (2021) 'From Farey fractions to the Klein quartic and beyond.', Ars Mathematica Contemporanea, 20 (1). pp. 37-50.


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.

Item Type:Article
Full text:(AM) Accepted Manuscript
Available under License - Creative Commons Attribution 4.0.
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Date accepted:19 September 2020
Date deposited:08 July 2021
Date of first online publication:14 July 2021
Date first made open access:08 July 2021

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