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From Farey fractions to the Klein quartic and beyond

Ivrissimtzis, Ioannis; Singerman, David; Strudwick, James

From Farey fractions to the Klein quartic and beyond Thumbnail


David Singerman

James Strudwick


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.

Journal Article Type Article
Acceptance Date Sep 19, 2020
Online Publication Date Jul 14, 2021
Publication Date 2021
Deposit Date Jul 7, 2021
Publicly Available Date Jul 8, 2021
Journal Ars Mathematica Contemporanea
Print ISSN 1855-3966
Publisher University of Primorska
Peer Reviewed Peer Reviewed
Volume 20
Issue 1
Pages 37-50
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