  In his 1878/79 paper "Ueber die transformation siebenter ordnung der
elliptischen functionen", 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 this idea from 7 to 11 as Klein attempted to do in his follow up
paper "Ueber die transformation elfter ordnung der elliptischen functionen", in
1879.
