2382 KB29 KB200820252021Ivrissimtzis, IoannisSingerman, DavidStrudwick, Jamesštevilka:1letnik:20str. 37-50DOI:10.26493/1855-3974.2046.cb6ISSN:1855-3966COBISSID_HOST:91101187URN:URN:NBN:SI:doc-7E400FT4enUniverza na Primorskem, Fakulteta za matematiko, naravoslovje in informacijske tehnologijeArs mathematica contemporaneaFarey tessellationFareyjevo tlakovanjeglavne kongruenčne podgrupeKlein quarticKleinov kvartikmodular groupmodularna grupaprincipal congruence subgroupsregular mapsregularni zemljevidiRiemann surfacesRiemannove ploskveFrom Farey fractions to the Klein quartic and beyond|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 ?$\mathrm{PSL}(2, 7)$? 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 1879V članku objavljenem leta 1878/79 je Klein prikazal svoj slavni 14-strani poligon, ki predstavlja t.i. Kleinov kvartik, Riemannovo ploskev z rodom enakim 3 in grupo avtomorfizmov ?$\mathrm{PSL}(2, 7)$?. Njegova konstrukcija in metoda prirejanja stranic sta precej zapleteni. Upoštevaje Fareyjev zemljevid reduciran po modulu 7 pokažemo, kako dobimo fundamentalni poligon te Kleinove ploskve z uporabo aritmetike. Zdaj so prirejanja stranic lahka in v bistvu ista kot v Kleinovem članku. Prav tako razširimo njegovo raziskavo s primera 7 na primer 11, kot je to storil tudi Klein v svojem naslednjem članku iz leta 1879TEXTznanstveno časopisjejournalsSlovenian National E-content AggregatorNational and University Library of SloveniaUniverza na Primorskem, Fakulteta za naravoslovje, matematiko in informacijske tehnologije