Farey Boat: Continued Fractions and Triangulations, Modular Group and Polygon Dissections

  • Sophie Morier-GenoudEmail author
  • Valentin Ovsienko
Survey Article


We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections appear when extending these theorems for elements of the modular group \(\mathrm{PSL}(2,\mathbb {Z})\). These polygon dissections are interpreted as walks in the Farey tessellation. The combinatorial model of continued fractions can be further developed to obtain a canonical presentation of elements of \(\mathrm{PSL}(2,\mathbb {Z})\).


Continued fractions Farey graph Polygon dissections Ptolemy rule Pfaffians Modular group 



We are grateful to Charles Conley, Vladimir Fock, Sergei Fomin, Alexey Klimenko, and Sergei Tabachnikov for multiple stimulating and enlightening discussions. We are grateful to the referee for a number of helpful remarks and suggestions. This paper was partially supported by the ANR project SC3A, ANR-15-CE40-0004-01.


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© Deutsche Mathematiker-Vereinigung and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu-Paris Rive GaucheSorbonne Université, Université Paris Diderot, CNRSParisFrance
  2. 2.Laboratoire de Mathématiques U.F.R. Sciences Exactes et NaturellesCentre national de la recherche scientifiqueReims cedex 2France

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