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

By Sophie Morier-Genoud, Valentin Ovsienko. Jahresbericht Der Deutschen Mathematiker-vereinigung, 2018

55 citations recorded by the source corpus.

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 PSL(2,Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{PSL}(2,\mathbb {Z})$\end{document}. 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 PSL(2,Z)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\mathrm{PSL}(2,\mathbb {Z})$\end{document}.

Subfields: Computational Theory and Mathematics, Discrete Mathematics and Combinatorics, Geometry and Topology

View Paper: https://www.semanticscholar.org/paper/d5f22d528a35fe44b26421a0c66f32d6b09157bd

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In Computational Theory and Mathematics

  1. Sophie Morier-Genoud
  2. Valentin Ovsienko

In Discrete Mathematics and Combinatorics

  1. Valentin Ovsienko
  2. Sophie Morier-Genoud

In Geometry and Topology

  1. Sophie Morier-Genoud
  2. Valentin Ovsienko
  3. Serge Tabachnikov

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