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Jahresbericht der Deutschen Mathematiker-Vereinigung

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14-02-2019 | Survey Article

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

Authors: Sophie Morier-Genoud, Valentin Ovsienko

Published in: Jahresbericht der Deutschen Mathematiker-Vereinigung | Issue 2/2019

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Abstract

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 \(\mathrm{PSL}(2,\mathbb {Z})\).

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…similar to Cheshire cat’s grin.

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Title: Farey Boat: Continued Fractions and Triangulations, Modular Group and Polygon Dissections
Authors: Sophie Morier-Genoud
Valentin Ovsienko
Publication date: 14-02-2019
Publisher: Springer Berlin Heidelberg
Published in: Jahresbericht der Deutschen Mathematiker-Vereinigung / Issue 2/2019
Print ISSN: 0012-0456
Electronic ISSN: 1869-7135
DOI: https://doi.org/10.1365/s13291-019-00197-7

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