Farey boat I. Continued fractions and triangulations, modular group and polygon dissections

2018/11/03 by Sophie Morier-Genoud, Morier-Genoud, Sophie, Valentin Ovsienko +1 · 5 citations
Mathematics · #Advanced Combinatorial Mathematics #Advanced Mathematical Identities #Combinatorics (math.CO) #FOS: Mathematics #Mathematics and Applications #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1811.01229

openalex publication_date 2018/11/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

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 PSL(2,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 PSL(2,ℤ).

Cited by

• SL₂-tilings with translational symmetry
• On a tree of rational functions related to continued fractions
• Frieze patterns in representation theory
• Friezes and continued fractions
• Local Statistics of the M_n-Dimer Model

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