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

TL;DR

This paper reinterprets classical results on continued fractions through combinatorial models involving polygon dissections and Farey tessellations, extending to elements of the modular group.

Contribution

It introduces a combinatorial framework for continued fractions and extends known theorems to the modular group using polygon dissections and Farey tessellations.

Findings

Reformulation of Conway and Coxeter's theorem in combinatorial terms

Extension of Series' theorem to the modular group

Development of a canonical presentation for PSL(2,Z) elements

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