Even-integer continued fractions and the Farey tree

TL;DR

This paper explores a geometric representation of even-integer continued fractions using a hyperbolic tessellation related to the Farey tree, providing new insights and proofs for fundamental theorems in the field.

Contribution

It introduces a novel geometric approach to even-integer continued fractions via the Farey tree, leading to new proofs and theorems in the area.

Findings

New geometric representation of even-integer continued fractions

Proofs of fundamental theorems using hyperbolic tessellation

Introduction of new theorems with familiar counterparts

Abstract

Singerman introduced to the theory of maps on surfaces an object that is a universal cover for any map. This object is a tessellation of the hyperbolic plane together with a certain subset of the ideal boundary. The 1-skeleton of this tessellation comprises the edges of an infinite tree whose vertices belong to the ideal boundary. Here we show how this tree can be used to give a beautiful geometric representation of even-integer continued fractions. We use this representation to prove some of the fundamental theorems on even-integer continued fractions that are already known, and we also prove some new theorems with this technique, which have familiar counterparts in the theory of regular continued fractions.