Geometry of Farey-Ford polygons

TL;DR

This paper investigates the geometric properties and asymptotic behavior of Euclidean and hyperbolic Farey-Ford polygons, which are constructed from Farey sequences and Ford circles, revealing new insights into their structure.

Contribution

It introduces two new sequences of polygons linked to Farey sequences and Ford circles, analyzing their geometric properties and asymptotic behavior in Euclidean and hyperbolic contexts.

Findings

Analysis of areas, side lengths, and slopes of the polygons

Asymptotic behavior of geometric properties studied

Insights into the structure of Farey-Ford polygons

Abstract

The Farey sequence is a natural exhaustion of the set of rational numbers between 0 and 1 by finite lists. Ford Circles are a natural family of mutually tangent circles associated to Farey fractions: they are an important object of study in the geometry of numbers and hyperbolic geometry. We define two sequences of polygons associated to these objects, the Euclidean and hyperbolic Farey-Ford polygons. We study the asymptotic behavior of these polygons by exploring various geometric properties such as (but not limited to) areas, length and slopes of sides, and angles between sides.