Plane geometry of $q$-rationals and Springborn Operations

TL;DR

This paper explores the geometry of $q$-rational numbers, introduces Springborn operations as quadratic Farey additions, and analyzes their geometric and algebraic properties, including deformed Farey triangulations and Markov numbers.

Contribution

It introduces Springborn operations on $q$-rationals, providing a quadratic analogue of Farey addition with geometric interpretation and applications.

Findings

Constructed deformed Farey triangulation and modular surface.

Interpreted $q$-rationals as circles similar to Ford circles.

Derived a $q$-deformed midpoint formula and a new $q$-deformation of Markov numbers.

Abstract

We study the geometry of $q$-rational numbers, introduced by Morier-Genoud and Ovsienko, for positive real $q$. In particular, we construct and analyse the deformed Farey triangulation and the deformed modular surface. We interpret every $q$-rational geometrically as a circle, similar to the famous Ford circles. Further, we define and study new operations on $q$-rationals, the Springborn operations, which can be seen as a quadratic version of the Farey addition. Geometrically, the Springborn operations correspond to taking the homothety centers of a pair of two circles. As an application, we derive a formula for the $q$-deformed midpoint of two Farey neighbors and we consider a new $q$-deformation of Markov numbers.