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Plane geometry of $q$-rationals and Springborn Operations

Perrine Jouteur, Olga Paris-Romaskevich, Alexander Thomas

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.

Plane geometry of $q$-rationals and Springborn Operations

Abstract

We study the geometry of -rational numbers, introduced by Morier-Genoud and Ovsienko, for positive real . In particular, we construct and analyse the deformed Farey triangulation and the deformed modular surface. We interpret every -rational geometrically as a circle, similar to the famous Ford circles. Further, we define and study new operations on -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.
Paper Structure (28 sections, 58 theorems, 145 equations, 12 figures)

This paper contains 28 sections, 58 theorems, 145 equations, 12 figures.

Table of Contents

  1. Introduction
  2. From $q$-integers to $q$-rationals to $q$-reals
  3. Definition(s) of $q$-rationals
  4. Extension of the deformation from $\mathop{\mathrm{\mathrm{PSL}}}\nolimits_2(\mathbb{Z})$ to $\mathop{\mathrm{\mathrm{PGL}}}\nolimits_2(\mathbb{Z})$.
  5. From $q$-rationals to $q$-reals and beyond
  6. Hyperbolic geometry and deformed Farey tesselation
  7. Classical Farey triangulation
  8. Symmetries of classical Farey triangulation
  9. $q$-Farey tesselation and $q$-modular surface
  10. Deformed Farey determinants and operations
  11. $q$-Farey determinant
  12. $q$-Farey operations
  13. Classical Springborn operations
  14. Motivation: Homothetic centers
  15. Regular pairs and Springborn operations
  16. ...and 13 more sections

Key Result

Proposition A

The fundamental domain of the $q$-deformed action of $\mathop{\mathrm{\mathrm{PGL}}}\nolimits_2(\mathbb{Z})$ on $\mathbb{H}^2$ is a hyperbolic "triangle" open towards infinity, with two vertices given by $\tfrac{i}{\sqrt{q}}$ and $\sigma=\frac{1+i\sqrt{3}}{2}$. It is a deformation of the triangle wi

Figures (12)

  • Figure 3.1: The ideal triangles of the classical Farey triangulation $\mathcal{F}$ attached to rationals with denominator at most $3$. Note that the rational points are not equidistant on the boundary.
  • Figure 3.2: The subdivided Farey triangulation, consisting in all geodesics between rationals of Farey determinant 1 (black) or 2 (red), up to the denominator at most $3$. This Figure is an refinement of the Figure \ref{['fig:farey_tesselation']}. The points appearing as intersections of three red lines belong to the orbit of $\sigma$ under the action of the modular group, while the intersections between red and black lines to the orbit of $i$.
  • Figure 3.3: Fundamental quadrilateral for the action of $\mathop{\mathrm{\mathrm{PGL}}}\nolimits_2(\mathbb{Z})$ on $\mathbb{H}^2$.
  • Figure 3.4: Representation of the $q$-disks corresponding to the numbers $-1, 1/2, 0, 1/2, 1, 2$ and $\infty$ are represented for the parameter $q$ specified to $q=0.45$. The "$q$-disk" corresponding to $\infty$ is a half-plane bordered by a vertical line $x=\frac{1}{1-q}$ in this representation. The convergence properties of $q$-rationals and $q$-irrationals from Theorem \ref{['Thm:convergence-q-numbers']} can be well-understood in this visualization.
  • Figure 3.5: Fundamental quadrilateral for the action of $\mathop{\mathrm{\mathrm{PSL}}}\nolimits_2(\mathbb{Z})$ on $\mathbb{H}_q^2$. The angles at $C$ and $D$ are equal to $\frac{\pi}{2}$. The point $B$ is obtained as the intersection of the circle centered at $\frac{1}{1-q}$ passing by $C$ and of the (Euclidean) line connecting $\frac{1}{1-q}$ and $A$.
  • ...and 7 more figures

Theorems & Definitions (132)

  • Proposition A: Proposition \ref{['Prop:q-fund-domain']}
  • Theorem B: Theorem \ref{['Thm:link-left-right-dF']} and Proposition \ref{['Prop:positivity-q-dF']}
  • Theorem C: Proposition \ref{['prop:homotheticformulas']} and Theorem \ref{['Thm:q-gcd']}
  • Theorem D: Theorem \ref{['Thm:charact-regularity']}
  • Theorem E: Theorem \ref{['Thm:main']}
  • Example 2.1
  • Definition 2.3: MGO-2020BBL
  • Example 2.4
  • Proposition 2.5: Leclere-Morier-GenoudJouteur
  • proof
  • ...and 122 more