Plane geometry of $q$-rationals and Springborn Operations
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866916008326856704 |
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| author | Jouteur, Perrine Paris-Romaskevich, Olga Thomas, Alexander |
| author_facet | Jouteur, Perrine Paris-Romaskevich, Olga Thomas, Alexander |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_04295 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Plane geometry of $q$-rationals and Springborn Operations Jouteur, Perrine Paris-Romaskevich, Olga Thomas, Alexander Quantum Algebra Combinatorics Dynamical Systems Geometric Topology 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. |
| title | Plane geometry of $q$-rationals and Springborn Operations |
| topic | Quantum Algebra Combinatorics Dynamical Systems Geometric Topology |
| url | https://arxiv.org/abs/2603.04295 |