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arxiv: 2603.04295 · v2 · pith:QCH2ZBNDnew · submitted 2026-03-04 · 🧮 math.QA · math.CO· math.DS· math.GT

Plane geometry of q-rationals and Springborn Operations

Perrine Jouteur , Olga Paris-Romaskevich , Alexander Thomas This is my paper

Pith reviewed 2026-05-15 16:39 UTC · model grok-4.3

classification 🧮 math.QA math.COmath.DSmath.GT
keywords q-rational numbersFarey triangulationFord circlesSpringborn operationshomothety centersMarkov numbersmodular surface
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The pith

q-rational numbers correspond to circles in a deformed Farey triangulation for every positive real q.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper develops a geometric model for q-rational numbers for any positive real parameter q. It constructs a deformed version of the Farey triangulation and the modular surface that keeps the classical incidence and adjacency relations intact. Every q-rational is realized as a circle in the plane, extending the classical Ford-circle construction. The authors introduce Springborn operations on these q-rationals that act as a quadratic analogue of Farey addition and are realized geometrically as the centers of homotheties between the associated circles. These operations yield an explicit formula for the q-deformed midpoint of Farey neighbors and produce a new q-deformation of Markov numbers.

Core claim

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.

What carries the argument

The circle representation of each q-rational inside the deformed Farey triangulation, where Springborn operations are realized as homothety centers of pairs of such circles.

If this is right

  • The deformed Farey triangulation and modular surface remain free of singularities for all positive real q.
  • Springborn operations supply a quadratic extension of classical Farey addition on the level of q-rationals.
  • An explicit algebraic formula exists for the q-deformed midpoint of any two Farey neighbors.
  • A new one-parameter family of q-deformed Markov numbers arises directly from iterated Springborn operations.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same circle-and-homothety picture may extend classical results on continued fractions or Diophantine approximation to the q-setting.
  • The deformed modular surface could be used to study q-analogues of hyperbolic geometry or circle packings.
  • The construction supplies a geometric route to other q-deformations that appear in cluster algebras or quantum Teichmüller theory.

Load-bearing premise

The q-deformation of the Farey triangulation preserves the classical incidence and adjacency relations for every positive real q without introducing singularities.

What would settle it

A positive real q for which two q-rationals that are adjacent in the deformed triangulation correspond to circles that are not tangent, or for which the homothety center of the pair fails to satisfy the algebraic rules of the Springborn operation.

Figures

Figures reproduced from arXiv: 2603.04295 by Alexander Thomas, Olga Paris-Romaskevich, Perrine Jouteur.

Figure 3.1
Figure 3.1. Figure 3.1: The ideal triangles of the classical Farey triangulation F attached to rationals with denominator at most 3. Note that the rational points are not equidistant on the boundary. The second construction of the Farey triangulation starts from the presentation of PGL2(Z) as a subgroup of Isom(H2 ) given by (see (2.6)): PGL2(Z) = ⟨s1, s2, s3 | s 2 i = (s1s2) 2 = (s1s3) 3 = 1⟩. Proposition 3.4. Consider the ide… view at source ↗
Figure 3.2
Figure 3.2. 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 [PITH_FULL_IMAGE:figures/full_fig_p011_3_2.png] view at source ↗
Figure 3.3
Figure 3.3. Figure 3.3: Fundamental quadrilateral for the action of PGL2(Z) on H2 . Corollary 3.11. For q ∈ R+, the group generated by Tq, Sq and Nqc is isomorphic to PGL2(Z). Proof. It is clear that the map Tq 7→ T, Sq 7→ S, Nqc 7→ Nc is a surjective group ho￾momorphism. We only have to show that its kernel is trivial, i.e. that there are not more relations among the generators than expected. This follows from Poincar´e’s theo… view at source ↗
Figure 3.4
Figure 3.4. Figure 3.4: Representation of the q-disks corresponding to the numbers −1, 1/2, 0, 1/2, 1, 2 and ∞ are represented for the parameter q specified to q = 0.45. The “q-disk” corresponding to ∞ is a half-plane bordered by a vertical line x = 1 1−q in this representation. The convergence properties of q-rationals and q-irrationals from Theorem 2.11 can be well-understood in this visualization. Proposition 3.14. For x, y … view at source ↗
Figure 3.5
Figure 3.5. Figure 3.5: Fundamental quadrilateral for the action of PSL2(Z) on H2 q . The angles at C and D are equal to π 2 . The point B is obtained as the intersection of the circle centered at 1 1−q passing by C and of the (Euclidean) line connecting 1 1−q and A. Corollary 3.17. If dF (x, y) ∈ {1, 2}, the inversion with respect to γ([x], [y]) is a sym￾metry of Q. We analyze more in detail the quotient H2 q /PSL2(Z), the con… view at source ↗
Figure 5.1
Figure 5.1. Figure 5.1: Inner (in purple) and outer (in orange) homothety centers. Observation 5.2. For many pairs a b , c d  ∈ Q2 , we find i ha b i , h c d i =  ab + cd b 2 + d 2 ♯ q and e ha b i , h c d i =  ab − cd b 2 − d 2 ♭ q . One of the main result of our work is to prove this observation for some pairs satisfying arithmetic conditions, that we call regular pairs. Theorem 6.2 covers the particular case of pair… view at source ↗
Figure 5.2
Figure 5.2. Figure 5.2: Springborn operations from Ford circles [PITH_FULL_IMAGE:figures/full_fig_p028_5_2.png] view at source ↗
Figure 5.3
Figure 5.3. Figure 5.3: Springborn operations from Ford-like circles orthogonal to the real line We are grateful to Boris Springborn, who mentionned an equivalent formulation using hyperbolic involutions: Proposition 5.16. Let C1, C2 be two circles centered on the real line, with disjoint interiors. The outer homothety center e(C1, C2) is the image of ∞ under the unique orientation-reversing isometry of H2 exchanging C1 and C2.… view at source ↗
Figure 6.1
Figure 6.1. Figure 6.1: A pair ([x], [y]) with dF (x, y) = 1, and its Farey and Spring￾born sum and difference. In blue, the geodesic γ between [x] and [y]. In green, the geodesic between [x ⊕F y] and [x ⊖F y]. Proof. Let us consider the pair a+c b+d , a−c b−d  , which also has Farey determinant 1 or 2. The inversion with respect to the geodesic γ between a+c b+d [PITH_FULL_IMAGE:figures/full_fig_p033_6_1.png] view at source ↗
Figure 7.1
Figure 7.1. Figure 7.1: The (oriented) rational Markov tree with the Springborn local rule, starting with 0 1 and 1 2 . Every vertex of such tree corresponds to a triple of Markov fractions - two parents and a child. A couple of parents a b , c d  has one child p q with p := ac+bd ad−bc and q := b 2+d 2 ad−bc . Lemma 7.1. Any two rational numbers in a rational Markov triple form an inner regular pair. Proof. The initial pair 0… view at source ↗
Figure 7.2
Figure 7.2. Figure 7.2: Snake graphs associated to the Markov fraction 12 29 . 1 0 1 1 0 1 1 2 1 3 2 5 3 7 5 12 7 17 12 29 [PITH_FULL_IMAGE:figures/full_fig_p044_7_2.png] view at source ↗
Figure 7.3
Figure 7.3. Figure 7.3: Triangulated polygon associated to 12 29 . More precisely, let µ be a Markov fraction and Sµ its triangular snake graph. This graph has an even number of triangles, so let group them two by two in order to create quadrilateral tiles, starting with the first two adjacent triangles. The continued fraction of µ is obtained by reading Sµ from bottom to top, with the following rules : ◦ replace the first tile… view at source ↗
read the original 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.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The paper studies the plane geometry of q-rational numbers for positive real q, constructing a deformed Farey triangulation and modular surface. It interprets each q-rational as a circle analogous to Ford circles, defines Springborn operations as a quadratic version of Farey addition realized via homothety centers of circle pairs, derives an explicit formula for the q-deformed midpoint of Farey neighbors, and introduces a new q-deformation of Markov numbers.

Significance. If the incidence relations are preserved without degeneracies, the work supplies a concrete geometric model for q-analogues that links combinatorial q-structures to circle packings and hyperbolic geometry. The Springborn operations and midpoint formula provide explicit, potentially parameter-free expressions that could be used to generate q-Markov numbers and test conjectures in q-deformed Diophantine approximation.

major comments (2)
  1. [Abstract / deformed Farey triangulation construction] Abstract and the construction of the deformed Farey triangulation: the claim that incidence and adjacency relations (including tangencies of q-Ford circles) are preserved for every q > 0 is load-bearing for the subsequent definition of Springborn operations via homothety centers, yet no explicit verification or curvature analysis is supplied to rule out degeneracies such as vanishing curvatures or overlapping centers for generic q.
  2. [Application to q-midpoint and q-Markov numbers] Application section on q-Markov numbers: the derived formula for the q-midpoint of two Farey neighbors is presented as a direct consequence of the Springborn operations, but the manuscript does not include the classical limit check (q → 1) or a table of numerical values confirming agreement with ordinary midpoints and Markov numbers.
minor comments (2)
  1. [Introduction / preliminaries] Notation for q-continued fractions and q-mediants is introduced without a dedicated preliminary subsection, making it difficult to track the transition from combinatorial to geometric definitions.
  2. [Figures] Figure captions for the deformed triangulation and circle packings lack explicit labels for the homothety centers and curvature values, reducing readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading of our manuscript and the constructive comments. We address each major comment below and plan to incorporate revisions to strengthen the paper.

read point-by-point responses

  1. Referee: [Abstract / deformed Farey triangulation construction] Abstract and the construction of the deformed Farey triangulation: the claim that incidence and adjacency relations (including tangencies of q-Ford circles) are preserved for every q > 0 is load-bearing for the subsequent definition of Springborn operations via homothety centers, yet no explicit verification or curvature analysis is supplied to rule out degeneracies such as vanishing curvatures or overlapping centers for generic q.

    Authors: We agree that an explicit verification would improve the robustness of the presentation. The preservation of incidence relations follows directly from the definitions of the q-circles' centers and curvatures in terms of the q-rationals, which by construction maintain the tangency conditions analogous to Ford circles for all q > 0, as the formulas ensure positive curvatures and non-coincident centers. To address this, we will add a dedicated paragraph or subsection providing the curvature analysis and confirming no degeneracies occur for generic q > 0. revision: yes

  2. Referee: [Application to q-midpoint and q-Markov numbers] Application section on q-Markov numbers: the derived formula for the q-midpoint of two Farey neighbors is presented as a direct consequence of the Springborn operations, but the manuscript does not include the classical limit check (q → 1) or a table of numerical values confirming agreement with ordinary midpoints and Markov numbers.

    Authors: We appreciate this suggestion for enhancing the application section. We will include the classical limit check demonstrating that as q approaches 1, the q-midpoint formula recovers the standard arithmetic mean, and add a table of numerical examples for small q-Markov numbers compared to their classical counterparts to confirm agreement. revision: yes

Circularity Check

0 steps flagged

No circularity: new geometry and operations built on externally introduced q-rationals

full rationale

The paper takes q-rationals as given from the external reference Morier-Genoud and Ovsienko, then constructs the deformed Farey triangulation, modular surface, circle interpretations, and Springborn operations as new objects on top of that definition. No equation or claim inside the paper reduces a derived quantity to a parameter fitted from the same data, renames an input as a prediction, or relies on a load-bearing self-citation whose content is itself unverified. The incidence-preservation statements are presented as consequences of the external q-rational definitions rather than tautologies internal to this manuscript.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 2 invented entities

The central claims rest on the prior definition of q-rationals for positive real q, together with standard properties of circles, homotheties, and triangulations in the plane.

axioms (2)
  • domain assumption q-rationals are well-defined for positive real q as introduced by Morier-Genoud and Ovsienko
    The paper takes this definition as given and builds the geometry and operations upon it.
  • standard math Homothety centers exist and are unique for pairs of circles in the plane
    Invoked when defining Springborn operations geometrically.
invented entities (2)
  • Deformed Farey triangulation no independent evidence
    purpose: Geometric structure for q-rationals
    Constructed as a deformation of the classical Farey triangulation.
  • Springborn operations no independent evidence
    purpose: Quadratic analog of Farey addition via homothety centers
    Newly defined operations on q-rationals.

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Reference graph

Works this paper leans on

31 extracted references · 31 canonical work pages · 1 internal anchor

  1. [1]

    Aigner.Markov’s Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings

    M. Aigner.Markov’s Theorem and 100 Years of the Uniqueness Conjecture: A Mathematical Journey from Irrational Numbers to Perfect Matchings. Springer International Publishing, 2013

    work page 2013

  2. [2]

    Aval and S

    J.-C. Aval and S. Labb´ e.q-analogs of rational numbers: from Ostrowski numeration systems to perfect matchings. Preprint, arXiv/2511.11290, 2025

    work page arXiv 2025

  3. [3]

    Bapat, L

    A. Bapat, L. Becker, and A. M. Licata.q-deformed rational numbers and the 2-Calabi-Yau cate- gory of typeA 2.Forum Math. Sigma, 11:41, 2023. arXiv/2202.07613

    work page arXiv 2023

  4. [4]

    Bittmann, P

    L. Bittmann, P. Jouteur, E. Kantarcı O˘ guz, M. Molander, and E. Yıldırım. A mirror deformation of Markov numbers. Preprint, arXiv/2602.14802, 2026

    work page arXiv 2026

  5. [5]

    Elzenaar, J

    A. Elzenaar, J. Gong, G. J. Martin, and J. Schillewaert. Bounding deformation spaces of Kleinian groups with two generators. Preprint, arXiv/2405.15970, 2024

    work page arXiv 2024

  6. [6]

    P. Etingof. Onq-real andq-complex numbers. Preprint, arXiv/2508.08440, 2025

    work page arXiv 2025

  7. [7]

    Evans, P

    S. Evans, P. Jouteur, S. Morier-Genoud, and V. Ovsienko. Onq-deformed Markov numbers. Cohn matrices and perfect matchings with weighted edges. Preprint, arXiv/2507.19080, 2025

    work page arXiv 2025

  8. [8]

    L. R. Ford. Fractions.Am. Math. Mon., 45:586–601, 1938

    work page 1938

  9. [9]

    G. H. Hardy and E. M. Wright.An Introduction To The Theory Of Numbers. Oxford University Press, 07 2008

    work page 2008

  10. [10]

    P. Jouteur. Symmetries of theq-deformed real projective line. Preprint, arXiv/2503.02122, 2025

    work page arXiv 2025

  11. [11]

    Kantarcı O˘ guz

    E. Kantarcı O˘ guz. Oriented posets, rank matrices andq-deformed Markov numbers.Discrete Mathematics, 348(2):114256, 2025. arXiv/2206.05517

    work page arXiv 2025

  12. [12]

    Kogiso.q-deformations andt-deformations of Markov triples

    T. Kogiso.q-deformations andt-deformations of Markov triples. Preprint, arXiv/2008.12913, 2020

    work page arXiv 2008

  13. [13]

    Kogiso, K

    T. Kogiso, K. Miyamoto, X. Ren, M. Wakui, and K. Yanagawa. Arithmetic onq-deformed rational numbers. Preprint, arXiv/2403.08446, 2024

    work page arXiv 2024

  14. [14]

    Labb´ e and M

    S. Labb´ e and M. Lapointe. Theq-analog of the Markoff injectivity conjecture over the language of a balanced sequence.Combinatorial Theory, 2(1), 2022. arXiv/2106.15886

    work page arXiv 2022

  15. [15]

    Labb´ e, M

    S. Labb´ e, M. Lapointe, and W. Steiner. Aq-analog of the Markoff injectivity conjecture holds. Algebraic Combinatorics, 6(6):1677–1685, 2024. arXiv/2212.09852

    work page arXiv 2024

  16. [16]

    Leclere.q-analogues des nombres r´ eels et des matrices unimodulaires : aspects alg´ ebriques, combinatoires et analytiques

    L. Leclere.q-analogues des nombres r´ eels et des matrices unimodulaires : aspects alg´ ebriques, combinatoires et analytiques. PhD thesis, https://theses.fr/2024REIMS019, 2024

    work page 2024

  17. [17]

    Leclere and S

    L. Leclere and S. Morier-Genoud.q-deformations in the modular group and of the real quadratic irrational numbers.Advances in Applied Mathematics, 130:102223, 2021. arXiv/2101.02953

    work page arXiv 2021

  18. [18]

    Leclere, S

    L. Leclere, S. Morier-Genoud, V. Ovsienko, and A. Veselov. On radius of convergence ofq-deformed real numbers.Moscow Mathematical Journal, 2021. arXiv/2102.00891

    work page arXiv 2021

  19. [19]

    Manecke and A

    S. Manecke and A. Thomas.q-deformed rational number fractal. Mathematical illustration, www.shadertoy.com/view/wX3cz2, 2025

    work page 2025

  20. [20]

    B. Maskit. On Poincar´ e’s theorem for fundamental polygons.Adv. Math., 7:219–230, 1971. Sci- encedirect/S0001870871800038

    work page 1971

  21. [21]

    Farey boat I. Continued fractions and triangulations, modular group and polygon dissections

    S. Morier-Genoud and V. Ovsienko. Farey Boat: Continued fractions and triangulations, modular group and polygon dissections.Jahresbericht der DMV, 121(2):91–136, 2019. arXiv/1811.01229

    work page internal anchor Pith review Pith/arXiv arXiv 2019

  22. [22]

    Morier-Genoud and V

    S. Morier-Genoud and V. Ovsienko.q-deformed rationals andq-continued fractions.Forum Math. Sigma, 8:55, 2020. arXiv/1812.00170

    work page arXiv 2020

  23. [23]

    Morier-Genoud and V

    S. Morier-Genoud and V. Ovsienko. Onq-deformed real numbers.Exp. Math., 31(2):652–660,

    work page

  24. [24]

    Ovsienko

    V. Ovsienko. Towards quantized complex numbers:q-deformed Gaussian integers and the Picard group.Open Commun. Nonlinear Math. Phys., 1:73–93, 2021. arXiv/2103.10800

    work page arXiv 2021

  25. [25]

    X. Ren. Onq-deformed Farey sum and a homological interpretation ofq-deformed real quadratic irrational numbers. Preprint, arXiv/2210.06056, 2024

    work page arXiv 2024

  26. [26]

    Ren and K

    X. Ren and K. Yanagawa. Transposes in theq-deformed modular group and their applications to q-deformed rational numbers. Preprint, arXiv/2502.02974, 2025

    work page arXiv 2025

  27. [27]

    C.-L. Simon. Linking numbers of modular knots.Geom. Topol., 29(6):3241–3270, 2025. arXiv/2211.05957. PLANE GEOMETRY OFq-RATIONALS AND SPRINGBORN OPERATIONS 51

    work page arXiv 2025

  28. [28]

    Springborn

    B. Springborn. The worst approximable rational numbers.J. Number Theory, 263:153 – 205, 2024. arXiv/2209.15542

    work page arXiv 2024

  29. [29]

    Sullivan

    D. Sullivan. The density at infinity of a discrete group of hyperbolic motions.Publica- tions Math´ ematiques de l’Institut des Hautes ´Etudes Scientifiques, 50:171–202, 1979. Num- dam/PMIHES 1979 50 171 0

    work page 1979

  30. [30]

    A. Thomas. Infinitesimal modular group:q-deformedsl 2 and Witt algebra.SIGMA, Symmetry Integrability Geom. Methods Appl., 20:paper 053, 16, 2024. arXiv/2308.06158

    work page arXiv 2024

  31. [31]

    A. Veselov. Markov fractions and the slopes of the exceptional bundles onP 2. Preprint, arXiv/2501.06779, 2025. Universit´e de Reims Champagne-Ardenne, CNRS, LMR, UMR 9008, Reims, France Email address:perrine.jouteur@univ-reims.fr CNRS, ICJ UMR 5208, ´Ecole Centrale de Lyon, INSA Lyon, Universit ´e Claude Bernard Lyon 1, Universit´e Jeann Monnet, 69622 Vi...

    work page arXiv 2025