Advertisement

Farey Boat: Continued Fractions and Triangulations, Modular Group and Polygon Dissections

  • Sophie Morier-GenoudEmail author
  • Valentin Ovsienko
Survey Article

Abstract

We reformulate several known results about continued fractions in combinatorial terms. Among them the theorem of Conway and Coxeter and that of Series, both relating continued fractions and triangulations. More general polygon dissections appear when extending these theorems for elements of the modular group \(\mathrm{PSL}(2,\mathbb {Z})\). These polygon dissections are interpreted as walks in the Farey tessellation. The combinatorial model of continued fractions can be further developed to obtain a canonical presentation of elements of \(\mathrm{PSL}(2,\mathbb {Z})\).

Keywords

Continued fractions Farey graph Polygon dissections Ptolemy rule Pfaffians Modular group 

Notes

Acknowledgements

We are grateful to Charles Conley, Vladimir Fock, Sergei Fomin, Alexey Klimenko, and Sergei Tabachnikov for multiple stimulating and enlightening discussions. We are grateful to the referee for a number of helpful remarks and suggestions. This paper was partially supported by the ANR project SC3A, ANR-15-CE40-0004-01.

References

  1. 1.
    Adamczewski, B., Allouche, J.-P.: Reversals and palindromes in continued fractions. Theor. Comput. Sci. 380, 220–237 (2007) MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Aigner, M.: Markov’s Theorem and 100 Years of the Uniqueness Conjecture. A Mathematical Journey from Irrational Numbers to Perfect Matchings. Springer, Cham (2013) zbMATHGoogle Scholar
  3. 3.
    Bergeron, F., Reutenauer, C.: \(SL_{k}\)-tilings of the plane. Ill. J. Math. 54, 263–300 (2010) zbMATHGoogle Scholar
  4. 4.
    Berstel, J., Reutenauer, C.: Noncommutative Rational Series with Applications. Encyclopedia of Mathematics and Its Applications, vol. 137. Cambridge University Press, Cambridge (2011) zbMATHGoogle Scholar
  5. 5.
    Boca, F.: Products of matrices [ 1 1 0 1 ] Open image in new window and [ 1 0 1 1 ] Open image in new window and the distribution of reduced quadratic irrationals. J. Reine Angew. Math. 606, 149–165 (2007) MathSciNetGoogle Scholar
  6. 6.
    Borwein, J., van der Poorten, A., Shallit, J., Zudilin, W.: Neverending Fractions. An Introduction to Continued Fractions. Australian Mathematical Society Lecture Series, vol. 23. Cambridge University Press, Cambridge (2014) CrossRefzbMATHGoogle Scholar
  7. 7.
    Bourgain, J., Kontorovich, A.: Beyond Expansion III: Reciprocal Geodesics. arXiv:1610.07260
  8. 8.
    Caldero, P., Chapoton, F.: Cluster algebras as Hall algebras of quiver representations. Comment. Math. Helv. 81, 595–616 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Conley, C., Ovsienko, V.: Rotundus: triangulations, Chebyshev polynomials, and Pfaffians. Math. Intell. 40, 45–50 (2018) MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Conley, C., Ovsienko, V.: Lagrangian configurations and symplectic cross-ratios. arXiv:1812.04271
  11. 11.
    Conway, J.H., Coxeter, H.S.M.: Triangulated polygons and frieze patterns. Math. Gaz. 57, 87–94 (1973), 175–183 MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Coxeter, H.S.M.: Frieze patterns. Acta Arith. 18, 297–310 (1971) MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Duke, W., Imamoglu, Ö., Tóth, Á.: Kronecker’s first limit formula, revisited. Res. Math. Sci. 5, 20 (2018) MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fomin, S., Zelevinsky, A.: Cluster algebras. I. Foundations. J. Am. Math. Soc. 15, 497–529 (2002) MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Fomin, S., Zelevinsky, A.: Cluster algebras. II. Finite type classification. Invent. Math. 154, 63–121 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Gekhtman, M., Shapiro, M., Vainshtein, A.: Cluster Algebras and Poisson Geometry. Mathematical Surveys and Monographs, vol. 167. American Mathematical Society, Providence (2010) zbMATHGoogle Scholar
  17. 17.
    Graham, R., Knuth, D., Patashnik, O.: Concrete Mathematics. a Foundation for Computer Science Addison-Wesley Publishing Company, Advanced Book Program, Reading (1989). xiv+625 pp. CrossRefzbMATHGoogle Scholar
  18. 18.
    Hall, R., Shiu, P.: The index of a Farey sequence. Mich. Math. J. 51, 209–223 (2003) MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Hardy, G.H., Wright, E.M.: An Introduction to the Theory of Numbers, sixth edn. Oxford University Press, Oxford (2008). Revised by D.R. Heath-Brown, J.H. Silverman. With a foreword by Andrew Wiles, 621 pp. zbMATHGoogle Scholar
  20. 20.
    Henry, C.-S.: Coxeter friezes and triangulations of polygons. Am. Math. Mon. 120, 553–558 (2013) MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Hirzebruch, F.E.P.: Hilbert modular surfaces. Enseign. Math. (2) 19, 183–281 (1973) MathSciNetzbMATHGoogle Scholar
  22. 22.
    Hirzebruch, F., Zagier, D.: Classification of Hilbert modular surfaces. In: Complex Analysis and Algebraic Geometry. Collected Papers II, pp. 43–77. Iwanami Shoten, Tokyo (1977) CrossRefGoogle Scholar
  23. 23.
    Karpenkov, O.: Geometry of Continued Fractions. Algorithms and Computation in Mathematics, vol. 26. Springer, Heidelberg, New York, Dordrecht, London (2013) zbMATHGoogle Scholar
  24. 24.
    Katok, S.: Coding of closed geodesics after Gauss and Morse. Geom. Dedic. 63(2), 123–145 (1996) MathSciNetCrossRefzbMATHGoogle Scholar
  25. 25.
    McMullen, C.: Uniformly Diophantine numbers in a fixed real quadratic field. Compos. Math. 145, 827–844 (2009) MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Morier-Genoud, S.: Coxeter’s frieze patterns at the crossroads of algebra, geometry and combinatorics. Bull. Lond. Math. Soc. 47, 895–938 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Morier-Genoud, S., Ovsienko, V.: Farey boat II. ℚ-deformations: \(q\)-deformed rationals and \(q\)-continued fractions. arXiv:1812.00170
  28. 28.
    Morier-Genoud, S., Ovsienko, V., Tabachnikov, S.: \(\mathrm{SL}_{2}(\mathbb {Z})\)-tilings of the torus, Coxeter–Conway friezes and Farey triangulations. Enseign. Math. 61, 71–92 (2015) MathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Morier-Genoud, S., Ovsienko, V., Schwartz, R., Tabachnikov, S.: Linear difference equations, frieze patterns and combinatorial Gale transform. Forum Math. Sigma, 2, e22 (2014) MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Ovsienko, V.: Partitions of unity in \(\mathrm{SL}(2,\mathbb {Z})\), negative continued fractions, and dissections of polygons. Res. Math. Sci. 5(2), 21 (2018) MathSciNetCrossRefGoogle Scholar
  31. 31.
    van der Poorten, A.J.: An introduction to continued fractions. In: Diophantine Analysis, Kensington, 1985. London Math. Soc. Lecture Note Ser., vol. 109, pp. 99–138. Cambridge Univ. Press, Cambridge (1986) CrossRefGoogle Scholar
  32. 32.
    Series, C.: The modular surface and continued fractions. J. Lond. Math. Soc. (2) 31(1), 69–80 (1985) MathSciNetCrossRefzbMATHGoogle Scholar
  33. 33.
    Ustinov, A.: A short proof of Euler’s identity for continuants. Math. Notes 79, 146–147 (2006) MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Zagier, D.: Nombres de classes et fractions continues. Astérisque, 24–25, 81–97 (1975). Journées Arithmétiques de Bordeaux, Conference, Univ. Bordeaux, 1974 MathSciNetzbMATHGoogle Scholar

Copyright information

© Deutsche Mathematiker-Vereinigung and Springer-Verlag GmbH Deutschland, ein Teil von Springer Nature 2019

Authors and Affiliations

  1. 1.Institut de Mathématiques de Jussieu-Paris Rive GaucheSorbonne Université, Université Paris Diderot, CNRSParisFrance
  2. 2.Laboratoire de Mathématiques U.F.R. Sciences Exactes et NaturellesCentre national de la recherche scientifiqueReims cedex 2France

Personalised recommendations