Journal of Mathematical Biology

, Volume 78, Issue 1–2, pp 1–19 | Cite as

Golden ratio and phyllotaxis, a clear mathematical link

  • François BergeronEmail author
  • Christophe Reutenauer


Exploiting Markoff’s theory for rational approximations of real numbers, we explicitly link how hard it is to approximate a given number to an idealized notion of growth capacity for plants which we express as a modular invariant function depending on this number. Assuming that our growth capacity is biologically relevant, this allows us to explain in a satisfying mathematical way why the golden ratio occurs in nature.


Modular group Markoff approximation theory Golden ratio Phyllotaxis 

Mathematics Subject Classification

51F15 11Y65 92C15 



We would like to thank Stéphane Durand and Christiane Rousseau for drawing our attention to the notion that: “it is because it is hard to approximate by rational numbers that the golden ratio plays a key role in phyllotaxis”. Our objective in this work was to give a precise mathematical meaning to this statement. We also thank Nadia Lafrenière and Caroline Series for useful suggestions.


  1. Adler I (1974) A model of contact pressure in phyllotaxis. J Theor Biol 45:1–79CrossRefGoogle Scholar
  2. Aigner M (2013) Markov’s theorem and 100 years of the uniqueness conjecture, a mathematical journey from irrational numbers to perfect matchings. Springer, BerlinzbMATHGoogle Scholar
  3. Atela P, Golé et C, Hotton S (2002) A dynamical system for plant pattern formation: a rigorous analysis. J Nonlinear Sci 12:641–676MathSciNetCrossRefzbMATHGoogle Scholar
  4. Bacher R (2014) On geodesics of phyllotaxis. Confluentes Math 6(1):3–27MathSciNetCrossRefzbMATHGoogle Scholar
  5. Couder Y, Douady S (1996a) Phyllotaxis as a dynamical self organizing process part I: the spiral modes resulting from time-periodic iterations. J Theor Biol 178:255–274CrossRefGoogle Scholar
  6. Couder Y, Douady S (1996b) Phyllotaxis as a dynamical self organizing process part II: the spontaneous formation of a periodicity and the coexistence of spiral and whorled patterns. J Theor Biol 178:275–294CrossRefGoogle Scholar
  7. Couder Y, Douady S (1996c) Phyllotaxis as a dynamical self organizing process part III: the simulation of the transient regimes of ontogeny. J Theor Biol 178:295–312CrossRefGoogle Scholar
  8. Coxeter HSM (1972) The role of intermediate convergents in Tait’s explanation for phyllotaxis. J Algebra 20:167–175MathSciNetCrossRefzbMATHGoogle Scholar
  9. Douady S (1998) The selection of phyllotactic patterns. In: Jean RV, Barabe D (eds) Symmetry in plants. World Scientific, Singapore, pp 335–358CrossRefGoogle Scholar
  10. Hermite C (1916) Sur l’introduction des variables continues dans la théorie des nombres. J Reine Angew Math 41:191–216MathSciNetGoogle Scholar
  11. Humbert G (1916) Sur la méthode d’approximation d’Hermite. J Math Pures App 7ème série 2:79–103zbMATHGoogle Scholar
  12. van Iterson G (1907) Mathematische und mikroskopisch-anatomische Studien über Blattstellungen, nebst Betrachtungen über den Schalenbau der Miliolinen, Verlag von Gustav Fischer in JenaGoogle Scholar
  13. Jacobs B (2014) On Hermite’s algorithm. Bachelor’s thesis, Utrecht UniversityGoogle Scholar
  14. Jean RV, Barabé D (1998) Symmetry in plants. World Scientific, SingaporeCrossRefzbMATHGoogle Scholar
  15. Leigh EG Jr (1983) The golden section and spiral leaf-arrangement. Trans Conn Acad Arts Sci 44:163–176Google Scholar
  16. Markoff AA (1879) Sur les formes quadratiques binaires indéfinies. Math Ann 15:381–496CrossRefzbMATHGoogle Scholar
  17. Markoff AA (1880) Sur les formes quadratiques binaires indéfinies (second mémoire). Math Ann 17:379–399MathSciNetCrossRefzbMATHGoogle Scholar
  18. Marzec C, Kappraff J (1983) Properties of maximal spacing on a circle related to phyllotaxis and to the golden mean. J Theor Biol 103:201–226MathSciNetCrossRefGoogle Scholar
  19. Okabe T (2012a) Systematic variations in divergence angle. J Theor Biol 313:20–41. arXiv:1212.3377
  20. Okabe T (2012b) Geometric interpretation of phyllotaxis transition. arXiv:1212.3112
  21. Refahi Y, Brunoud G, Farcot E, Jean-Marie A, Pulkkinen M, Vernoux T, Godin C (2016) A stochastic multicellular model identifies biological watermarks from disorders in self-organized patterns of phyllotaxis. eLIFE.
  22. Reutenauer C (2018) From Christoffel words to Markoff numbers. Oxford University Press, OxfordCrossRefzbMATHGoogle Scholar
  23. Ridley JN (1986) Ideal phyllotaxis on general surfaces of revolution. Math Biosci 79(1):1–24MathSciNetCrossRefzbMATHGoogle Scholar
  24. Serre J-P (1970) Cours d’arithmétique. Presses universitaires de France, PariszbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Département de mathématiquesUniversité du Québec à MontréalMontrealCanada

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