Golden ratio and phyllotaxis, a clear mathematical link
- 126 Downloads
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.
KeywordsModular group Markoff approximation theory Golden ratio Phyllotaxis
Mathematics Subject Classification51F15 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.
- 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
- Jacobs B (2014) On Hermite’s algorithm. Bachelor’s thesis, Utrecht UniversityGoogle Scholar
- Leigh EG Jr (1983) The golden section and spiral leaf-arrangement. Trans Conn Acad Arts Sci 44:163–176Google Scholar
- Okabe T (2012a) Systematic variations in divergence angle. J Theor Biol 313:20–41. arXiv:1212.3377
- Okabe T (2012b) Geometric interpretation of phyllotaxis transition. arXiv:1212.3112
- 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. https://doi.org/10.7554/eLife.14093