On geodesics of phyllotaxis

Bacher, Roland

Confluentes Mathematici, Tome 6 (2014) no. 1, pp. 3-30.

Seeds of sunflowers are often modelled by $n⟼{\varphi }_{\theta }\left(n\right)=\sqrt{n}{e}^{2i\pi n\theta }$ leading to a roughly uniform repartition with seeds indexed by consecutive integers at angular distance $2\pi \theta$ for $\theta$ the golden ratio. We associate to such a map ${\varphi }_{\theta }$ a geodesic path ${\gamma }_{\theta }:{ℝ}_{>0}⟶{\mathrm{PSL}}_2\left(\mathbb{Z}\right)\setminus ℍ$ of the modular curve and use it for local descriptions of the image ${\varphi }_{\theta }\left(\mathbb{N}\right)$ of the phyllotactic map ${\varphi }_{\theta }$.

Classification : 92B99, 11H31, 52C15
Mots clés : Lattice, hyperbolic geometry, phyllotaxis, sunflower-map

[1] J.W. Anderson. Hyperbolic Geometry, Springer, 2005. | DOI | MR | Zbl

[2] L. and A. Bravais. Essai sur la disposition des feuilles curvisériées, Ann. Sci. Naturelles (2), 7:42–110, 1837.

[3] H.S.M. Coxeter. The role of intermediate convergents in Tait’s explanation for phyllotaxis, J. of Alg., 20:167–175, 1972. | DOI | MR | Zbl

[4] G.H. Hardy, E.M. Wright. An Introduction to the Theory of Numbers, Oxford University Press, 1960 (fourth edition). | DOI | Zbl

[5] G. van Iterson. Mathematische und mikroskopisch-anatomische Studien über Blattstellungen nebst Betrachtungen über den Schalenbau der Miliolinen, Gustav Fischer, Jena, 1907. | DOI | Zbl

[6] R.V. Jean, D. Barabé (editors). Symmetry in Plants, Series in Mathematical Biology and Medecine, vol. 4, World Scientific, 1998. | DOI | Zbl

[7] A. Ya. Khinchin. Continued fractions, The University of Chicago Press, Chicago, 1964. | DOI | Zbl

[8] L.S. Levitov. Energetic Approach to Phyllotaxis, Europhys. Lett., 6:533–539, 1991. | DOI

[9] Mathoverflow: http://mathoverflow.net/questions/3307/can-a-discrete-set-of-the-plane-of-uniform-density-intersect-all-large-triangles.

[10] R.V. Jean, D. Barabé (editors). Symmetry in plants, World Sci. Publishing, River Edge, NJ, 1998. | DOI | Zbl

[11] F. Rothen, A.-J. Koch. Phyllotaxis, or the properties of spiral lattices. I Shape invariance under compression, J. Phys. France, 50:633–657, 1989. | DOI | MR

[12] J-F. Sadoc, J. Charvolin, N. Rivier. Phyllotaxis: a non conventional solution to packing efficiency in situations with radial symmetry, Acta Cryst. A, 68:470–483, 2012. | DOI

[13] C. Series. The Geometry of Markoff Numbers, Math. Int., 7(3):20–29, 1985. | DOI | MR | Zbl

[14] J-P. Serre. Cours d’arithmétique, Presses Universitaires de France, 1970. | Zbl

[15] D.W. Thompson. On Growth and Form, Dover reprint (1992) of second ed. (1942) (first ed. 1917). | DOI | Zbl

[16] H. Vogel. A better way to construct the sunflower head, Math. Biosc., 44:179–189, 1979. | DOI