On geodesics of phyllotaxis
Confluentes Mathematici, Volume 6 (2014) no. 1, p. 3-27

Seeds of sunflowers are often modelled by nϕ θ (n)=ne 2iπnθ leading to a roughly uniform repartition with seeds indexed by consecutive integers at angular distance 2πθ for θ the golden ratio. We associate to such a map ϕ θ a geodesic path γ θ : >0 PSL 2 () of the modular curve and use it for local descriptions of the image ϕ θ () of the phyllotactic map ϕ θ .

DOI : https://doi.org/10.5802/cml.10
Classification:  92B99,  11H31,  52C15
Keywords: Lattice, hyperbolic geometry, phyllotaxis,sunflower-map
@article{CML_2014__6_1_3_0,
     author = {Bacher, Roland},
     title = {On geodesics of phyllotaxis},
     journal = {Confluentes Mathematici},
     publisher = {Institut Camille Jordan},
     volume = {6},
     number = {1},
     year = {2014},
     pages = {3-27},
     doi = {10.5802/cml.10},
     language = {en},
     url = {http://www.numdam.org/item/CML_2014__6_1_3_0}
}
Bacher, Roland. On geodesics of phyllotaxis. Confluentes Mathematici, Volume 6 (2014) no. 1, pp. 3-27. doi : 10.5802/cml.10. http://www.numdam.org/item/CML_2014__6_1_3_0/

[1] J.W. Anderson. Hyperbolic Geometry, Springer, 2005. | MR 2161463 | Zbl 1077.51008

[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. | MR 295187 | Zbl 0225.10033

[4] G.H. Hardy, E.M. Wright. An Introduction to the Theory of Numbers, Oxford University Press, 1960 (fourth edition). | Zbl 0086.25803

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

[6] R.V. Jean, D. Barabé (editors). Symmetry in Plants, Series in Mathematical Biology and Medecine, vol. 4, World Scientific, 1998. | Zbl 0926.92024

[7] A. Ya. Khinchin. Continued fractions, The University of Chicago Press, Chicago, 1964. | MR 161833 | Zbl 0117.28601

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

[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. | Zbl 0926.92024

[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. | MR 997212

[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.

[13] C. Series. The Geometry of Markoff Numbers, Math. Int., 7(3):20–29, 1985. | MR 795536 | Zbl 0566.10024

[14] J-P. Serre. Cours d’arithmétique, Presses Universitaires de France, 1970. | MR 255476 | Zbl 0376.12001

[15] D.W. Thompson. On Growth and Form, Dover reprint (1992) of second ed. (1942) (first ed. 1917). | Zbl 0063.07372

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