On geodesics of phyllotaxis

Roland Bacher[1]

  • [1] Université Grenoble Alpes, Institut Fourier (CNRS UMR 5582), 38000 Grenoble, France

Confluentes Mathematici (2014)

  • Volume: 6, Issue: 1, page 3-27
  • ISSN: 1793-7434

Abstract

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Seeds of sunflowers are often modelled by n ⟼ ϕ θ ( n ) = n e 2 i π n θ leading to a roughly uniform repartition with seeds indexed by consecutive integers at angular distance 2 π θ for θ the golden ratio. We associate to such a map ϕ θ a geodesic path γ θ : ℝ > 0 ⟶ PSL 2 ( ℤ ) ∖ ℍ of the modular curve and use it for local descriptions of the image ϕ θ ( ℕ ) of the phyllotactic map ϕ θ .

How to cite

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Bacher, Roland. "On geodesics of phyllotaxis." Confluentes Mathematici 6.1 (2014): 3-27. <http://eudml.org/doc/275496>.

@article{Bacher2014,
abstract = {Seeds of sunflowers are often modelled by $n\longmapsto \varphi _\theta (n)=\sqrt\{n\}e^\{2i\pi n\theta \}$ 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 $\gamma _\theta : \mathbb\{R\}_\{&gt;0\}\longrightarrow \mathrm\{PSL\}_2(\mathbb\{Z\})\backslash \mathbb\{H\}$ of the modular curve and use it for local descriptions of the image $\varphi _\theta (\mathbb\{N\})$ of the phyllotactic map $\varphi _\theta $.},
affiliation = {Université Grenoble Alpes, Institut Fourier (CNRS UMR 5582), 38000 Grenoble, France},
author = {Bacher, Roland},
journal = {Confluentes Mathematici},
keywords = {Lattice; hyperbolic geometry; phyllotaxis; sunflower-map; lattice},
language = {eng},
number = {1},
pages = {3-27},
publisher = {Institut Camille Jordan},
title = {On geodesics of phyllotaxis},
url = {http://eudml.org/doc/275496},
volume = {6},
year = {2014},
}

TY - JOUR
AU - Bacher, Roland
TI - On geodesics of phyllotaxis
JO - Confluentes Mathematici
PY - 2014
PB - Institut Camille Jordan
VL - 6
IS - 1
SP - 3
EP - 27
AB - Seeds of sunflowers are often modelled by $n\longmapsto \varphi _\theta (n)=\sqrt{n}e^{2i\pi n\theta }$ 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 $\gamma _\theta : \mathbb{R}_{&gt;0}\longrightarrow \mathrm{PSL}_2(\mathbb{Z})\backslash \mathbb{H}$ of the modular curve and use it for local descriptions of the image $\varphi _\theta (\mathbb{N})$ of the phyllotactic map $\varphi _\theta $.
LA - eng
KW - Lattice; hyperbolic geometry; phyllotaxis; sunflower-map; lattice
UR - http://eudml.org/doc/275496
ER -

References

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  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. 
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