Phyllotaxis via shell buckling

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__P. Shipman__

University of Arizona

The leaves on a plant, scales on a pine cone, or stickers on a cactus, are organized in patterns of restricted variety. Common patterns of phyllotaxis are the arrangement of leaves in whorls or Fibonacci numbers of spirals. That simple dynamical rules can produce patterns as seen on plants has been demonstrated [1], but the plants' mechanism of choosing the positions of new leaves at the shoot apex is still to be understood. Recent work has investigated a mechanical model of phyllotaxis, viewing the plant as an elastic shell whose growth results in shell buckling, thus forming the primordial bumps on the plant surface that eventually develop into new leaves [2]. The key new observation presented here is that the strain energy of shell buckling--as measured by the product of the Airy stress and Gaussian curvature of the deformed shell's surface--plays a dominant role is the choice of the optimal buckling configuration. Due to the cubic component of this energy, it is combinations of triads of periodic deformations whose wavevectors add to zero that minimize the total elastic energy. Also, the elastic energy is a sensitive function of the triad of wavevectors that is chosen. The prevalence of the Fibonacci sequence and the golden angle in plant patterns is a natural consequence of these observations. Furthermore, this framework provides a relation between the whorled and spiral patterns seen on plants.

References:

1. S. Douady and Y. Couder.
Phyllotaxis as a Physical Self-Organizing Growth Process. *Physical
Review Letters* **68**, 2098-2101,
1992.

2. P. Green, C.S. Steele,
and S. C. Rennich. Phyllotactic
Patterns: A Biophysical Mechanism for their Origin. *Annals of Botany* **77**, 515-527, 1996.

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