Physica D: Nonlinear Phenomena
We study an iterative process modeling growth of phyllotactic patterns, wherein disks are added one by one on the surface of a cylinder, on top of an existing set of disks, as low as possible and without overlap. Numerical simulations show that the steady states of the system are spatially periodic, lattices-like structures called rhombic tilings. We present a rigorous analysis of the dynamics of all configurations starting with closed chains of 3 tangent, non-overlapping disks encircling the cylinder. We show that all these configurations indeed converge to rhombic tilings. Surprisingly, we show that convergence can occur in either finitely or infinitely many iterations. The infinite-time convergence is explained by a conserved quantity.
Disk packing, Phyllotaxis, Rhombic tiling, Attractor, Dynamical system
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*Published in Physica D, https://doi.org/10.1016/j.physd.2019.132278 ©2019. This manuscript version is made available under the CC-BY-NC-ND 4.0 license http://creativecommons.org/licenses/by-nc-nd/4.0/
Golé, Christophe and Douady, Stéphane, "Convergence in a Disk Stacking Model on the Cylinder" (2019). Mathematics and Statistics: Faculty Publications, Smith College, Northampton, MA.