Convergence in a disk stacking model on the cylinder

Abstract

We study an iterative process modelling 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.

Phyllotaxis as geometric canalization during plant development

“…Then the disappearance of a blue contact parastichy leads to the disappearance of a red step in the front, and a (6, 6) mode. But it is quickly followed by a blue triangle transition, making a red step reappear and a new blue parastichy, back to (6, 7), as is common in the convergence of irregular pattern toward a rhombic tilling (Golé and Douady, 2019)). Finally, a last pentagon transition leads to the symmetric (6,6) pattern.…”

Phyllotactic patterning of gerbera flower heads

“…Snow and Snow ( 14 , 15 ) refined this hypothesis by postulating that new primordia are inserted as soon as space becomes available for them within the growing apical meristem. The resulting process is commonly abstracted as an iterative addition—also referred to as accretion ( 16 , 17 ) or stacking ( 7 , 18 )—of new elements at the distal boundary of a spiral lattice ( Fig. 1 A ).…”