Phyllotactic order from radial growth of spatial symmetry-breaking instabilities in supercritical conditions
Phys. Rev. E 114, 044202 – Published 2 October, 2026
DOI: https://doi.org/10.1103/njhc-tl7p
Abstract
Phyllotactic patterns, i.e., the arrangements of organs around a stem in some plants, represent not only fascinating geometries but also a source of inspiration for advanced biomimetic technological applications. Recently, spatial symmetry-breaking instabilities have been shown to feature, in some cases, phyllotactic symmetries when the radial growth of the pattern is slaved to a reactive front. We show here that phyllotactic ordering can also emerge when a homogeneous system is unstable with respect to a spatially self-organizing instability, and pattern formation is triggered by a localized perturbation that propagates radially. To do so, we numerically investigate the emergence of phyllotactic patterns from two different spatial symmetry-breaking instabilities coupled to radial growth in supercritical conditions: a far-from-equilibrium Turing instability studied here with the Brusselator model and equilibrium spinodal decomposition modeled by the Cahn-Hilliard equation. In both cases, radial expansion of the perturbations leads to the formation of concentric rings that subsequently fragment into spots, generating highly symmetric phyllotactic arrangements. The resulting patterns belong to the family of whorled modes, characterized by equal numbers of clockwise and counterclockwise parastichies. The morphologies are robust under variations of the characteristic wavelength and exhibit self-similarity when rescaled by this intrinsic length scale. A geometric analysis reveals that phyllotactic order arises from a space-optimization hexagonal lattice packing, as confirmed by Voronoi diagrams and representations in rescaled polar coordinates. Despite the common lattice structure, the two instabilities may select different orientations of propagation with respect to the radial direction. These results confirm the radial growth of spatial symmetry-breaking systems as a universal mechanism able to generate phyllotactic order.
