Phyllotactic Structures in Radially Growing Spatial Symmetry Breaking Systems

Phyllotactic patterns, where elements such as leaves, seeds, or droplets arrange along alternate spirals, are fascinating examples of complex structures encountered in nature. In botany, their symmetries develop when a new primordium periodically grows in the largest gap left between the previous one and the apex. Experiments using ferrofluid droplets have shown that phyllotactic patterns can also spontaneously form when identical elements repulsing each other are periodically released at a given distance from an injection center and are advected radially at a constant speed. Here, we show that phyllotactic structures can also genuinely develop in the large class of spatial symmetry breaking systems with an intrinsic wavelength in the case of radial growth. The constraint of maintaining a fixed wavelength between spots while expanding radially either diffusively or advectively generalizes the concept of temporal release of repulsing agents in botany to new classes of systems. We explore this on three different systems: numerically on two models describing reaction-driven phase transitions and spatial Turing patterns, respectively, and experimentally on chemical precipitation patterns. This paves the way to engineer new complex self-organized structures in a panoply of different systems, ranging from spinodal decomposition, chemical, biological or optical Turing structures, and Liesegang patterns, to name a few.