A variational approach to physical and plant phyllotaxis

Phyllotaxis, the arrangement of plant organs, is also observed in certain physical systems, particularly those involving magnetic forces. In both cases, it arises from interactions among a large number of elements - mathematically well defined for magnetic systems, but only inferred, in the case of plants, from inhibition models. Among the latter, we have previously shown that our model corresponds closely to the evolution of auxin concentration within meristems. In this paper, we develop a continuous Lagrangian scheme that correctly approximates the results of current models aggregating these multiple interactions. Its potential landscape is isomorphic to the van Iterson diagram, and the convergence of spiral phyllotaxis paths toward irrational angles, such as the golden angle, is driven by the arithmetic of continued fractions. As density increases, the oscillations of each spiral path contract by a renormalization factor at successive generations of convergents, making the landscape asymptotically self-similar (discrete scale invariant). This Lagrangian optimization may explain the evolutionary success of spiral phyllotaxis long before the emergence of angiosperms, in which it is the dominant pattern today. By contrast, whorled phyllotaxis patterns, corresponding to rational angles, are unstable below a threshold plastochron ratio, which we calculated for each mode.