Phyllotaxis as a Dynamical Self Organizing Process Part I: The Spiral Modes Resulting from Time-Periodic Iterations

Abstract

This article is the first of a series of three in which the various phyllotaxic modes are shown to result from successive iterations of two possible simple dynamical systems. In this first part the hypotheses put forward by Hofmeister (1868) for the formation of primordia are re-examined and shown to form the rules of such a system. By means of a physics experiment and a numerical simulation, it is demonstrated that this system gives rise to the self organization of the spiral phyllotactic structures. The dynamic is controlled by only one parameter, G, equivalent to Richard's plastochrome ratio (1951), which characterizes growth. The diagram representing the values of the divergence angle as a function of the plastochrome ratio resembles that obtained geometrically by van Iterson (1907). In the present results, however, only one branch of solutions is continuous for all values of G; this difference with the geometrical results is important. The predominance of Fibonacci order in botany is related to this continuity: during the ontogeny, because of the continuous decrease in G, only this branch is followed. In this framework the build-up of such complex structures as the inflorescence of compositae is described.