Some analytical results about two physical models of phyllotaxis
If one looks at the seeds of a sunflower, the florets in a daisy or the scales of a pineapple, one is struck not only by the regular lattice they form, but also by the nearly systematic appearance of Fibonacci numbers in these patterns. Early botanical observations showed that the regularity of the phyllotactic pattern is directly related to the cellular differentiation process. Each budding leaf (or scale, or floret) appears on a ring shaped region (the meristem) surrounding the top of the stem. As the plant grows, the leaves are advected away radially from the meristem. The macroscopic structure of the leaves is therefore determined by this initial microscopic process. A first question to ask is whether the appearance of Fibonacci numbers or, more generally, of noble numbers as divergences, is due to some geometrical constraints. For if we represent the set of leaves by a lattice of points, it is possible to centre non-overlapping tangent circles at each point only if the lattice fulfills some conditions. Let us represent for instance cylindrical phyllotactic patterns by a two-parameter family of plane periodic lattices. The set of lattices for which the construction of a lattice of tangent circles of constant radius is possible corresponds to a tree in the parameter plane. We will describe the remarkable properties of such a tree. Let us mention however that there is exactly one path in the tree leading to any irrational divergence on the axis \(G=0\). In this sense all divergences are equally possible and the question of the appearance of Fibonacci numbers in nature still remains unresolved. What happens if the hard-core interaction between leaves introduced by tangent circles is replaced by an arbitrary repulsive potential? How can we modify in this case the geometrical constraint of tangency of circles? Recently \textit{L. S. Levitov} [Phys. Rev. Lett. 66, 224 ff (1991); Europhys. Lett. 14, 533 ff (1991)] and \textit{S. Douady} and \textit{Y. Couder} [Phys. Rev. Lett. 68, 2098 ff (1992); La Recherche 24, 26ff (1993)] gave an answer to these questions. They studied two physical systems, either theoretically or experimentally, whose main characteristic, as they emphasize, is a hierarchical selection of noble numbers. Levitov as well as Douady and Couder pointed out the analogy of these systems with phyllotaxis. The aim of this paper is to give some theorems concerning the selection of noble numbers in these problems.
scientific article; zbMATH DE number 778988