A renormalization approach to the universality of scaling in phyllotaxis

•Self similarity in Fourier transformed dynamics of Thornley’s model of phyllotaxis.•Construction of a renormalization theory for Thornley’s model of phyllotaxis.•Universality of scaling and divergence selection in Thornley’s model of phyllotaxis.

Phyllotaxis, i.e. the arrangement of plant organs like leaves, florets, scales, bracts etc. around a shoot, stem, or cone, is often highly regular. Across the plant kingdom phyllotaxis shows not only qualitatively, but also quantitatively identical features, like the occurrence of divergence angles close to noble irrationals. In a previous study (Reick, 2012) a mechanism has been identified that explains the selection of these particular divergence angles on the basis of self-similarity and scaling, numerically found in the bifurcation diagrams of simple dynamical models of phyllataxis. In the present paper, by constructing a renormalization theory, the universality of this scaling is proved for a whole class of models, prototypically represented by Thornley’s model of phyllotaxis (Thornley, 1975). The renormalization is constructed from another self-similarity found numerically for the Fourier transform of the abstract potential governing the mutual inhibition of primordia. Surprisingly, the resulting renormalization transformation is already known from the treatment of the quasiperiodic transition to chaos but operates here on a different function space. It turns out that the fixed points of the renormalization transformation are characterized by divergences of the form Θ(κ)=1/τ(κ)Θ(κ)=1/τ(κ), where, written as continued fraction, τ(κ)=[κ;κ,κ,…]τ(κ)=[κ;κ,κ,…], κ∈N+κ∈N+. To show the universality of the scaling, it is demonstrated that the fixed points are unstable and that the associated scaling factors α(κ)=−(τ(κ))2α(κ)=−(τ(κ))2 and β(κ)=τ(κ)β(κ)=τ(κ) are exactly those that were numerically found in (Reick, 2012) to rule the selfsimilarity of the bifurcation structure. Thereby, the present paper puts forward an explanation for the universal appearance of certain phyllotactic patterns that is independent of physiological detail of plant growth.