MSC Seminar. 04/03/2025. 11h30. Jonathan Swinton (Manchester, UK): “Beyond the golden ratio: new results and current challenges in Fibonacci phyllotaxis”

Mathematicians (notably including Alan Turing) have long been interested in the surprisingly frequent presence of Fibonacci numbers, like 34 and 55, in counts of plant forms. Older and often semi-mystical theories based on properties of the golden ratio have been displaced over the last thirty years by a compelling and mathematically rigorous, if largely untested, hypothesis based on bifurcation theory. Developmental and molecular biologists have largely ignored these analyses, old and new, for mostly valid reasons including a lack of confrontation between models and data. But there is no gene encoding the number 55: no satisfying molecular theory of plant morphogenesis seems possible without such a mathematical biology.

I’ll give a survey of the classical and modern theory of mathematical phyllotaxis, touching on connections to hyperbolic mathematics from the era of Poincaré, but concentrating on the modern applied mathematics of model testing. I’ll address the lack of testable predictions by presenting the first large scale quantitative evaluation of the ability of Schwendener disk-stacking models to generate the spiral patterns seen in a dataset of sunflower seedheads. The rich dynamics of these conceptually simple models are a fertile area for mathematical study, allow quantitative comparison with existing morphological data, can be related to and generate new hypotheses in molecular biology, and above all provide explanations of complex biological form that cannot be reduced to expression of a single gene. Phyllotaxis (still) has the potential to become a persuasive example of the need for mathematics in developmental biology.

Background reading: A lovely introduction to this subject can be found in Do Plants Know Math, by Stéphane Douady et al from Princeton University Press; this talk is based on my new undergraduate textbook, Mathematical Phyllotaxis, forthcoming this year from Springer Nature, with additional material from a recent Arxiv preprint: Disk-stacking models are consistent with Fibonacci and non-Fibonacci structure in sunflowers.