Fibonacci numbers in plants, such as in sunflower spiral counts, have long fascinated mathematicians. For the last thirty years, most analyses have been variants of a Standard Model in which plant organs are treated as point nodes successively placed on a cylinder according to a given function of the previous node positions, not too close or too far away from the existing nodes. These models usually lead to lattice solutions. As a parameter of the model, like the diameter of the cylinder, is changed, the lattice can transition to another, more complex lattice, with a different spiral count. It can typically be proved that these transitions move lattice counts to higher Fibonacci numbers. While mathematically compelling, empirical validation of this Standard Model is as yet weak, even though the underlying molecular mechanisms are increasingly well characterised.
In this talk I'll show a gallery of Fibonacci patterning and give a brief history of mathematical approaches, including a partially successful attempt by Alan Turing. I'll describe how the classification of lattices on cylinders connects both to a representation of $SL(2,Z)$ and to applications through defining the constraint that any model must satisfy to show Fibonacci structure. I'll discuss a range of such models, how they might be used to make testable predictions, and why this matters.
From 2011 to 2017 Jonathan Swinton was a visiting professor to MPLS in Oxford in Computational Systems Biology. His new textbook Mathematical Phyllotaxis will be published soon, and his Alan Turing's Manchester will be republished by The History Press in May 2022.