The Fundamental Theorem of Phyllotaxis revisited

TL;DR

This paper revisits and completes the Fundamental Theorem of Phyllotaxis, clarifying its application to spiral counts and angles in plant patterns, which is crucial for understanding Fibonacci structures in nature.

Contribution

It provides a complete and corrected statement and proof of the Fundamental Theorem of Phyllotaxis, addressing an important special case previously overlooked.

Findings

Corrected the theorem for an important special case

Provided a complete proof of the theorem

Clarified the relationship between spiral counts and angles

Abstract

Jean's `Fundamental Theorem of Phyllotaxis' (\emph{Phyllotaxis: a systematic study in Plant Morphogenesis}, CUP 1994) describes the relationship between the count numbers of observed spirals in cylindrical lattices and the horizontal angle between vertically successive spots in the lattice. It is indeed fundamental to observational studies of phyllotactic counts, and especially to the evaluation of hypotheses about the origin of Fibonacci structure within lattices. Unfortunately the textbook version of the theorem is incomplete in that it is incorrect for an important special case. This paper provides a complete statement and proof of the Theorem.