13 February 2021

Sunflower heads and other parabolic spirals

My interest in sunflower heads arose as part of my crusade against golden ratio misconceptions. Most of the claims involving the golden ratio are easily dismissed, but sunflowers are in a different league. There is no denying that they prefer Fibonacci numbers (see here), and this calls for an explanation. Botanists like Ridley (reference below) assure us that a fixed divergence angle, let alone a 'golden' one, is not the answer (see also here), but what is the answer then? In our paper (here) we contribute something to this fascinating question. Our text is self-contained, but more context is found in

1. Helmut Vogel, A Better Way to Construct the Sunflower Head, Mathematical Biosciences 44 (1979), 179-189.
2. J.N. Ridley, Packing Efficiency in Sunflower Heads, Mathematical Biosciences 58 (1982), 129-139.
3. Ivan Niven, Herbert S. Zuckerman  and Hugh L. Montgomery, An introduction to the theory of numbers, Fifth Edition, 1991, Chapter 7.
4. Oskar Perron, Die Lehre von den Kettenbrüchen, Band I. Dritte Auflage, 1954.

Ridley was the main source of inspiration.

It will turn out that the key mathematical notion is that of continued fraction, that Fibonacci and Lucas numbers impose themselves for botanical reasons, and that the golden ratio is not mentioned. As a matter of fact, there is no need to mention any irrational number. A FIDIPAS (finite discrete parabolic spiral) is completely characterised by three positive integers: besides the number N of points, the numerator p and denominator q of the fraction 𝛿 which determines how much of a turn is needed to get from one point to the next.

From the paper we only display the very last illustration, where a real sunflower of some 920 seeds (#56 of this collection) is compared to its mathematical counterpart of 920 points. Everything about the spirals is perfectly and quantitatively predictable.