22 March 2021

Mathematical equation of a sunflower head

This is sunflower #56 (image43.jpeg) from the archive Annotated seedhead images (here), mirrored horizontally.

I counted the number of seeds and found some 920 of them (near the rim, there is some ambiguity). I then had GeoGebra produce 920 points on a parabolic spiral whose parameters required some meticulous trial and error.

Superposing the natural sunflower and its mathematical counterpart shows how good the fit is. The white point is the pole of the spiral.

Feel free to experiment with my GeoGebra file, which is at your disposal here. Don't be too fussy though. In our real sunflower, there are 55 spirals turning one way and 89 the other way, but their curvature, as seen in the photo, is not constant. For proof, see image below. The left arc, marked "1", has been fitted as well as could be to the underlying "real spiral", which is one of 55. It has then been placed without deformation in two other places, where the fit is very poor. 

Now this may be a photographic deformation rather than a defect of the flower head. I therefore subjected the photo to some transformations in GIMP. Below is the superposition of the mathematical spiral and the adapted photo. 

Note that the 920 angular displacements represent more than 350 complete turns. Now the distance between the windings of a parabolic spiral decreases with every turn, and so the individual windings in our spiral soon cease to distinguishable, see image below. 

A nice feature of a parabolic spiral (indeed, one of the reasons for its introduction in botanical models) is that the mean density of the points added on it is constant, and so the discrete image (second image in this post) is entirely homogeneous, though the spiral itself is completely compressed. The angle is such that points added on this totally compressed spiral are as far from each other as possible!

Among mathematical botanists the logarithmic spiral (of Nautilus Pompilius fame) seems to be preferred over the parabolic one, though the former spiral requires excluding the core of the sunflower head, where it leads to absurd implications. Vogel's parabolic spiral (introduced in 1979, hereneeds no such interventions, as our example clearly shows. Moreover, Ridley's theorem (here), also using the parabolic spiral, is the only one I know that proves mathematically that the golden angle is the most efficient one — for an infinite sunflower that is (not so for finite ones; for those, see here).

Our divergence angle required some fine-tuning of magnitude 0.00001, the effect of which is quite noticeable. Mathematical botanists agree on the fact that extreme precision in the divergence angle is required to produce the aspects (well, some of the aspects) observed in real sunflowers. They disagree in their conclusions though. Some (Ridley 1982, here) feel that plants are simply not capable of iterating some 1000 times an angle with that precision, while others (Okabe 2012, here) are convinced they miraculously do.