11 May 2018

Fibonacci's tricky botany (3)

  1. H. 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. J.N. Ridley, Ideal Phyllotaxis on General Surfaces of Revolution, Mathematical Biosciences 79 (1986), 1-124.
*

Since Vogel's paper [1], the sunflower head is usually modeled as a parabolic (or Fermat) spiral, in which the radius is proportional to the square root of the polar angle. On this spiral, the sunflower seeds are represented by points that are added by increasing the polar angle by some fixed amount called the divergence angle. As all mathematical models, this one too has its limitations. First, there is no such thing as a fixed divergence angle in nature; plants are lousy geometers, just look at the clumsy pentagons or hexagons they produce. Second, real sunflowers add their seeds not at the outer rim but in the center, pushing everything outward.

A parabolic spiral is as tightly wound as a ship's rope. Here is one with 45 points added after some fixed divergence angle.
The primary spiral is rarely shown, and if it's absent the human eye simply connects nearby points and creates its own secondary spirals, like the three below. These are more than just optical illusions: in each one each point is followed by the nearest of its successors. In a way they can be regarded as the fastest communication lines between the center and the edge.

The most famous examples of these secondary spirals are the Fibonacci spirals in golden sunflowers. If the divergence angle is φ2 times a complete turn, the human eye detects a Fibonacci number of spirals turning one way and, with some luck, the previous Fibonacci number of spirals turning the opposite way. Here a few of the countless confirmations of this striking phenomenon are gathered. (For exceptions, see here.)

Seeing Fibonacci numbers arise in sunflowers: where has the sound principle Nature does not make jumps gone? Sunflowers display 21 spirals, but not 22, 23, etcetera until yes! 34. Some jump, from 21 to 34, no?

Clearly, this calls for an explanation. We start with the purely mathematical golden sunflower, which is infinite in size and governed by a golden divergence angle. Many ready-made explanations of the Fibonacci phenomenon can be found scattered around. Most of these are simple but wrong, and we challenge you to find one that is correct and convincing. After that, see our own proof in Fibonacci numbers in the golden sunflower.

This is the resulting figure for a golden sunflower with 1000 points:


Our 1000 points occur in zones, which we separated by circles. From the center out we meet
  • grey: 16 scattered points
  • red: 13 "spirals" (actually segments) turning clockwise 
  • blue: 21 spirals turning counterclockwise
  • purple: 34 spirals turning clockwise
  • green: 55 spirals turning counterclockwise
  • white: 89 spirals turning clockwise (under construction, only the first 120 points being available if the total number is 1000)
The fatter points are those that you obtain if you pass from the very first point to its nearest successor and then repeat this as far as you can.

It is very clear now how a continuous process can result in apparent jumps from one Fibonacci number to the next. Near the outer edge, the 55 green spirals are visually very striking because the next generation is still embryonic, while the previous generation is far behind. Adding points, the white 89 generation with its spirals turning in the opposite direction will slowly come into being, then start to dominate over the green 55 generation, which may however be observed for some time by extrapolation outside its habitat. At some point in the process the two successive generations create impressions of comparable strength, leaving the eye in doubt as to which paths are the shorter ones. After that, the new generation will unambiguously dominate, and the whole process will repeat.

In our mathematical sunflower, the Fibonacci phenomenon arises as a result of the golden divergence angle. This cannot be the case in real sunflowers. It is botanically impossible that a flower repeat a constant angle of 137.5° a 1000 or 2000 times in succession with the precision that is required to generate convergents with the denominators that are observed. Even slight variations in the angle destroy the Fibonacci order. If the golden angle is modified at each step by alternatively +0.00001 and -0.00001 of a turn you get this:
At the edge we count 17 double spirals, which yields the Fibonacci number 34 all right, but the 55 structure has completely gone. And if the perturbation is alternatively 0.001 and -0.001 (still tiny, 1 part in 1000) even 34 vanishes, leaving us with 21 spokes:




Generally, for two successive denominators r,s to be observable in this mathematical fixed-angle model, the angle must be constant within an absolute value of at most 1/(rs). For real sunflowers, this implies the following precision to generate the values observed in the outermost spirals:

for 21,34   : 0.00140
for 34,55   : 0.00053
for 55,89   : 0.00020
for 89,144 : 0.00007

and for the giant sunflower supposedly displaying 144,233: 0.00002. Now go into your garden and look at a flower with five petals. Geometrically speaking, each of the five angles should be 72°, but they're unequal and far from the exact value. And our giant sunflower would have to apply some 5000 times in succession an angle with a precision of 0.00002! Come on. There must be some Fibonacci phenomenon in its own right that generates these numbers. As far as we know, Ridley [2] is the only one to clearly admit this and to have sought and found [3] such a Fibonacci model.

In real sunflowers Fibonacci spirals, however they are generated, are undeniably real. In the same way, flowers in general do prefer a Fibonacci number of petals. Somehow this may be related to Fibonacci spirals being the fastest communication lines between the center and the edge.


* * *