06 November 2016

Fibonacci's tricky botany (2)

(Sequel of part 1)

2. Contrary to popular belief: golden packing is not optimal

Many people believe that the golden section is mysteriously at work in plants. A popular piece of evidence is the packing of seeds in sunflower heads, which is claimed to be optimal and to rely on the golden angle. This in turn would explain Fibonacci patterns. All of this is wrong.

First, the notion of a constant divergence angle, golden or not, is nothing else but a mathematical artifact (see here). The "constant divergence assumption" is expressly mentioned as one of the two elements leaving even the best phyllotaxis models somewhat idealized [R1986, p.2].   

Second, if efficient packing is the aim, Mother Nature has made the wrong choice. Hexagonal close packing, for instance, has an efficiency that is 40% higher than is obtainable by spirals [R1982,p.132]. Also, evolution doesn't create optimal solutions; it settles for any working solution, which it tries to stabilize at all price. 

Third, among spiral packings, the golden divergence angle provides an optimum only for infinitely large flower heads consisting of seeds of equal area [R1982,p.133]. Of course, all flower heads are finite, and for any of these there are spiral packings more efficient than the golden one [R1982,p.137]. Before giving examples we need the following definitions.

(1) is, of course, our beloved Mystic Number, (2) is its complement (or, which is the same thing, its square) and the angle in (3) is the golden angle γ. A sunflower head consisting of N seeds is mathematically modeled by the spiral given in (5). Its shape depends on the divergence angle α, which in (4) is expressed as a fraction f of the circle. (The golden angle corresponds to f=1-φ.) If d is defined by (6), the efficiency of the packing is given by (7). In [R1982, p.133] it is proved that the absolute maximum of η for infinite spirals is given in (8), and that it is then attained only if the divergence angle is golden. For finite spirals, it is quite possible to have η>ηmax, see table below. Efficiency (effg for the golden angle, eff for the angle adeg) is expressed as a percentage of ηmax.

(In case you want to see for yourself, here is the Geogebra file.) To assess what "more efficient" means, consider the two packings below, for N=500. The first is our champion beating the gold chap, while the second is an anonymous competitor, performing very well in spite of his angle—a good 99°—being completely different from the champion's. The red diamonds are the points with the smallest distance, determining the efficiency.

No doubt we could sell the last one to any golden numberist, claiming its divergence is the magic 137.50° angle—while it's nothing of the sort! People are impressed with how homogeneous a packing the golden angle achieves, but they are unaware of homogeneous packings with entirely different angles. They would be equally impressed with 99.51° (above, 96.5% of ηmax) or even 77.96° (below, 93.8% of ηmax).

In case you want to do a systematic search yourself: the Geogebra command

Execute[Join[Sequence[Join[{{"If[pct>80,SetValue[P,Length[P]+1,pct]]", "If[pct>80,SetValue[F,Length[F]+1,frac]]", "frac=frac+0.0001"}}],i,1,1000]]]

yields a list P with all percentages above 80 and the list F with the corresponding fractions. The search starts from the actual value of frac, and performs 1000 steps of 0.0001. (Adapt as you wish.)

3. Packing efficiency does not explain Fibonacci patterns

Singling out the golden packing among several of comparable efficiency requires a fine-tuning far beyond what biological processes can achieve. Two expert voices:

packing efficiency fluctuates so wildly with changes in the divergence angle 
that it can only be regarded as a consequence, 
not a cause, of Fibonacci phyllotaxis 
[R1982,p.139 and p.130]

patterns with small Fibonacci numbers do not exhibit optimal packing 
and the tighter packing observed in larger Fibonacci spirals must be 
a secondary consequence of some other process.

Courtesy of Geogebra, everybody can see for himself how wildly packing efficiency changes with the angle. The packing in the last image above drops from 96.50% to 55.16% if the angle is increased by no more than 0.0001 of a turn!

In 1982 the conclusion was

Explanations for Fibonacci phyllotaxis are extremely complicated, 
and the riddle is by no means solved.

Thirty-five years later the state of the art given in [P2015] has the telling title

Phyllotaxis: Some progress, but a story far from over

The overall conclusion must be that, while some Fibonacci mechanism is undeniably at work in botany (though less universally or unambiguously than generally believed), a convincing explanation is still lacking. Some alleged "explanations" are even busted at first sight. Let's consider leaves arranged around a stem, as in the picture below [J1984,p.4].

The sixteen successive leaves have been numbered 0,1,...,15. If the numbers differ by 8, the leaves are more or less superimposed (0 and 8, 1 and 9, etc.), and you get from one to the other by 3 complete clockwise turns around the stem. This pattern displays the Fibonacci numbers 3 and 8. Golden numberists seek the explanation in the fictitious golden angle, whose irrationality would avoid that lower leaves are in the shadow of leaves above. This doesn't make any sense, because the sun is only rarely in the zenith, and in major parts of the world it never is. Moreover, leaves are known to easily adapt their position to get better light conditions. In short, it's no argument at all. As such, it is typical of

the Pythagorean mysticism 
plaguing many scientific and popular expositions
 [C2006, p.4]

Actually, it's rather Plato who is to blame. In Timaeus (55 C) he associates, in obscure terms, the regular dodecahedron with the universe, and from this one might conclude that the golden section (the essential ingredient in the dodecahedron) reflects the aesthetic taste of the divine creator.

Plato worshipping mathematical beauty,

incarnated in—what else?—


* *

Thanks to abakus and Roman Chijner for solving my Geogebra question.