*really*fascinating. Not being a botanist, I will not try to mimic the expert terminology used in the references below. Many of these refer to

*phyllotaxis*, which is the arrangement of leaves, seeds and other botanical elements.

References

[C1904] A.H. Church,

*On the Relation of Phyllotaxis to Mechanical Laws,*Williams & Norgate, 1904.

[C2006] Todd J. Cooke,

*Do Fibonacci numbers reveal the involvement of geometrical imperatives or biological interactions in phyllotaxis?*Botanical Journal of the Linnean Society 150 (2006), 3-14

[J1984] Roger V. Jean,

*Mathematical Approach to Pattern & Form in Plant Growth,*Wiley & Sons, 1984 (relevant pages here)

[M1976] P.P. Majumder and A. Chakravarti,

*Variation in the number of ray- and disc-florets in four species of Compositae*, Fibonacci Quartely 14 (1976), 97-100.

[P2015] Matthew F. Pennybaker, Patrick D. Shipman and Alan C. Newell,

*Phyllotaxis: Some progress, but a story far from over,*Physica D, 2015.

*[R1982] J.N. Ridley,*

*Packing Efficiency in Sunflower Heads*, Mathematical Biosciences 58 (1982), 129-139

[R1986] J.N. Ridley,

*Ideal Phyllotaxis on General Surfaces of Revolution*, Mathematical Biosciences 79 (1986), 1-24

[S2016] Donald E. Simanek,

*Fibonacci Flim-Flam,*2016

[S2016] Jonathan Swinton and Erinma Ochu,

*Novel Fibonacci and non-Fibonacci structure in the sunflower: results of a citizen science experiment,*2016.

***

Before entering the subject proper, one caveat:

*spotting Fibonacci numbers does not mean*

*that the golden section has been spotted.*

The former deals with a short sequence of small integers, while the latter is an irrational number involving √5. It's not hard to conceive physical processes in which something is obtained as the sum of the last two somethings, but it's very hard to imagine nature trying to implement √5. Many solutions in nature involve some (approximate) symmetry, but this by no means implies any awareness of the underlying mathematical structure. Creatures have round eyeballs not because π=3.1415... is their favourite transcendental number, but because spherical symmetry offers some evolutionary advantage. Shells grow in logarithmic spirals not because their inhabitants cherish e=2.7182... but because their growth factor is proportional to their size. Likewise, creatures don't have four legs because of √2, nor have plants three leaves because of √3, or five petals because of the golden section. If a child can produce a beautiful five-pointed star totally unaware of any mathematics, why wouldn't Nature be as capable?

**1. Fibonacci numbers: preferred, but not exclusive**

The sequence of Fibonacci numbers is

0,1,1,2,3,5,8,13,21,34,55,89,144,...

each new number being the sum of its two predecessors. You might prefer to start with 1,1 (loosing 0) or with 1,2 (losing 0 and skipping the first 1). As anyone can observe in his own garden: flowers seem to prefer a Fibonacci number for their number of petals. This is not a universal law, but a statistical

*tendency*allowing numerous variations and exceptions. Whole lists can be established of non-Fibonacci flowers; here you can admire some flowers with 4, 6, 8, 14 and 16 petals. But even in those flowers classified as Fibonacci-petaled, Platonic truths give way to statistical blur as soon as larger samples are considered.
[C2006,p.10] has counted three samples, each 100 specimens, of Asteraceae variants.

Each time, the number of petals is normally distributed, the mean values being 12.82, 16.52 and 25.68 respectively. Only the first of these reveals a Fibonacci number, namely 13, and no more than 60% of the flowers got the number 'right'!

And [M1976] gives the statistics for four samples of Compositae. All have a number of petals centered around Fibonacci numbers: 5, 8, 8 and 21, respectively. The fourth is by far the most popular of all Fibonacci plants: the sunflower. While statistically having 21 petals, there is considerable variation to be observed in this sample of some 1000 specimens, and only some 35% can boast the 'right' number, 21.

Sunflowers are famous for yet another feature: the seeds in their heads are arranged in two sets of spirals, one clockwise, the other counterclockwise. (Usually, one of these sets is more conspicuous than the other.) Common lore has it, that the numbers of these spirals are a pair of successive Fibonacci numbers. The factual truth is far removed from this Platonic ideal. In [S2016] 657 sunflowers were examined, implying that 657*2=1314 spirals could be expected. Of these, only 768 (58%) could be unambiguously counted, and among these numbers only 565 (74%) were a Fibonacci number. All kinds of variants and deviations were observed, including 'quasi-regular' heads, in which no spirals could be distinguished at all. In those sunflowers which did have two recognizable sets of spirals, only 66% of the pairs consisted of successive Fibonacci numbers [S2016, table 3].

The same phenomenon is observed throughout the botanical realm: a preference for Fibonacci pairs, combined with a great tolerance for everything deviant. Among the non-Fibonacci pairs observed we find the following. (The last three are in [S2016, fig.4, fig.1 and section 4]; the others in [C1904, p.73, p.197, p.158 and fig.30].)

(3,4)

(4,5)

(5,6)

(6,6), (6,7), (6,8)

(7,7), (7,8), (7,9)

(8,9)

(11,12)

(12,13), (12,15)

(13,13), (13,14), (13,15), (13,16)

(14,15)

(15,15)

(16,16)

(22,23)

(34,53)

(42,68)

(54,87).

Some of these are best regarded as failed attempts at some Fibonacci pair, missing the target by 1 or 2. In other cases, the pairs occur in sequences behaving like Fibonacci (each term the sum of the two preceding terms) but starting differently. We leave you to figure out which of the above observations fit in the sequence

1,3,4,....

1,4,5,...

1,5,6,...

1,8,9,...

or in

2,4,6,... (which is Fibonacci doubled)

or perhaps in

2,5,7,... [J1984, p.5].

For the time being, mother Nature does

The same phenomenon is observed throughout the botanical realm: a preference for Fibonacci pairs, combined with a great tolerance for everything deviant. Among the non-Fibonacci pairs observed we find the following. (The last three are in [S2016, fig.4, fig.1 and section 4]; the others in [C1904, p.73, p.197, p.158 and fig.30].)

(3,4)

(4,5)

(5,6)

(6,6), (6,7), (6,8)

(7,7), (7,8), (7,9)

(8,9)

(11,12)

(12,13), (12,15)

(13,13), (13,14), (13,15), (13,16)

(14,15)

(15,15)

(16,16)

(22,23)

(34,53)

(42,68)

(54,87).

Some of these are best regarded as failed attempts at some Fibonacci pair, missing the target by 1 or 2. In other cases, the pairs occur in sequences behaving like Fibonacci (each term the sum of the two preceding terms) but starting differently. We leave you to figure out which of the above observations fit in the sequence

1,3,4,....

1,4,5,...

1,5,6,...

1,8,9,...

or in

2,4,6,... (which is Fibonacci doubled)

or perhaps in

2,5,7,... [J1984, p.5].

For the time being, mother Nature does

*not*seem fully convinced of the advantages of Fibonacci numbers, and prefers to play around for a while. Yet there must be some hidden Fibonacci mechanism at work: plants outgrowing one Fibonacci pair hurry to the next Fibonacci pair, skipping everything in between.* The world record is held by 1 giant sunflower showing off an impressive (144,233) pair. [J1984,p.2]***Update January 10, 2018.**Writing in the last paragraph 'Plants outgrowing one Fibonacci pair hurry to the next Fibonacci pair' I was under the spell of an optical illusion. The 'spirals' we tend to see at the outer border of a sunflower head are created in the human brain, where points are joined with the nearest points to form some connected chain. It can be mathematically explained why the number of these chains is a denominator of a convergent to the angle. For the golden angle, these are Fibonacci numbers. Moving away from the center bigger denominators become available and after a fuzzy transition one 'sees' more chains than in the previous phase (their number is now the next Fibonacci number) turning in the opposite direction. The changes in direction result from the fact that convergents are alternatively too small and too big approximations to the angle. In fact this growth process has no jumps at all (*natura non facit saltus*, as everybody knows), and there is no hurrying from one set of Fibonacci numbers to the next. Below is a nice illustration of the illusion of spiral-like chains turning one way, the next set in the opposite direction. Without our superimposed colours and circles, transitions would be fuzzy.