Phyllotaxis
the spirals are counted, not admired
The claims, measured

Fibonacci is a branch, not a law

Fibonacci counts come from one branch of the model. The Lucas branch — which the same rule reaches at a different growth rate — gives 47 and 76, and neither is a Fibonacci number. The sequence is a consequence of an angle rather than a property of plants.
6 min read 6 figures State the claim, state the testThe angle is an output

“Sunflower spirals are Fibonacci numbers” is the single most repeated fact about mathematics in nature. It is usually true and it is not a law, and the difference is the whole of this essay.

The spiral counts four different divergence angles produceFibonacci counts come from one angle. The Lucas angle — which the same dynamical model reaches on a different branch — gives 47 and 76, and neither number is a Fibonacci number.golden 137.51°55 and 89FibonacciLucas 99.50°47 and 76Lucas151.14°50 and 81neither77.96°37 and 60neithercounted from the pointsone sequence per branch
Fig. 1 Four heads at four divergence angles, with the counts taken from the points. One gives Fibonacci numbers, one gives Lucas numbers, and two give sequences with no name at all.

Where the Fibonacci numbers come from

They are not in the plant. They are the denominators of the best rational approximations to the golden ratio.

The parastichy numbers of a spiral lattice are the offsets that give short hops, and an offset m gives a short hop exactly when m·δ is close to a whole number of turns — which is to say, when there is a fraction with denominator m close to δ. The best such fractions are the convergents of the continued fraction, and for the golden ratio those denominators are 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.

So Fibonacci numbers appear in a golden-angle head for the same reason they appear in the continued fraction of the golden ratio, which is that they are the same sequence. The plant has not chosen them; they came with the angle.

What other angles give

Change the angle and the sequence changes, because a different number has different convergents.

The Lucas angle, near 99.5°, is the other attractor the dynamical model reaches, and its convergent denominators are the Lucas numbers: 1, 3, 4, 7, 11, 18, 29, 47, 76. A head built at that angle counts 47 and 76 in the band this site measures, and neither is a Fibonacci number.

Lucas phyllotaxis is not hypothetical. It is a real, repeatedly documented minority pattern in sunflowers and other plants, and any account that treats Fibonacci as a law has nothing to say about it.

Other angles give sequences that are not named at all — the convergents of whatever number they are — and a head at 151.14° or 77.96° counts perfectly well. Nothing about the pattern is defective; it is simply not on a famous branch.

The retelling

How a consequence became a law is worth tracing, because the mechanism is general.

The golden branch is broad. A wide range of growth rates lands on it, so most plants that produce spiral phyllotaxis produce the golden angle, so most counts come out Fibonacci. A survey therefore finds Fibonacci overwhelmingly, and “overwhelmingly” becomes “always” in the retelling, and “always” becomes a law.

Then the law needs explaining, and the explanation offered is usually that the plant benefits from the Fibonacci numbers — which inverts the causation entirely. Nothing benefits from a Fibonacci number. A plant benefits, if at all, from not stacking its primordia in rows, and the numbers are what falls out of the angle that achieves it.

What an honest survey would report

The distribution, with the exceptions counted rather than discarded.

Published counts of large numbers of real seed heads do find a Fibonacci majority and a real minority that is not — Lucas pairs, doubled Fibonacci pairs, and heads that cannot be classified at all because the counts are ambiguous or the head is damaged.

That last category is larger than one might expect, and this site’s own measurements say why: near a transition radius three offsets compete, so which pair is counted depends on where the counter worked. A survey with no radius protocol is partly measuring its own methodology.

None of that is a reason to doubt the majority result. It is a reason to state it as a majority.

The version worth keeping

Fibonacci numbers appear in seed heads because a broad attractor of a simple repulsion rule sits at the golden angle, and the golden angle’s rational approximations have Fibonacci denominators.

That sentence contains the mechanism, the reason for its prevalence, and the reason for the numbers, and it makes the exceptions unsurprising instead of embarrassing. It is also considerably more interesting than the version it replaces, which is the usual outcome when a mystery is traded for an explanation.

What the model settles on, against how fast the meristem growsA broad golden branch, a transition, and then the two-whorl regime at exactly half a turn. 8 of 27 converged settings land within 4° of the golden angle; 15 land more than 20° away.1001251501750.50011.50growth parameter Gangle the model settles on (°)goldentwo-whorlfilled: converged · hollow: still wanderingone rule, one knob
Fig. 2 The diagram the constant is one branch of. Sweeping the growth parameter gives a golden branch, a transition and a two-whorl regime at half a turn.
The two spiral families a counter finds between 0.55 and 0.95 of the radius21 spirals one way and 34 the other, found from the point positions alone — the counter is never told the divergence angle.21 and 34 spiralscounted, not assumed
Fig. 3 The counted spiral families drawn as the polylines joining every mth point — a fact about neighbours rather than a fitted curve.
The spiral counts, band by band, in one headThe same flower gives 13,21, 21,34, 34,55, 55,89 at different radii. A caption saying "34 and 55 spirals" is a statement about one annulus.0255075406080fraction of the head's radiusparastichy numbers found in a band there1000 primordia at the golden angle4 different pairs
Fig. 4 The counts band by band in one head. Three different pairs from one point set at one angle.
Continued fractions: why one number resists approximationA large partial quotient means a very good rational approximation just ahead of it. The golden ratio's are all 1, the smallest they can be, all the way down.golden ratio[1; 1, 1, 1, 1, 1, 1, 1, 1, …]best approximations 2/1 3/2 5/3√2[1; 2, 2, 2, 2, 2, 2, 2, 2, …]best approximations 3/2 7/5 17/12π[3; 7, 15, 1, 292, 1, 1, 1, 2, …]best approximations 22/7 333/106 355/113partial quotientsall ones is the extreme case
Fig. 5 The arithmetic underneath the sequences. A large partial quotient means a very good rational approximation just ahead of it.
How nearly each angle is a simple fraction of a turnA dip means a rational approximation and a lattice with visible rows. The golden angle's floor is 0.008; the rational angle reaches 0e+0.00.2000.400204060denominator qhow nearly q × angle is a whole turn (0 = exactly)golden 137.508°137.3°135° = 3/8distance to the nearest whole turnno dips means no rows
Fig. 6 How nearly each candidate angle is a simple fraction of a turn. A dip means a good rational approximation, and a good rational approximation means visible rows.

Where this goes next

1001251501750.50011.50growth parameter Gangle the model settles on (°)goldentwo-whorlfilled: converged · hollow: still wanderingone rule, one knob The bifurcation diagram Sweep the one parameter of the rule and the settled angle traces a diagram — a broad golden branch, a transition, and a two-whorl regime at exactly half a turn. The famous constant is one branch of it, which is a more useful thing to know than the constant. 21 and 34 spiralscounted, not assumed Counting the spirals Almost every claim about phyllotaxis is a claim about how many spirals run through a pattern, and the count is almost never done. It can be done from the points alone, by machinery that is never told what angle built them — and then a count of 34 is evidence rather than a restatement. 00.2000.400100120140160divergence angle (°)resistance to rational approximation (higher is more irrational)1/√5 — the bound137.508° — 0.4381500 angles on a 0.05° grid, plus the golden angle exactlythis claim is sharp The claim that survives Of the three famous assertions about this subject, one is out by a factor of two, one is true of a branch rather than of plants, and one is right — in a sharper form than the version usually told, and about arithmetic rather than about packing.