Fibonacci is a branch, not a law

“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.

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.

The sequence, and where it actually lives #

The Fibonacci numbers in a seed head are not in the seeds. They are in the continued fraction of the golden ratio, and they arrive in the pattern through a chain of three steps that is worth setting out in order.

Step one. The divergence angle is the golden angle, because that is where a repulsion rule settles over a broad range of growth rates.

Step two. The parastichy numbers of a spiral lattice are the denominators of the good rational approximations to its divergence angle — an offset m gives short hops exactly when there is a fraction with denominator m close to the angle.

Step three. The convergent denominators of the golden ratio are 1, 2, 3, 5, 8, 13, 21, 34, 55, 89. That is the Fibonacci sequence, by definition of the continued fraction whose partial quotients are all 1.

So the sequence enters at step three, from arithmetic, and the plant contributes step one. Nothing anywhere counts.

What the Lucas branch looks like #

Not hypothetical, and worth being concrete about.

The Lucas angle is 360·(5−√5)/10 = 99.502°. Its continued fraction convergent denominators are 1, 3, 4, 7, 11, 18, 29, 47, 76, 123 — the Lucas numbers — and a head built at that angle counts 47 and 76 in the band this site measures.

Neither of those is a Fibonacci number, the head looks perfectly ordinary, and the spirals are as clean as a golden-angle head’s. A reader shown the two side by side without labels would not be able to say which was which.

Lucas phyllotaxis occurs in real plants at a low but consistent rate — a few per cent in surveys of sunflowers, and more commonly in some other genera. Any account that treats Fibonacci as a law has nothing whatever to say about those plants.

What a survey would need to do #

If the question is “how often is a real seed head Fibonacci”, the answer depends on choices that are rarely stated, and this site’s measurements say which.

At what radius was it counted? The counts change across a head, so a protocol that counts wherever the spirals are clearest is partly measuring where spirals are easy to trace.

What was done with the ambiguous cases? Near a transition, three offsets compete and the pair is genuinely undetermined. Discarding those heads as unclassifiable biases the result toward the clean cases, and clean cases are the ones well inside a regime.

What counts as Fibonacci? A pair like 34 and 55 is unambiguous. A pair like 4 and 7 is Lucas. A pair like 6 and 10 is double-Fibonacci — a bijugate pattern with two parallel spiral systems — and whether that counts as Fibonacci is a convention rather than an observation.

None of that makes a survey impossible. It makes the protocol part of the result, and a reported percentage without one is not comparable with another.

The inversion worth naming #

The commonest form of the claim gets the causation exactly backwards.

It says: plants use Fibonacci numbers, and Fibonacci numbers are good for packing, so the plant benefits.

Nothing benefits from a number. What a plant packing many primordia into a disc benefits from is not stacking them in rows, which means avoiding divergence angles near simple fractions — and that is a property of the angle, not of the counts. The counts are downstream of the angle, and they are downstream of it through pure arithmetic with no biology in the chain at all.

Reversing that produces a mystery where there is none, and the mystery is the reason the subject has the reputation it has.

Double Fibonacci, and what a bijugate pattern is #

One family of exceptions deserves its own paragraph, because it looks like a counterexample and is not.

Some heads count 6 and 10, or 10 and 16, or 16 and 26 — pairs that are twice consecutive Fibonacci numbers. Those are bijugate patterns: the plant has two parallel spiral systems offset by half a turn, so every count doubles.

Geometrically this is a lattice with a non-coprime parastichy pair, and it is exactly the case the angle recovery declines to answer: when the two counts share a factor, no single divergence angle produces the pattern, because it is really two interleaved patterns.

So a bijugate head is not evidence against the Fibonacci story and it is not evidence for it either. It is a different structure, and treating a count of 10 and 16 as a failed Fibonacci pair rather than as a doubled one is a classification error that a survey has to decide about explicitly.

The honest summary #

Fibonacci parastichy numbers are common, explicable and not universal.

Common, because the golden branch of the dynamical model is broad and a wide range of growth rates lands on it.

Explicable, because the counts are the convergent denominators of the divergence angle and the golden ratio’s are the Fibonacci numbers.

Not universal, because other branches exist, produce other sequences, and occur in real plants at rates that are small and consistently non-zero.

Every clause of that is checkable and every clause is more informative than the version it replaces. The version it replaces has a mystery in it, and the mystery was manufactured by leaving out the middle step.

Why this is the claim worth correcting #

Of the three assertions this site measures, this is the one that does the most damage, and it is worth saying why.

The nautilus claim is simply false and correcting it costs nothing — nobody’s understanding of anything rests on it.

The packing claim is a loose version of something true — right about how far a head keeps its organs apart, for every noble angle alike, and wrong about its cells — and the correction improves it.

The Fibonacci claim is different because it is usually right, which makes the exceptions invisible and the mechanism unnecessary. A reader who believes plants use Fibonacci numbers has an account that works for nineteen heads in twenty and has no way to notice the twentieth, and no reason to ask where the numbers come from. The claim is self-sealing in a way the other two are not.

Replacing it with the three-step chain — a rule settles on an angle, the angle has convergents, the convergents are the counts — costs a sentence and makes the exceptions expected rather than embarrassing. That is the trade this site keeps proposing, and this is the case where it buys the most.

What to take away #

Fibonacci numbers in a sunflower are a consequence of arithmetic applied to an angle that a physical process settles on. They are not a plant’s arithmetic, they are not selected for, and they are not universal.

The interesting fact is not the sequence. It is that a rule with no numbers in it lands on an angle whose approximation properties are extremal, and that this can be demonstrated in a dish of oil with no biology in it.

How common is the exception #

The honest question after “it is not a law” is “how often does it fail”, and there are numbers.

Surveys of divergence angles across seed plants put roughly 90–95% of spiral species on the Fibonacci branch. The Lucas branch — 99.5°, counts 4, 7, 11, 18, 29, 47, 76 — accounts for a couple of per cent, and is well documented in particular groups: some cacti, some sunflower cultivars, Aeonium.

Rarer still are bijugate patterns, where the counts are Fibonacci numbers doubled (2·8 and 2·13, say). These come from a meristem that produces primordia in pairs, and they are a different phenomenon rather than a different angle — the lattice is two interleaved copies.

And there are anomalous series that fit no named sequence, which are exactly what the model’s other branches predict should exist at low frequency.

So the popular claim is a good description of the common case and a bad description of the rule. “Almost always Fibonacci, sometimes Lucas, occasionally something else, and the model produces all of them” is not much longer and is correct.

The branches are more numerous than the named ones #

“Anomalous series that fit no named sequence” is the vaguest clause in the summary, and it can be made specific, because the rule’s own settled runs have been counted.

Sweeping starting angles and rises and counting every run that settles gives seventeen distinct parastichy pairs, and they sort into about ten additive sequences — each one a ladder with a limit divergence of its own, computable from its first two terms. The Fibonacci and Lucas branches are two of the ten. The others are ordinary in every respect except that nobody has named them: 1, 4, 5, 9, 14 with a limit near 78°; 2, 5, 7, 12, 19 with a limit near 151°.

So “something else” is not a residue category. It is at least eight further branches, and the rule reaches several of them more often than it reaches the Lucas one — the 151° branch alone takes more runs than either named ladder.

That does not change the field frequencies, which are what plants do rather than what the rule can do. It changes what the frequencies are evidence about: a survey finding 90 per cent Fibonacci is not reporting that the rule prefers one ladder, since the rule visibly does not. It is reporting something about which basins a real meristem starts in.

Why one branch dominates #

If the branches are all attractors of the same rule, the frequencies need explaining, and the explanation is about basins rather than about optimality.

Starting the model from a nearly empty meristem — one or two primordia, no established pattern — the golden branch is reached from most initial angles. The Lucas branch is reached from a narrower set. The basin of attraction of the golden branch is simply bigger, and a plant beginning from an unstructured meristem is sampling initial conditions from something like a broad distribution.

That is a much better explanation than “Fibonacci is optimal”, for two reasons. It predicts the observed frequencies, which an optimality argument does not — an optimality argument predicts that everything should be on the best branch. And it predicts which alternative should be second most common, since the Lucas branch has the next largest basin.

It also makes the exceptions ordinary. A cactus on the Lucas branch is not defective or unusual; it started somewhere slightly different.

What the counter would have to do to be lying #

The claim that the Lucas angle gives 47 and 76 rests on a counter handed only coordinates, so it is worth asking what would have to be true for it to produce those numbers spuriously.

It would have to prefer offsets near 47 and 76 for reasons unrelated to the point positions, which is testable: run it on a golden-angle head of the same size and it returns 34 and 55.

It would have to be tuned to the answer, which it cannot be — it has no list of expected sequences, no reference to the divergence angle, and the same code path serves every figure on the site.

Or the head itself would have to be wrong, which the angle recovery checks independently: fed the counts and the radius, it returns 99.502°, which is the angle the head was built at and which the recovery was never shown.

Two pieces of machinery that share no code agreeing on a number neither was told is the strongest form of evidence available here, and it is the arrangement used throughout.

The claim, restated so it can be checked #

Putting it together, the version worth repeating is:

A spiral seed head’s parastichy numbers are the denominators of the best rational approximations to its divergence angle, measured in an annulus at a stated radius. Most plants sit on a dynamical branch whose angle is 137.5°, whose approximation denominators are the Fibonacci numbers; a few sit on a branch at 99.5°, whose denominators are Lucas numbers; the same one-parameter model produces both.

Every clause of that is a measurement someone can make. Count the spirals, state the radius, recover the angle, and compare against the model’s branches.

The popular version — “sunflowers use Fibonacci numbers” — has no clause anyone can check, which is why it has survived a century of counterexamples.