How often is it Fibonacci
“Sunflower spirals come in consecutive Fibonacci numbers.” The claim is everywhere, it is usually stated without a qualifier, and it is the reason most people have heard of phyllotaxis at all.
The foundation essays already softened it in two ways. The counts change with radius, so a pair is a statement about an annulus. And Fibonacci is a branch rather than the output — the dynamical model reaches the Lucas angle too, and real plants with Lucas phyllotaxis exist.
Neither of those is a frequency. This essay asks the question that is actually being made: given a divergence angle, how likely is it that the pattern’s counts are consecutive Fibonacci numbers?
Asking the geometry rather than a field survey
The honest way to answer “how often” is a survey of real plants, and good ones exist. They are hard to do — a count needs a stated radius or a stated position on a stem, and most published counts have neither — and this site cannot do one.
What it can do is ask the same question of the geometry. Fix a rise. Sweep the divergence across the whole available range. At each divergence, compute the dominant parastichy pair, and classify it: consecutive Fibonacci, consecutive Lucas, whorled — meaning the two counts share a factor — or none of those.
The result is a measure: the fraction of the circle of possible divergences that produces each kind of pair. It is not a prediction about plants, because plants are not uniformly distributed over divergence angles. It is the thing one has to know before a statement about plants means anything, which the geometry supplies for nothing and which a claim about biological preference is implicitly comparing against.
The numbers
At a rise of 0.12 — a coarse pattern, few elements per turn, a young shoot with long internodes:
Fibonacci 67.4% · Lucas 27.1% · whorled 4.9% · other 0.6%
At 0.05:
Fibonacci 40.0% · other 29.8% · whorled 19.2% · Lucas 11.0%
At 0.02:
other 44.4% · whorled 27.7% · Fibonacci 24.1% · Lucas 3.8%
At 0.008 — a fine pattern, a pine cone or a seed head:
other 48.4% · whorled 35.5% · Fibonacci 14.7% · Lucas 1.5%
The Fibonacci share falls by a factor of four and a half as the pattern gets finer. The share of pairs with no name at all rises from under one per cent to nearly half.
What is being counted, exactly
The census is a simple thing and it is worth being precise so the number is not read as more than it is.
Fix a rise. Step through 1,200 divergence angles evenly spaced from 0° to 180° — half a turn, since and are mirror images and produce identical counts. At each, compute the lattice vectors for offsets 1 to 140, take the two shortest, and classify the resulting pair.
Whorled first: the two counts share a factor, so the pattern is rows and not spirals. Fibonacci next: the pair is two consecutive members of 1, 1, 2, 3, 5, 8, 13, 21… Lucas: two consecutive members of 1, 3, 4, 7, 11, 18, 29… Other: everything else.
The order matters. Checking whorled first means a pair like 2/4 is classed as a whorl rather than falling into “other”, which is correct — it is a different kind of object, not an unusual spiral. And the four categories are exhaustive and mutually exclusive, which the build checks by requiring the shares to sum to exactly 1.
That check is not a formality. The first version of the classifier had Fibonacci ahead of whorled, and 2/4 — being 2 and 4, neither of which is adjacent to the other in the Fibonacci sequence — fell through to “other” while 1/2 and 2/3 were caught. The shares still summed to 1, so the sum check passed. What caught it was the whorled share coming out implausibly low at fine rises, where whorls should be common.
Why it falls
The reason is in the van Iterson plane. At a large rise there are very few possible pairs, because only small offsets can be short: at 0.12 essentially every divergence gives 1/2, 1/3, 2/3 or a whorl, and three of those four are consecutive Fibonacci or Lucas pairs by virtue of being small.
As the rise falls, the plane divides. Each region splits at its fork into two narrower ones, so the number of distinct pairs available grows and each individual region shrinks. The Fibonacci regions shrink along with everything else, and since there is one Fibonacci region at each rise and an ever-growing number of others, its share of the circle falls.
So the high Fibonacci share at coarse rises is not evidence of anything. It is what comes of having four options, two of which are called Fibonacci.
The finding this reverses
The natural reading of “Fibonacci is common in plants” is that Fibonacci pairs are somehow favoured — that the geometry is generous towards them, or that they are more stable, or occupy more room.
The measurement says the opposite. At the rises where real seed heads and cones sit, a Fibonacci pair is what an arbitrary divergence gives about one time in seven, and it gets rarer the finer the pattern. Whatever makes real plants Fibonacci is working against a geometry that becomes steadily less generous towards it.
That is a much sharper statement of the problem than “why 137.5°”, and it has an answer already on the site. From the tree: a pattern that begins coarse — where Fibonacci pairs are two thirds of everything — and then never jumps branch as its rise falls stays Fibonacci all the way down, however narrow its region becomes.
Which is to say the frequency is not about the geometry being kind at fine scales; it is about continuity from a coarse start. A plant is Fibonacci because it was Fibonacci when it was small and nothing made it leave, not because Fibonacci is where a fine pattern naturally lands.
What “other” contains
Nearly half the circle at fine rises produces pairs belonging to no named sequence: 4/9, 7/12, 5/13, and hundreds more.
These are not exotic. They are the ordinary pairs of the ordinary branches of the tree, and every one of them is the parastichy pair of a perfectly good cylindrical lattice. There are simply a great many branches, most of which nobody has named because no common plant is on them.
Two named sequences and an unbounded number of unnamed ones is the actual structure, and the naming reflects which branches real plants occupy rather than anything about the geometry. A tree where the branches were named after their limits would have Fibonacci, Lucas, and then an infinite family of noble numbers with no popular names at all.
What the census would have to be to be about plants
It is worth spelling out what a real frequency claim needs, because the gap between this measurement and that one is where most of the confusion lives.
A statement like “most plants are Fibonacci” is a statement about a distribution over species, weighted somehow, at some stage of development, counted at some position. Every one of those has to be pinned down before the sentence has a truth value.
Which plants. Weighted by species, by individuals, or by the number of leaves? The three give different answers, and the popular claim is usually smuggling in “the plants people photograph”.
At what stage. The counts change through a shoot’s life, so a survey of mature specimens and a survey of young ones are different surveys.
Counted where. On a disc the pair depends on the radius; on a cone it depends on the position along the axis. Only on a stem is the count a property of the object, which is one more reason the cylindrical literature is the one with a usable comparative record.
None of that is available here. What is available is the baseline: at a given rise, this much of the parameter space gives Fibonacci. Whatever the properly-specified biological frequency turns out to be, it has to be compared against that number rather than against an unstated assumption that Fibonacci is what the geometry hands over.
What “whorled” contains, and why it grows
The whorled share climbs from 5% to 35% as the rise falls, and this is worth a note because it is not obviously right.
A whorled pair is one whose counts share a factor, which happens when the divergence is near a simple fraction of a turn. At a coarse rise the only fractions with small enough denominators to matter are a half and a third, so the whorled set is small. As the rise falls, larger denominators come into range — quarters, fifths, sevenths — and each brings its own band of divergences producing rows rather than spirals.
So finer patterns are more susceptible to whorling, not less. That is the opposite of the intuition that a fine pattern is more “spiral”, and it is a consequence of the same region-splitting that shrinks the Fibonacci share.
It also means the classification is doing real work. A pair like 6/10 is not a failure of the counter; it is a lattice with two rows of five, and the recovery correctly refuses to extract a single divergence angle from it.
The two ways of being wrong about a frequency
The claim being examined has two failure modes and they pull in opposite directions, which is part of why it survives.
Over-claiming. “All sunflowers have Fibonacci spirals” is refuted by any counterexample, and counterexamples exist — Lucas phyllotaxis is documented, whorled arrangements are common, and surveys of real material that state their counting radius find a non-trivial fraction of non-Fibonacci heads.
Under-specifying. “Fibonacci numbers occur remarkably often in plants” is nearly unfalsifiable, because “remarkably” is comparing against an unstated baseline. This essay’s contribution is to supply a baseline, and the direction it points is the interesting part: the baseline is low at the scales of real material, so the observed frequency is more remarkable rather than less.
That second point is worth being clear about, because a measurement that makes a popular claim look better is not the usual outcome on this site. The census does not undermine the observation that plants are overwhelmingly Fibonacci. It sharpens it, by removing the explanation that the geometry was generous.
What is asserted, and what it can reject
The build requires the four shares to sum to exactly 1 at every rise the figure draws, which catches a divergence classified twice or not at all.
Separately, the census is required to change across the slider’s range — the same discipline every draggable figure on this site is under, since a slider whose readout does not move demonstrates a claim not happening. Here that requirement is doing real work: a classifier with a bug that put everything in one bucket would produce a perfectly stable figure and a completely wrong essay.
What the census cannot reject is a claim about plants, and nothing in the check pretends otherwise. It is a measurement of a parameter space, gated as a measurement of a parameter space.
What this is not
Three things the census does not establish.
It is not a claim about plants. Divergence angles in real plants are not uniformly distributed — they cluster hard near 137.5°, which is the observation the whole subject starts from. The census describes the geometry a plant is choosing within, not the choice.
It is not a probability. Calling 14.7% “the probability that a plant is Fibonacci” would require plants to pick divergences uniformly at random, which they emphatically do not. The number is a measure on a parameter space, and its use is as a baseline: the thing a preference has to be a preference over.
It does not depend on the rise being read as time. The four rises are four patterns, not four moments of one. The developmental reading — that a shoot’s rise falls as it matures, so it travels from the first row of numbers towards the last — is a separate claim, and it is made in the ladder essay with its own support.
What survives about the golden angle
Given how much of this essay is subtraction, it is worth saying plainly what is left, because it is not nothing and it is the sharpest thing in the subject.
The golden angle is the number hardest to approximate by rationals — 0.4377 on the site’s measure against 0.3306 for the best of 1,500 sampled, approaching Hurwitz’s bound of . A lattice built on a near-rational divergence develops visible radial rows and an empty wedge that grows without bound as the pattern fills; a lattice on the golden angle does not.
That is a real and sharp distinction, and none of the frequency measurement touches it. What the frequency measurement does is separate two claims that are usually run together: this angle has a special arithmetic property — true — and this angle is what the geometry hands over — false, and increasingly false as patterns get finer.
The first is why a plant on the Fibonacci branch stays well ordered as it grows. The second was never the reason for anything.
The classification could be wrong and would say so
The census depends on a classifier that decides whether a pair is consecutive Fibonacci, consecutive Lucas, whorled, or other, and the build requires the four shares to sum to exactly 1. That catches the failure this arrangement invites: a divergence counted twice, or a pair matching two categories, or a pair matching none.
The classifier’s rules are in that order for a reason. Whorled is checked first, because a non-coprime pair like 2/4 would otherwise be read as neither Fibonacci nor Lucas and land in “other”, which is true and unhelpful — a whorled pattern is a different kind of object, not an unusual spiral.
One genuine softness is worth admitting. The pair 1/2 counts as consecutive Fibonacci, and at a rise of 0.12 it alone is 57% of the circle. That is arithmetically correct and rhetorically generous, since a pattern with one spiral one way and two the other is not what anybody means by Fibonacci phyllotaxis. Excluding it would push the coarse-rise Fibonacci share from 67% down to about 11%, and would strengthen this essay’s argument rather than weaken it. It is left in because the honest thing is to take the definition at face value and note where it flatters the claim being examined.