The counts change with radius

Count the spirals near the middle of a sunflower and count them again near the rim, and the numbers are different. Not slightly different, and not because the counting was sloppy — a different pair of Fibonacci numbers, reliably, with a clean transition in between.

This is not obscure. It is the first thing anyone counting a real head runs into, and it is almost never mentioned in the captions that quote a pair of numbers as though a flower had one.

Why it happens #

The reason is in the geometry of the lattice, and it is short.

A point at radius r has its neighbours at a set of index offsets. For each offset m, the hop from a point to the one m places earlier has two components: an angular part, which is the fractional part of m·δ turns multiplied by the circumference 2πr, and a radial part, which is roughly m divided by 2r in model units.

The angular part grows with r. The radial part shrinks with r. So the offset that gives the shortest hop is not the same at every radius: small offsets win near the centre, where the circumference is small and the radial cost dominates; large ones win at the rim.

The offsets that win are exactly the denominators of the good rational approximations to the divergence angle — the convergents of its continued fraction. For the golden angle those denominators are the Fibonacci numbers, which is where Fibonacci enters the subject at all. As the radius grows the pattern climbs that list, one convergent at a time.

Where the transitions fall #

The crossover between offset m and offset n happens where their hop lengths are equal, and that is a solvable equation. Setting the two hypotenuses equal gives a radius that depends on the fractional parts of m·δ and n·δ, and on nothing else.

For a golden-angle head, that means the transition radii are fixed by the angle and the model’s scale, and they are predictable before the pattern is drawn. Doubling the number of primordia moves every transition outward in a computable way.

Two consequences follow that matter for anyone counting real material.

A head with more seeds shows higher counts. Not because a larger flower is somehow more Fibonacci, but because a larger head reaches further up the same list of convergents. A count is as much a measure of head size as of anything else.

Two counts from one flower are not a contradiction. They are two observations from two annuli, and both are correct.

The ambiguous bands #

At the transitions three offsets have nearly equal hops, and the count there is genuinely undetermined.

Measured in one head at one radius, offset 55 gives a hop of 1.68 mean spacings, offset 34 gives 2.01, and offset 89 gives 2.10. Two of those three will be picked out by any counter, and which two depends on details with no information in them — where exactly the band was placed, how the median was taken, how the eye happened to travel.

This is worth stating because it sets a floor on how precise a spiral count can be. A published pair with no radius attached is ambiguous by construction, and a disagreement between two counters of the same flower may be a disagreement about annuli rather than about arithmetic.

What this does to the published record #

The literature on real phyllotaxis counts is more careful than the popular one, and the popular one has a specific problem that this geometry explains.

A photograph captioned “34 and 55 spirals” has usually been counted where the spirals are easiest to see, which is a particular annulus — typically the outer half, where the seeds are large and well ordered. That is a legitimate measurement of that annulus.

What it is not is a property of the flower. And a survey that collects such counts and reports the distribution of pairs is, without meaning to, partly measuring the distribution of head sizes and partly measuring where people find spirals easiest to trace.

The fix is not difficult and it is rarely applied: state the radius, or state the count as a function of it.

Rising phyllotaxis #

The changing counts have a name in the botanical literature — rising phyllotaxis — and there it means something slightly different from what is drawn here, in a way worth separating.

In this site’s figures the divergence angle is constant and the counts change purely because the geometry of the lattice changes with radius. In a real growing plant the counts also rise, and there the primordia are added over time at a changing ratio of primordium size to meristem size. Both effects produce rising counts and they are not the same mechanism.

Distinguishing them takes more than a photograph of a finished head. It takes a time series, and the fact that a static model reproduces the rising counts without any developmental change at all is a caution rather than a result: reproducing a pattern is not explaining it, and here is a case where two quite different processes leave the same trace.

Why this is the first real finding #

Of everything on this site, this is the one that needed nothing but doing the measurement properly.

There is no clever mathematics in it. The counting machinery was built to check something else — whether the divergence angle could be recovered — and pointing it at successive annuli of the same head took an afternoon. What came out is a structural feature of every spiral lattice, visible in every sunflower, and absent from almost every account of them.

That is a fair description of what happens when a subject is taught through captions. The captions are copied, the observation behind them is not repeated, and a feature that anyone counting for themselves would notice in an hour goes unmentioned for decades.

The practical upshot #

For anyone counting real material, three rules follow.

State the radius. A count without one is ambiguous by construction.

Expect a change across the head, and treat two different pairs from one flower as two observations rather than as a contradiction.

Expect larger heads to give larger counts, and do not read that as a difference in kind. It is the same angle further up the same list of approximations.

What a published count is a statement about #

The practical consequence comes first, because it changes how the literature reads.

A caption saying “34 and 55 spirals” is a statement about one annulus of one head. It is not a property of the species, not a property of the individual, and not even a property of the whole flower being photographed. Move the counting band inward and the same flower gives 21 and 34.

That makes many published counts incomparable. Two papers reporting different pairs for the same species may be in complete agreement about the plant and disagreeing only about where they counted, and since the counting radius is almost never reported there is no way to tell.

The fix is not difficult and it is worth stating: report the radius, in units of the head radius, along with the counts. A pair without a radius is a pair with an unstated free variable.

The transition is visible, and it is not a defect #

Between two counting regimes the pattern has a genuinely mixed region, where the two candidate offsets give hops of nearly equal length and neither family dominates.

In a photograph this looks like the spirals “breaking up” or “going wrong”, and it is sometimes described that way. It is neither. It is the crossover, it happens at a computable radius, and a head without one would be the surprising thing.

Three regimes and two transitions in a single head is the normal case for a sunflower of ordinary size. Larger heads show more. The number of transitions is roughly the number of Fibonacci convergents whose denominators fall below the point count, which is a statement one can check by counting bands rather than by counting spirals.

Why the offsets are Fibonacci numbers #

The essay has said the winning offsets are the continued fraction convergents’ denominators. That deserves its own paragraph, because it is where the famous sequence enters and it enters for a reason that has nothing to do with plants.

An offset m gives a short angular hop when m·δ is close to a whole number of turns — that is, when m/1 approximates the rotation well. Over all m up to some bound, the ones that do this best are exactly the denominators of the best rational approximations to δ, which is a theorem about continued fractions rather than an observation about lattices.

For δ = the golden angle, φ’s continued fraction is all 1s, so its convergents are ratios of consecutive Fibonacci numbers and their denominators are 1, 2, 3, 5, 8, 13, 21, 34, 55, 89.

For the Lucas angle the same argument gives Lucas numbers, and the model reaches that branch too. For a divergence angle with a large partial quotient, the list skips — and the pattern skips regimes with it, which is a prediction one can make before drawing anything.

So Fibonacci is not in the plant. It is in the arithmetic of the angle the plant ends up with, and it would be a different sequence for a different angle.

Counting the model instead of the picture #

The band analysis here is done by the same counter used everywhere else on this site: handed coordinates, never the angle, asked which index offsets give the shortest hops among the points in a radial band.

That matters more than usual for this result, because the claim is that one point set gives four different answers. If the counter had access to the divergence angle it could have derived the expected pair analytically and the demonstration would be circular.

It does not, and the four pairs it returns are four independent measurements on four subsets of the same coordinates. The transitions between them can then be compared against the radii predicted from the hop-length equation, and they agree — which is two routes to the same numbers, the arrangement this site prefers wherever it is available.

What this does to the popular claim #

Putting the radius dependence together with the branch structure leaves the familiar statement with very little of itself.

“Sunflowers have 34 and 55 spirals” has, on inspection, two unstated variables and one false universal. The radius is unstated, and changing it changes the pair. The species and individual are unstated, and the counts scale with head size. And “Fibonacci” is a claim about the branch, which is usual rather than universal.

What survives is: at a stated radius in a given head, the counts are the two adjacent denominators from the continued fraction of that head’s divergence angle. Everything famous about the claim is in the last clause, and it is a statement about a number rather than about a plant.

That is not a debunking. It is the same claim with its variables bound, and it is strictly more useful — it predicts the transitions, the head-size dependence and the Lucas exceptions, none of which the loose version can say anything about.

Counting a real head, practically #

Since the whole point is that the measurement is doable, here is what doing it requires.

Choose an annulus and mark it. A band about a tenth of the head radius wide, at a stated fraction of the way out, is enough to have a well-defined pair in it and narrow enough not to straddle a transition.

Count both families in that band, following spirals that pass through it rather than ones that start or end inside it.

Record the band’s radius as a fraction of the head radius, and estimate the total primordium count from the density — a count in a small square, scaled.

Those three numbers are what the angle recovery needs, and they turn a photograph into a divergence angle with an uncertainty. A pair of counts alone will not do it, which is the practical reason the radius has to be reported and not merely the pedantic one.

Rising phyllotaxis, and what it is not #

The upward climb through the convergents has a name in the botanical literature — rising phyllotaxis — and it is worth separating the two things the term covers.

Within one head, the counts rise with radius because the lattice geometry changes as the circumference grows. That is what this essay measures, and it needs no change in the plant at all: one angle, one point set, three answers.

Through development, a plant’s phyllotaxis can also rise because the growth parameter itself changes as the meristem enlarges relative to the primordia. That is a different mechanism and it happens over time rather than across a radius.

The two produce similar-looking sequences and are routinely conflated. Distinguishing them requires either following one plant through development or measuring across a radius at one instant, and the second is much easier — which is a reason to be careful about reading developmental claims off a photograph of a mature head.

The finding, in one sentence #

If this essay has one thing to carry away, it is that a spiral count is a measurement with a location, and a location is not optional.

Every other result on this site inherits that. The angle recovery needs the radius as an input and returns nothing useful without it. The comparison across branches is only meaningful between counts taken at comparable places. The packing statistics have the same problem in a different coordinate — they depend on head size, which is the same dependence seen from outside rather than inside.

The general form: a quantity measured on a structure that varies across itself is not a property of the structure until the place is stated. That is unremarkable in most of science and is routinely dropped in this subject, which is why a century of published counts is harder to compare than it should be.

Where the transitions actually are #

The founding essays established that the counts change and located the transitions by counting bands. Expansion produced the number they are at, from a different model entirely.

A disc is a cylinder whose rise falls with radius. Between one primordium and the next, Vogel’s model advances c²/2r radially on a circle of circumference 2πr, so the rise — the advance per element in circumferences, which is a cylindrical lattice’s second parameter — is

$$h(r) = \frac{c^2}{4\pi r^2}$$

A falling rise climbs the Fibonacci ladder, whose transitions sit at computable rises. Inverting gives the radii, with no fitted quantity: transitions a factor of φ ≈ 1.618 apart in radius, each one advancing the pair by one rung.

Checked against a blind counter run on sixteen bands of a disc, the prediction agrees in fifteen — and the one disagreement straddles a predicted transition, which is where a count is genuinely ambiguous.

So the phenomenon this essay reports is not a peculiarity of seed heads. It is what any lattice does when its rise falls, and a disc is simply the geometry in which the rise falls fastest.

Which is why the radius was asked for, and what it is actually worth #

This page is the origin of a request this collection carried for a long time: since the counts change with radius, every published count is a statement about an annulus, so a survey should record the counting radius.

The reasoning is right and the work after it measured what the field is worth, which is less than the prominence implied. Given the pair, adding the rise its counting radius implies narrows the divergence angles consistent with the report by a factor of 1.10 — a tenth, at every rung tried.

The field stays on the list and its justification changes. It is needed for comparing specimens counted in different regimes, and for the transition positions along an axis where the position is the measurement. It is not what recovers an angle, and the sentence on this page that leads to it should be read as being about which count a counter gets rather than about what that count then settles.