Phyllotaxis
the spirals are counted, not admired
The pattern itself

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.
9 min read 5 figures Counted, not admiredThe round trip

Look at a sunflower and spirals appear. Follow one with a finger and it curves out to the rim; follow one going the other way and it does the same. Count them and the numbers are usually 34 and 55, or 55 and 89, or some other pair of consecutive Fibonacci numbers.

That is the observation the whole subject rests on, and it is worth asking what exactly is being counted.

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. 1 The two families a counter finds in one head, drawn as the polylines joining every 34th and every 55th point in birth order. They are not curves fitted to the picture — each is a statement about which points are neighbours.

What a spiral is, in a pattern made of points

A spiral in a seed head is not a curve. It is a chain of near-neighbours.

Take the point born 34 places before any given point. In a golden-angle head it sits very close by — much closer than a point 20 or 45 places earlier. Join every point to the one 34 places before it and the result is a set of 34 chains, each winding out from the centre. Those are the spirals a person traces.

So a spiral family is labelled by an offset: the number of places back in birth order that lands one on a near neighbour. The parastichy numbers of a pattern are the offsets that do this, and everything about counting spirals is about finding them.

That reframing is what makes counting a computation rather than an impression. A curve fitted to a photograph is a matter of judgement; the question of which index offset gives the shortest hop is not.

The counter, and what it is not told

The arrangement matters more than the algorithm, so it is worth being explicit.

One function in lib/lattice.js builds a head from a divergence angle. A different function is handed an array of coordinates and asked which offsets give short hops. It does not receive the angle. It does not receive the model. It does not know that Fibonacci numbers are the expected answer, and it would return 47 and 76 just as readily.

The method: for each candidate offset m, measure the distance from every point to the point m places before it, take the median, and divide by the local spacing so the answer does not depend on scale. Plot that against m and the parastichy numbers appear as local minima — offsets that give hops much shorter than their neighbours do.

Requiring a local minimum rather than simply a short hop is the part that does the work. Offset 1 always gives a shortish hop; so does offset 2. Without the local-minimum requirement the counter would report 1 and 2 for every pattern ever made, which is true and useless.

The bug that only a second route could find

The first version of this counter returned the two smallest offsets among the minima rather than the two shortest.

For a golden-angle head that means it reported 21 and 34, when the offsets that actually give the closest neighbours at that radius are 34 and 55. Both are genuine parastichy families — 21 is a real spiral family in that head — so the drawing looked entirely correct. Twenty-one spirals were drawn and there were twenty-one of them.

Nothing in the figure showed the error. What showed it was the angle recovery, which takes a pair of counts and asks which divergence angle makes those two offsets the closest. Fed 21 and 34 at that radius it refused: no angle does. The refusal was correct, and it was the only thing in the system that could tell the difference.

That is the argument for the second route in one paragraph. A count that is merely plausible cannot be distinguished from a count that is right by looking at the picture the count produced.

Four heads, and the divergence angle recovered from eachThe counter is shown the points and nothing else. The worst recovery across the four is 0.012°.the first of the fourused to buildcountsrecovered137.508°55 · 89137.520°99.502°47 · 7699.500°151.100°31 · 81151.105°77.960°37 · 6077.960°worst error 0.012°counts in, angle outthe recovery never sees the angle
Fig. 2 The second route. Four heads built at stated angles, the counts taken from the points, and the angle recovered from the counts — worst error twelve thousandths of a degree.

Showing that the counter rejects

An assertion that has never failed on anything is not a test, and on this subject that matters more than usual: a counter that always returned Fibonacci numbers would confirm every claim ever made about sunflowers.

So the build runs it on a lattice built at 45°, which is a rational angle and produces eight radial rows and no spirals at all. The counter returns 8 and 16 — the row count and its double — and not a Fibonacci pair anywhere.

That single check is what makes every other count on this site worth reading. Without it, “the counter found 34 and 55” is a statement about the counter.

Why the answer depends on where one look

The counter takes a radius band as well as a point set, and it has to, because the answer genuinely depends on it.

Near the centre of a head the points are sparse and the short offsets win: 13 and 21, or 21 and 34. Further out, the same head gives 34 and 55, then 55 and 89. The transitions happen at radii that can be computed, and they are the subject of the next essay.

This is not a defect of the method. It is the pattern being honest about itself: a spiral lattice does not have a parastichy pair, it has a parastichy pair at a radius. Every published count of a sunflower is a count in whichever annulus the counter happened to work in, and the practice of quoting one pair per flower obscures a real and computable structure.

Ambiguity at the transitions

Between one regime and the next there is a band where three offsets have almost the same hop length, and there the count is genuinely ambiguous.

At one radius in the head measured here, offset 55 gives a hop of 1.68 in units of the mean spacing, offset 34 gives 2.01, and offset 89 gives 2.10. Which pair the eye picks out — or the counter — depends on details that carry no information.

That ambiguity forced a change in the machinery. The angle recovery originally required the observed pair to be the two shortest offsets; near a transition it refused patterns it had counted correctly. It now asks for both to be among the shortest three, and reports a wider interval when the constraint is looser. The tolerance is not a fudge — it is the ambiguity of the pattern, made explicit.

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. 3 Where the ambiguity lives. Each band of the same head gives its own pair, and the transitions between them are where three offsets compete.

What the counting does not settle

Two things, and both are usually assumed.

It does not say the counts are Fibonacci. It says what they are. That they usually come out Fibonacci in a golden-angle head is a consequence of the angle, and heads built at other angles give other sequences — Lucas numbers, or nothing named at all.

It does not say anything about real flowers. Everything here is measured on generated patterns, where the truth is available. That is deliberate: the site’s job is to establish what a count means and what it can be trusted to reveal, and a photograph would add authority the argument has not earned. What the counting establishes is that the operation is well defined and testable, which is the precondition for anyone doing it on real material and being believed.

The general shape of it

The pattern this essay repeats is the one the whole fleet is built on, and it is worth naming.

An observation that is usually made by eye — there are 34 spirals — is restated as a computable quantity: offset 34 gives a local minimum of the median hop length in this annulus. The restatement is more work, it is less evocative, and it is the difference between a claim that can be checked and one that can only be repeated.

Then the computation is shown to reject, on input where the right answer is known to be different. Then it is checked against a second, independently-shaped computation that could disagree. Only after all three does a number get quoted in a caption.

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. 4 Why the offsets that win are the ones they are. A parastichy number is a denominator at which the divergence angle is nearly a whole number of turns, so the counts are the denominators of the angle’s good rational approximations.
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. 5 And what the counter returns at other angles. Fibonacci numbers are what a golden-angle head gives; the Lucas angle gives Lucas numbers, and other angles give sequences with no name.

What counting is worth

The operation is unglamorous and it is what everything else on this site is built on.

The angle can be recovered because the counts carry it. The transitions across a head are visible because the count can be repeated at different radii. The Fibonacci claim can be tested because other angles can be counted too, and their counts are not Fibonacci.

None of that is available to a reader who takes the count from a caption.

Where this goes next

divergence 137.508°closest pair 1.60 × mean spacing A head is a set of points The nth primordium at n times an angle, and a radius of root n. Two lines of arithmetic produce a sunflower head, which is either remarkable or suspicious depending on how carefully the claim is stated — and stating it carefully is most of the work. the first of the fourused to buildcountsrecovered137.508°55 · 89137.520°99.502°47 · 7699.500°151.100°31 · 81151.105°77.960°37 · 6077.960°worst error 0.012°counts in, angle outthe recovery never sees the angle Recovering the angle from the counts Build a head at a stated divergence angle, forget the angle, and get it back from the spiral counts alone. Four angles, worst error twelve thousandths of a degree — and the only thing that crossed between the two halves was a list of coordinates. 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 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.