Phyllotaxis
the spirals are counted, not admired
The pattern itself

The counts change with radius

The same head gives 21 and 34 near the middle, 34 and 55 further out, and 55 and 89 at the rim. The transitions are at computable radii, and a photograph captioned with one pair is a statement about one annulus rather than about a flower.
7 min read 5 figures Counted, not admiredOne parameter, one family

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.

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. 1 Bands across one generated head, counted independently. Three different pairs appear at three different radii, from one point set built at one angle.

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.

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. 2 The same head with the counting band drawn. Moving it changes the answer, and the figure lets the band move so that the change can be watched rather than described.

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.

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. 3 The list the pattern climbs. The offsets that win at successive radii are the denominators of the successive good rational approximations to the divergence angle — Fibonacci numbers, for the golden angle.
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. 4 And why the transitions matter for the recovery. A pair of large counts pins the angle far more tightly than a pair of small ones, so a count taken near the rim carries more information than one taken near the middle.
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. Each convergent of the continued fraction is a better approximation than the last, and its denominator is the next parastichy number the growing head reaches.

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.

Where this goes next

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. 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. 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.