Phyllotaxis
the spirals are counted, not admired
The pattern itself

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.
8 min read 4 figures The round tripCounted, not admired

A generator that only ever runs forwards proves nothing about itself. Feed it an angle, get a pattern; the pattern looks like a sunflower; there is no way to tell from it whether the machinery did what it claimed.

So the machinery runs backwards as well. A head is built at a stated angle, the angle is set aside, and a second piece of code — which has never been shown it — is handed the spiral counts and asked which divergence angle produces them.

It gets the angle back to about a hundredth of a degree.

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. 1 Four heads at four stated angles, the counts taken from the points, and the recovered angle beside each. The worst error across the four is 0.012°.

The relation it uses

The recovery rests on one piece of lattice geometry.

In a Vogel head the point i sits at radius √i, so stepping j places back moves a point by two things at once. Around the disc it moves by the fractional part of j·δ turns, which is an arc of 2πr times that fraction. Outward it moves by roughly j/2r in model units, because the radius is a square root.

The hop for offset j is the hypotenuse of those two, and the parastichy numbers are the offsets with the shortest hops. So given observed counts m and n at a known radius, the recovery asks: for which divergence angles are m and n among the shortest? That has a genuine interval of solutions, and the centre of the interval is the answer.

Both quantities the relation needs are read off the point set rather than taken from the model. The radius is the middle of the band that was counted. The scale comes from the density, since Vogel’s model puts an equal area around every point.

The cost function that had no geometry in it

The first version scored candidate angles by how nearly m·δ and n·δ landed on whole turns, without the radial term.

It was minimised at δ ≈ 0. At zero every multiple is a whole number of turns, so the score is perfect — and the “lattice” is a single radial line with every point on top of every other. The recovery confidently returned 0.0018° for a golden-angle head.

The mistake is worth naming because it is a common shape. The angular condition is necessary for an offset to be a parastichy number and it is not sufficient: the offset also has to be small enough that the radial hop does not dominate. Leaving out the term that grows with j left a cost function whose minimum was a degenerate configuration, and degenerate configurations are exactly where an incomplete cost function goes.

Adding the radial term fixes it, and it fixes it for a reason rather than by tuning: the term is not a penalty introduced to avoid a bad answer, it is the second component of a distance that was always there.

When the recovery refuses

Two cases, and both are informative rather than annoying.

Non-coprime counts. If the two counts share a factor, no single divergence angle is recoverable. That is not a failure — it means the pattern is whorled rather than spiral. A divergence angle near a simple fraction of a turn produces radial rows, and a head at 137.0° counts as 21 and 42, because 137.0 is within a whisker of 960/7. Twenty-one and forty-two share a factor of 21, and what the pattern really is is seven rows of something. The recovery says so and stops.

No consistent angle. If the counts cannot both be among the shortest offsets at that radius for any angle, the recovery refuses. That is what caught the counter returning the two smallest offsets rather than the two shortest — fed 21 and 34 at a radius where the true neighbours are 34 and 55, no angle works, and the refusal was the only signal that anything was wrong.

The interval, and why it is reported

The recovery returns a width as well as a centre, and the width matters.

Two small counts constrain the angle loosely; two large ones constrain it tightly. A pair like 8 and 13 pins the angle to within a degree or so; a pair like 55 and 89 pins it to a few hundredths. That is a real property of the problem — a coarse pattern carries less information about its generating angle than a fine one — and reporting a single number without it would overstate what a count can say.

It also sets a limit on what any measurement of a real plant can establish. A pine cone with 8 and 13 parastichies is consistent with a range of divergence angles a degree wide, and no amount of care in counting narrows it. Distinguishing 137.5° from 137.9° needs a head with high counts, which means a large one.

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. 2 The other side of the same coin. The counts are the denominators of the good rational approximations to the angle, so the finer the pattern, the deeper into the approximation sequence it has gone — and the more precisely it identifies the number.

What the round trip proves

The agreement is to about a hundredth of a degree across four quite different angles, and it is worth being precise about what that establishes.

It proves the forward generator and the backward recovery are consistent, which is a real thing to have proved. A sign error in the lattice geometry, a confusion between turns and degrees, or an off-by-one in the birth order would all break it, and every one of those leaves a pattern that looks exactly like a pattern.

It does not prove that Vogel’s model describes any plant. Two consistent implementations of the same wrong idea agree perfectly. That is why the site carries checks that come at the geometry from other directions — the counter shown to find rows in a rational lattice, the packing measured against several criteria that disagree, the angle produced by a dynamical model that was never told it.

And it does not prove anything about a photograph. Recovering an angle from a real head would need the counts, the radius they were taken at, and the model’s scale, and real heads are noisy in all three.

The same shape, three times over

This is the third instance of a pattern that has now appeared across the fleet often enough to be worth naming.

Generate something from a stated parameter. Forget the parameter. Recover it by a computation shaped differently from the one that made it, with the original hidden behind an interface narrow enough to inspect. Compare.

A crystallography site generates a pattern, forgets which symmetry group produced it, and rediscovers the group. A perspective site draws a box from a camera and recovers the camera’s focal length from the drawn edges. Here a head is built from an angle and the angle comes back out of the spiral counts.

In each case the value is not in the forward computation, which is easy. It is in the fact that a second computation with a different shape had to arrive at the same answer with the first one’s workings hidden — and in each case the round trip found a bug that nothing else did.

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. 3 The input to the recovery. The counts come from the polylines joining every 34th and every 55th point, which is a fact about which points are neighbours rather than a curve fitted to a picture.
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. 4 And why the recovery has to be told where the counts came from. The pair depends on the annulus, and the lattice geometry the recovery uses depends on the radius.

What it costs

Very little, which is worth saying because the case for round trips is usually made on principle rather than on price.

The forward generator is two lines of trigonometry. The recovery is a scan over candidate angles with a hypotenuse at each, and it runs in a few milliseconds. Both run at build time on every figure that uses them, so a change to the lattice geometry that breaks the correspondence stops the site building rather than producing a figure that is wrong in a way nobody can see.

That is the argument in one sentence. The failure being guarded against is not a crash; it is a pattern that looks entirely convincing and was generated by a rule with a bug in it, which is the ordinary condition of illustrations in this subject.

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. 0255075406080fraction of the head's radiusparastichy numbers found in a band there1000 primordia at the golden angle4 different pairs 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. 05e+51e+62e+60100200300angle around the boundary (°)repulsion from what is theregrowth 0.40 · 14 elements in playthe minimum is where the next one goes The angle is an output 137.5° is not a constant of nature. It is where a rule settles — a rule that places each new element as far as it can from the ones already there, contains no reference to the golden ratio, and reaches the same answer from starting angles a hundred and sixty degrees apart.