The organ that was taken away

Everything this collection measures is read off a pattern that was finished before the reading started. A head of florets, a stem of leaves, a list of divergence angles taken off either: the plant did whatever it did, and the instrument arrives afterwards and counts.

This collection has already found out exactly what that costs. It had offered one observable — a comb in the autocorrelation of the divergence sequence — as evidence that a plant computes its pattern rather than merely having one, and then built an arrangement with no rule in it anywhere that reproduces the comb, the second comb, and the parastichy pair on eight seeds out of eight. What was left was a ratio: the rule divides the correlation between its two combs differently from a transported disturbance, and that difference is a constraint rather than a discriminator.

This essay does the other kind of experiment. Remove one organ from a settled stem and place the next one against what is left.

Why an intervention is a different kind of evidence #

The two accounts agree about the arrangement and disagree about where it came from, which is precisely the situation an observation cannot settle and an intervention can.

Under the placement rule, organ i goes where the repulsion from the organs already present is least. Its azimuth is a function of theirs. Delete one of them and the function has a different argument, so it has a different value: the rule must answer, and what it answers is computable before the experiment is run.

Under the transported-error account, organ i sits at exactly i times the divergence angle, plus an error it inherited from the organs m and n places back. The divergence is a constant of the plant. The error is a mistake. Neither term is a function of whether some earlier organ exists, so deleting one changes which errors the later organs inherit — the pattern’s mistakes are rearranged — and moves no organ’s intended position at all.

So the prediction is not a number one model fits better than the other. It is a number one model has and the other has none. That asymmetry is what makes this worth a day rather than an hour.

The experiment, stated before its result #

A stem is grown at a fixed rise of 0.005 circumferences per organ until it has settled: 400 organs, a divergence of 137.84°, a scatter of eight hundredths of a degree, and a counted pair of 8 and 13. Then two runs continue from that same history. The control places the next organ against every organ present. The other places it against the same history with one organ deleted.

The organ deleted is identified by how far back it is: one place back is the most recent, thirteen places back is older. Everything else is identical — same rise, same heights, same rule, same arithmetic, same sample grid. The number reported is how far apart the two runs put the organ that comes next.

It is worth saying what is not free here, because the answer’s value depends on it. The exponent of the repulsion is Douady and Couder’s inverse cube, and every exponent above about 1.06 gives the same lattice, the same ladder and the same transitions — so the prediction does not depend on it. The neighbourhood is six local spacings of recent organs, and the recency window was later replaced by a stated cut-off, which found the lattice survives out to a few spacings whichever shape the cut-off has. The rise and the divergence are measured off the stem being cut.

There is nothing left to choose. The number below is a prediction, not a fit.

The result is a step, and its edge is a count #

Removing an organ moves the next one if and only if the organ removed is one of the most recent thirteen. Not approximately, and not by a decaying amount.

Thirteen is the larger of the two parastichy numbers at this rise. That could be a coincidence of one rung, so it was measured at two more.

Three rungs, three pairs, three boundaries: 5, 8, 13. The number of organs whose removal moves the next one is the larger parastichy number, every time.

Which makes this a parastichy count obtained by intervention. It uses no angles, no coordinates, and no counting of spirals. A botanist ablates primordia one at a time, notes for each whether the next primordium came out where it was going to anyway, and the number of organs for which the answer is no is n. Every existing method on this site — counting the rows in a photograph, recovering the divergence from positions, reading the two combs out of a sequence of angles — needs the finished pattern and a measurement of it. This needs a pair of forceps and a yes-or-no.

Why the boundary is the larger parastichy number #

The organs whose removal matters are the ones with an exposed edge: the growing tip’s own boundary. A cylindrical lattice’s boundary holds exactly n organs, because n is the number of contact rows the surface is divided into at that rise. An organ older than that is behind the front; its vacancy is enclosed by organs that are still there, and the next primordium is placed too far away for the hole to reach it.

The claim has a number in it, and the number checks. The organs within one local spacing of the tip are the ones a new organ can reach: the stem gains one organ per rise h, the spacing between neighbours on a surface holding one organ per h of height is √h, and so a band one spacing deep holds √h/h = 1/√h organs. At the three rises measured that is 14.1, 8.8 and 5.6.

The larger parastichy number is the same quantity. A cylindrical lattice’s two parastichy numbers multiply to about 1/2h — 8 × 13 = 104 against 100, 5 × 8 = 40 against 38.5, 3 × 5 = 15 against 15.6 — and their ratio is the golden ratio, so n = √(φ/2h) = 0.9/√h: 12.7, 7.9 and 5.0.

The two routes do not give quite the same number, and the difference is worth keeping rather than rounding away. A band one spacing deep holds 1/√h organs — 14.1, 8.8 and 5.6 at the three rises. The lattice arithmetic gives 0.9/√h, or 12.7, 7.9 and 5.0. The boundaries actually measured are 13, 8 and 5. Against the first estimate they are out by 1.14, 0.77 and 0.59; against the second by 0.27, 0.11 and 0.03. The golden factor is not a tidying convenience, then. It is the difference between an estimate that is consistently a little high and one that lands on the measured integer at every rung tried.

That is a small enough margin to be worth pricing. Ten per cent separates the two estimates, so they disagree about the answer only where an integer happens to sit between them — which is the case at all three rises here and would not be at every rise. A stem whose front came out at fourteen would tell them apart; a stem at a rise where both estimates round the same way would not, and no amount of care with the forceps would change that. Three rungs is three chances for the finer estimate to have been wrong and it was not, which is a claim with a size on it rather than a claim of exactness.

The front’s width and the larger parastichy number are the same number, up to that factor of nine tenths, because both are the number of organs that fit around the circumference at the tip. That is why an intervention counts the spirals, and it is not an analogy — it is the same 1/√h, arrived at from the geometry of the boundary in one case and from the arithmetic of the lattice in the other.

This also explains the shape of the step, which is not flat inside the front. The displacement is 138° at one place back, 84° at two, 53° at three, 168° at four — and 2.6° at thirteen. The large values are not noise: they are the next organ going into the vacancy rather than leaning towards it, which is what an argmin does with a hole in a lattice. Remove the most recent organ and the next primordium appears where that one was, a displacement of exactly one divergence angle. Remove one further back and the hole is lower down, so filling it costs the new organ some height it cannot give up, and it lands at a compromise between the vacancy and its own preferred site. By thirteen places back the hole is nearly buried and the compromise is nearly nothing.

The size of the effect, which is what makes it an experiment #

Tens of degrees is not a subtle prediction. It is worth putting beside what the angle sequence costs to read.

This site’s own readout of the parastichy pair from a list of divergence angles needs about 900 organs on one stem, a protractor good to about four tenths of a degree, and a plant whose disturbance sits inside a window between too quiet to read and too disordered to have a lattice. It then refuses for four different reasons that it cannot tell apart.

The intervention’s smallest signal — 2.6°, at the far edge of the front — is six times the precision that readout demands, and every other offset is between ten and four hundred times it. The measurement a plant would have to support is not “where exactly did the next primordium form”, but “did it form where the undisturbed sequence says, or a quarter of a turn away”.

What the rival predicts, which is nothing #

It is worth computing rather than asserting, because the strength of the test is that the rival’s answer is zero by construction and not by fitting.

In the transported-error account each organ’s position is fixed before the plant starts: azimuth i times the divergence, plus an inherited error. Deleting organ i − 8 removes one term from the sums that produce later organs’ errors, so the later errors change — they are re-drawn, not shifted. Their mean is zero. Their spread is the plant’s own scatter, half a degree.

So the experiment is a one-sided test with an effect size of tens of degrees against a null of tenths of one. There is no coupling, no exponent and no neighbourhood width in the account that could be tuned to produce a displacement, because nothing in it is a function of which organs are present.

What the azimuth grid decides, and it is not this #

The rule takes its minimum over a finite number of sampled azimuths, and the whole measurement here is a displacement in degrees, so the grid is a fair thing to be suspicious of. That study was asked for earlier here and never done.

At 1,536 azimuths and at 4,608 the front is thirteen organs wide both times, and every displacement inside it agrees between the two grids to within 0.23° — one sample step of the coarser grid.

What the study did turn up is a caution about the site’s own usual setting. At 384 azimuths, the grid every flat run in this collection uses, the noiseless rule has no golden lattice to perturb at all: it falls into a repeating cycle of divergences at 185.8° before anything is removed. The site’s flat runs are all noisy ones, and it is the noise that keeps them off it — which nobody knew, and which is worth knowing before anything reaches for 384 again.

Outside the front, the number reported is a bound #

The step’s inside is measured and its outside is not, and the difference is in the instrument rather than in the stem.

Inside the front the displacements run from 2.6° to 168°, on a stem whose organs sit 25° apart and whose settled run-to-run scatter is eight hundredths of a degree. Those are measurements. The smallest of them is thirty times the scatter of the stem it was read off, and the largest is most of a turn.

Outside the front the largest displacement anywhere is 0.47°, and the azimuth grid the run samples has 384 steps of 0.94°. Half a step is exactly where a displacement of nothing lands when the two runs happen to round to adjacent samples. So 0.47° is not the size of a residual effect. It is the largest number the instrument can return while the true displacement is zero, and reporting it as a measurement of a small effect would be reporting the grid.

The convergence study is what turns that from an assumption into a check. At 1,536 azimuths and at 4,608 — four and twelve times the setting the flat runs here use — the front is thirteen organs wide both times, so refining the grid did not promote anything outside it into a signal.

The honest statement is therefore asymmetric, and the asymmetry runs the useful way. Removing an organ inside the front moves the next one by an amount measured in tens of degrees, against a null the rival account fixes at zero by construction. Removing one outside the front moves it by less than the resolution of the experiment, which at the finest grid tried is under a quarter of a degree. That is a bound, not a zero. An account predicting a small graded tail beyond the front — a weak reach that falls off rather than stopping — is not excluded by what is drawn here, and would be excluded, or found, by a finer grid and a longer stem. It is the one open question this measurement leaves that better arithmetic alone cannot close.

What this does not settle #

Three things, and stating them is what keeps an intervention from being sold as a proof.

It is a prediction, not an observation. No plant has been ablated here. What has been established is that the two accounts make incompatible predictions about an experiment that is within reach of a dissecting microscope and a needle — which is a different and better position than the one this thread was left in, where they made the same prediction about everything measurable.

A real apex is not a cylinder with prescribed heights. In the model an organ’s height is fixed by the rise and only its azimuth is free, which is the same idealisation every cylindrical result in this collection rests on and is usually harmless. Here it is doing more work than usual: part of the reason the next organ goes into the vacancy rather than towards it is that it cannot lower itself to fill the hole properly and must compromise in azimuth alone. On an apex where the plastochron and the radial position are both free, the compromise has another dimension to spend, and the displacement in azimuth could be smaller than the model’s. What would not change is the boundary: an organ behind the front cannot be reached whatever the new organ is free to do.

And a real apex regrows. A removed primordium leaves tissue, and the tissue divides. The prediction that survives that translation is the one this essay’s boundary is about — whether the next organ appears at its undisturbed site or in the vacancy, and how many organs back one must go before removal stops mattering — and it survives it because both halves are about the organs, not about what is between them.

And the front’s width is a prediction about a count, so it inherits the count’s own difficulties. A plant whose rise puts it near a transition has two pairs contending, and this experiment has not been run there.

Two organs, and a parameter that is a control #

The intervention has a two-organ version, and what recommends it is not that it does more damage. It has a second parameter — the gap between the organs removed — and that parameter has a setting at which the experiment must reproduce this one.

It does: with the second organ behind the front, the displacement agrees with the single cut’s to within a degree at three offsets. Inside the front it swings by two hundred and forty degrees, so the parameter is not idle.