Phyllotaxis
the spirals are counted, not admired
What a plant might be doing

A comb is evidence of a rule

Build the same lattice kinematically — every node at an exact multiple of the divergence, an independent error on each azimuth, no feedback anywhere — and the spectrum is empty. The photograph is identical and the parastichy pair is identical. The comb is not a property of the arrangement.
17 min read 9 figures Form is not mechanismState the claim, state the testTwo routes, one number

Worth reading first: What a mechanism would have to show · The sequence has a memory · Where the model stops.

Every model on this site is a model of form. A lattice that matches a sunflower does not show that the plant computes one, and the essays say so wherever it matters — a divergence angle recovered to a hundredth of a degree is a statement about an arrangement, not about a meristem. The honest limit has been stated so often that it is worth asking what would lift it. What could a measurement of a finished plant say about the process that made it?

This essay is one answer, and it arrives as a control rather than as a proposal.

The control

Take the stem the previous essays read. It settles at a divergence of 137.826° and is held at a rise of 0.005; the position counter says its parastichy pair is 8 and 13; the readout on its angle sequence returns the same two.

Now build a second arrangement that agrees with it on all of that and shares none of its history. Node i goes at exactly i times 137.826°, at exactly i times the rise. Then displace each azimuth by an independent draw of half a degree. There is no rule: nothing is minimised, no node sees any other, and the error at node forty has no relationship to the error at node forty-eight.

The two arrangements are the same lattice. Their divergence angles agree, their rises agree, their parastichy pairs agree, their scatter agrees. A counter shown their coordinates cannot tell them apart, and neither can a photograph.

The same lattice with no rule in itA cylindrical lattice at a divergence of 137.826° and a rise of 0.005, built by placing node i at exactly i times the divergence and then displacing each azimuth independently by 0.5°. Its photograph is the photograph of the stem in the figure beside it and its parastichy pair is the same pair. The largest comb mean in it is 0.03 against a sampling band of 0.07, and the readout refuses.-0.50000.500125810131621242629lag, in internodescorrelation between a divergence and the one that many internodes laterno comb clears the bandlargest mean 0.03 · band 0.07sampling bandkinematic lattice · 759 divergences · 0.5° of independent scattergenerated from a stated rule, not drawn to look right
Fig. 1 The kinematic arrangement’s angle spectrum, on the same axes and against the same sampling band as the stem the rule built. Nothing clears it. The largest comb mean anywhere in the thirty lags is 0.02 against a band of 0.11, and the readout refuses — on every scatter tried, on every seed.
Two combs, at a rise of 0.005The autocorrelation of 760 divergence angles from one stem held at a rise of 0.005. The filled teeth are the lags at multiples of 8; the open teeth are the second comb, at the same spacing offset by 5. Reading the spacing off the first and the offset off the second gives the pair 8 and 13, which is what the position counter reports for the same stem — from angles alone, with no coordinate anywhere in the calculation.-0.50000.500125810131621242629lag, in internodescorrelation between a divergence and the one that many internodes later816245132129spacing 8 · offset 5pair 8/13 — counter says 8/13sampling bandone stem · 760 divergences · disturbance 0.25generated from a stated rule, not drawn to look right
Fig. 2 The stem the rule built, for comparison. Same rise, same divergence, same pair, same length of sequence. The main comb averages 0.64 and the second 0.50.

Nought point nought two against nought point six four. The comb is not in the lattice. It is in the order the angles were made in, and that is the one thing a finished pattern does not carry and a history does.

What makes the difference

A kinematic lattice is a description. It says where the organs are, and the errors in it are errors of the description — each organ’s departure from its ideal position is drawn on its own.

The rule’s stem is a process. Each node is placed where the repulsion from the ones already there is least, so a node that ends up to one side of its ideal position changes the profile the next few nodes compute against. The departure at node i is therefore not independent of the departure at node i + 8: the second was computed in a world containing the first.

The rule, 24 steps in, at a growth of 0.35The next primordium goes where the repulsion is least — the marked minimum at 327°. Nothing in the rule refers to any particular angle.05e+51e+61.5e+60100200300angle around the boundary (°)repulsion from what is theregrowth 0.35 · 14 elements in playthe minimum is where the next one goes
Fig. 3 The rule itself, part way through a run: the next element goes wherever the repulsion from the ones already placed is least. Nothing in it names an angle, and nothing in it names a spiral. What it does have that a description does not is that each choice is made in the presence of the previous ones.

That dependence is the whole of the signal, and it has a definite arithmetic shape: a node’s influence is largest on the nodes that have it as a near neighbour, which are m and n places later. So the correlation appears at those lags and their sums, which is the comb.

An arrangement can have the neighbours without the influence. That is exactly what the kinematic lattice is, and it is why the control is decisive rather than merely suggestive: the geometry is held fixed and the process is removed, and the signal goes with the process.

Why this is a mechanism claim and not a form claim

The site’s standing division is between what a pattern is and what made it, and almost every measurement here has been on the first side. The counting is about form. The recovery of a divergence angle from spiral counts is about form. The ladder, the transitions, the forks, the closed-form branch points: all form.

A comb in a sequence of divergence angles is on the other side. It cannot be computed from the arrangement, because two arrangements identical in every geometrical respect differ in it. What it distinguishes is whether the organs were placed one at a time in the presence of the ones already there — which is the substantive content of every mechanism proposed for phyllotaxis, from Hofmeister’s rule to auxin depletion.

That is worth being careful about, because it is easy to overclaim. The comb does not say the plant runs Douady and Couder’s rule, or any particular rule. It says that whatever produced the sequence had a memory with a delay in it, and that the delay is the parastichy number. A model in which each primordium’s position depends on the two nearest existing primordia will produce it. A model in which positions are laid down all at once, or read off a pre-existing template, will not.

The angle between the first two peaks, for five starting disordersThe same equations, the same ring, five different starting perturbations: 177°, 47°, 109°, 151°, 47°. A spiral needs the angle between successive elements to be the same one each time, and a stationary ring does not supply that.137.5°disorder 3177°8 peaksdisorder 747°8 peaksdisorder 11109°7 peaksdisorder 19151°7 peaksdisorder 2347°7 peaksthe angle a spiral would need to repeatfive runs, identical equationsspread 129°
Fig. 4 The alternative that the comb rules out, drawn from this site’s own mechanism half: a ring of cells running a reaction–diffusion system, whose peaks appear over a short window rather than one at a time. Five starting disorders give first-to-second angles spread across more than a hundred degrees, so the ring has no divergence angle to speak of. An arrangement made that way has no order for a comb to live in.

The three scatters, and why the control is not a scale artefact

The obvious objection to a control is that it was run at the wrong amplitude — that a kinematic lattice with the right amount of scatter would produce a comb after all, and the one tested happened to miss it.

It was run at a quarter of a degree, at half, and at one degree, which brackets the pattern’s own scatter of 0.70° and covers the range in which the rule’s stems are readable at all. The main comb mean is 0.02, 0.03, 0.02, 0.05 and 0.00 across five seeds, and it is the same at every scatter, because it is the same computation: independent errors have no correlation at any lag whatever their size, and the only thing the amplitude changes is the size of the numbers being uncorrelated.

That is the honest reason the control is decisive rather than lucky. It is not a measurement that happened to come out near zero. It is a measurement that has to come out near zero, and the value of running it is that the machinery — the same pairFromAngles, the same thirty lags, the same three clearance tests — is shown returning nothing when there is nothing.

Only noise that arrives before the choice can change what is chosenIntact runs only, from the whole amplitude sweep. Placement noise displaces the node after the rule has picked an azimuth: 14 runs, none of which changed branch at any amplitude that left a lattice. Field noise perturbs the energy profile the rule picks over, so it can move the minimum into a neighbouring gap: 1 of 17 did.the rule: compute the energy round the circle, take its minimum, place the nodefield noiseperturbs the energy, before16intact runs kept the branch1changed branchplacement noisedisplaces the node, after14intact runs kept the branch0changed branch — none didthe one that moved: 8/13 at 137.8°, 1.31° of scatter31 intact runs of 481 of 17 against 0 of 14
Fig. 5 The three places noise can enter a placement rule, from the phase that separated them. Two of the three arrive before the rule’s choice and change what is chosen; one arrives after and does not. A kinematic lattice is the limiting case of the third, with the rule taken away entirely.

The gate it gives a mechanism claim

The reason to want this is that mechanism claims in phyllotaxis are usually argued from agreement of form: a model produces spiral counts that look like a sunflower’s, so the model is what the sunflower does. This site has spent four phases showing how weak that inference is. Three criteria give three different best angles. A whorled lattice at a rational angle beats the golden angle on hexagonality and on area-evenness. The nautilus is out by a factor of two on a claim everybody repeats. Agreement of form is cheap because many processes make the same forms.

A comb is not cheap, because it is a statement about a sequence and most processes do not produce one. So the useful form of the result is a test a mechanism has to pass:

  • a model that places organs one at a time in the presence of the existing ones predicts a comb at the parastichy number, and predicts its harmonics;
  • a model that lays a pattern down as a field, or that reads it from a template, predicts no comb;
  • and the two are distinguished by a measurement on one stem, not by an aesthetic judgement about which picture looks more like a plant.
60 cells with a carrier that pumps auxin up the gradientEach short line is one cell's polarisation — the neighbour it pumps towards, which is always the richer one. 12 peaks come out, at a contrast of 94%, from a start that was uniform to within 6%.12 peaks60 cells, transport up the gradient12 peaks, contrast 94%
Fig. 6 The transport model this site uses as its mechanism half: carriers that move auxin towards whichever neighbour already has more of it, so a cell slightly ahead drains the ones beside it. It produces peaks with a selected spacing and it makes them on a ring, all at once. Whether a real apex works this way or one organ at a time is the question a comb answers and a picture of the finished pattern does not.

What is still assumed

Three things, and none is small.

That a real plant’s departures are as large as the model’s. The comb is measured on stems disturbed by a tenth to four tenths of a degree of azimuth per node, giving a recorded scatter of half a degree to one degree. If a real apex is far quieter than that, the sequence is too near a constant to autocorrelate and the test returns nothing — which is a refusal, not a negative. The window is measured in its own essay and it has two ends.

That the reading error is small enough. The comb dilutes by the ratio of variances, so a protractor error comparable to the pattern’s own scatter erases it. A quarter of a degree per organ costs nothing; three quarters of a degree costs a factor of four in the length of stem required.

And that the rule’s delay really is the parastichy number in a plant, which is the substantive assumption the whole thread rests on and the one a measurement would test. In the model it is a theorem about the arrangement: node i’s near neighbours are i ± m and i ± n, so a rule that acts on neighbours has those delays. In a plant it is a hypothesis about which existing primordia a new one responds to — and if the answer turned out to be only the single nearest, the comb would have one tooth and no second class, which is a distinguishable outcome rather than a hidden failure.

A sixfold neighbourhood, and nothing to diluteThe prediction was that a rule with fewer neighbours would convert a jostle into divergence scatter more efficiently. Across a sixfold widening the ratio sits between 0.86 and 0.89, and the one point that differs is the narrowest, at 0.80 — smaller, where the prediction wanted larger. The row underneath is why: past four spacings the rule builds the identical lattice, internode for internode, so there is no neighbourhood left to widen.00.50012346912how far the rule looks, in units of the local spacingscatter a jostle adds, over the scatter the same displacement adds after the choiceequal damage0.80 — the wrong wayinternodes that differ between one neighbourhood and the next422→3113→4none4→6none6→9none9→123 runs per point · window 32–190 nodeseach disagreement is one grid sample
Fig. 7 The question the last of those assumptions turns on, from the phase that measured it: past about four spacings the rule builds an identical lattice, so the neighbourhood the model has is much smaller than the neighbourhood the loop sums over. A comb read off a real plant would be a measurement of that neighbourhood rather than an assumption about it.

What a real measurement would look like

The specification is short enough to write out, and writing it out is the point of a control: a result that cannot be turned into an instruction is a result about a program.

Find a stem with a long unbranched stretch — the essay on windows says how long, and the answer is about two hundred and fifty internodes on a shoot slow enough that its parastichy pair does not change across them. Mark the azimuth of each leaf scar to a quarter of a degree, in order, from the bottom up. Difference the list. Autocorrelate the differences out to thirty lags. Sort the lags that clear the band by their remainder on division by the spacing of the strongest comb.

If two residue classes come out, the plant placed its organs one at a time in the presence of the ones already there, and the spacing and the offset are the parastichy pair — which can then be checked against a photograph of the same stem, by a person counting spirals with a finger, with no model in it anywhere.

If nothing clears the band, the answer is one of four things and the sequence alone does not say which: the shoot was too fast, the plant was too quiet, the plant was too disturbed, or the window was in the wrong place. That ambiguity is the subject of its own essay and it is the honest end of this thread rather than a detail.

What the experiment costs, in internodesThe combined sampling band of two autocorrelations falls as one over the root of the sequence length. The difference to be resolved is 0.76 — between noise that arrives before the primordium is placed and noise that arrives after — so the count needed is 56 internodes on a single stem. Every other open question in this collection is priced in tens of specimens.00.2500.5000.750100200300internodes counted on one stemsmallest difference in correlation the count can resolvethe difference to resolve — 0.7656 internodesmatched at 0.75° of scatterone stem, counted once
Fig. 8 What the earlier statistic in this thread cost in internodes, which is the number the specification above is an increase on. The pair is dearer than the single number by a factor of four, and the reading error is dearer still.

The one thing the control cannot do

A control shows that the signal is absent when the process is absent. It does not show that the signal is present whenever the process is present, and the difference is exactly the difference between a sufficient and a necessary condition.

Concretely: there are placement rules that would produce no comb. A rule with no memory at all — each organ placed at a fixed angle from the last, with an independent error — is a process, and its sequence is uncorrelated at every lag, because nothing about node i is available to node i + 8. So “no comb” does not mean “no rule”. It means no rule with a delay in it.

That is a weaker conclusion than the one it is tempting to draw and it is the correct one. The comb’s absence is uninformative; its presence is what carries the evidence, and it carries it because a set of independent errors cannot manufacture two residue classes at the parastichy numbers by accident. The three clearance tests exist to make that “cannot” quantitative, and the kinematic control is where they were shown returning nothing on data that has nothing in it.

Two stems at 0.75° of scatter, one angle at a timeThe divergence of each node, with the slow climb of the ladder removed. Both stems scatter by about 25.97° and a botanist would report them identically. The jostled one wanders in runs — its lag-one correlation is 0.70 — and the one with placement noise alternates about the mean at 0.21, because a displacement of one node enters two consecutive divergences with opposite signs.jostle noise — correlation 0.70placement noise — correlation 0.21243 nodes each, both at 25.97° of scattercorrelations 0.70 and 0.21
Fig. 9 Two stems with the same recorded scatter and different processes behind them, angle by angle. A ruler reports the same number for both; what separates them is the order, which is the same distinction this essay’s control makes in its strongest form.

The shape of the result

The comb belongs with the small number of measurements on this site that are about process rather than shape, and it is the only one of them that could in principle be made on a specimen in a herbarium: a list of divergence angles up a single stem, in order, to a quarter of a degree.

Everything else the site can offer a mechanism is a constraint on what the mechanism has to reproduce — a ladder, a set of transitions, a Fibonacci share that falls as the pattern gets finer. Those are demanding and they are still statements about form. This one asks a different question: not what did the plant make, but did the plant make it one organ at a time.

That the answer is available at all is a consequence of the previous essay’s arithmetic, and it was not what that essay was looking for.

There is one more thing the control settles, quietly, and it is about this site rather than about plants. Four phases of work here have been careful to say that a model reproducing a form is not evidence that a plant runs the model, and the carefulness has occasionally read as a disclaimer attached to results that were doing perfectly well without it. It is not a disclaimer. It is a statement about what class of measurement the site had been making, and the way to see that is to find a measurement in the other class.

A kinematic lattice and a rule’s stem agree on every quantity this site has measured in four phases — divergence, rise, parastichy pair, transitions, contact families, side-count distribution, hop lengths. Every one of those instruments returns the same answer on both. That is not a weakness in the instruments; it is the content of the disclaimer, made concrete. Form is what they measure, and two things with the same form have the same form.

The comb is the first quantity here that separates them, and the separation is not marginal — 0.64 against 0.02, on the same sampling band. Which means the division the site has been asserting since its foundation phase is now a division it can demonstrate, on two arrangements it built itself, with the same code reading both.

Where this goes next

-0.50000.500125810131621242629lag, in internodescorrelation between a divergence and the one that many internodes later816245132129spacing 8 · offset 5pair 8/13 — counter says 8/13sampling bandone stem · 760 divergences · disturbance 0.25generated from a stated rule, not drawn to look right The second comb The autocorrelation of a divergence sequence has peaks at the smaller parastichy number and at every multiple of it. It also has a second set of peaks, at the same spacing, offset by the difference of the pair — so a list of angles with no coordinate in it returns both numbers rather than one. 120° — a third of a turn3 and 6 — whorled137.51°2 and 3 — Fibonaccirise 0.055 in both panelsthe counts decide, not the eye What a mechanism would have to show This site says of every model it draws that reproducing a pattern is not explaining it. That is easy to repeat and hard to make precise. Here it is made precise — a list of what an account of phyllotaxis would have to establish, with each item marked according to whether the models on this site establish it. 0501000.2500.5000.7501growth parameter Gspread of the last 30 steps (°) — 0 means settledfilled dark: convergedthe usable range is stated, not implied Where the model stops Below a growth parameter of about 0.18 this implementation does not converge — the settled angle wanders over a hundred degrees however long the run. That is the range where the literature says the interesting behaviour lives, and it is worth a figure rather than a quietly chosen axis.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

  • A harmonic is a step taken twice — both name autocorrelation, divergence angle, equilibrium, lattice, measurement, parastichy pair, the placement rule, self correction
  • The test a plant could settle — both name autocorrelation, discrimination, divergence angle, evidence, falsifiability, measurement, noise, self correction
  • A shoot too fast to remember — both name autocorrelation, divergence angle, equilibrium, measurement, noise, the placement rule, self correction
  • The memory was the rise — both name autocorrelation, divergence angle, equilibrium, measurement, noise, the placement rule, self correction
  • What a refusal does not say — both name autocorrelation, discrimination, divergence angle, evidence, falsifiability, measurement, noise
  • What one angle says about the next — both name autocorrelation, discrimination, divergence angle, measurement, noise, the placement rule, self correction

Named objects

A flat tag is an object no other essay names yet.

AutocorrelationDiscriminationDivergence angleEquilibriumEvidenceFalsifiabilityLatticeMeasurementMechanismMeristemModel scopeNoiseParastichy pairThe placement ruleSelf correction