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

The ratio was never about the rule

One phase ago the comb was retracted as evidence that a plant computes its pattern, and one quantity was exempted from the retraction: the ratio of the two combs, which a placement rule and a transported disturbance divide differently. Drive seven disturbances through the same rule and the ratio spans 0.45 to 1.09. The exemption does not hold, and the angle sequence has nothing left.
19 min read 9 figures Form is not mechanismState the claim, state the testCounted, not admired

Worth reading first: Errors that pass between organs · A disturbance with a memory · What a mechanism would have to show.

The previous phase retracted a claim and kept a number.

The claim was that a comb in the autocorrelation of a divergence sequence is evidence that a plant computes its pattern rather than merely having one. It failed because an arrangement with no rule in it — a lattice whose organs sit at exact multiples of the divergence, with errors inherited from the organs eight and thirteen places back — reproduces the comb, the second comb and the counted pair on every seed.

The number was the ratio between the two combs: 0.65 for the placement rule against 1.30 for a transport weighted by distance. A factor of two, and the last thing standing.

Which arrangements carry a comb, and what each one reportsThe largest comb mean in five arrangements at a rise of 0.005, all read by the same instrument at the same length, with the sampling band of 0.073 marked. Only the first is a placement rule; the other four are kinematic lattices with no rule in them, differing from one another only in how their azimuth errors are structured. Independent errors and errors with a memory leave nothing to read. A repeating error puts up a comb and names a partner that is not the lattice's. Errors inherited from the contact neighbours reproduce both the comb and the pair.three sampling bandsthe placement rule0.6428/13independent errors0.031refusedan error with a memory0.014refusedan error that repeats0.4338/10, 8/12errors passed between neighbours0.5538/13one rule, four kinematic latticesgenerated from a stated rule, not drawn to look right
Fig. 1 The five arrangements the previous phase compared, all of which agree about every qualitative observable it had. The rightmost has no rule in it; the leftmost is the rule. The ratio was the only quantity that told them apart.

This essay withdraws the exemption.

The experiment

Take the placement rule — unmodified, the site’s own — and drive it with seven different disturbances. Every one arrives the same way, as a jostle: each organ is placed exactly where the rule says, and then the organ moves, so the disturbance is invisible in what a botanist measures and present in every decision the rule makes afterwards.

Every one is normalised to the same displacement per organ. Every one leaves a lattice standing at the amplitude used. The rule is identical in all seven, the rise is identical, the seeds are identical.

Only the correlation structure of the disturbance changes.

The ratio follows the disturbance, not the ruleThe ratio of the second comb to the main comb on stems grown by the placement rule and jostled by seven different disturbances, all at 0.25° of displacement per organ and all on the same rule. Independent errors and errors with a memory return 0.76–0.81, which is the value this site measured for the rule. A periodicity at the smaller parastichy number takes it down to 0.45; errors inherited from the contact neighbours take it up to 1.09, most of the way to the 1.24 a transported disturbance gives with no rule in it at all. So the quantity separates arrangements by how their errors are related, not by whether anything computed the positions.second comb ÷ main comb, at 0.25° of displacementthe rule, 0.79no rule at all, 1.24independent0.80a memory, ρ = 0.50.78a memory, ρ = 0.90.76a memory, ρ = 0.970.81repeating every 80.45inherited, a = 0.51.02inherited, a = 0.71.095 stems a row · rise 0.005generated from a stated rule, not drawn to look right
Fig. 2 The ratio of the second comb to the main comb, on seven arrangements that all contain the same placement rule. The two dashed marks are the rule’s own value and the value the same readout gives on a kinematic lattice with no rule in it at all.
the disturbance, with the rule underneath it ratio
independent errors 0.80
a memory, ρ = 0.5 0.78
a memory, ρ = 0.9 0.76
a memory, ρ = 0.97 0.81
a periodicity every 8 organs 0.45
inherited from the contacts, a = 0.5 1.02
inherited from the contacts, a = 0.7 1.09
no rule at all, inherited errors 1.24

The last row is the kinematic forgery at equal coupling to its two contact neighbours; weighted by distance, as a d⁻³ interaction would weight them, it gives 1.29. Both numbers are the rival’s and the difference between them is not what this essay turns on.

A placement rule gives anything from 0.45 to 1.09 depending on what is disturbing it. The quantity that was supposed to identify the process spans most of the range the identification was made over, without the process changing at all.

Two of the rows deserve a second look, because they are the ones that make the result a measurement rather than a demonstration.

The periodicity moves it the other way. A disturbance that repeats every eight organs — the smaller parastichy number — strengthens the main comb, which collects the multiples of eight, and leaves the second comb alone. Main 0.77 against the rule’s own 0.59; second 0.34 against 0.47; ratio 0.45. So the quantity is not merely pushed upward by anything correlated: it follows the disturbance in both directions, which is what says it is tracking the disturbance rather than being degraded by it.

And the memories move it hardly at all. 0.78, 0.76 and 0.81 against the undisturbed 0.80, across correlation coefficients from 0.5 to 0.97. A memory is the shape everyone means by “correlated noise”, it is the one the previous phase tested and cleared, and it is the one that turns out not to matter. Had this experiment been run with memories alone it would have confirmed the exemption.

What this does to the discriminator

The reading of the previous phase’s number was: near 0.65 is consistent with a rule, and near or above 1.30 is evidence against one. It was written into the survey specification in exactly those words.

A plant that both computes its organ positions and passes a fraction of each organ’s displacement to the organs it touches reads at 1.02 to 1.09. That is a plant with a rule in it, reading in the band that was to count against a rule.

And such a plant is not a contrivance. It is the likeliest thing a plant is. Organs in a meristem are packed at the density where they touch, they grow while in contact, and a displacement in one is not obviously lost before it reaches the next. The previous phase made that argument itself, at length, as the reason its forgery was not a contrivance — and then exempted the ratio from it.

The exemption is withdrawn. There is no reading of the ratio that counts against a placement rule.

A lattice with an error inherited from the two contact neighboursThe autocorrelation of 759 divergence angles from a kinematic lattice at a divergence of 137.8261° and a rise of 0.005, with 0.5° of scatter on each azimuth. There is no placement rule anywhere in it: node i is put at exactly i times the divergence and then displaced. The only thing that differs between this figure and the control is the structure of the displacement — here, an error inherited from the two contact neighbours at coupling 0.7. The largest comb mean is 0.514 against a sampling band of 0.073, and the readout returns 8/13.-0.50000.500125810131621242629lag, in internodescorrelation between a divergence and the one that many internodes later816245132129reads 8/13 · no rule in itmain 0.514 · band 0.073the shaded strip is the sampling bandkinematic lattice · 759 anglesgenerated from a stated rule, not drawn to look right
Fig. 3 The arrangement the ratio was meant to exclude. What this essay adds is that a stem which does contain the rule, disturbed the way that arrangement’s errors are disturbed, produces nearly the same number.

It is worth being exact about how far the reading has to move, because “the discriminator fails” can mean anything from a slight overlap to a complete collapse.

The rule’s own value is 0.80 and the forgery’s is 1.24, a gap of 0.44. The contact-transported jostle sits at 1.02 and 1.09, which is half to two thirds of the way across that gap. It does not reach the forgery’s value.

So the honest statement is not that a rule and a forgery are indistinguishable by the ratio. It is that a rule with a plausible disturbance lands in the middle of the gap, which means no threshold placed anywhere in it separates the two hypotheses without also separating a rule from itself. A discriminator whose two classes are not separated by any threshold is not a discriminator, and that is a stronger statement than an overlap.

What the ratio does measure

Something, and it is worth stating precisely because a retraction that leaves nothing behind is usually an overcorrection.

Read down the table. The three arrangements whose disturbance is independent or nearly so give 0.76 to 0.81. The one whose disturbance repeats at the smaller parastichy number gives 0.45. The two whose disturbance is passed between contact neighbours give 1.02 and 1.09, and the arrangement whose errors are entirely inherited from contacts, with no rule anywhere, gives 1.24.

The quantity is monotone in something, and the something is how much of the disturbance is shared between contact neighbours. That is a real property of a plant, nobody has measured it, and the ratio measures it.

So the correct statement of what a comb and its ratio are worth is:

A comb rules out independent errors. Established two phases ago, unaffected.

The ratio says how the errors are related. Below about 0.6, something in the disturbance repeats at the smaller parastichy number. Between about 0.7 and 0.9, the errors are independent or share a common mode. Above one, they are transmitted between contact neighbours.

And neither says whether anything computed the positions.

That third line is what this phase costs and it is worth reading twice, because it is the whole of the thread’s original ambition.

A memory manufactures nothingThe largest comb mean found in a kinematic lattice whose azimuth errors are an AR(1) process, against the coefficient of that process, over eight seeds at each point. The dashed line is where the rule's own stems sit, at 0.64; the shaded strip is three sampling bands. Every point is inside the strip — 0.028, 0.026, 0.022, 0.014, 0.015 at ρ = 0.3, 0.5, 0.7, 0.9, 0.97 — and the readout returns nothing on 40 runs out of 40. A correlated error is not a periodic one.00.2000.4000.6000.3000.5000.7000.9000.970how strongly each error remembers the last, ρthe largest comb mean anywhere in the thirty lagsthe rule's own stems: 0.64three sampling bands0.0280.0260.0220.0140.015kinematic lattice · AR(1) errorgenerated from a stated rule, not drawn to look right
Fig. 4 The one control the previous phase got right and this phase confirms from the other side. A memory manufactures no comb on a kinematic lattice, and driven through the rule it leaves the rule’s own ratio alone — the only disturbance shape tested that does neither.

The prediction that was written down, and how it failed

This measurement was asked for a phase ago, with a prediction attached:

A jostle whose displacements are correlated from organ to organ is a different object, because the rule responds to it — the prediction written down here is that a self-correcting rule shortens whatever correlation length it is given, so this phase’s negative result should hold a fortiori.

The first half is right in a form the phase did not anticipate. The rule does damp a correlated disturbance, and hard: a lattice survives three times the displacement when the organs share it, and records a third of the scatter, both because the rule works on differences and a shared displacement has none.

The second half does not follow from the first and is wrong. Damping the common part of a disturbance leaves the structured part untouched, and it is the structured part the readout is sensitive to. A contact transport is almost entirely structure — correlated at two lags and nowhere else — so the rule damps almost none of it, and what reaches the readout is what the forgery would have sent.

The failure is instructive rather than embarrassing, and its shape is common in this collection: an argument about a mechanism was used to predict the value of a statistic, and the statistic turned out to be sensitive to something the argument did not mention.

Three disturbances, three places to get inThe rule reads its neighbours, builds a profile of the energy at every azimuth, takes the least of it, and records a position. field noise enters at the profile; jostle noise enters at the neighbours; placement noise enters at the record. Two of the three are upstream of the choice and can change which minimum is taken; the third is downstream and never can.upstream of the choicethe neighboursalready placedthe profileenergy by azimuththe choicethe least of itthe recordwhat a ruler readsfield noisejostle noiseplacement noiseone rule, three entry pointsthe order is the argument
Fig. 5 The three entry points, from the phase that separated them. This thread adds a second axis — what shape the disturbance has — and the phase plan’s prediction was made along the first axis about a quantity that lives on the second.

It is worth naming what would have had to be true for the exemption to hold, since it was not an unreasonable thing to believe.

The exemption rested on the ratio being a property of the placement: the rule prefers the shorter index offset even when the longer one is the geometrically nearer neighbour, which no falloff explains and which the previous phase reported as measured rather than derived. If that preference were a fact about how the rule chooses, it would survive any disturbance the rule could still work through — and the ratio would be a signature of the mechanism in the way a spectral line is a signature of an element.

What the measurement says is that the preference is not that kind of fact. It is the ratio of two correlations in the output, and correlations in the output inherit correlations in the input. The rule contributes its share and the disturbance contributes its share, and at the amplitudes a plant would have, the disturbance’s share is comparable.

What is left

The angle sequence has been this collection’s most productive instrument. It gave the parastichy pair from angles alone, then a second comb and therefore both members of the pair, then a check on the rate from two overlapping windows. Every one of those is a verdict and every one survives.

What it has never given, and now demonstrably cannot, is a statement about process. Three attempts:

The comb. Retracted last phase: a transported disturbance forges it.

The ratio. Retracted here: the rule’s own value moves with the disturbance into the forgery’s range.

And the rung dependence, which is not a third attempt so much as the second one’s other flank: even holding the disturbance fixed, the ratio depends on where the plant sits between two transitions, and near a boundary it reaches the forgery’s value on its own.

The ratio is a U across every rung, and its floor is the number that was reportedThe ratio of the second comb to the main comb, on five stems at each of 15 rises spanning two rungs, against the ladder's own coordinate for where each rise sits inside its rung. Both rungs give the same shape: a floor of 0.71 and 0.79 about two thirds of the way up, climbing towards the transition at either end. The dashed line is a transported disturbance with no rule in it at 1.29, which does not vary with the rise at all — a kinematic lattice's angle sequence has no rise in it. Where the rule's curve crosses that line the two accounts are indistinguishable.1200.2500.5000.7501where the rise sits in its rung — 0 at the next transition down, 1 at the last onesecond comb ÷ main comb5/8 rung8/13 runga transported disturbance, no rulethe floor, 0.795 stems a point · 1152 azimuthsgenerated from a stated rule, not drawn to look right
Fig. 6 The other half of the same problem. Two variables the ratio depends on that have nothing to do with the mechanism, and one — the plant’s own disturbance structure — that cannot be measured independently.

The reason all three failed is the same and it was visible from the start. The divergence sequence is a record of where organs ended up. Two processes that put organs in the same places leave the same record, and the previous phase’s forgery is an existence proof that a process with no rule in it can put them in the same places. Everything since has been an attempt to find a second-order statistic that distinguishes them, and second-order statistics of the same arrangement are the same statistics.

Could a fourth attempt work?

It is fair to ask whether some other statistic of the angle list could do what these three could not, and the answer looks like no for a reason that can be stated rather than merely suspected.

The forgery is a lattice with a history in its errors. Its organ positions are decided in advance and its mistakes are computed in order, each from the mistakes of the organs it touches. So it has everything a placement rule has except the placement: an order of production, a neighbour graph, and errors that propagate along it.

Any statistic of a divergence sequence is a function of the sequence of azimuth errors, because the divergence is the constant angle plus a difference of consecutive errors and the constant contributes nothing to any centred statistic. So a statistic that distinguished the two accounts would have to distinguish error sequences generated by transport along the contact graph from error sequences generated by a rule minimising against displaced neighbours — and the second phase of this thread’s work has been finding that these are the same kind of object.

The rule’s errors and the forgery’s errors both propagate along the contact graph, in the order of production, with a coupling. The rule’s coupling is implicit in the minimisation and the forgery’s is a parameter. What is different is that the rule’s coupling is negative and self-correcting where the forgery’s is positive — which shows up in the lag-one correlation, and lag one is the one place where the two do differ.

That is worth someone’s afternoon and it is not this phase’s, because the lag-one statistic has its own problem: it is dominated by measurement error, and this collection established two phases ago that any error in a recorded position enters two consecutive divergences with opposite signs and puts a large negative value there. A discriminator that lives at lag one is a discriminator that measures the protractor.

The third identifiability failure

Put beside its predecessors, this result is less of a surprise than it feels.

The measurement phase found that the interaction range is not identifiable from a finished pattern: every exponent above about 1.06 gives the same lattice, the same ladder and the same transitions, so a pattern cannot say how far its rule reaches.

The mechanism-02 phase found that the cut-off shape is not identifiable either: two shapes with different mechanisms behind them, put on a common scale, disagree about where the lattice ends by two thirds, and nothing in the arrangement says which one is right.

And now the existence of a rule is not identifiable from the angle sequence.

Three results with one shape: a model parameter that seemed to matter, measured against everything a finished arrangement records, and found not to be recoverable from it. What all three have in common is that the arrangement is an attractor — the end state of a process, which by construction has forgotten most of the process.

The general lesson is unglamorous and worth having: the more strongly a mechanism converges on its pattern, the less its pattern says about the mechanism. A rule that reliably produces the golden angle from any start is a rule whose output cannot report that it was that rule.

What replaces it

An intervention, which is the rest of this phase.

Remove one organ from a settled apex. The placement rule has to answer, because the neighbourhood it minimises over has changed: the next organ moves by between 2.6° and 168°, and the number of organs whose removal matters is the larger parastichy number. The transported-error account cannot answer, because in it no organ’s position was ever a function of which organs exist.

The next organ moves for the last 13, and for no othersOne row per organ removed, counted back from the tip of a stem at a rise of 0.005 whose counted pair is 8 and 13. Removing any of the last 13 moves the next organ by 2.6° to 167.6°; removing an older one moves it by at most 0.47°, which is under the azimuth grid. The boundary is at 13, and 13 is the larger parastichy number — so the experiment counts the spirals without measuring an angle.organ removed, counted back from the tiphow far the next organ moves, in degrees1138.0°284.4°353.4°4167.6°529.3°6101.7°7120.7°816.4°9165.2°1056.7°1181.1°12140.6°132.6°— the front ends here140.0°150.0°160.5°rise 0.005 · pair 8/13generated from a stated rule, not drawn to look right
Fig. 7 The replacement. A step function whose edge is a count, an effect thirty times the plant’s own noise, and a rival that predicts nothing at any offset.

That is a different kind of evidence and it is the kind the question needed all along. A finished pattern is compatible with any process that could have made it; an intervention asks the process a question while it is still running.

Take away the organ eight places back, and the next one goes into the holeThe last 34 organs of a stem at a rise of 0.005, unrolled. The open circle is the organ removed — eight places before the tip. The ring at the top is where the rule puts the next organ with every organ present; the filled mark beside it is where the rule puts it with that one missing. The two are 16.4° apart, against a local spacing of 25°, and the vacancy itself is 22.7° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.moved 16.4°open ring: where the next organ was going · filled: where it goes · large open circle: the organ removedrise 0.005 · cut 8 back · height ×6generated from a stated rule, not drawn to look right
Fig. 8 The intervention itself. One organ removed, and the next one placed against what is left — which is the first experiment in this collection that could not have been done on a photograph.
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 The standing lesson of the thread, reached once more and for the last time from this direction: two arrangements can agree in everything a finished pattern records and differ in what made them. Three statistics have now been offered as the exception and none of them was.

Where this goes next

-0.50000.500125810131621242629lag, in internodescorrelation between a divergence and the one that many internodes later816245132129reads 8/13 · no rule in itmain 0.514 · band 0.073the shaded strip is the sampling bandkinematic lattice · 759 anglesgenerated from a stated rule, not drawn to look right Errors that pass between organs An organ's neighbours are the ones eight and thirteen places back — that is what a parastichy pair is. So a disturbance transmitted by contact is correlated at exactly the two lags the readout examines, and it does not have to be told them. Driven into a lattice with no rule in it, it returns the counted pair on eight stems out of eight. 1200.2500.5000.7501where the rise sits in its rung — 0 at the next transition down, 1 at the last onesecond comb ÷ main comb5/8 rung8/13 runga transported disturbance, no rulethe floor, 0.795 stems a point · 1152 azimuthsgenerated from a stated rule, not drawn to look right The ratio was the floor of a curve The last phase left one number standing between a placement rule and a transported disturbance, measured at one rise, with the explanation that the geometry there happens to favour the larger parastichy number. Swept across two rungs the number is a U — a floor of about 0.79 two thirds of the way up a rung, climbing past 2.8 as a transition approaches — and the geometry is flat exactly where the curve is steepest. moved 16.4°open ring: where the next organ was going · filled: where it goes · large open circle: the organ removedrise 0.005 · cut 8 back · height ×6generated from a stated rule, not drawn to look right The organ that was taken away Every observable this site has is read off an arrangement that was finished before the reading began, and the last phase showed what that costs. So remove one primordium from a settled stem and place the next one against what is left. The rule has to answer. The rival account cannot, because in it no organ's position was ever computed from its neighbours.

Shares its objects with

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

  • The comb was never the rule — both name autocorrelation, discrimination, evidence, falsifiability, honest limits, identifiability, measurement, mechanism, null model, the placement rule, transport
  • What a forgery has to know — both name autocorrelation, discrimination, evidence, honest limits, identifiability, measurement, noise, null model, parastichy pair, the placement rule, transport
  • A comb is evidence of a rule — both name autocorrelation, discrimination, divergence angle, evidence, falsifiability, measurement, mechanism, noise, parastichy pair, the placement rule
  • A periodicity is not a lattice — both name autocorrelation, discrimination, divergence angle, ensemble, evidence, identifiability, measurement, noise, null model, parastichy pair
  • The control a survey would need — both name autocorrelation, discrimination, evidence, falsifiability, honest limits, identifiability, measurement, null model, parastichy pair, transport
  • The disturbance that travels — both name autocorrelation, discrimination, divergence angle, ensemble, honest limits, measurement, noise, parastichy pair, the placement rule, transport

Named objects

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

AutocorrelationDiscriminationDivergence angleEnsembleEvidenceFalsifiabilityHonest limitsIdentifiabilityMeasurementMechanismNoiseNull modelParastichy pairThe placement ruleTransport