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

What the rule does to a drift

A placement rule was supposed to leave no slow wander in a divergence sequence, because its errors are corrections rather than inheritances. Driven by a disturbance that drifts, it leaves a larger one than a lattice with no rule in it at all — while cutting the per-organ scatter by more than half. The rule removes what is relative between neighbours, and a drift is not.
20 min read 11 figures Form is not mechanismState the claim, state the testOne parameter, one family

Worth reading first: A disturbance with a memory · The sequence has a memory.

There is a sentence in this collection that reads, of a placement rule: its errors are corrections rather than inheritances, and its divergence sequence is anticorrelated at lag one rather than drifting. It was written to say that a rule produces neither of the two signatures a transported disturbance produces, which would make a stem carrying one of them evidence against the rule.

The first half is right and measured. The second half is a claim about a rule under a disturbance, and which disturbance was not specified. Under independent noise the rule produces no drift. Under a drifting disturbance it produces more drift than the same disturbance produces with no rule anywhere.

The measurement

The same coloured disturbances that this collection drives into kinematic lattices are driven through the rule instead, as a jostle: each organ is placed exactly where the rule says and then moves, so the disturbance is invisible in what is measured and present in every decision the rule makes afterwards. Every stream is normalised by its own standard deviation, so a quarter of a degree means a quarter of a degree of displacement per organ in every row.

Through the rule, the drift survives and the inheritance still does notHow much of a divergence sequence's variance survives being averaged over blocks, on stems the rule grew. The vertical quantity is B² times the variance of the block means divided by the variance of the sequence, which is one at every block size for independent errors — the arithmetic is normalised for a *differenced* stream, since a divergence is the difference of two organs' errors. A line that climbs is a sequence with power at frequencies below one per block. The disturbances that remember the last error climb to 46 at a block of 64. The ones inherited between touching organs do not climb at all — 1.51 and 1.90 at the same block — although their own deviates carry ×— and ×— an independent stream's variance in exactly this statistic. What they do instead is dig a hole: at block sizes of 8 and 13, which are the offsets they couple at, the statistic falls to 0.06 and 0.12.10.11024813163264block size, in organsvariance of the block means, against independent errorsindependenta memory, ρ = 0.9a memory, ρ = 0.97inherited, a = 0.5inherited, a = 0.7900 organs · jostled at 0.25° · 3 stems eachgenerated from a stated rule, not drawn to look right
Fig. 1 How much of a divergence sequence’s variance survives being averaged over blocks, on stems the placement rule grew. A flat line at one is a sequence with no low-frequency power in it.

Independent noise gives 0.91 at a block of sixty-four, which is the flat line the prediction wanted. A memory with coefficient 0.97 gives 46.47. On a kinematic lattice — the same disturbance, added to positions with no rule involved — the same block gives 30.03.

So the rule does not remove a drift. It multiplies the drift’s visibility by about half again.

The scatter goes the other way

The same runs, read as a scatter, look like the rule doing exactly what it is supposed to do. Under independent noise the divergence sequence scatters by 0.553°. Under the drifting disturbance it scatters by 0.246° — less than half — at the same per-organ displacement.

That is not a contradiction; it is the two halves of one mechanism.

The rule places each organ at the least of a repulsion profile summed over its neighbours. A displacement that all the neighbours share moves the whole neighbourhood without changing its shape, and the least of a translated profile is the same place relative to the neighbourhood. What a placement rule responds to is the part of a disturbance that is relative between neighbours — which is its high-frequency part.

A drift is nearly the same displacement for every organ in a window of a few dozen. It is almost entirely non-relative, so the rule absorbs almost none of it and corrects almost none of it. Independent noise is entirely relative, so the rule corrects a large share of it.

What a lattice survives depends on the colour of the disturbance, sixfoldfive stems for each of seven disturbances at each of six displacements, every stream normalised by its own measured spread so that a displacement of half a degree is half a degree in every row. A filled mark is a stem that still has a lattice — a divergence scatter under 2° — and an open one is a stem that does not. Independent errors survive to 0.5°; errors that remember the last one to 1.5°; errors inherited from the contact neighbours only to 0.25°. The number beside each row is the scatter a protractor would record where the lattice is standing, and it is the quantity that explains the table: what destroys a lattice is not how far an organ moves, but how far it moves relative to the organs it is placed against.displacement per organ0.25°0.5°1°1.5°2°3°independent0.57–1.03°a memory, ρ = 0.50.38–0.68°a memory, ρ = 0.90.28–0.98°a memory, ρ = 0.970.26–1.31°repeating every 80.63–0.93°inherited, a = 0.50.97–0.97°inherited, a = 0.71.15–1.94°scattera latticeno lattice left5 stems a cell · rise 0.005generated from a stated rule, not drawn to look right
Fig. 2 The same mechanism seen through survival: at equal per-organ displacement, the amplitude a lattice survives depends on the colour of the disturbance by a factor of six. A shared displacement is nearly free and a relative one is not.

Why that makes the wander bigger and not merely undiminished

The statistic is a ratio. Its numerator is the variance of the block means, which the drift dominates and the rule barely touches; its denominator is the variance of the sequence, which the rule cuts by more than half.

A ratio whose numerator is untouched and whose denominator is halved goes up. That is the whole of it: 30.03 becomes 46.47 because the sequence it is measured against got quieter, not because the drift got louder.

This is worth stating carefully because it inverts the intuition that a self-correcting system is harder to read. A self-correcting placement rule is an excellent high-pass filter, and a high-pass filter improves the visibility of anything slow that survives it. The rule makes a drift easier to detect, not harder.

A lattice or a wreck, with nothing in betweenEvery run in the sweep, at both kinds of noise. The line at 6° is where a run stops being counted as a lattice, and nothing lands near it: the intact runs reach 1.97° and the destroyed ones start at 8.06°, a factor of 4.1 away.00.50011.5001234567amplitude, by stepscatter, log₁₀ degrees6°intactno latticeplacementfield48 runs · both kindsan empty factor of 4.1 at the cut
Fig. 3 Divergence scatter against the displacement that produced it, by colour. The relative part is what sets the scatter, which is why disturbances of equal size land in different places on this axis.

The filter, written out

It is worth being precise about “high-pass filter”, because the phrase is doing explanatory work and could be doing it loosely.

Take the rule’s response to a disturbance that is a pure oscillation in the organ index — every organ displaced by an amount that goes round a cycle every L organs. If L is short compared with the neighbourhood the rule sums over, the neighbours’ displacements at any moment are spread over a whole cycle, they push in different directions, and the shape of the neighbourhood is genuinely distorted: the rule sees it and responds. If L is long compared with the neighbourhood, every organ in the window is displaced by nearly the same amount, the neighbourhood translates, and the rule sees nothing.

The neighbourhood at these rises is about ninety organs deep. So the crossover is somewhere in the tens of organs, which is exactly where the wander statistic’s block sizes are, and exactly where a memory with a coefficient of 0.97 puts most of its power: its correlation falls to a half at about twenty-three organs.

That gives the two effects a common cause and a common scale. The scatter falls because the fast part is removed; the block-mean variance survives because the slow part is not. Both are consequences of the neighbourhood having a depth, and a rule with a much shorter reach would show neither.

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. 4 The neighbourhood the rule actually sums over, and where in it a disturbance has to act to be felt. The depth of that window is what sets the scale separating what the rule corrects from what it passes through.

The inherited disturbance, through the rule

The other half of the table is the shape that has no drift to begin with, and it behaves as the arithmetic says it should.

An inherited disturbance at a coupling of 0.7 gives 1.90 at a block of sixty-four through the rule — near one, as on a kinematic lattice — and at the two contact offsets it gives 0.06 and 0.12. The hole is deeper than the kinematic version’s 0.22 and 0.21, for the same reason the memory’s plateau is higher: the denominator has shrunk.

So the two signatures separate more sharply through the rule than without it. A sequence from a rule-grown stem shows a drift as a larger number and a comb as a deeper hole, both because the rule has removed the part of the disturbance that is neither.

The disturbance with the largest wander leaves none in the sequenceHow much of a divergence sequence's variance survives being averaged over blocks, on kinematic lattices. The vertical quantity is B² times the variance of the block means divided by the variance of the sequence, which is one at every block size for independent errors — the arithmetic is normalised for a *differenced* stream, since a divergence is the difference of two organs' errors. A line that climbs is a sequence with power at frequencies below one per block. The disturbances that remember the last error climb to 32 at a block of 128. The ones inherited between touching organs do not climb at all — 1.06 and 0.83 at the same block — although their own deviates carry ×5 and ×49 an independent stream's variance in exactly this statistic. What they do instead is dig a hole: at block sizes of 8 and 13, which are the offsets they couple at, the statistic falls to 0.22 and 0.21.10.11024813163264128block size, in organsvariance of the block means, against independent errorsindependenta memory, ρ = 0.5a memory, ρ = 0.9a memory, ρ = 0.97inherited, a = 0.5inherited, a = 0.7shared once, a = 0.76000 organs · 6 runs eachgenerated from a stated rule, not drawn to look right
Fig. 5 The same curves without the rule, on lattices with the disturbance simply added. The shapes are the same and the departures from one are smaller — the rule sharpens the contrast rather than creating it.
The damage is the sharing; the forgery is the historyThree disturbances of the same size, measured four ways. The two left columns are stems grown by the placement rule and jostled at 0.25° per organ: a disturbance shared between the contact neighbours scatters the lattice by 0.71° against white noise's 0.57°, and one inherited from them — the same sharing, passed on again at every organ — by 0.97°. The two right columns are kinematic lattices with no rule in them at all, where the whole question is what a disturbance can manufacture. The inherited one returns the pair on 8 seeds of 8 with a main comb of 0.205 against a band of 0.073; the shared one, at the same coupling and the same scatter, returns it on 1 and makes a comb of 0.099, which is the band. So sharing an error with the organs you touch does the damage, and only passing it on and on forges the evidence.scatterthrough the rulecomb ratiothrough the rulemain combforged, no ruleseeds agreeingforged, no ruleindependent0.57°0.80——shared once0.71°0.910.101/8inherited0.97°1.020.218/8same coupling, same offsets 8 and 13, same eight seedsrule at 0.25° a organ · forged at the same couplinggenerated from a stated rule, not drawn to look right
Fig. 6 What an inherited disturbance costs a lattice at the displacement used here. Its damage is concentrated at the two offsets the rule’s profile is dominated by, which is the same fact that puts the hole in the wander statistic at those two block sizes.

The rule’s own dip at eight

One detail in the table is not about any disturbance. Under independent noise the rule’s stems give 0.20 at a block of eight, against 0.91 at sixty-four and 0.75 at two.

Eight is the smaller parastichy number of the arrangement these stems carry. The rule’s own placements are correlated at the contact offsets, because that is where its neighbours are; an organ placed against a neighbourhood inherits a little of the arrangement of that neighbourhood, and the arrangement repeats at eight and thirteen.

That is a small, real signal, and it is the reason the hole cannot be read as proof of a transported disturbance on its own. A stem with no inheritance at all still digs a shallow hole at its own contact offsets. The inherited disturbance digs a much deeper one — 0.06 against 0.20 — but the null is not one and a test built on the assumption that it is would be a test with an inflated rate of false positives.

The memory of a divergence sequence, at 0.75° of scatterWith no noise at all the lag-one correlation is 0.54: the rule corrects itself, so a lattice arrives with a memory in it. Matched at the same recorded scatter, placement noise leaves -0.04, jostle noise leaves 0.65, field noise leaves 0.50. The band is ±0.13, which is what an uncorrelated sequence of this length gives.-0.25000.2500.500123456lag, in nodescorrelation between a divergence and the one that many nodes laterno noiseplacement noisejostle noisefield noisesampling band3 runs each · 243 divergences per runmatched at 0.75° of scatter
Fig. 7 The rule’s own signature at short lags: strong anticorrelation at lag one, which is the restoring force. The dip at a block of eight in the wander statistic is the same structure read at a longer scale.

What this does to the observable

The proposal this collection had made was that a rule produces no drift, so a drifting sequence is evidence against a rule. That is now wrong in a specific way, and the corrected version is more useful.

A drift in a divergence sequence says the disturbance drifts. It says nothing about whether a placement rule is operating, because a placement rule passes a drift through — and it does not even weaken the reading, since a rule-grown stem shows the drift more clearly.

What separates a rule from a transport is still the thing this collection has been using: the anticorrelation at lag one, which is a restoring force and cannot be produced by an error inherited from neighbours. The wander is orthogonal to that question and belongs to a different one.

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. 8 The comb ratio by colour, at the displacement used throughout. This is the statistic that does separate the rule from a transport, and nothing in this essay touches 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 52.26° 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 0.2160 nodes each, both at 52.26° of scattercorrelations 0.70 and 0.21
Fig. 9 Two stems matched on the quantity a botanist would report — the scatter of the divergences — and differing in the colour of the disturbance behind them. Matching on the visible quantity is what makes a comparison between colours a comparison at all.

A prediction with a shape

The measurement gives an experiment something to look for that is not a single number.

If a shoot is placed by a local rule and its errors are dominated by a drifting environment, its divergence sequence should be quiet at short scales and loud at long ones — a small scatter organ to organ, and block means over a hundred organs varying much more than that scatter would allow. A shoot whose errors are independent should be loud at short scales and quiet at long ones, in the same sequence.

That is a shape rather than a threshold, and shapes are cheaper to trust. It also comes with a warning attached: a shoot with a small organ-to-organ scatter is usually read as a well-behaved shoot, and this says the opposite can be true — a small scatter may be a rule doing a great deal of correcting against a large disturbance, and the size of that disturbance is only visible at long scales.

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. 10 The three places a disturbance can enter a placement rule, as this collection distinguishes them. The jostle used here is the one that leaves the rule’s arithmetic untouched and moves the organ afterwards, which is why it can be compared with a lattice that has no rule in it.

Reading a real sequence with this in mind

Suppose a sequence of divergences from a real shoot, several hundred organs long, measured to a fraction of a degree. What the two statistics would say together:

A small organ-to-organ scatter and a flat variance-time curve is a quiet shoot with a quiet environment, and it says nothing about mechanism at all.

A small scatter and a climbing curve is the interesting case. It says the shoot’s placements are being corrected against something large and slow, and the size of that something can be read off the plateau: for a disturbance with a memory, the curve levels at one over one minus its own correlation coefficient, so the plateau names the memory’s length.

A large scatter with a flat curve and holes at the contact offsets is the inherited case this collection has spent three essays on, and the holes are what would say so.

A large scatter with neither is independent error, which is the null.

Four readings from two statistics, and the point of setting them out is that three of the four were unavailable while the wander was believed to be the comb’s other half. Getting the observable wrong cost a hypothesis; getting it right supplies three.

Where each kind's lattice gives wayThe largest amplitude at which every run still has a lattice, and the scatter it produces there. The amplitudes are incomparable — field 0.015 (fraction of the barrier), jostle 1 (degrees of azimuth), placement 0.8 (degrees of azimuth) — and the scatters agree to 19%. The boundary belongs to the pattern rather than to the disturbance: a lattice fails at about a degree and a half of scatter, and which of three mechanisms produced it does not move where.field1.64°intact to 0.015, broken by 0.02jostle1.72°intact to 1, broken by 1.4placement1.42°intact to 0.8, broken by 13 runs per amplitudescatters 19% apart
Fig. 11 How much disturbance a lattice survives, by the kind of disturbance and by where it enters. The readings above are only usable on a shoot whose lattice is intact, and this is the boundary of that.

What this does not say

It does not say the rule amplifies a drift. The drift’s own size in the sequence is unchanged; what changes is everything else. The ratio goes up because the denominator goes down, and the essay is careful to give both numbers.

It does not say the rule fails to correct. It corrects a great deal — more than half the scatter, at this displacement — and what it corrects is precisely the part that would destroy the lattice. A drift does not destroy a lattice, which is the same fact from the other side and is measured elsewhere here.

It does not say a drifting disturbance is realistic. An autoregressive disturbance with a coefficient of 0.97 has a memory about thirty organs long, and nothing here says a meristem has one. It is a shape chosen to be maximally different from the inherited one, and its value is as a contrast.

It does not say the two disturbances are equally plausible. Nothing here argues that a meristem is more likely to face a drift than an inheritance. The comparison is between two shapes with the same per-organ size, chosen because they are opposite, and its value is in what a measurement could distinguish rather than in what a plant is likely to have.

And it does not say the hole at eight is negligible. Under independent noise the rule’s own stems give 0.20 there. Any test using the hole as evidence has to be calibrated against that and not against one.

What was actually claimed, and what replaces it

The sentence at the top of this essay has to be replaced rather than qualified, so here is both halves side by side.

Claimed: a placement rule produces neither a comb nor a drift, so a sequence carrying either is evidence against a rule.

True: a placement rule produces no comb and no drift of its own, and a sequence carrying either is evidence about the disturbance rather than about the placement. A rule passes a drift through with its visibility increased; a rule does not manufacture the correlations at the contact offsets that make a comb, and that half of the claim is unaffected and is measured elsewhere here.

The difference between the two versions is the phrase “of its own”, and it is the difference between a statement about a mechanism and a statement about a mechanism under a stated input. Every claim in this collection about what a rule produces has that ambiguity available to it, and this one is the second to fall into it.

The check that would refuse it

Three assertions, taken on the same rule at the same displacement.

A stem grown under independent noise has to carry no wander — under two at the widest block. That is the half of the original prediction that survives, and it is the calibration for the rest.

A stem grown under the drifting disturbance has to carry at least eight times that. The factor is large because the effect is: forty-six against nine tenths.

And the drifting stem’s per-organ scatter has to be the smaller of the two. That is the assertion that makes the essay’s explanation testable rather than decorative: if the rule were simply passing everything through, the quieter sequence would be the one with less wander, and it is the one with more.

Where this goes next

in the disturbanceblocks of 100in the divergencesblocks of 128the horizontal rule is what independent errors giveindependent×0.930.96a memory, ρ = 0.9×16.2410.25inherited, a = 0.7×49.090.83shared once, a = 0.7×2.550.956000 organs · 6 runs eachgenerated from a stated rule, not drawn to look right A difference forgets a drift This collection proposed a second observable and priced it as free: if a plant's errors are inherited between touching organs, the divergence sequence should carry a slow wander as well as a comb. The disturbance with the largest wander of any built here leaves none at all in the sequence, because a divergence is a difference and differencing is what removes a drift. scatterthrough the rulecomb ratiothrough the rulemain combforged, no ruleseeds agreeingforged, no ruleindependent0.57°0.80——shared once0.71°0.910.101/8inherited0.97°1.020.218/8same coupling, same offsets 8 and 13, same eight seedsrule at 0.25° a organ · forged at the same couplinggenerated from a stated rule, not drawn to look right What the sharing costs a lattice A disturbance inherited from the contact neighbours destroys a stem's lattice at half the displacement independent noise needs, and it moves the comb ratio a fifth of the way to a forgery's. Take the inheritance out and keep the sharing, and the damage stays and most of the ratio shift goes — so the two effects have different causes. placement noise, at 1°2.00°field noise, at 0.015 of the barrier1.68°no noise at all0.64°largest divergence scatter still holding a latticethe two differ by 0.33° — a fifth of what either toleratesand by 2.9× more than a noiseless run scatters65 nodes per rung · 10 runs per amplitude2.00° against 1.68° Two degrees of scatter A lattice tolerates about two degrees of wander in its divergence angle, and two kinds of noise sharing no code agree on the number to within a third of a degree. It is not a constant: carried finer, the same stem survives 0.8°, and the tolerance tracks the band of angles that produce its pair at all.

Shares its objects with

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

  • The disturbance that travels — both name autocorrelation, ensemble, honest limits, measurement, noise, the placement rule, repulsion, self correction, tolerance, transport
  • A disturbance that is not passed on — both name autocorrelation, ensemble, evidence, honest limits, measurement, noise, null model, summary statistic, transport
  • Errors that pass between organs — both name autocorrelation, evidence, honest limits, measurement, noise, null model, the placement rule, self correction, transport
  • The ratio was never about the rule — both name autocorrelation, ensemble, evidence, honest limits, measurement, noise, null model, the placement rule, transport
  • What a forgery has to know — both name autocorrelation, evidence, honest limits, measurement, noise, null model, the placement rule, repulsion, transport
  • A disturbance the organs share — both name autocorrelation, ensemble, honest limits, measurement, noise, the placement rule, self correction, tolerance

Named objects

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

AutocorrelationEnsembleEvidenceHonest limitsMeasurementNegative resultNeighbourhoodNoiseNull modelThe placement ruleRepulsionSelf correctionSummary statisticToleranceTransport