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

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.
22 min read 11 figures State the claim, state the testForm is not mechanismTwo routes, one number

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

An earlier essay here ended with a proposal, stated in two sentences and offered as a bargain:

If a plant’s disturbances are inherited, the divergence sequence carries a slow wander as well as a comb: block means over a hundred organs vary several times more than independent errors would allow. If they are merely shared between touching organs, the sequence has teeth at the contacts and no wander.

It was priced as the cheap half. A comb costs about nine hundred organs at half a degree of protractor precision; a variance ratio is not a spectral feature and needs no extra measurements, so a stem long enough for the comb is more than long enough for both. Two statistics off one sequence, and they disagree about the hypothesis.

The proposal is wrong, and the arithmetic that shows it is one line that was never written down.

The line that was never written down

A disturbance acts on positions. Organ i sits at i divergences plus an error eᵢ, and the divergence between consecutive organs is therefore

δ + (eᵢ − eᵢ₋₁)

A divergence is a difference. And differencing a series is precisely the operation that removes its low-frequency power: a drift moves eᵢ and eᵢ₋₁ by nearly the same amount, and the difference between them does not see it.

The wander is real. It is in the disturbance. The divergence sequence — the one thing about a plant a botanist can actually record — is the one place it is not.

The wander is in the disturbance and not in what a plant lets you measureEach disturbance measured twice, in the same statistic. On the left, the variance of the block means of the disturbance's own deviates, over blocks of 100, as a multiple of what independent draws would give; on the right, the same quantity for the divergence sequence those deviates produce, over blocks of 128. The left column is what this site measured when it proposed a slow wander as a second observable. The right column is what a botanist would have: a divergence is the difference of two organs' errors, and differencing is exactly the operation that removes power at low frequencies. The disturbance inherited between touching organs goes from ×49.1 — the largest here — to 0.83, which is what independent errors give. The one with a memory in time keeps most of its own.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
Fig. 1 Each disturbance measured twice in the same statistic: on its own deviates, where the earlier proposal measured it, and on the divergences those deviates produce, which is what a measurement would have.

The two columns

Block means over a hundred, as a multiple of what independent draws would give:

disturbance in the disturbance in the divergences
independent ×0.93 0.96
a memory, ρ = 0.9 ×16.24 10.25
inherited at the contacts, a = 0.7 ×49.09 0.83
shared once, no history ×2.55 0.95

The disturbance with the largest wander of any shape this collection has built leaves none in the sequence. A disturbance with a memory in time keeps most of its own. The two columns are not two versions of one measurement; they reorder the disturbances almost completely.

That reordering is the whole finding, and it is why the proposal was not merely imprecise. It named the transport as the shape that would show a wander, and the transport is the shape that shows none.

Why the transport in particular

The reason is more specific than “differencing removes drift”, and it is worth having because it says what a wander is evidence of.

Write the statistic properly. For a differenced stationary stream the variance of a block mean of B values is 2σ²(1 − ρ(B))/B², so the quantity that is one for independent errors is

W(B) = B²·Var(block means)/Var(divergences) = (1 − ρ(B))/(1 − ρ(1))

where ρ is the autocorrelation of the disturbance. Measured by two routes that share no arithmetic — one a variance of block means of the angles, the other a lagged product of the deviates — the two agree everywhere to inside the sampling error of the noisier of them, across seven disturbances and eight block sizes.

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. 2 How much of a divergence sequence’s variance survives being averaged over blocks, on lattices with the disturbance added and no rule anywhere. A flat line at one is a sequence with no low-frequency power.

So W(B) is the autocorrelation function, evaluated at the block size instead of at a lag. It is not a second statistic at all.

And that settles the transport. A contact transport puts its correlation in a residue class — at eight, thirteen, twenty-one, and at their sums and differences, and at nothing else. Its autocorrelation at a general large lag is near zero, so W is near one, and no amount of coupling changes that. Nothing whose correlation lives in a residue class can produce a wander, which is a statement about arithmetic rather than about this particular model.

Same correlation at the contacts, and only one of them has a historyThe autocorrelation of each disturbance against lag, over 40,000 draws at a coupling of 0.5. Both are correlated at 8 and 13 — the two contact offsets of a stem at this rise — and at 5, their difference, which is where the second comb comes from. The inherited disturbance, in which an organ takes a share of what its neighbours were displaced by, also carries power at 16, 21, 26, 29, 34: every sum and difference of the two offsets, because an error that enters it is passed on again and again. The shared disturbance, in which an organ takes a share of the fresh deviates drawn for those neighbours, carries nothing past the two. The number on the right is how far each one's block means wander: over 100 organs the inherited stream's block means have 5.5 times a white stream's variance and the shared one's 2.2.inherited againcoupling 0.5drift ×5.5shared oncecoupling 0.51235813162124262934drift ×2.2correlation against laglag, in organs40,000 draws · offsets 8 and 13generated from a stated rule, not drawn to look right
Fig. 3 Where an inherited disturbance puts its correlation. At the two contact offsets, at their difference, and at combinations of them — a set of lags rather than a decaying tail, which is exactly what a comb is made of and exactly what a wander is not.

The number the identity supplies

An identity is worth more when it predicts something with no free parameters, and this one does. For a disturbance that remembers the last error with coefficient ρ, the autocorrelation is ρ^B, so W climbs to 1/(1 − ρ) and stops.

Measured against the closed form at five coefficients: 1.38 against 1.43, 1.90 against 2.00, 3.15 against 3.33, 9.51 against 10.00, and 30.94 against 33.33. That is 0.93 to 0.97 of the closed form at every coefficient — a uniform low bias, which is what a variance of block means does in a finite run, rather than a disagreement about the shape.

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, 0.022, 0.014 at ρ = 0.3, 0.7, 0.9 — and the readout returns nothing on 24 runs out of 24. A correlated error is not a periodic one.00.2000.4000.6000.3000.7000.900how strongly each error remembers the last, ρthe largest comb mean anywhere in the thirty lagsthe rule's own stems: 0.64three sampling bands0.0280.0220.014kinematic lattice · AR(1) errorgenerated from a stated rule, not drawn to look right
Fig. 4 The same disturbance seen through the readout this collection uses for a comb. A memory manufactures no comb at any coefficient, which is the other half of the separation: the shape with a wander has no comb and the shape with a comb has no wander.

What the statistic is, in detail

It is worth writing the recipe out, because a variance ratio has more than one plausible normalisation and the wrong one hides the whole result.

Take the divergence sequence. Cut it into consecutive blocks of B, take the mean of each block, and take the variance of those means. Divide by the variance of the sequence itself, and multiply by B squared.

The square is the part that has to be right. For an ordinary series — one that is not a difference of anything — block means have variance falling as 1/B, and the natural normalisation is a single factor of B. A divergence sequence’s block means fall as 1/B² instead, because the block mean telescopes: the mean of eᵢ − eᵢ₋₁ over a block is the last error minus the first, divided by B — a difference of two values rather than a sum of B of them.

Normalised with a single B, every disturbance here produces a tidy curve falling as 1/B and nothing is distinguishable from anything. Normalised with B², independent errors give a flat line at one and everything else is a departure from it. The two normalisations differ by a factor that is the same for every series, so neither is wrong; only one of them puts the interesting structure where an eye can see it.

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 10 at a block of 128. The ones inherited between touching organs do not climb at all — 0.83 at the same block — although their own deviates carry ×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 = 0.9inherited, a = 0.76000 organs · 6 runs eachgenerated from a stated rule, not drawn to look right
Fig. 5 Three of the seven disturbances on their own, so the shapes are legible: independent errors flat at one, a memory climbing to a plateau, an inherited disturbance flat at one with a hole at the offsets it couples at.

What the two observables actually separate

The proposal had the right instinct and the wrong pairing. There are two observables, they are almost independent, and they separate two different kinds of inheritance:

  • A comb says the disturbance is correlated at the contact offsets and at combinations of them. It is produced by an error passed between touching organs and passed on again, and — as this collection has already had to concede — it is produced by nothing else that has been tried.
  • A wander says the disturbance is correlated at every lag, with a tail that decays slowly. It is produced by a memory in time: an apex whose conditions drift, a measurement whose zero moves, a shoot growing through a changing season.

A sequence with a comb and no wander is a shoot whose errors are shared between neighbours and not carried through time. A sequence with a wander and no comb is a shoot in a drifting environment with independent local errors. A sequence with both is a shoot with both, and there is no reason it should not have.

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. 6 The comb as this collection reads it, on the inherited disturbance at the coupling used throughout. The teeth are at the contact offsets and their combinations, which is the structure the wander statistic is blind to and this one is built for.
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 8/13the shaded strip is the sampling bandone stem · 760 divergences · disturbance 0.25generated from a stated rule, not drawn to look right
Fig. 7 The same readout on a stem the rule grew, for comparison. Both statistics in this essay are computed on sequences of this kind — a settled arrangement with a disturbance in it — so the difference between them is not a difference of subject.

What is left of the wander, and it is not nothing

Two things survive, and both are sharper than what they replace.

A hole at the comb’s own lags. An inherited disturbance drives W down where the block size is one of the offsets it couples at: 0.22 at a block of eight and 0.21 at thirteen, against 1.00 and 0.99 for independent errors. That is a signature with the right sign for a one-sided test, and it is free, since the lags are the ones the comb already names. It is not a wander; it is the comb, read through a different window.

The identity itself is a check. Two routes to one number, agreeing across seven disturbances, is the sort of thing this collection puts weight on — and here it is doing more than reassurance. It says that anybody who computes a variance ratio on a divergence sequence and reports it as new information has computed the autocorrelation, and should say so.

Both statistics, on the same stems, at a rise of 0.005Five seeded stems at each disturbance, held at a fixed rise. Bars are how many returned the pair the position counter finds; open portions are refusals. The pair comes out from 0.1 to 0.25, and across that whole range the lag-one correlation of the *same* sequences is -0.33, -0.58, -0.59 — decisive, negative and flat. There is no trade between the two: one stem supplies both. Below the window the sequence has locked onto the sampling grid and is a cycle rather than a sample; above it there is no lattice left, at 116° of scatter.012345-1.70-1.30-1-0.824-0.602-0.398-0.2220disturbance amplitude, degrees of azimuth per nodestems out of five returning the counted pair0.020.050.10.150.250.40.61lag-one correlation = 0lag one, on the same sequencesrise 0.005 · 5 stems per point · bars are the pair, line is lag onefilled where the pair agrees with the position counter
Fig. 8 Two statistics of one sequence as this collection has used them elsewhere. The lesson from this essay applies to any such pair: two numbers off one series are only two measurements if they are not functions of each other, and that has to be checked rather than assumed.

What it costs an experiment, and what it buys

The proposal’s accounting has to be redone, and it comes out better in one place and worse in another.

Worse: there is no free second statistic. A divergence sequence yields one autocorrelation function, and the comb and the wander are two windows onto it. An experimenter who measures a sequence long enough for a comb has not thereby measured a second thing that discriminates the same hypotheses.

Better: the hole at the contact offsets is a genuinely cheap addition, and it is cheaper than the comb it comes from. A comb is read by scoring a whole set of lags against a sampling band and requires the pair to be known or recovered; the hole is a single ratio at a stated block size, and the block sizes to use are the two parastichy numbers, which a counting of the shoot already supplies. At a coupling of 0.7 it is 0.22 and 0.21 against a null of one.

And better again in a way the proposal did not anticipate: the wander is now an observable for a hypothesis nothing here could previously address. Whether a meristem’s errors drift — whether the placement conditions of organ five hundred resemble those of organ four hundred more than those of organ one — is a question about the plant’s environment and its own development rather than about contacts, and a variance-time curve on a long enough sequence answers it.

What the pair costs, at a rise of 0.005Five seeded stems at each length, read at four protractor errors. With no reading error the pair needs 250 internodes — against the sixty the single parastichy number costs. At 0.25° per organ it needs 250; At 0.5° per organ it needs 400; At 0.75° per organ it needs 1100. The pattern's own scatter here is 0.70°, so the last of those is a reading error larger than the signal being read.0123451502504007601.1e+3internodes measured on one stemstems out of five returning the counted pairno reading error0.25° per organ0.5° per organ0.75° per organrise 0.005 · disturbance 0.25 · pattern scatter 0.70°generated from a stated rule, not drawn to look right
Fig. 9 What reading a sequence costs in organs and in protractor precision, as this collection has priced it. Nothing in that accounting changes; what changes is how many independent answers the sequence yields once it has been paid for.
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. 10 The same curves on stems the placement rule grew rather than on lattices with errors added. The retraction holds there too — and something happens to the memory that has its own essay.

How the mistake was made

Worth setting out, because the shape of it is general.

The wander was measured, correctly, on the disturbance — block means of the deviates, ×5.5 at one coupling and ×55 at another, reported as evidence that the inherited disturbance has a history and the shared one does not. That measurement stands and it was the right one for the question it answered, which was about the structure of the disturbance.

The error was in the sentence that carried it across. “The divergence sequence carries a slow wander” was written as though the disturbance and the sequence were the same series with a scale factor between them. They are not, and the operation between them is exactly the one that annihilates the quantity being carried across.

Nothing about the model changed. What changed is that somebody asked the statistic to be computed on the other side of a difference operator, and it took a few minutes.

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. 11 The rule’s own signature in a divergence sequence: strong anticorrelation at lag one, which is a restoring force rather than an inheritance. That reading was made on the sequence rather than on the disturbance, and it is unaffected.

What this does not say

It does not say the wander is not there. It is there, in the disturbance, at the sizes previously measured. What it is not is observable in the sequence a plant hands over.

It does not say the earlier measurement was wrong. The block-mean variances of the streams are recomputed here and agree: ×0.93 for white, ×49.09 for the inherited disturbance at a coupling of 0.7, ×2.55 for the shared one. Only the claim about what those numbers imply for a sequence has gone.

It does not say the comb is unaffected. The comb is a different statistic and this essay leaves it exactly as it was, including the concession that it is evidence of a re-transmitted disturbance rather than of a placement rule.

And it does not say a wander is uninformative. A divergence sequence with a wander in it says something specific and testable — that the disturbance has a memory in time — and this collection had no observable for that before. It is a different question from the one it was proposed for.

A general form of the mistake

The failure has a name and it is worth attaching, because this collection is going to keep proposing observables.

An observable is a function of what can be measured. The disturbance is not what can be measured; the positions are, and a divergence sequence is a particular function of them. Every claim of the form “if the plant is doing X, the measurement will show Y” therefore has a step in it where a property of the hidden thing is carried across an operator to a property of the visible thing — and that step can annihilate exactly the property being carried.

The operators this collection has between hidden and visible are all of that kind. Positions are differenced to get divergences. Divergences are wrapped modulo a turn. A jugate divergence is reduced modulo a fraction of a turn. Cell areas are taken relative to a background. Each of them destroys some property of what went in, and each of them is applied without comment because it is what a measurement is.

The cheap protection is to compute the proposed statistic on the visible side before writing the sentence. It costs a few minutes and it is the difference between an observable and a sentence about one.

The check that would refuse it

Four assertions, and the first two are the retraction.

Independent errors have to give one at every block size, within a tenth. That is the calibration, and without it every other number here is a ratio against nothing.

The inherited disturbance has to have many times an independent stream’s variance in its own block means and no low-frequency power in the divergences it produces. Both halves are required together: either on its own is consistent with a mistake in one of the two computations.

The identity has to hold, at every shape and every block size, to within the sampling error of the block-mean variance — which is a tolerance that widens with the block, because a variance over forty-six numbers is a fifth of itself uncertain and a fixed tolerance would either fail on sampling at the wide blocks or pass anything at the narrow ones.

And a memory has to leave a wander several times what independent errors give, with a curve that climbs. Without that half the essay is consistent with the statistic being unable to detect anything at all.

Where this goes next

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 The forgery needs a history A disturbance passed between touching organs manufactures the comb, the second comb and the parastichy pair on an arrangement with no rule in it — which is why the comb stopped being evidence. Give the organs the same correlation with no accumulation in it and the forgery collapses: one seed in eight returns a pair, and the comb is the noise floor. inherited againcoupling 0.5drift ×5.5shared oncecoupling 0.51235813162124262934drift ×2.2correlation against laglag, in organs40,000 draws · offsets 8 and 13generated from a stated rule, not drawn to look right A disturbance that is not passed on The disturbance that forges every observable on this site does two things at once — it correlates an organ's error with its contact neighbours', and it hands that error on to be handed on again. Every result about it has been unable to say which half did the work. This is the control that takes the second half away and keeps the first. 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 The second statistic was the first The experiment this collection has been specifying was priced as two readings off one sequence, the second of them free. The two readings turn out to be one function looked at twice, so the specification loses a statistic — and gains a cheaper one, a warning about how observables get priced, and a question it could not previously ask.

Shares its objects with

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

  • Matching instead of correcting — both name artefact, claim testing, ensemble, evidence, honest limits, identifiability, measurement, negative result, null model, summary statistic
  • What the rule does to a drift — both name autocorrelation, ensemble, evidence, honest limits, measurement, negative result, noise, null model, summary statistic, transport
  • The ratio was never about the rule — both name autocorrelation, ensemble, evidence, honest limits, identifiability, measurement, noise, null model, transport
  • A periodicity is not a lattice — both name artefact, autocorrelation, ensemble, evidence, identifiability, measurement, noise, null model
  • What a forgery has to know — both name autocorrelation, evidence, honest limits, identifiability, measurement, noise, null model, transport
  • A period the grid invented — both name artefact, claim testing, honest limits, measurement, negative result, refusal, summary statistic

Named objects

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

ArtefactAutocorrelationClaim testingEnsembleEvidenceHonest limitsIdentifiabilityMeasurementNegative resultNoiseNull modelRefusalSummary statisticTransportUntested claim