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

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.
21 min read 11 figures State the claim, state the testForm is not mechanismOne parameter, one family

Worth reading first: A disturbance with a memory · Errors that pass between organs · The sequence has a memory.

One disturbance on this site does all the damage. Give a stem’s organs errors that each one inherits from the two organs it touches — the ones m and n places back, which is what a parastichy pair is — and an arrangement with no placement rule in it reproduces the comb in the divergence sequence, the second comb, and the pair itself on eight seeds of eight. That result retired the comb as evidence of a rule, and a later measurement retired the ratio of the two combs as well.

Everything in that thread was measured with one implementation, and the implementation does two separable things.

Two properties, one stream

The transported disturbance is a recursion. Organ i takes a share of what organs i−m and i−n were displaced by, and those displacements were themselves inherited from organs further back. So the stream has:

Structure. Its errors are correlated at the two contact offsets, which is the hypothesis about plants — an organ’s disturbance is transmitted by the organs it touches.

History. An error that enters the stream is still in it hundreds of organs later, spread across every sum and difference of the two offsets, and the stream wanders slowly because those contributions accumulate.

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 0.5. The largest comb mean is 0.166 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 0.166 · band 0.073the shaded strip is the sampling bandkinematic lattice · 759 anglesgenerated from a stated rule, not drawn to look right
Fig. 1 What the transported disturbance manufactures. Driven into a lattice whose positions were never computed from anything, it produces the comb the site had been reading as evidence that a plant computes its pattern — which is the result the whole thread turns on.

Nothing measured so far can say which of the two is responsible for what. The hypothesis about plants is the first; the second arrived with it because a recursion is the easiest way to write the first down.

The control, and what makes it one

The control is a moving average over the same offsets. Organ i takes a share of the fresh deviates drawn for organs i−m and i−n — the innovations, not the displacements those organs ended up with.

That keeps the structure and removes the history. It is one line of arithmetic different from the transport, and the difference is which array is read from.

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. 2 The two streams’ correlations at each lag, over forty thousand draws at a coupling of a half. Both carry the contact offsets, eight and thirteen, at 0.42 and 0.41 against 0.31 and 0.30. Past those two the shared stream is flat at zero and the inherited one carries power at sixteen, twenty-one, twenty-four, twenty-six, twenty-nine and thirty-four.

Three properties make it a control rather than a different experiment.

It carries the same lags. At a coupling of a half the transport has correlations of 0.418 and 0.414 at the two contact offsets and the average has 0.308 and 0.303. Same lags, same sign, same order of magnitude.

It carries their difference too. Both are correlated at lag five, which is thirteen minus eight: 0.176 for the transport and 0.126 for the average. That matters because the second comb — the observable that carried this site’s last discriminator — lives at that offset. Organ i and organ i−5 both contain the innovation from organ i−13, in both streams, for the same reason.

And it carries nothing else. Past the two offsets the average’s correlation is 0.000, 0.002, 0.008, 0.002, 0.004, 0.006 at lags 16, 21, 24, 26, 29 and 34 — zero to three decimal places at every one — where the transport has 0.175, 0.298, 0.081, 0.176, 0.170 and 0.172.

Why a recursion was the natural way to write it, and what that cost

It is worth asking how the two properties came to be welded together in the first place, because the answer is not carelessness.

The hypothesis is about a plant: a primordium that is displaced pushes the organs it touches, so its error appears in theirs. Writing that down, the obvious sentence is organ i’s error is a share of the errors of the organs it touches, plus its own — and the errors of the organs it touches are the errors they actually have, which include what they inherited. That is a recursion, and it is the honest transcription of the hypothesis.

The moving average is a different sentence: organ i’s error is a share of the disturbances that arrived at the organs it touches, plus its own. As a model of a plant it is worse, because it supposes an organ can feel what happened to its neighbour without feeling where its neighbour ended up.

But the results this thread produced were not about plants. They were about what a class of disturbance can manufacture, and the class was defined by a property — correlation at the contact offsets — that both sentences have. Every conclusion was stated in terms of that property, and only one of the two objects was ever run.

So the cost was not a wrong model. It was a conclusion phrased in terms of a property when it depended on a property the object also happened to have.

A periodicity reports a different partner every timeeight kinematic lattices, differing only in the seed of their disturbance, each read by the same instrument. The disturbance repeats every 8 organs at a weight of 0.7: it puts a strong comb at spacing 8 — 0.43 against a band of 0.07 — and the partner it names is 8/10, 8/12 across the 8 stems and never 8/13, which is what the position counter finds in every one of them. There is no placement rule in any of these arrangements.stemwhat the angles say18/10not the lattice's pair2refused3refused48/10not the lattice's pair58/12not the lattice's pair6refused78/12not the lattice's pair8refusedthe positions say 8/13kinematic lattice · error of period 8generated from a stated rule, not drawn to look right
Fig. 3 The same readout applied to a disturbance that simply repeats every eight organs. It does manufacture a comb, and the pair it reports is an accident of its own period rather than the lattice’s — read on more than one stem it convicts itself, which is what makes it a different kind of rival from the inherited one.

How much history each one has

“History” can be measured rather than described. Take the mean of a hundred consecutive draws, and then the variance of those block means across the stream. For independent draws that variance is the stream’s own variance divided by a hundred; anything above that is power at frequencies below one per hundred organs — which is what a slow wander is.

At a coupling of a half the transport’s block means carry 5.5 times a white stream’s variance and the average’s carry 2.2. At a coupling of 0.7 the transport’s carry fifty-five and the average’s 2.6.

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. 4 The same coupling seen through the readout the site uses. The transported disturbance’s power is not confined to the two offsets it was built from — it appears at combinations of them, which is what a stream with a memory of its own outputs does and what a comb is made of.

The average’s 2.2 is not one, and it should not be: correlating at lags eight and thirteen does put some power at low frequencies. What it does not do is grow with the coupling, because there is nothing to accumulate.

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. 5 The other correlated disturbance this site has tested, for comparison. An error that remembers the last one is the ordinary meaning of “correlated noise”, and it manufactures nothing at any coefficient — the arithmetic of why is in its own essay, and it comes down to a decaying tail having no teeth.

The arithmetic of why lag five is shared

The difference of the two offsets is where the second comb lives, and both streams carrying it is not an accident of the implementation. It falls out of the construction in one line, and it is worth doing because the second comb is the observable that survived longest.

In the moving average, organ i’s error contains the innovation of organ i−8 and that of organ i−13. Organ i−5’s error contains the innovations of organs i−13 and i−18. The two share the innovation at i−13, so their errors are correlated — at lag five, with a coefficient that follows from the couplings alone.

The same holds in the transport, with more terms, because the inherited quantities contain everything the innovations do and more.

What that means for the comparison is that neither stream has been given an advantage at the observable being tested. The second comb has a mechanism in both, and if the average manufactures no second comb it will not be for want of the correlation that produces one.

The same lattice with no rule in itA cylindrical lattice at a divergence of 137.822° 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 0.04 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 0.04 · band 0.07the shaded strip is the sampling bandkinematic lattice · 759 divergences · 0.5° of independent scattergenerated from a stated rule, not drawn to look right
Fig. 6 The two teeth on a forged stem: the main comb at the smaller parastichy number and the second at the difference of the pair. Both streams here have the correlations that could put those teeth there; whether both actually do is a measurement rather than a deduction.

What the two streams look like beside each other

Correlations at lags are an austere way to describe a disturbance, and the two objects are easier to hold in mind as behaviour.

The shared stream is jumpy. Each organ’s displacement is mostly its own fresh deviate with a fraction of two older ones added, and those older ones are themselves independent draws, so nothing persists. Watch a hundred organs and the mean of them is close to zero; watch the next hundred and it is close to zero again. It has texture at two spacings and no tendency.

The inherited stream drifts. An organ that is displaced to the left passes part of that to the two organs it touches; they pass part of what they now have to the organs they touch, and so on up the stem, so a single early disturbance leaves a bias that decays over hundreds of organs rather than tens. Watch a hundred organs and their mean is somewhere; watch the next hundred and it is somewhere else, five times further from zero than chance would put it.

That is the entire difference, and it is why the two can be told apart by a statistic as crude as the variance of a block mean. It is also why the inherited stream is the more dangerous one for an arrangement: a lattice can absorb an organ being displaced, and what it cannot absorb is every organ in a stretch being displaced the same way.

Three kinds of noise, matched at 0.75° of divergence scatterThe amplitudes differ — field 0.0056 (fraction of the barrier), jostle 0.15 (degrees of azimuth), placement 0.18 (degrees of azimuth) — and are in different units, so they cannot be compared directly. What can be compared is what they produce, and matched here they are within 27% of one another. Everything a finished pattern records about its noise is shared between the three.field — before the choice0.92°amplitude 0.0056jostle — before the choice0.70°amplitude 0.15placement — after it0.79°amplitude 0.183 runs each, at the amplitude that reaches 0.75°27% apart on the ruler
Fig. 7 Why the distinction has to be made at equal recorded scatter. Two disturbances of the same nominal size can leave a stem with quite different scatters, because what a protractor records is the displacement of an organ relative to its neighbours rather than its displacement — and a shared disturbance moves neighbours together.

And it stays the same size

A control that is bigger than the thing it controls for is not a control. The transport has a problem here that had to be corrected once already: its recursion inherits from two lagged terms and divides by the normalisation of one, so at a coupling of 0.7 its deviates have a standard deviation of 2.01 rather than one. Comparing at equal nominal amplitude compares a disturbance with one twice its size.

The average has unit variance by construction, at every coupling: 1.006 measured at 0.5, 0.7 and 0.9. Its two inherited terms are independent of its own fresh term and of each other, so the variances add to exactly what they were built to add to.

Every comparison in this thread is made at equal measured spread, with each stream divided by its own standard deviation, so this is bookkeeping rather than a result — but it is bookkeeping that has gone wrong here before.

Transported errors report the same pair every timeeight kinematic lattices, differing only in the seed of their disturbance, each read by the same instrument. The disturbance at each node is inherited from the nodes 8 and 13 places back, at a coupling of 0.5. Every stem returns 8/13, which is the pair the positions give and the pair the placement rule's own stems give. There is no placement rule in any of these arrangements.stemwhat the angles say18/13the lattice's own pair28/13the lattice's own pair38/13the lattice's own pair48/13the lattice's own pair58/13the lattice's own pair68/13the lattice's own pair78/13the lattice's own pair88/13the lattice's own pairthe positions say 8/13kinematic lattice · inherited errorgenerated from a stated rule, not drawn to look right
Fig. 8 The readings the transported disturbance produces on eight independent seeds, which is what makes it a forgery rather than a coincidence: it returns the same pair every time, at the strength a real stem’s comb has.

A limit of the transport that the average does not share

Worth recording because it constrains where the comparison can be made at all. The transported stream’s recursion is stable only while its coupling is small enough: at 0.9 the two inherited terms feed back hard enough that the variance grows without bound and the stream has no standard deviation to normalise by.

So the transport can be driven at 0.5, 0.6 and 0.7, and not much further. The average can be driven at any coupling below one, and is, in the essay that follows — which turns out to matter, because the obvious objection to that essay’s result is that the control was simply not pushed hard enough.

The couplings each one can be run at

A last practical difference, which turns out to matter for the comparisons the control is for.

The inherited stream’s recursion is stable only while its coupling is small. At 0.5 and 0.7 it produces a stream with a finite variance; at 0.9 the two inherited terms feed back hard enough that the variance grows without bound, and the stream has no standard deviation to normalise by at all.

The shared stream has no such limit. Its variance is the sum of the squares of its three coefficients, whatever the coupling, because none of its terms depends on its own past. It can be run at 0.9 or at 0.99 and it is the same size as ever.

That asymmetry is why the essays that follow can answer the obvious objection to their results. When a control fails to reproduce an effect, the first question is whether it was simply driven too weakly — and here the control can be driven harder than the thing it is a control for.

What this is not

It is not a hypothesis about a plant. No process transmits a displacement between two organs without transmitting the displacement they themselves received; an organ that pushes its neighbour pushes it from wherever it actually is. The average is a deliberately unphysical object built to hold one property fixed while removing another, which is what a control is for.

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. 9 Where this site’s disturbances enter the machinery. The shared and inherited streams are both jostles — a displacement applied to an organ after the rule has chosen where it goes — so they enter at the same place and differ only in how they are generated.

It is not a claim that the transport is wrong. The transported account is the physically natural one and it remains the strongest rival to the placement rule. What the control makes possible is to say which of its two properties each of its successes depends on.

And it is not a new observable. The readouts are the ones the site already has: the comb, the second comb, their ratio, the recorded scatter, and whether a counter shown the positions agrees. Nothing about the instruments changes; the only new thing is a second disturbance to point them at.

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. 10 The readout itself, on a stem the rule made. The main comb sits at the smaller parastichy number and the second at the difference of the pair, and everything in this thread is a question about what can put those two teeth there.

What it is for

Two questions, each with its own essay.

Does the forgery need the history? The transported disturbance manufactures a comb on a lattice with no rule in it. If the average does too, then correlation at the contacts is enough and the forgery is about the neighbour graph. If it does not, the forgery needs re-transmission, which is a stronger claim about a plant than merely having neighbours that share errors.

Does the damage need the history? A transported disturbance costs a lattice more than an independent one of the same size — a stem survives half as much displacement. If the average is equally harmful, the damage is done by the correlation at the contacts; if it is harmless, the damage was the accumulation.

The two answers turn out to be different, which is the useful outcome: one of the transport’s two properties is responsible for one of its two effects, and the other for the other.

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. 11 Both answers in one table, and the shape of the result. Through the rule, the shared disturbance sits between white noise and the inherited one; on a lattice with no rule in it, it manufactures nothing while the inherited one manufactures everything.

The check

Three assertions run on the streams themselves, before anything is driven through anything.

The first requires the two to agree at the contact offsets to within 0.15 in correlation, and to agree at their difference — so a control that had quietly lost the structure would fail rather than produce a clean and meaningless negative.

The second requires the average to have no power past the two offsets and the transport to have some, which is the property being isolated.

The third requires the transport’s block means to wander at least twice as much as the average’s. That is the one number the whole comparison rests on, and it is asserted at the coupling the comparisons are made at rather than at the coupling where the difference is most flattering.

Where this goes next

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 The disturbance that travels If a lattice survives three times the displacement when the organs share it, then a disturbance passed between the organs that actually touch should be the gentlest of all — it is correlated at exactly the offsets the rule places against. It is the harshest. Half the displacement destroys what independent noise leaves standing, and the reason separates two things that had been one. -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. 0.4000.6000.80011.201.40-0.30100.1760.3010.4770.699how much more strongly the error is inherited from the 8-neighbour than from the 13-neighbourthe second comb's strength as a fraction of the main comb'sthe placement rule: 0.65equal combs1:21:11.5:12:13:15:1at 3:1 the ratio is 0.75kinematic lattice · 3 seeds a pointgenerated from a stated rule, not drawn to look right What a forgery has to know A lattice with transported errors reproduces the comb and the pair, so one quantity is left: the two combs' relative strength. Weighted by distance the forgery puts more in the second comb than the first; the rule does the opposite. It matches only if the coupling is turned three to one towards the further neighbour, which no falloff supplies.

Shares its objects with

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

  • The forgery needs a history — both name artefact, autocorrelation, discrimination, ensemble, evidence, falsifiability, honest limits, lattice offset, measurement, mechanism, noise, null model, parastichy pair, transport
  • The ratio was never about the rule — both name autocorrelation, discrimination, ensemble, evidence, falsifiability, honest limits, measurement, mechanism, noise, null model, parastichy pair, transport
  • A periodicity is not a lattice — both name artefact, autocorrelation, discrimination, ensemble, evidence, lattice offset, measurement, noise, null model, parastichy pair
  • The comb was never the rule — both name autocorrelation, discrimination, evidence, falsifiability, honest limits, measurement, mechanism, null model, transport
  • The control a survey would need — both name autocorrelation, discrimination, evidence, falsifiability, honest limits, measurement, null model, parastichy pair, transport
  • What the sharing costs a lattice — both name artefact, discrimination, ensemble, honest limits, measurement, noise, parastichy pair, summary statistic, transport

Named objects

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

ArtefactAutocorrelationDiscriminationEnsembleEvidenceFalsifiabilityHonest limitsLattice offsetMeasurementMechanismNoiseNull modelParastichy pairSummary statisticTransport