Phyllotaxis
the spirals are counted, not admired
Stems and cones

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.
19 min read 10 figures State the claim, state the testOne parameter, one familyTwo routes, one number

Worth reading first: Errors that pass between organs · A disturbance with a memory · Two degrees of scatter.

The disturbance that forges this site’s observables also damages the arrangements it is applied to, and the two facts have always travelled together. A stem jostled by errors inherited from its contact neighbours loses its lattice at half the displacement that independent errors need, and its comb ratio moves a fifth of the way towards the value a forgery gives.

The control that separates the disturbance’s structure from its history says the two effects have different causes, and that the more interesting of them belongs to the less interesting half.

The experiment

Three disturbances driven through the placement rule itself — not into a kinematic lattice, but into stems whose every organ is placed where the repulsion from the existing ones is least. Independent errors, errors shared once with the contact neighbours, and errors inherited from them. Same amplitude per organ, each stream divided by its own measured spread so that a quarter of a degree means a quarter of a degree in every row.

A lattice survives about 1.6° of scatter, whichever way the noise arrivesThe largest divergence scatter at which a run is still a lattice, from each kind of noise at the largest amplitude that leaves one. The two routes share no code below the placement rule: one displaces the node after the choice, the other perturbs the energy the choice is made over. They agree to 0.66°.placement noise, at 1°1.97°field noise, at 0.015 of the barrier1.31°no noise at all0.64°largest divergence scatter still holding a latticethe two differ by 0.66° — a fifth of what either toleratesand by 2.6× more than a noiseless run scatters65 nodes per rung · 3 runs per amplitude1.97° against 1.31°
Fig. 1 The tolerance two different kinds of disturbance leave, measured on the same rule. All three disturbances here are jostles — the rule chooses where the organ goes and the organ is then displaced — so nothing about the rule changes between the rows, and the comparison is a comparison of disturbances.

The damage is the sharing

Take the amplitude at which each disturbance still leaves every stem with a lattice — a divergence scatter under two degrees, on five stems.

disturbance survives to
independent errors 0.5°
shared once 0.25°
inherited 0.25°

The shared disturbance is exactly as harmful as the inherited one. Both cost a lattice half of what it withstands from independent noise of the same size.

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 survival table this row belongs to. A disturbance’s colour decides what a lattice withstands, over a range of six; errors correlated at the contact offsets are at the harsh end, and errors that merely remember the last one are at the gentle end.

That locates the harm where the earlier control put it. The two contact organs dominate the minimum the rule is taking; correlating their displacements makes them push the same way instead of partly cancelling, and the next organ is displaced further. Whether those correlated displacements were themselves inherited makes no difference to that — the rule sees the neighbours’ positions, not their provenance.

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. 3 The tolerance itself, measured across the kinds of noise the site distinguishes. About two degrees of recorded scatter, whichever way the disturbance gets in — which is the number the survival table is read against.

Why a correlated disturbance is harsher at all

The result that shared and inherited disturbances are equally harmful is easier to hold on to with the mechanism in view, and the mechanism is short.

The rule places an organ at the minimum of the repulsion from the organs already there, and at any settled rise two families dominate that sum: the organs m and n places back, whose hops are the two shortest. The minimum sits between them.

If those two organs are displaced independently, their displacements partly cancel in the position of the minimum — one pushes it one way, the other the other, and the expected shift is smaller than either. If they are displaced together, they push the same way and the minimum moves by the full amount. The next organ inherits that as a genuine displacement, and its own displacement is added to it.

So a correlation between the two contact neighbours converts partial cancellation into addition. Nothing in that argument involves where the correlation came from — only that it is there when the rule takes its minimum — which is exactly why the control and the original are equally harmful.

The earlier off-contact control is the other half of the same argument, from the other direction: correlate a disturbance at offsets the rule is not dominated by — seven and eleven at this rise, neither of them a short hop — and the damage disappears, leaving a stem as intact as white noise leaves it. Together the two controls say the harm is neither correlation as such nor inheritance as such, but correlation at the offsets that hold the minimum up.

A stem gathers neighbours linearly; a growing disc barely gathers them at allOn a cylinder of circumference 1 with a rise of 0.005, the nodes within distance d number 2d/0.005 once d exceeds one turn — a fitted exponent of 1.011 and 400 per unit against the 400 the geometry fixes. In the disc model an element of age k sits at radius e^(0.4k), so each doubling of the distance adds the same two or three neighbours rather than twice as many.123-0.50000.50011.50distance from the node, log₁₀neighbours, log₁₀a stemslope 1a disca logarithmrise 0.05 · 4000 nodes · meristem growth 0.4slope 1.013 against slope 1
Fig. 4 The two families that dominate the minimum at this rise, with their separations. Everything about why a correlated disturbance is harsh is in the fact that these two terms carry most of the sum — and everything about why the off-contact control is harmless is in how little the rest of it carries.

And the scatter follows the same line

At a fixed displacement of a quarter of a degree per organ, the recorded scatter of the surviving stems is 0.57° for independent errors, 0.71° for shared ones and 0.97° for inherited ones.

The shared disturbance sits between the two, closer to white. So the damage is not entirely the structure — the inheritance adds to it — but the structure accounts for the step that matters, since 0.71° is already most of the way to the harsh end in survival terms and the survival boundary falls between 0.25° and 0.5° for both.

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 1.77° and the destroyed ones start at 7.06°, a factor of 4.0 away.-0.50000.50011.5001234567amplitude, by stepscatter, log₁₀ degrees6°intactno latticeplacementfield48 runs · both kindsan empty factor of 4.0 at the cut
Fig. 5 How recorded scatter grows with the disturbance driving it. The quantity a protractor reports is the displacement of an organ relative to its neighbours, which is why two disturbances of the same size can leave different scatters — and why every comparison here is repeated at equal recorded scatter as well.

The amplitudes in between

The survival table is coarse — it reports the largest amplitude at which every stem of a colour still has a lattice, from a list that doubles — so it is worth saying what the boundary looks like between its steps.

At a quarter of a degree, every stem of every colour survives: five of five standing for white, shared and inherited alike, with scatters of 0.57°, 0.71° and 0.97°. At half a degree, white still has five of five and the two correlated disturbances do not. Past that the failures are not marginal — a stem that loses its lattice does not scatter by three degrees, it scatters by tens, because the pattern has stopped being a lattice rather than become a worse one.

So the boundary is between 0.25° and 0.5° for both correlated disturbances and above 0.5° for white, and nothing in the tables sits near a threshold. That matters for the claim being made: “the same survival” is not two numbers that happen to round together, it is two disturbances failing in the same interval while a third does not.

It also sets what a finer sweep would buy. Halving the step would put the boundary within a fifth of a degree for each colour and might separate the shared from the inherited one — the scatters at a fixed amplitude do differ, so the survival boundaries probably differ a little too. That is five times the runs for a refinement of a number that is already doing its job, which is why the coarse table is what is here.

A lattice survives about 1.6° of scatter, whichever way the noise arrivesThe largest divergence scatter at which a run is still a lattice, from each kind of noise at the largest amplitude that leaves one. The two routes share no code below the placement rule: one displaces the node after the choice, the other perturbs the energy the choice is made over. They agree to 0.66°.placement noise, at 1°1.97°field noise, at 0.015 of the barrier1.31°no noise at all0.64°largest divergence scatter still holding a latticethe two differ by 0.66° — a fifth of what either toleratesand by 2.6× more than a noiseless run scatters65 nodes per rung · 3 runs per amplitude1.97° against 1.31°
Fig. 6 The tolerance boundary measured for two kinds of noise that enter the rule at different points. It comes out at about the same recorded scatter for both, which is the result that lets a survival table be read as a statement about the pattern rather than about the disturbance.

The ratio is mostly the history

The comb ratio — the second comb divided by the main one — was this site’s last discriminator between a rule and a forgery, and it was retired when it turned out to follow the disturbance’s colour on arrangements that all contain the same rule. The control says which half of the colour it follows.

disturbance comb ratio
independent errors 0.80
shared once 0.91
inherited 1.02

The shared disturbance moves the ratio by 0.11 and the inherited one by 0.22. But those three rows are at three different recorded scatters, and the ratio is known to move with scatter — so the comparison a protractor could make is against a white run at the same scatter.

Drive white noise at 0.35° per organ and it records 0.77° of scatter, against the shared disturbance’s 0.71°, and gives a ratio of 0.86.

So of the shared disturbance’s 0.11, about 0.06 is what any disturbance of that recorded size would produce, and 0.05 is the structure. Of the inherited one’s 0.22, the same accounting leaves most of it unexplained by scatter — a white run at its scatter would be beyond the amplitudes at which stems survive at all.

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. 7 The table this row extends. Seven disturbances through the same rule, and the ratio runs from below the rule’s own value to the forgery’s — which is what took the observable out of service. The shared disturbance’s place in it is near the rule’s end.

Which is a small rehabilitation and not a rescue. The ratio still moves with the disturbance and still cannot be quoted as a property of the mechanism. What the control adds is that the movement is mostly driven by the property that also forges the comb — the inheritance — rather than by the structure a plant is most likely to have.

An accounting note about equal sizes

Every row above is at “the same size”, and that phrase has caused trouble in this thread once already, so it is worth spelling out what it means here.

Each disturbance is a stream of deviates multiplied by an amplitude in degrees. The streams are not automatically the same size: the inherited one’s recursion draws from two lagged terms and normalises as though it drew from one, so at a coupling of 0.7 its deviates have a standard deviation of 2.01 rather than one. An earlier comparison in this thread was made without correcting for that, and the result was a control that was described as being at one scatter and was at another.

So every stream here is divided by its own measured standard deviation, over twenty thousand draws, before the amplitude is applied. A displacement of a quarter of a degree is a quarter of a degree of root-mean-square displacement per organ in all three rows.

Recorded scatter is then a result rather than an input, which is the right way round: the whole point of the comparison is that disturbances of equal size leave different scatters, because what a protractor records is relative displacement.

Two effects, two causes, and they are not the same

Set the two results beside each other, because the pattern is the point.

what is affected which half of the disturbance does it
the damage to a lattice the structure — shared and inherited alike
the forgery of a comb the history — inherited only
the shift in the comb ratio mostly the history, partly the scatter

A disturbance shared between touching organs is dangerous to a pattern and useless for forging evidence about it. A disturbance that is handed on and on is dangerous and forges everything.

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. 8 The whole comparison in one table: two columns through the rule and two on a lattice with no rule in it. The left half separates the shared disturbance from white noise; the right half separates it from the inherited one. Neither half separates it from both.

That asymmetry has a consequence for how a plant’s own noise should be thought about. The kind of coupling everybody would grant — touching organs push each other — is the kind that costs a lattice most of what it costs and manufactures none of what was being read as evidence. The kind that manufactures evidence is a stronger hypothesis about the tissue, and it is also the kind that would destroy the pattern fastest.

What a botanist would see, and what they would not

The three rows of this comparison are three different disturbances producing three recorded scatters, and the recorded scatter is the only one of the three quantities a photograph carries.

A stem jostled by independent errors at a quarter of a degree records 0.57°. One jostled by shared errors of the same size records 0.71°. One jostled by inherited errors records 0.97°. So a botanist measuring the scatter of a real stem’s divergences and finding, say, 0.7° cannot tell whether they are looking at a plant with a large independent noise or a smaller correlated one — the two are the same number.

That is the same difficulty the whole thread has been circling, and it is why the observables here are sequence statistics rather than spreads. A spread is one number and the accounts differ by two: what the disturbance is, and how it is shared.

The consequence for an experiment is a design note rather than a result. Reporting a scatter is nearly free and settles nothing; reporting the autocorrelation of the sequence at a handful of lags costs the same measurements and separates accounts that a scatter cannot. Any survey of divergence sequences should therefore publish the sequence, not a summary of it — a point this site has now arrived at from four different directions.

The awkward corollary

Put those two together and the transported account has an internal tension worth stating.

To forge the comb, the inheritance has to be strong: at a coupling of 0.5 it manages it, and below about 0.4 there is no comb at all. But a coupling that strong is also what makes the disturbance harsh, and the harshness is what destroys a lattice at half the amplitude white noise needs. A plant whose errors propagate strongly enough to produce the observable would be a plant whose lattice is fragile.

The numbers do not quite close that argument — a stem at a quarter of a degree survives at either coupling — but they narrow the window. The account needs an inheritance strong enough to populate the whole residue class of lags and weak enough not to accumulate the pattern away, and the two requirements pull in opposite directions.

Two stems at 0.75° of scatter, one angle at a timeThe divergence of each node, with the slow climb of the ladder removed. Both stems scatter by about 25.97° and a botanist would report them identically. The jostled one wanders in runs — its lag-one correlation is 0.70 — and the one with placement noise alternates about the mean at 0.21, because a displacement of one node enters two consecutive divergences with opposite signs.jostle noise — correlation 0.70placement noise — correlation 0.21243 nodes each, both at 25.97° of scattercorrelations 0.70 and 0.21
Fig. 9 Two stems at the same recorded scatter and different disturbances, which is what the argument above is about. What a photograph shows is the scatter; what separates the accounts is how the errors got there.

What this does not say

It does not say the shared disturbance is what a plant has. It is a control, and an unphysical one: no process passes a displacement between two organs without passing the displacement they themselves received.

It does not restore the comb ratio. The ratio moved with the colour; it still does; what has changed is the account of which property of the colour moves it.

What the sequence sees that the scatter cannotEach point is an ensemble at one amplitude, placed at the scatter it produces. A stem at three quarters of a degree of scatter has a lag-one correlation near zero if its noise arrived after the primordium was placed, and near 0.7 if it arrived before — and no measurement of the scatter can tell those apart. The separation closes above about a degree, because what the other two kinds preserve is the correlation of a lattice.-0.20000.2000.4000.6000.5000.75011.251.50divergence scatter, in degrees — the one quantity a plant offerscorrelation between one divergence and the nextplacement noisejostle noisefield noise4 runs per point · band ±0.13every point is a lattice
Fig. 10 The statistic the rule does own: the anticorrelation of consecutive divergences, which is a restoring force and not a property any of these disturbances has. Whatever colour the noise is, a stem grown by the rule carries this and a kinematic one does not.

And it does not settle the discrimination. After all of this the position is that a comb needs a propagating disturbance, a propagating disturbance is harsher than a shared one, and a placement rule leaves a signature in the lag-one correlation that neither has. Three statements, none of which is a clean observable, and the clean observable of the previous work is the intervention rather than any reading.

The check

Two assertions carry the essay and they are separately falsifiable.

The first requires the shared and inherited disturbances to break a lattice at the same amplitude, and both to break it before white noise does. A control that was merely gentler would fail the first half; one that was harmless would fail the second.

The second requires the ratios to be ordered — white below shared below inherited — and requires the matched white run to account for more of the shared disturbance’s shift than of the inherited one’s. That is the claim that the ratio belongs mostly to the history, stated so that a measurement in which the scatter explained everything, or nothing, would stop the build.

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. 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 The ratio was never about the rule The comb has already been retracted here 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.

Shares its objects with

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

  • A disturbance the organs share — both name artefact, divergence angle, ensemble, equilibrium, honest limits, lattice, measurement, measurement error, noise, the placement rule, tolerance
  • A rule that cannot heal a hole — both name artefact, divergence angle, ensemble, equilibrium, honest limits, lattice, measurement, noise, parastichy pair, the placement rule, tolerance
  • The disturbance that travels — both name discrimination, divergence angle, ensemble, honest limits, lattice, measurement, noise, parastichy pair, the placement rule, tolerance, transport
  • The grid was in the number — both name artefact, discrimination, divergence angle, ensemble, honest limits, measurement, measurement error, noise, the placement rule, tolerance
  • A front with no middle — both name artefact, divergence angle, ensemble, equilibrium, honest limits, lattice, measurement, parastichy pair, the placement rule
  • The block is the count it was cut from — both name artefact, divergence angle, ensemble, equilibrium, honest limits, lattice, measurement, parastichy pair, the placement rule

Named objects

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

ArtefactDiscriminationDivergence angleEnsembleEquilibriumHonest limitsLatticeMeasurementMeasurement errorNoiseParastichy pairThe placement ruleSummary statisticToleranceTransport