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

The corner that does not move

Read as degrees of drift getting through rather than as a ratio, and compared seed by seed, the deep and shallow rules change hands. The share that goes to the deeper rule climbs from twenty-three per cent under white noise to ninety-four at a correlation length of a hundred organs — and the crossing sits at two or three organs whether the two rules differ by a factor of four or sixty-one.
33 min read 29 figures State the claim, state the testOne parameter, one familyTwo routes, one number

Worth reading first: A disturbance with a memory · Fitting the exponent · How far a primordium reaches.

A rule with a deep neighbourhood was supposed to correct a slow drift and let a fast one through; a rule with a shallow one, the reverse. The crossover between them was supposed to sit where the drift’s correlation length matched the depth. Swept across a factor of sixty in the depth, the measurement found no crossover at any depth and the deep rule passing more rather than less.

Two things have changed since. The neighbourhood has been counted four ways and they all give the same ordering, so the result is not an artefact of the definition. And the organs at the top of the rule’s profile turn out to be the same five at every exponent, so a sweep of the exponent was never sweeping the thing a filter argument is about.

Four ways to count the rule's neighbourhood, and one orderingHow many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by two different definitions. Counting the organs that carry nine tenths gives 182, 120, 30, 8, 3. Counting the organs that carry the weighted mean lag gives 68, 45, 21, 14, 11. They disagree about the size of the neighbourhood by a factor of 10 — the widest moves by 61 times across the sweep and the narrowest by 6.1 — and every one of them falls as the exponent rises. So no choice among them changes which rule is the deeper, and no redefinition can turn a result about the depth around.31030100nine tenthsweighted mean lag1.52346falloff exponent of the placement ruleorgans in the neighbourhoodrise 0.005 · 5 exponents · top lags 13, 8, 5, 21, 26 at every onegenerated from a stated rule, not drawn to look right
Fig. 1 Where the two changes leave the sweep: a neighbourhood that moves by sixty organs by one measure and six by another, with the same ordering under both.

This essay changes the quantity being read, and a corner appears.

The ratio has the scatter in its denominator

The statistic the earlier sweep reported is a ratio: how much a drift moves the mean of a block of divergences, against how much independent draws would. It is the right statistic for asking how much of a drift survives the rule, and it has a denominator.

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. 2 The statistic, drawn. How far the block means wander as the block grows, against what independent draws would give.

That denominator is the scatter of the divergences themselves, and the scatter depends on the exponent for reasons that have nothing to do with drift: a shallow rule makes a noisier lattice. Across the sweep the scatter runs from 0.205° at the deepest to 0.321° at the shallowest — a factor of 1.57 — while the drift getting through, measured in degrees, moves by a factor of 1.20.

The wander climbs because its denominator fallsThree quantities across the same sweep of the rule's depth, each divided by its own value for the shallowest rule so that they share an axis. The wander rises by a factor of 3.7 as the neighbourhood goes from 3 organs to 182. The scatter between neighbouring divergences falls by 1.57, because that is the part of a disturbance a placement rule corrects and a deeper rule corrects it better. What is left — how many degrees of slow drift actually reach the divergences — changes by 1.20, from 1.52 to 1.82 degrees. The rule barely filters a drift at any depth.012342.262.081.480.9030.477organs in the neighbourhoodagainst the shallowest rulewanderscatterdrift througha drift correlated over 33 organs · 3 seeds a pointdrift through: 1.52° to 1.82°
Fig. 3 The two moving in opposite directions. Most of what the ratio reported was the denominator, which is a fact about the lattice rather than about the drift.
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 latticeplacementfield64 runs · both kindsan empty factor of 4.1 at the cut
Fig. 4 The denominator on its own: how much a stem’s divergences scatter under a disturbance, which is what a shallow rule does more of at every colour.

So the quantity to compare is degrees of drift getting through — the ratio undone, the scatter multiplied back in. That is the quantity a reader of a real stem would measure, and it is the one this essay uses.

What “drift getting through” is

The quantity deserves a definition, because two are in play and the whole result turns on which is read.

Take the divergences of a stem grown under a disturbance. Split them into blocks of sixty-four and take the mean of each block. If the disturbance had no memory, those block means would scatter by the divergence scatter divided by eight; a drift moves them further. The wander is how much further, as a ratio to what independent draws would give.

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 wander as the block size grows, on lattices with a stated colour of disturbance and no rule in them. This is the quantity before the rule is applied to it.

Degrees through is the wander’s square root multiplied by the scatter of the divergences themselves — which undoes the normalisation and gives the actual size, in degrees, of the slow movement in the block means.

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. 6 Why the two can disagree: two runs matched on scatter can carry very different amounts of drift, and two runs carrying the same drift can have very different scatter.

Both are legitimate. The wander answers “how much of the drift survived, relative to the noise floor”; degrees through answers “how big is the slow movement anybody would measure”. The first has the noise floor in its denominator and the noise floor is a strong function of the exponent, which is what makes it the wrong one for a comparison across exponents.

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. 7 The floor in question: how much a stem’s divergences scatter, which is set by the rule and not by the disturbance.

Paired seeds, because the spread is larger than the effect

Degrees through is noisy. At one setting of the rule and one colour of disturbance, six random seeds give answers spanning a factor of 1.4 to 2.6. The difference between two rules at one colour is smaller than that.

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 48.38° 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 0.2170 nodes each, both at 48.38° of scattercorrelations 0.70 and 0.21
Fig. 8 Why a mean of six is not enough: two runs of the same rule with different seeds differ by more than two rules differ from each other.

Comparing two means with that much spread under them compares two numbers whose ranges overlap. So the comparison is paired: the same disturbance — the same seed, the same sequence of displacements — is given to both rules, and what is recorded is which of the two let more of it through.

One seed, two rates, two laddersBoth stems begin as forty nodes of Lucas lattice at a rise of 0.12. At 65 nodes per rung the divergence stays at 99.5° and the counts walk 1/3 → 3/4 → 4/7 → 7/11. At 131 it leaves for 137.7° and walks 1/3 → 2/3 → 3/5 → 5/8 → 8/13 instead.10011012013014011.502falling rise, as −log₁₀divergence the stem is producing (°)137.51°, Fibonacci99.50°, Lucas65/rung → 7/11131/rung → 8/13seeded at 99.50°, rise 0.127/11 against 8/13
Fig. 9 The pairing, in a neighbouring experiment: one seed driving two settings, so that the difference between them is not a difference between two draws.
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. 10 And the reason pairing is worth the trouble here: the quantity carries a large seed-to-seed component that a paired design removes and an unpaired one does not.

Then the comparisons are pooled. Every pair of exponents whose neighbourhoods differ by at least a factor of three — eight pairs out of the ten available — is run at six seeds, giving forty-eight paired comparisons at each colour.

The corner

correlation length 0 1.4 2.8 4.5 9.5 99.5
share to the deeper rule 23% 29% 56% 65% 81% 94%
Which rule passes more drift — exponent 1.5 against exponent 6The same disturbance is given to two placement rules, one with a falloff exponent of 1.5 and one of 6, and the bar counts how many of six seeds let more of it through under the deeper rule. Above the line means the deeper rule passed more. Reading left to right the disturbance is given a longer memory, from white noise to a correlation length of 99 organs. The two change hands: the shallower rule passes more up to a correlation length of 1.4 organs and the deeper one from 2.8 organs onward. The comparison is made seed by seed rather than between two averages, because the spread between seeds at one setting is larger than the difference being measured.63036white1/61.42/62.84/64.55/69.56/699.56/6seeds agreeing — up: the deeper rule passed morecorrelation length of the disturbance, in organsexponent 1.5 against 6 · 182/3 organs · 6 seedsgenerated from a stated rule, not drawn to look right
Fig. 11 One pair of rules across the colours, seed by seed. Below the line means the shallower rule let more through. The slider walks every pair of exponents whose neighbourhoods differ by at least a factor of three.

Against white noise the shallower rule lets more drift through, at thirty-seven comparisons out of forty-eight. At a correlation length of a hundred organs the deeper rule does, at forty-five out of forty-eight. The share rises at every step in between, and it crosses a half between correlation lengths of 1.4 and 2.8 organs.

Which rule passes more drift — exponent 3 against exponent 4The same disturbance is given to two placement rules, one with a falloff exponent of 3 and one of 4, and the bar counts how many of six seeds let more of it through under the deeper rule. Above the line means the deeper rule passed more. Reading left to right the disturbance is given a longer memory, from white noise to a correlation length of 99 organs. The two change hands: the shallower rule passes more up to a correlation length of 9.5 organs and the deeper one from 2.8 organs onward. The comparison is made seed by seed rather than between two averages, because the spread between seeds at one setting is larger than the difference being measured.63036white3/61.40/62.84/64.52/69.53/699.56/6seeds agreeing — up: the deeper rule passed morecorrelation length of the disturbance, in organsexponent 3 against 4 · 30/8 organs · 6 seedsgenerated from a stated rule, not drawn to look right
Fig. 12 The narrowest pair in the comparison, at a factor of under four between the two neighbourhoods. The same shape, and the crossing in the same place.

That is a corner. The earlier sweep could not see it because the ratio it was reading has the scatter in its denominator, and the scatter moves the other way with the exponent — enough to hold the ratio’s ordering fixed across the whole range of colours.

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. 13 The colours the sweep runs over: a disturbance with no memory at one end and one whose correlation outlasts the front at the other.

Why eight pairs and not ten

Ten pairs of exponents are available and two of them are excluded, by a rule stated before the comparison rather than after it.

The exclusion is on the ratio between the two neighbourhoods: a pair is compared only if one is at least three times the other. Exponents 1.5 and 2 describe neighbourhoods of 182 and 120 organs, a factor of one and a half, and exponents 4 and 6 describe 8 and 3, a factor of two and two thirds. Those two pairs are comparisons between rules that are very nearly the same rule.

Four ways to count the rule's neighbourhood, and one orderingHow many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by two different definitions. Counting the organs that carry nine tenths gives 182, 120, 30, 8, 3. Counting the organs that carry ninety-nine hundredths gives 284, 263, 149, 54, 9. They disagree about the size of the neighbourhood by a factor of 10 — the widest moves by 61 times across the sweep and the narrowest by 6.1 — and every one of them falls as the exponent rises. So no choice among them changes which rule is the deeper, and no redefinition can turn a result about the depth around.31030100300nine tenthsninety-nine hundredths1.52346falloff exponent of the placement ruleorgans in the neighbourhoodrise 0.005 · 5 exponents · top lags 13, 8, 5, 21, 26 at every onegenerated from a stated rule, not drawn to look right
Fig. 14 The quantity the exclusion is stated on, and beside it the stricter share: the neighbourhoods, and how far apart consecutive exponents put them.

What they do is what nearly-identical rules should do. The 1.5-against-2 comparison gives four, two, four, three, three and four seeds out of six across the colours — a coin toss at every one, with no trend. Including it would have added noise to the pooled share and, more importantly, would have added a pair with no crossing to a claim about where crossings sit.

Which rule passes more drift — exponent 1.5 against exponent 3The same disturbance is given to two placement rules, one with a falloff exponent of 1.5 and one of 3, and the bar counts how many of six seeds let more of it through under the deeper rule. Above the line means the deeper rule passed more. Reading left to right the disturbance is given a longer memory, from white noise to a correlation length of 99 organs. The two change hands: the shallower rule passes more up to a correlation length of 2.8 organs and the deeper one from 1.4 organs onward. The comparison is made seed by seed rather than between two averages, because the spread between seeds at one setting is larger than the difference being measured.63036white1/61.44/62.83/64.55/69.55/699.55/6seeds agreeing — up: the deeper rule passed morecorrelation length of the disturbance, in organsexponent 1.5 against 3 · 182/30 organs · 6 seedsgenerated from a stated rule, not drawn to look right
Fig. 15 One of the pairs that is included, for comparison with what an excluded one looks like: a clear trend rather than a coin toss.

The exclusion also cuts the other way and is worth checking for that. If the crossing were an artefact of which pairs were chosen, the eight would not agree — and they do: every one has the shallower rule ahead at white noise, every one has the deeper rule ahead at the longest colour, and all eight cross within the first ten organs.

Which rule passes more drift — exponent 3 against exponent 6The same disturbance is given to two placement rules, one with a falloff exponent of 3 and one of 6, and the bar counts how many of six seeds let more of it through under the deeper rule. Above the line means the deeper rule passed more. Reading left to right the disturbance is given a longer memory, from white noise to a correlation length of 99 organs. The two change hands: the shallower rule passes more up to a correlation length of 2.8 organs and the deeper one from 9.5 organs onward. The comparison is made seed by seed rather than between two averages, because the spread between seeds at one setting is larger than the difference being measured.63036white2/62.83/69.55/699.56/6seeds agreeing — up: the deeper rule passed morecorrelation length of the disturbance, in organsexponent 3 against 6 · 30/3 organs · 6 seedsgenerated from a stated rule, not drawn to look right
Fig. 16 The last of the eight, over a shorter list of colours. Which colours are sampled changes nothing about where the two change hands.

And it does not move with the depth

This is the claim that separates a corner from a filter. A filter’s corner sits at its own scale, so two rules whose neighbourhoods differ by a factor of sixty should change hands somewhere quite different from two whose neighbourhoods differ by four.

They do not.

Every one of the eight pairs has the shallower rule ahead against white noise and the deeper rule ahead at the longest correlation. Every one of them changes hands inside the first ten organs of correlation. The pair whose neighbourhoods are 182 organs and 3 crosses between 1.4 and 2.8; the pair whose neighbourhoods are 120 and 30 crosses between 2.8 and 4.5.

Which rule passes more drift — exponent 1.5 against exponent 6The same disturbance is given to two placement rules, one with a falloff exponent of 1.5 and one of 6, and the bar counts how many of six seeds let more of it through under the deeper rule. Above the line means the deeper rule passed more. Reading left to right the disturbance is given a longer memory, from white noise to a correlation length of 99 organs. The two change hands: the shallower rule passes more up to a correlation length of 1.4 organs and the deeper one from 2.8 organs onward. The comparison is made seed by seed rather than between two averages, because the spread between seeds at one setting is larger than the difference being measured.63036white1/61.42/62.84/69.56/699.56/6seeds agreeing — up: the deeper rule passed morecorrelation length of the disturbance, in organsexponent 1.5 against 6 · 182/3 organs · 6 seedsgenerated from a stated rule, not drawn to look right
Fig. 17 The widest pair available, at a factor of sixty-one between the two neighbourhoods, crossing where the narrowest pair crosses.
Which rule passes more drift — exponent 2 against exponent 3The same disturbance is given to two placement rules, one with a falloff exponent of 2 and one of 3, and the bar counts how many of six seeds let more of it through under the deeper rule. Above the line means the deeper rule passed more. Reading left to right the disturbance is given a longer memory, from white noise to a correlation length of 99 organs. The two change hands: the shallower rule passes more up to a correlation length of 2.8 organs and the deeper one from 4.5 organs onward. The comparison is made seed by seed rather than between two averages, because the spread between seeds at one setting is larger than the difference being measured.63036white1/61.43/62.83/64.54/69.54/699.54/6seeds agreeing — up: the deeper rule passed morecorrelation length of the disturbance, in organsexponent 2 against 3 · 120/30 organs · 6 seedsgenerated from a stated rule, not drawn to look right
Fig. 18 And the pair whose two neighbourhoods are 120 organs and 30. If the crossing tracked the depth, these two figures would look nothing like each other.

The crossing is also shorter than every neighbourhood in the comparison. It sits at two or three organs; the neighbourhoods it is comparing are 182, 120, 30, 8 and 3 organs wide by the nine-tenths count, and 68, 45, 21, 14 and 11 by the weighted mean. Only one of the ten numbers is anywhere near the crossing.

Four ways to count the rule's neighbourhood, and one orderingHow many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by four different definitions. Counting the organs that carry half the profile gives 33, 12, 3, 2, 1. Counting the organs that carry nine tenths gives 182, 120, 30, 8, 3. Counting the organs that carry ninety-nine hundredths gives 284, 263, 149, 54, 9. Counting the organs that carry the weighted mean lag gives 68, 45, 21, 14, 11. They disagree about the size of the neighbourhood by a factor of 10 — the widest moves by 61 times across the sweep and the narrowest by 6.1 — and every one of them falls as the exponent rises. So no choice among them changes which rule is the deeper, and no redefinition can turn a result about the depth around.131030100300half the profilenine tenthsninety-nine hundredthsweighted mean lag1.52346falloff exponent of the placement ruleorgans in the neighbourhoodrise 0.005 · 5 exponents · top lags 13, 8, 5, 21, 26 at every onegenerated from a stated rule, not drawn to look right
Fig. 19 The neighbourhoods, against a crossing at two or three organs. A filter whose corner sits sixty times below its own width is not filtering.

The disturbance, and what its correlation length means

The knob being swept is the memory of the jostle, and it is worth saying exactly what is varied and what is held.

Each organ is displaced by a small amount before the rule places it. The displacements are drawn from a first-order autoregressive process: each one is a fixed fraction of the last plus a fresh draw. That fraction is the knob. Its correlation length — the number of organs over which the disturbance forgets itself — runs from zero at white noise to about a hundred at the largest setting used here.

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. 20 The disturbances themselves, and where they enter: each organ displaced before the rule places it, with the same amplitude throughout and a memory that runs from none to about a hundred organs.

The amplitude is held fixed at a quarter of a degree per organ, and it is held fixed after the disturbance is divided by its own measured standard deviation. Without that division the correlation coefficient would change the size of the disturbance as well as its colour, and a sweep of the colour would be partly a sweep of the amplitude.

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. 21 Why the normalisation matters: the amount of disturbance is a strong lever on everything downstream, so it has to be the thing held rather than the thing that drifts.
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. 22 And the check that the runs are runs: whether the pattern survives at all across the amplitudes and colours the sweep uses.

So what varies across the columns of the table is one thing — how far into the past a displacement remembers — with the size of the displacement, the rise, the grid, the run length and the block size all fixed.

What the corner is, if it is not the depth

The measurement stops here and the reading is short, so it is worth marking where one ends and the other begins.

Measured: the two rules change hands at a correlation length of two or three organs, at every separation of depth tried, and the crossing does not move with the depth.

Which rule passes more drift — exponent 2 against exponent 6The same disturbance is given to two placement rules, one with a falloff exponent of 2 and one of 6, and the bar counts how many of six seeds let more of it through under the deeper rule. Above the line means the deeper rule passed more. Reading left to right the disturbance is given a longer memory, from white noise to a correlation length of 99 organs. The two change hands: the shallower rule passes more up to a correlation length of 2.8 organs and the deeper one from 4.5 organs onward. The comparison is made seed by seed rather than between two averages, because the spread between seeds at one setting is larger than the difference being measured.63036white0/61.43/62.83/64.55/69.55/699.56/6seeds agreeing — up: the deeper rule passed morecorrelation length of the disturbance, in organsexponent 2 against 6 · 120/3 organs · 6 seedsgenerated from a stated rule, not drawn to look right
Fig. 23 One more pair, for the record. Eight of them behave this way and none behaves otherwise.

Read: the scale that does not move with the exponent is the contact scale. The five organs carrying the largest terms of the profile are the same five at every exponent, so whatever the rule is holding on to is held at the same distance throughout. A disturbance correlated over fewer organs than that is something the rule can average away regardless of how far its tail reaches; one correlated over more is something every version of the rule carries.

Which offsets give short hops, at a rise of 0.005The two lowest points are at 8 and 13, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.00.2000.4005101520index offsetmedian hop between node i and node i+m81326 nodes, 20 offsets triedshortest at 8 and 13
Fig. 24 The scale in question: the steps of the contact families, which are what the largest terms of the profile are, and which the exponent does not move.

That reading is consistent with everything measured and it is not established by it. Two organs is also roughly the shortest correlation that is distinguishable from none at these run lengths, so a crossing there is consistent with “any memory at all changes the sign” as well as with “the contact scale sets it”. Separating those two would need a sweep of the rise, which moves the contact scale while leaving everything else alone.

Every transition as the rise fallsThe pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.381, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820.0.4000.6000.8001-2-1.50-1log₁₀ of the rise between nodes (falling to the right is the plant growing)log₁₀ of the larger parastichy number2/33/55/88/13500 rises, shortest vectors recomputed at eachratio 0.3819 against 1/φ² = 0.3820
Fig. 25 The sweep that would separate them, and is not made here: the rise moves the contact numbers from three to thirty-four while the rule stays exactly the same.

What a sign test can and cannot say

The statistic is deliberately coarse — it throws away how much more one rule passed and keeps only which — so it is worth being clear about the price.

What it buys is robustness. A ratio of two noisy quantities has a distribution with tails; a count of which of two paired numbers is larger does not care how long the tails are. With six seeds a pair gives six comparisons and the sampling noise is a binomial with a known shape, so a five-one split at one colour is informative and a three-three split is not, without any assumption about how the underlying quantity is distributed.

Every open question here needs under 34 specimensThe sample size at which each comparison reaches 90 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives.plants show consecutive Fibonacci pairs far more…4and more often even than a coin weighted to a half14a conifer cone's rings are spaced as a cone rather…1multijugate patterns are a real minority rather than…34against 14.7%, if the truth is 90%needs: the pair, at a stated rungagainst 14.7%, if the truth is 50%needs: the pair, at a stated rungagainst φ² = 2.62, if the truth is φ^(2/1.88) = 1.67needs: three ring positions, to ±3%against 2%, if the truth is 15%needs: the pair; the whorl's symmetryspecimens neededexact binomial · α = 0.05 · power 0.91 to 34 specimens
Fig. 26 The general form of the arithmetic: how many independent comparisons a difference of a given size needs before it is distinguishable from a coin.

What it costs is size. The table says which rule passes more at each colour and says nothing about by how much. From the means, the answer is “not much”: at the longest correlation the deep rule passes 1.83° and the shallow one 1.44°, a factor of 1.27, and at white noise 0.39° against 0.58°, a factor of 1.51. Those are real differences and they are small beside the seed-to-seed spread, which is exactly why the pairing was needed.

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. 27 And the spread that swamps it: two runs of one rule differing by more than two rules differ from each other.

So the corner is established as a change of sign and not as a magnitude. Anybody wanting the size of the effect would need many more seeds; anybody wanting to know whether it changes sign, and where, has what they need here.

What has been repaired

The earlier result said there was no corner. There is one; it was in a quantity with a moving denominator and was invisible in the ratio.

The earlier result also said the deep rule passes more drift than the shallow one. That stands, and now has a boundary on it: it is true past a correlation length of about three organs and false below it.

The wander climbs because its denominator fallsThree quantities across the same sweep of the rule's depth, each divided by its own value for the shallowest rule so that they share an axis. The wander rises by a factor of 2.8 as the neighbourhood goes from 3 organs to 182. The scatter between neighbouring divergences falls by 1.43, because that is the part of a disturbance a placement rule corrects and a deeper rule corrects it better. What is left — how many degrees of slow drift actually reach the divergences — changes by 1.18, from 1.09 to 1.29 degrees. The rule barely filters a drift at any depth.01232.261.480.477organs in the neighbourhoodagainst the shallowest rulewanderscatterdrift througha drift correlated over 9 organs · 3 seeds a pointdrift through: 1.09° to 1.29°
Fig. 28 The claim that survives, with its new boundary. Past a few organs of memory the deeper rule really does pass more.

And the prediction the whole thread started from — that the crossover tracks the neighbourhood — is refuted for a second time and more cleanly than before. It is not that no crossing exists. It is that the crossing exists, is sharp, and sits in the same place for rules whose neighbourhoods differ by a factor of sixty-one.

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 — 0.031, 0.026, 0.022, 0.014, 0.015 at ρ = 0, 0.5, 0.7, 0.9, 0.97 — and the readout returns nothing on 40 runs out of 40. A correlated error is not a periodic one.00.2000.4000.60000.5000.7000.9000.970how strongly each error remembers the last, ρthe largest comb mean anywhere in the thirty lagsthe rule's own stems: 0.64three sampling bands0.0310.0260.0220.0140.015kinematic lattice · AR(1) errorgenerated from a stated rule, not drawn to look right
Fig. 29 The null this thread was built on and which still holds: a disturbance’s memory changes nothing a counter reports, however much of it there is. What it changes is a quantity nobody would read off a specimen, and that is where the corner is.

Where this goes next

131030100half the profilenine tenthsweighted mean lag1.52346falloff exponent of the placement ruleorgans in the neighbourhoodrise 0.005 · 5 exponents · top lags 13, 8, 5, 21, 26 at every onegenerated from a stated rule, not drawn to look right The nearest organ is not the nearest neighbour Rank the terms of the sum the rule minimises and read off which organs the biggest ones belong to. At every falloff exponent from 1.5 to 6 the answer is the same five: lags 13, 8, 5, 21 and 26. The organ placed immediately before is not among them, and counting the neighbourhood in organs was the wrong unit. 02550750102030how many organs the disturbance stays correlated overwander at a block of 64182 organs120 organs30 organs8 organs3 organsneighbourhood measured as the organs carrying 90% of the profileno crossover at any depth The drift goes the other way A rule that corrects what its neighbourhood shares should let through any disturbance slower than its own reach, and should suppress anything faster — a crossover, tracking the depth. Swept over a neighbourhood that changes by a factor of sixty, there is no crossover anywhere, and the deep rule passes nearly four times as much as the shallow one. The prediction is not weakly supported; it is backwards. 012342.262.081.480.9030.477organs in the neighbourhoodagainst the shallowest rulewanderscatterdrift througha drift correlated over 33 organs · 3 seeds a pointdrift through: 1.52° to 1.82° What the ratio was hiding The statistic that says a rule sharpens a drift rises by a factor of nearly four across a sweep of the rule's depth. Undo the normalisation and ask instead how many degrees of drift actually reach the divergences, and the answer changes by a fifth. Nearly all of the effect was in the denominator, and the denominator is the thing the rule is good at.

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AutocorrelationControlDriftEnsembleHonest limitsThe range of the interactionMeasurementNegative resultNeighbourhoodNeighbourhood depthNoiseThe placement rulePredictionSelf correctionSummary statistic