Phyllotaxis
the spirals are counted, not admired
Concept

Neighbourhood — where it appears

The set of organs a placement rule actually responds to. It is set by how fast the repulsion falls rather than by how many terms are summed, and confusing the two produces a sweep in which nothing changes and the sameness reads as robustness.

Named by 17 essays across 2 fields — each of them below, with the objects they name alongside it.

Three cut-offs at the same nominal width of 3 spacings. The weight the interaction is multiplied by, against distance. They halve at 2.08 (exponential), 2.50 (gaussian), 3.00 (hard) spacings — so a rule described as "cut off at 3 spacings" is three different rules until the falloff is named. Every later figure is read in half-weight radii for that reason.

A neighbourhood is a hypothesis

Every simulation of this kind stops summing somewhere. The earlier work found that where it stops decides what pattern comes out — so the stopping place is not a detail of the program but a claim about how far a primordium's influence reaches, and it should be written down as one.

emergence · The range of the interaction Through the rule, the drift survives and the inheritance still does not. How 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.

What the rule does to a drift

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

mechanism · Noise colour The landscape the rule chooses over, at a cut-off of 3 spacings. One height of an ideal lattice, swept around the circle. The exponential cut-off hands the rule a smooth landscape; the hard one hands it a landscape with steps, because a neighbour enters the sum as the candidate slides past it. Halving the sample resolution multiplies the largest jump between neighbouring points by 1.99 on the smooth curve and by 1.19 on the hard one — which is the definition of the difference, since a smooth function's steepest step is bounded by its derivative and a discontinuity's is not. An argmin taken over steps is pinned to the steps.

A hard edge is not a falloff

The prediction was that cutting the neighbourhood at three spacings would reproduce the pattern truncation had manufactured. It does — if the cut is smooth. A hard cut at the same distance produces no pattern at any width, and the reason is that it is the only one of the three whose neighbour set depends on where the candidate is.

emergence · Truncation The loop bound is not the neighbourhood. The wander a placement rule leaves in its divergences, against how many recently placed organs the rule sums over, at four correlation lengths of the disturbance driving it. The loop runs from 15 organs to 85 and nothing moves: the largest change along any line is smaller than the change between random seeds at one setting. That is the shape a parameter has when it is not binding, and it is the same shape a robust result has, which is why the sweep is drawn with the seed spread rather than reported as a number.

The window was not the neighbourhood

A placement rule corrects what is relative between neighbours and passes what moves them all together, so how much of a slow disturbance gets through should depend on how deep the neighbourhood is. The obvious knob is how many organs the rule sums over. Swept across a factor of six, it changes nothing at all — and a parameter that is not binding produces exactly the flat sweep a robust result produces.

mechanism · Noise colour A deeper rule passes more of a drift, not less. The wander left in a stem's divergences, against how many organs its disturbance stays correlated over, for rules whose neighbourhoods run from 3 organs to 182. The prediction under test said a rule should pass a drift once the drift outlasts its neighbourhood, so the shallow rules should be the leaky ones and each line should turn where its own depth is crossed. Every line rises smoothly and the deepest rule is the highest of them at every correlation length — 82 against 22 at the longest drift. There is no crossover anywhere in the sweep.

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.

mechanism · Noise colour Two shapes, two ranges, one contrast. The exponential's lattice ends at 3.63 spacings and the gaussian's at 2.25 — ranges 47% apart — and at those two ranges the contrast is 5.74 and 6.09, 6% apart. The band is what the exponent route leaves: 3.98 at p = 1, where the uncut rule makes nothing, and 7.63 at p = 1.25, where it makes a lattice.

Two shapes, one threshold

Read in the same unit, an exponential falloff and a gaussian one disagree about where the lattice ends by half. The quantity they agree on turns out to be one the earlier work measured for an unrelated reason — and it agrees with a bracket left by a sweep of a completely different parameter.

emergence · Falloff exponent Which lattices survive a fifth of a degree of noise. The share of runs that still have a lattice. The prediction was that a cut-off at this range would be as fragile as the truncation it replaces; it is not. Stating the neighbourhood as a function of distance did not merely make the old result honest — 100% of runs survive against 33%, at a scatter an inverse-cube rule cannot be told from.

The fragility belonged to the window

A pattern that exists only because the rule cannot see far was expected to be held together by that cut, and to fall over when nudged. It does — while the cut is a loop bound. Written down as a falloff at the same range, the same rule keeps every run under the same nudge, at a scatter an inverse-cube rule cannot be told from.

emergence · Noise amplitude Four ways to count the rule's neighbourhood, and one ordering. How 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.

Four ways to count a neighbourhood

How deep does the placement rule look? Counting the organs that carry nine tenths of its profile gives 182 down to 3 as the falloff steepens. Counting the ones that carry half gives 33 down to 1. The weighted mean lag gives 68 down to 11. The four disagree by an order of magnitude about the size and agree exactly about the order.

mechanism · Noise colour Four ways to count the rule's neighbourhood, and one ordering. How many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by three 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 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.

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.

mechanism · Noise colour Which rule passes more drift — exponent 1.5 against exponent 6. The 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.

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.

mechanism · Noise colour What a removal costs the next organ. One mark per wrecked cut in the census: how far the first organ placed after the removal ended up from where the control put it. The rows split by which organ was taken. Removing a direct chain-neighbour of the growing tip — an organ at a multiple of one of the two counted numbers — moves the next organ by between 8.9 and 30.7 degrees. Removing anything else inside the front moves it by between 62.8 and 167.6. Nothing lands between the two groups and the ratio across the gap is 2.05, so the line is a gap rather than a threshold. Taking away a neighbour is the cheap removal, which is the opposite of what the words suggest.

Removing a neighbour costs least

Take away an organ that is a direct chain-neighbour of the growing tip and the next organ moves by under thirty-one degrees. Take away anything else inside the front and it moves by at least sixty-three. Thirty cuts, two groups, a factor of two between them and nothing in the gap.

mechanism · Ablation Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 5 places back and the organ 8 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 26.3° and 4.9°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 164.1°, against 31.2° for the two effects added, so the interaction is +132.9°. The slot is not two independent walls.

Both walls of the slot

The growing tip sits between its two chain-neighbours. Removing either alone is a cheap removal on all six lattices — 2.3° to 41.7°. Removing both together throws the next organ past the expensive line on three of them, and the interaction runs from −25.8° to +132.9°.

mechanism · Both walls Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 5 places back and the organ 8 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 25.8° and 12.0°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 12.0°, against 37.7° for the two effects added, so the interaction is -25.8°. The slot is not two independent walls.

A removal that changes nothing

On one of the six lattices, taking away both walls of the slot moves the next organ 11.953125° — which is exactly, to the last digit, what taking away the larger wall alone moves it. The smaller wall's removal contributes nothing at all when the larger one is already gone.

mechanism · Both walls Which hops survive one wall, the other, and both. One row per lattice. The last three columns are the lags whose hop the cut stem still holds, unchanged from a control that shares its history — the measurement that identifies what a wrecked stem has become. Removing a single wall always leaves something standing, which is what every single-organ cut in this collection does. Removing both leaves nothing at all on two of six lattices, including the coarse rung that no single removal can wreck. A stem that keeps no rigid hop is not a wrecked lattice with a slip in it; it is a stem that is no longer a lattice.

The rung that two organs wreck

On the coarse 3/5 stem both walls of the slot heal when removed alone and wreck when removed together — and the wreck keeps no rigid hop at all. Two of the six pairs in the design end at a destination single removals almost never reach.

mechanism · Both walls Which lag survives, at every lattice and every offset. A row for each of twelve lattices and a column for each offset an organ is removed at, with the lag whose hop survived written in the cell. 30 cells never repair. The lag left standing is one of the two contact families at 29 of them, and at 17 it is the second shortest step on the lattice rather than the shortest, which is what refuses the reading that a rule holds its nearest neighbours. The ringed cells are the five the offset rule gets wrong. Two lattices carry no cells at all because no single removal wrecks them.

Not the shorter of the two

If a damaged stem keeps one contact family standing, the obvious guess is that it keeps the nearer one. Across thirty wrecked offsets that is true twelve times and false seventeen, and on one lattice the two steps differ by a quarter of a per cent — where the words shorter and longer are doing no work at all.

emergence · Ablation The deeper rule against the disturbance's memory, at three contact scales. How many of six seeds agree that the deeper rule passed more drift, swept across the correlation length of the disturbance, at three rises. The rule, the amplitude and the run length are identical in every panel; only the rise differs, and with it the contact numbers — 3 and 5, then 5 and 8, then 8 and 13. The shapes are not the same: 3/5 is crossing, 5/8 is no corner, 8/13 is crossing. A corner that sat at a fixed number of organs would look the same in all three, and it does not.

The corner moves with the rise

The corner was either the contact scale or simply any memory at all, and nothing in the thread had ever varied the rise — the one knob that moves the contact numbers while leaving the rule, the amplitude and the run length alone. Swept over it, the comparison does not keep its shape.

emergence · Noise colour Where the deeper rule changes hands, and where it never does. One row per rise, with the contact numbers on the left and what the comparison does on the right. At the coarsest the deeper rule wins at both ends and loses in the middle; at the middle rise it wins throughout; at the finest it crosses once, from losing to winning. The contact scale runs over a factor of 2.6 across these three rises and the behaviour is not a translation of one curve — it is three different curves. That refutes a corner fixed at a short correlation, and it does not by itself establish one that tracks the contacts.

The panel with no corner

Sweeping the rise gave three shapes where one was expected, and the middle one is the informative panel: at the 5/8 contact scale the deeper rule wins at every correlation and there is no crossing to locate. That is either a fact about the lattice or a fact about the pair of exponents, and one measurement separates them.

emergence · Noise colour

Named alongside it

The objects these essays reach for when they reach for this one.

The placement ruleThe range of the interactionMeasurementNegative resultHonest limitsNoiseRepulsionControlNearest neighbourNeighbourhood depthLattice offsetParastichy pair

All concepts