Two shapes, one threshold

Once the placement rule’s neighbourhood is a stated falloff rather than a loop bound, the obvious question is how far it can reach before the pattern stops. That question has a clean answer, and the answer is not the one it looks like.

The disagreement #

Two falloffs with mechanisms behind them. The exponential, $e^{-d/L}$, is the steady state of something produced at a source that diffuses and decays — the shape the usual account of auxin depletion implies. The gaussian, $e^{-(d/L)^2}$, is what a briefly-produced signal looks like as it spreads without decaying. They are the two standard stories, and a paper choosing between them would be making a claim about the chemistry.

Sweep each one’s range and ask, at every width, what share of an ensemble of grown stems still has a lattice. Both give a clean boundary: a lattice below, none above.

The boundaries are in different places. Read in half-weight radii — the distance at which the weight has fallen to a half, which is the unit built precisely so that two shapes can be compared — the exponential holds a lattice out to about 3.75 spacings and the gaussian only to about 2.25. Fifty per cent apart, against a sweep whose own resolution is about a sixth.

This was not the expected result. The unit was invented on the assumption that it would reconcile them — that the disagreement was a parameterisation artefact, that half-weight radii would put both shapes on one axis, and that the conclusion would be a tidy only the range is identifiable, and the shape is decoration. That conclusion would have been convenient. A range is the kind of thing an experiment could hope to recover, and a shape is not.

Two other scales, and both are worse #

Before concluding that the shape genuinely matters, it is worth checking that half-weight radius is not merely a poor choice among several — that the claim is about the shapes and not about how hard somebody looked for a scaling.

Two other scale-free readings fall out of the shapes’ own integrals, with nothing fitted.

The mean distance a unit of weight sits at, over a two-dimensional density, is $(\sqrt\pi/2)L \approx 0.89L$ for the gaussian and $2L$ for the exponential. Converted at the two critical widths, those come out more than four times apart — worse than the half-weight reading by a factor of eight.

The effective neighbour count, $\pi\rho L^2$ against $2\pi\rho L^2$, is the number of neighbours the rule effectively sees. At criticality those are also further apart than the half-weight radii.

So of three natural scales, half-weight radius is the best and it still leaves the shapes fifty per cent apart. The assertion in the library checks exactly this: the gap has to exceed a third and the two alternative scalings have to be wider still. A claim that nothing reconciles two curves is otherwise a claim about the author’s patience, and this one is a measurement.

What does agree #

The quantity that turns out to be invariant is one this site measured a phase ago while arguing about something else entirely.

The previous phase asked what the falloff exponent controls, and rejected the tidy answer — that it controls whether the energy converges — in favour of one that survived being tested: it controls contrast. Park a candidate one rise above a node in the middle of a static lattice, sweep it around the circle, and record how much the energy contributed by each ring of neighbours varies over the sweep. nearRatio is the first ring’s variation over the second’s. A high value means the rule is placing a node with respect to its immediate neighbours; a low one means it is placing it in a broad shallow landscape assembled from dozens of them, and a lattice is exactly the first thing.

Measure that with each cut-off in place, at each shape’s own critical range:

Fifty per cent apart in range and six per cent apart in contrast. Two shapes that disagree completely about how far the rule may see agree closely about how much of the profile’s variation the nearest ring has to own.

And it is not a coincidence of two curves crossing #

Two monotone curves crossing a threshold will always agree at the threshold; that is what a threshold means. What makes this a measurement rather than an arithmetical remark is that the threshold was established somewhere else first.

The exponent route gives an independent estimate. Sweep the falloff exponent with no cut-off at all: at $p = 1$ the rule makes nothing, and at $p = 1.25$ it makes a lattice every time. Measure the contrast at those two exponents on the same lattice with the same shells:

• $p = 1$, no lattice — contrast 3.98
• $p = 1.25$, a lattice — contrast 7.63

The cut-off route’s answer, 5.7–6.1, sits inside that bracket. Two completely different ways of making the rule more local — steepening the falloff everywhere, or truncating a shallow one — cross the same line, and the second route pins it to a few per cent where the first could only put it inside a factor of two.

That is the shape of result this collection exists to produce. The exponent sweep and the cut-off sweep share no parameters, no code path and no failure mode: one varies an exponent with an unrestricted neighbourhood, the other holds the exponent at 1 and varies a weight function. Their answers agree, and the agreement is about a quantity neither sweep was designed to measure.

Why the shapes differ where they do #

It is worth working out why the gaussian gives up first, because the reason is visible in the two functions and explains the direction of the effect rather than merely its size.

At equal half-weight radius the two curves cross, by construction, at the half-weight point. What differs is everything either side of it. Near zero the gaussian is flat — $e^{-u^2} \approx 1 - u^2$, a parabola with zero slope at the origin — while the exponential is already falling linearly, $1 - u$. So over the first ring of neighbours, where the profile’s variation comes from, the gaussian weights the near and the middling almost equally and the exponential does not.

Far out the ordering reverses: past the half-weight point the gaussian collapses much faster than the exponential, which is the property everybody thinks of first and which is not the property that matters.

Put those together. Contrast is the near ring’s variation over the next ring’s, so what sets it is the relative weighting of the first two rings, and there the exponential is the more discriminating of the two at equal half-weight radius. It can therefore afford a longer nominal reach before its contrast falls to the critical value. The gaussian’s flat top costs it exactly that margin.

This also says why the far-field behaviour — the part of a falloff a modeller usually worries about — is nearly irrelevant here. Two shapes with wildly different tails and similar near-field slopes would put their boundaries in nearly the same place; two shapes with identical tails and different near-field slopes would not. The tail is where the physics is usually discussed and the first ring is where the pattern is decided.

What this costs a survey #

The consequence for anybody hoping to measure this on a plant is the uncomfortable one, and it is worth stating without softening.

How far inhibition reaches is the quantity a biologist could imagine getting at: image the apex, perturb a primordium, see how far the effect is felt. It is a length, and lengths are measurable. It does not determine whether there is a lattice. A reach of three spacings is inside the pattern-forming region for an exponential falloff and outside it for a gaussian, and the difference between those two hypotheses is a difference in shape — how the influence declines, not how far it goes — which is far harder to measure and is the half nobody reports.

So the model’s prediction is conditional on something a paper would not usually state. “Inhibition falls off over about three cell diameters” is compatible with the model making a lattice and with it making nothing, and a reader who took the range alone as a test would find the model confirmed or refuted by their choice of functional form.

The one consolation is that the quantity that is invariant is also, in principle, measurable — contrast is a statement about how much of a primordium’s local environment its nearest neighbours account for, and that is a question about an image rather than about a chemical. It is not a measurement anybody has made. But it is the one this thread says would settle the case, and knowing which measurement would settle a case is most of the value of having a model.

The control that keeps this honest #

Everything above is about a long-ranged rule, where a neighbourhood is a question. At an inverse cube — the exponent the site’s other figures use, and the one the original ferrofluid experiment obeys — the interaction has fallen by three orders of magnitude within a couple of spacings on its own. A cut-off anywhere in the swept range removes nothing that was contributing.

The check is that it changes nothing: at $p = 3$, every cut-off range from 1.5 to 4 half-weights gives a lattice, and every one of them lands within a degree of where the uncut rule does. If that failed, every number above would be about the cut-off code rather than about the range of the rule.

It is a dull assertion and it is the one that makes the rest of the essay a measurement. A model with three new parameters can be made to say almost anything; what stops it is a region where the new parameters are known to be irrelevant, and a requirement that the model be unchanged there.

The sweep’s own limits, stated #

Three things about the measurement are worth putting on the record, because each of them could be mistaken for a result.

The grid. Ranges are swept in steps of half a spacing, so a boundary is located to about a quarter of a spacing and the interpolated figures — 3.75 and 2.25 — carry a precision the grid does not have. They are reported to two decimal places because the arithmetic produces them, and the claim they support is a fifty per cent gap, which is six times the grid step. Anything finer would be a claim about the sweep.

The ensemble. Each point is a handful of runs separated by a fifth of a degree of placement noise, which is a tenth of the scatter a lattice is measured as surviving. Separating members by their starting angle instead does not work, and the previous phase found out the expensive way: at a coarse rise the only arrangement is 1/2, so every history collapses onto it within four nodes and five “independent” runs return the same divergence to three decimal places. A share of five identical runs is a share of one.

The gaussian’s tail. Its sweep is not perfectly monotone — a couple of the widest points show a run or two surviving where the middle of the sweep shows none. Those are marginal runs at ranges well past the boundary, and the boundary is taken at the first crossing rather than the last for that reason. It is stated here rather than smoothed away, because a non-monotone tail in a share statistic is usually a sign that a threshold is being crossed twice, and here it is a sign that a few runs at a wide range happen to keep a pattern their seeds gave them.

None of the three changes any conclusion. All three are the kind of thing that, unstated, turns a reader’s reasonable question into a suspicion.

What the numbers say, together #

Three facts, in the order they were found:

1. Two falloff shapes, read in the same unit, put their lattice boundaries fifty per cent apart, and no natural rescaling of the shapes brings them closer.
2. At those two boundaries the near-shell contrast agrees to six per cent.
3. That contrast lies inside the bracket left by a sweep of the falloff exponent, which shares nothing with the cut-off sweep except the rule being swept.

There is a fourth fact that belongs with them, and it is about what was almost concluded. The half-weight radius was introduced to make the shapes comparable, and when it failed to reconcile them the first instinct was to look for a better unit — to keep searching until some transformation put the two boundaries on top of each other, and then report that transformation as the invariant. That search would have succeeded: there is always some monotone reparameterisation that maps one critical value onto another, and with two shapes and one free function it takes a few minutes to find one.

It would also have been worthless. A quantity fitted to make two numbers agree predicts nothing, because it was constructed from those two numbers and has no other content. What makes contrast different is that it was not fitted here at all: it was defined a phase earlier, for the exponent question, with these shapes not yet in existence — and its critical value was bracketed there. Being told the answer before asking the question is the only thing that separates an invariant from a curve fit, and it is worth naming as a general test.

The reading is that locality is a single quantity with two knobs on it. The exponent and the range are not independent hypotheses about the model; they are two ways of moving one number, and the number has a critical value near six on this site’s scale. Below it there is no lattice; above it there is one, and above it the lattice is largely insensitive to how it got there — which is why an inverse cube with a wide neighbourhood and an inverse first power with a narrow one produce divergence angles four hundredths of a degree apart.

That last insensitivity is worth dwelling on, because it is the reason the collection’s earlier essays were not wrong to ignore all of this. Every figure on this site that draws a golden-angle lattice was made at an inverse cube with a generous window, comfortably inside the region where the pattern does not care — and the pattern it draws is the same pattern the narrow-cut-off rule draws. The neighbourhood question does not change any of the site’s results about lattices, counts, ladders or transitions. What it changes is what those results are evidence for: they are statements about a rule that is local, rather than statements about repulsion in general, and until this thread nothing said which.

That is a more useful statement than either sweep alone could make, and it is the kind of thing that only shows up when two parameters that were introduced for different reasons are measured against the same quantity.