Two degrees of scatter

Almost every number on this site is about geometry. The divergence angle of a lattice, the rises at which its counts change, the share of angles that give a Fibonacci pair — all of them are properties of an arrangement of points, and none of them is a property of a plant.

This essay has one that is closer to being both, and it arrived as a by-product of asking a different question. It also arrived with a headline that turned out to be wrong in an instructive way, and the correction is the better half.

The measurement, and the agreement #

Stems are grown from a fixed seed at a fixed rate while a noise amplitude is turned up, in two independent ways. Placement noise displaces each node after the rule has chosen where to put it, by an angle. Field noise perturbs the energy profile the rule chooses over, as a fraction of the barrier between the best azimuth and the typical one. Ten runs at each amplitude, because one noisy run is an anecdote.

Every run is then measured for two things: whether the blind counter still finds a parastichy pair at the top, and how much the divergence angle wanders over the last quarter of the stem.

The two amplitudes are incomparable. One is an angle and the other a fraction of an energy, and the numbers that bound them differ by two orders of magnitude — a degree of placement noise against 0.015 of field noise. There is no conversion between them that does not go through the model.

What can be compared is the quantity a plant would be measured by. Take, for each kind, the largest divergence scatter observed in a run that was still a lattice. Read that way the two agree closely: 2.00° from placement noise and 1.68° from field noise, against 0.64° in a run with no noise in it at all.

That agreement is the reason to take the number seriously. The two routes share nothing below the placement rule itself — one perturbs the answer and one perturbs the question, one is applied after the minimum is taken and the other before — so if the tolerance were an artefact of either implementation there is no reason they would land in the same place.

Getting the statistic right took two goes #

The first version compared the mean scatter of each ensemble at the amplitude where the pattern was about to fail, and put the two kinds five degrees apart. That number was meaningless, and the error is worth recording because the symptom looked like a result.

At the amplitude where half the runs have been destroyed, the ensemble mean is an average of runs at 1.9° and runs at 30°. It describes neither population. Worse, it moves with the proportion destroyed rather than with anything about the survivors, so what it was really reporting was how abruptly each kind’s ensemble collapsed — a fact about the amplitude grid.

What survives is a property of the runs that survived. The corrected statistic — the largest scatter among intact runs anywhere in the sweep — takes the two kinds from five degrees apart to a third of a degree apart, and it is the statistic the gate holds.

Why the failure is a cliff rather than a slope #

The tolerance is sharp because the thing being tolerated is discrete.

A parastichy pair is not a quantity. It is a statement about which nodes are neighbours of which — that node i is closest to i ± 8 and i ± 13 — and a neighbour relation is either there or it is not. Blur the positions and the relation survives intact until the blur becomes comparable with the gaps between the candidate neighbours, at which point it is not slightly wrong but replaced.

That is also why nothing lands in the middle. A pattern at three degrees of scatter would be one whose neighbour graph is being scrambled about half the time, and a neighbour graph scrambled half the time does not settle at half-scrambled. Each node is placed among the ones already there, so a misplaced node is surrounded and repaired by its successors while the majority of the lattice is intact — and once the majority is not intact, nothing repairs anything. There is a restoring force in one direction and a positive feedback in the other, and no equilibrium between them.

Measured, the two populations are separated by a factor of four with nothing in the gap. The gate requires that gap before it will sort anything by the cut, which is what keeps this a measurement rather than a filing decision.

The noiseless run does not scatter zero, and the reason matters #

Before the tolerance means anything, the baseline has to. A stem grown with the noise amplitude set to zero produces a divergence sequence with a scatter of 0.64°, not 0.00°, and it is worth knowing what that six tenths of a degree is made of, because it is the floor everything else is measured against.

Two things contribute and they are different in kind.

The azimuth is chosen on a grid. The rule samples the energy at 384 points around the circumference and takes the smallest, so the answer is quantised to 360/384 = 0.94° in the worst case and about half that typically. That is arithmetic, not phyllotaxis, and it puts an irreducible floor under any scatter this model can report. Raising the sample count lowers it and costs time linearly; 384 was chosen for the sweeps because the tolerance being measured is several times it.

And the divergence genuinely wanders near a transition. The previous phase measured this directly: a growing stem does not hold one angle, it produces a sequence that drifts by a degree or two and comes back, and the drift is largest exactly where the counted pair changes. That is real behaviour of the rule rather than a numerical artefact — at a transition two arrangements are nearly equally good, and the rule is choosing between them with very little to go on.

The consequence for this essay is a limit on resolution rather than an error. A tolerance of 0.83°, measured at the finest configuration, is only a little over the grid’s own floor, and it should be read as at most that rather than as a measurement of the lattice. The coarser configurations, at 2.89° and 4.12°, are several times the floor and are measurements.

How much precision the angle needs, from the other direction #

There is a second way to ask how much wander a lattice can take, and this collection has been asking it since its foundation phase without connecting the two.

The site’s argument about the golden angle is an arithmetic one. What makes 137.5° special is that it is the hardest number to approximate by a rational, so no family of parastichies ever lines up into rays; the essays measure that resistance and find 0.4377 against 0.3306 for the best of 1,500 sampled angles. That is a statement about an angle held exactly.

A plant does not hold an angle exactly, and the tolerance measured here says how far from exact it can be. Put the two together and the question becomes: at a given rise, how far can the divergence stray before the pattern is a different lattice rather than a noisy version of the same one?

That question has an answer computed elsewhere in this phase, from geometry alone: a pair (m, n) is the two shortest families over a band of divergence angles about 221°/mn wide. At 8/13 that band is 2.12°. So a stem holding 8/13 has, in the static picture, about two degrees of room either side before it is holding something else — and the measured tolerance at that pair is 2.89°.

The two calculations have nothing in common. One grows stems with a rule that mentions no angle and counts them blind; the other sweeps a lattice parameter and asks which two families are shortest. That they land within a factor of one and a half of each other is the best evidence available that the tolerance is a fact about the arrangement rather than about the model.

The correction: it is not two degrees, it is two degrees there #

The result above is stated for one configuration — a stem carried down to a rise of 0.004, which ends on 8/13. Nothing in it says the tolerance is a property of lattices in general, and the obvious next question is whether it is.

It is not.

Carry the same stem to a coarser end rise and it ends on 5/8 and survives 4.12°. Carry it finer, to 0.0015, and it ends on 8/13 again but survives 0.83°. A factor of five across a range of end rises that is well inside what a real organ covers.

So “about two degrees” is a scale, not a constant. What is invariant is the agreement between the two kinds of noise at a given configuration; the number itself is a property of how fine the arrangement is.

Once stated that way it is not surprising, and there is a quantity already on this site that it can be compared with. A pair (m, n) is the two shortest families over a band of divergence angles about 221°/mn wide — measured in a different corner of this collection, from the geometry alone, with no dynamics anywhere near it. A lattice on a higher rung is one whose defining arrangement holds over a narrower range of angles, so a pattern on a higher rung should tolerate less wander. The band for 5/8 is 5.53° and for 8/13 is 2.12°; the tolerances measured at those two pairs are 4.12° and 2.89°.

Those are the same order and the same direction, and they are not the same number. The correspondence is offered as the shape of the explanation rather than as a fit — the third configuration ends on the same pair as the second and tolerates a third as much, which no band width explains, and the most likely reason is that a longer stem accumulates more chances to fail. A tolerance that is part geometry and part exposure is what one would expect and it is not resolved here.

What it means for anybody hoping to measure this #

There is a natural thing to want from a number like this: a threshold, measured on plants, that would say whether a given apex is noisy enough to matter. Two things stand in the way, and both are worth saying plainly.

The number depends on where it is measured. A tolerance quoted without the rung it was measured at is a tolerance quoted without half its content. On a real organ that means the counting band has to be stated, which is the same demand this collection has been making of published counts for three phases — and, as it happens, one whose usefulness a later essay in this phase revises.

And it is not a sufficient statistic for the noise. Two stems measured at 1.5° of scatter — comfortably inside the tolerance, visibly noisier than a clean lattice — do not have the same set of futures, because the scatter records how much the answer moved and not whether the question was disturbed. That distinction decides whether a pattern can change branch at all, and it is invisible in the scatter. It is the subject of the next essay in this ladder.

There is a third obstacle which is not about the model at all. A ruler measures the angle between two visible organs on a real stem, and that includes whatever has happened since they were placed: torsion, differential growth on one flank, the deformation of a node that has carried a leaf for a season. The model’s divergence is the azimuth at the moment of placement. The gap between them is not noise in the sense used here; it is a systematic transformation applied afterwards, and it would appear in the measurement in exactly the units the tolerance is stated in.

Does a head tolerate the same thing? #

Everything above is measured on a stem, and a stem is the geometry where the question is cleanest: the rise is the same the whole way up in a static lattice, so a pattern holds one pair and one divergence and the scatter is a scatter about a constant.

A disc is not like that, and the difference is not a detail. On a Vogel head the rise falls as one over the radius squared, so the pair changes with radius — 13/21, 21/34, 34/55 across one head — and the pattern is passing through transitions continuously rather than sitting on a rung. If the tolerance at a rung is set by the width of the band the rung occupies, then a head is a stem being carried through the ladder without ever settling, and the relevant tolerance would be whatever the local rung supplies at the radius being measured.

That is a prediction and it is not tested here, for a reason worth stating. Growing a disc with noise means running the disc form of the placement rule, and the disc model on this site drifts outward exponentially — which the kernel work in this same phase shows makes its neighbourhood behave completely differently from a stem’s. Carrying a tolerance measured on one geometry to the other would be exactly the kind of unexamined transfer this collection exists to catch.

What can be said is the direction. A count taken far out on a head sits on a high rung, whose angular band is narrow, so the divergence there should tolerate less scatter than a count taken near the middle — and the two counts are on the same plant, made by the same apex, with the same noise. Anybody measuring divergence scatter on a real capitulum would therefore need the radius as well as the angles, and would expect the answer to fall as they worked outward.

What survives #

Stated carefully, the result is this. A phyllotactic lattice fails abruptly rather than gradually, at a divergence scatter that depends on which rung it is holding, and the failure point is the same whichever way the noise arrives. At 8/13 it is around two degrees; at 5/8 twice that; carried deeper, less.

The part worth carrying forward is not the number but its shape: the pattern is a discrete arrangement, discrete arrangements do not degrade, and the amount of disorder one absorbs before it stops existing is set by how narrow the arrangement’s own margin was to begin with.

There is a small methodological result beside it, and it generalises past this subject. The first version of this measurement compared ensemble means and reported a difference of five degrees where the corrected statistic reports a third of one. Nothing was wrong with the runs; what was wrong was averaging over a population that had two modes in it, at exactly the amplitude where the proportions in those modes were changing fastest. A mean taken across a threshold measures the threshold’s position, not the quantity. That is easy to write and hard to see, because the number that comes out is smooth, plausible, and moves in the direction the story expects.