A ring cannot make a spiral
A reaction–diffusion ring selects a spacing, reliably, and the spacing is the one its dispersion relation predicts. That is the previous essay, and it is a genuine result.
It is also not phyllotaxis. Spiral phyllotaxis is not a statement about how far apart elements are; it is a statement about the angle between successive ones, and successive presupposes an order.
The natural objection to a ring model is therefore that its peaks all appear at once, so there is no order and no angle. That objection is worth making precisely, because in the form usually stated it is wrong.
The peaks do not all appear at once
Run the integration and record, for each peak in the final pattern, the moment its value first crosses a threshold set from the finished field.
On a ring of circumference 1.4 the crossings happen between and , against a total formation time of 384. So the peaks emerge over about a fifth of the time the pattern takes to form, which is a real window and not an instant. There is a first peak, a second, a last.
The picture in one’s head — the whole ring buckling into eight bumps simultaneously — is not what the equations do. The instability grows out of the starting disorder, and the places where the disorder happened to be most favourable get going first.
So the naive objection fails. A ring does produce an order.
But it is not the same order twice
Change the starting disorder and run again. Same equations, same ring, same parameters, different pseudo-random seed.
The angle between the first two peaks to cross the threshold comes out at 177°, then 47°, then 109°, then 151°, then 47°, then 133°.
A spread of 129° across six runs, from a mechanism that would need to produce the same number every time for the word “divergence angle” to mean anything.
That is the measurement, and it is what settles the question. The peaks do arrive in an order, and the order is a record of the noise. Nothing in the chemistry favours any angular relationship between the first two, because nothing in the chemistry distinguishes any position on the ring from any other — the equations are rotationally symmetric and the initial condition is the only thing that breaks the symmetry.
How the emergence times are measured
The measurement has a trap in it, and avoiding the trap is most of the work.
The obvious approach is to pick a threshold, run the integration, and record when each cell first exceeds it. The trouble is choosing the threshold. Too low and every cell crosses during the initial transient, in an order set by the random starting values rather than by the instability. Too high and only the tallest peaks ever cross, so the sample is selected on the outcome. Either way the number that comes out is a function of the choice, and the choice is made by whoever wants a particular answer.
So the threshold is set from the finished pattern. The integration is run once to completion, the peaks in the final field are located, and the threshold is placed halfway from the uniform steady state to the highest final value. Then the run is repeated from the same seed — deterministically, from the same pseudo-random sequence — and the crossing time of each eventual peak is recorded.
That ordering matters. A threshold chosen in advance is a parameter; a threshold read off the answer and then applied to a replay is a measurement of the same object it was derived from. It is a slightly awkward two-pass arrangement and it is the reason the emergence figure costs two integrations per seed rather than one.
The result does not depend on where in the upper half the threshold sits. Moving it from a half to a third or two-thirds of the way up changes the individual times and leaves the spread across seeds essentially unchanged, which is the quantity the claim rests on.
Why this is the right test
There is a weaker test that would have been easier and would have proved much less: run once, look at the angle, observe that it is not 137.5°.
That would show nothing. One run gives one number; any number is not 137.5° with probability one; and a mechanism that gave a reproducible 109° would be enormously interesting and would fail such a test exactly as a mechanism that gave nothing.
The informative quantity is the spread across starting conditions, and it is the one the check is written on: the build requires the six angles to span more than 30°. A mechanism that produced a repeatable angle — any angle — would fail that check, and would deserve to, because it would have discovered something.
This is the same discipline the site applies to its geometric models. The dynamical model’s claim that the golden angle is an attractor is only worth anything because it is checked from two unrelated starting conditions and reaches the same place from both. Here the same test is run and the answer is the opposite one, and the test is what makes the two comparable.
What a ring makes is a whorl
The botanical name for the arrangement a ring produces is a whorl: some number of elements at one level, spaced around, with no ordering among them.
That is a real pattern and a common one. Many plants put leaves on in twos, threes or fours at a node, and for them the ring account is not missing anything — the elements of a whorl genuinely do arise together, and the question “at what angle from the previous one” genuinely has no answer within a whorl.
So the limitation is a limitation only relative to the spiral case. A ring model is an adequate account of whorled phyllotaxis and an inadequate account of spiral phyllotaxis, and the difference between the two patterns is exactly the thing the ring lacks.
The lattice essays give that difference an arithmetic form. On a cylinder, a whorled pattern is one whose parastichy counts share a factor — the elements of each turn are level with one another — and a spiral one is one whose counts are coprime. So “whorl or spiral” is not a qualitative distinction; it is a divisibility question about two integers a counter can extract from the coordinates.
Why the count varies by one and the angle by everything
The two quantities measured across the six runs behave so differently that it is worth being explicit about where each one’s variation comes from, since “one is noisy and one is not” is not an explanation.
The count is set by the dispersion relation and a fitting constraint. The chemistry picks a wavelength; a whole number of peaks has to fit around the ring; and the count is that wavelength divided into the circumference, rounded. On a ring of 1.4 the ratio is about 7.6, so the count is 7 or 8 depending on which way a particular run rounds — and which way it rounds is exactly the thing the starting disorder can influence, since a slightly favourable arrangement can support one extra peak or one fewer.
So the variation in the count is a rounding, and its size is one. It cannot be larger, because a run that produced 5 or 11 peaks would be at a wavelength the chemistry does not select.
The angle has no such constraint. Once the count is fixed, the peaks are equally spaced by the wavelength, but where the set of them sits on the ring is free — the equations are rotationally symmetric, so every rotation of a solution is a solution. And which member of the set gets going first is likewise free.
That is the difference. One quantity is pinned by a physical scale with a discreteness on top; the other is pinned by nothing at all.
What has to be added
If a stationary ring cannot make a spiral, what is missing?
The answer, in one word, is motion. Specifically: the elements have to leave. A spiral arrangement arises when the region where new elements can form is continuously vacated by the old ones, so that each new element is placed in the gap left by the ones before it, and the gap moves.
That is precisely the Douady–Couder rule the emergence field is built on: existing elements drift outward as the meristem grows, and the next appears where the repulsion from what is left is least. The growth parameter in that model — how fast elements move away — is the same quantity as the rise on a cylinder and the plastochron ratio in the botanical literature.
Which gives a clean statement of the division of labour. The chemistry supplies a spacing. The motion supplies an order. Neither alone produces a spiral, and the parameter that decides whether the result is a whorl or a spiral is the one that says how fast elements leave relative to how fast they form.
The rate, and what it decides
That last point deserves to be made as a number rather than a sentence.
If elements form much faster than the region vacates, several appear before anything has moved, and they appear together: a whorl. If they form much slower, each has the region to itself and lands opposite the last: alternate leaves, a divergence of 180°. In between, each new element sees a few predecessors at different distances, and the angle it settles on is the interesting one.
On the cylinder that ratio is the rise, and the ladder is what happens as it falls. In the dynamical model it is the growth parameter , and the bifurcation diagram is what happens as it falls. Those two figures are pictures of the same quantity, arrived at by different arithmetic, and putting them side by side is the subject of a later essay.
The ring, by construction, has that ratio at infinity: nothing ever leaves. So it sits at the extreme end of the parameter that decides the question, which is why it gives the answer it gives.
The peak count is the stable thing
It is worth putting the two quantities side by side, because the contrast is the cleanest statement of what a ring mechanism is good for.
Across the six runs the peak count is 7 or 8 — one unit of variation, and that unit is the discreteness of fitting a whole number of peaks around a fixed circumference rather than any instability in the mechanism. The angle between the first two peaks varies by 129°, which is most of the range available to it.
Same equations, same runs, same measurements. One quantity is reproducible and the other is not, and which is which is not a matter of how carefully anything was measured.
That division is exactly the division of labour set out above. A ring determines the spacing, which fixes the count; it determines nothing about arrangement, and the emergence order is the first thing about arrangement one can ask for.
It also suggests what a claim about a mechanism should look like. “This model produces phyllotaxis” is not a checkable statement. “This model reproduces the peak count to within one across starting conditions, and the divergence angle not at all” is, and the second is what the arithmetic supports.
What a botanist would say about all this
The distinction being drawn here is not new to botany, and it has vocabulary.
An arrangement in which several elements arise together at one level is whorled; one in which they arise singly is spiral or alternate. Which of the two a species has is a stable taxonomic character, and there are lineages that switch between them, sometimes within a single plant as it matures.
That last observation is the interesting one for a model, because it means whorled and spiral are not deeply different states requiring different machinery. Something continuous changes and the arrangement flips. The models here agree: the parameter is the rate at which elements leave the competent region relative to the rate at which they form, and it is the rise on a cylinder and the growth parameter in the dynamical model.
So the ring’s failure to make a spiral is not a failure to model plants. It is a correct model of one end of a continuum that real plants occupy the whole of, and the thing it is missing is the parameter that moves along it.
What this does not show
Two readings to head off, since a negative result invites both.
It does not show that reaction–diffusion is the wrong account of phyllotaxis. It shows that a stationary one-dimensional reaction–diffusion system is, and that is a statement about the reduction rather than about the chemistry. A reaction–diffusion system on a growing two-dimensional apex, with the pattern advected outwards as the apex expands, is a different object and can produce spirals — Meinhardt built such models and they work.
It does not show that the noise is a nuisance to be removed. The scattered angles are not measurement error; they are the correct answer to the question asked. A system whose outcome is set by its initial condition is reporting that the outcome is not determined by the mechanism, and the useful response is to find what else the real system has, not to average over seeds until the scatter goes away.
The second reading is the more tempting one and it is worth resisting explicitly, because averaging six scattered angles gives 111°, which is a number, and a number in a table looks like a result.
The same test, run on something that passes it
A negative result is only as good as the positive control beside it, and there is one on the site.
The dynamical model is asked exactly this question in its own gate: run it from a seed angle of 0.31 of a turn and again from 0.77, two starting conditions with nothing in common, and require that the settled angles agree to within 3°. They do, at a growth parameter of 0.40 — both land within three degrees of 137.5°.
So the two models are subjected to the same test with the same tolerance and give opposite answers. That is what makes the ring’s failure informative rather than merely a report that a simulation was noisy: the test is known to be passable, by machinery in the same repository, run the same way.
The difference between the two is not sophistication. The dynamical model is a cruder object in most respects — a sampled boundary, an inverse-cube repulsion, a finite window of remembered elements. What it has that the ring does not is that its elements leave, and that is the entire difference.
The measurement in one line
Six starting conditions, one set of equations, one ring. The angle between the first two peaks to appear: 177°, 47°, 109°, 151°, 47°, 133°. Spread 129°.
The peak count over those same six runs is 7 or 8 — stable to within one. So the mechanism is reproducible in exactly the quantity the dispersion relation predicts and irreproducible in the quantity phyllotaxis is about, which is as clean a division as this subject offers.
A ring selects how far apart things are. It does not select where they go.