The pattern the cut leaves behind

The previous essay left five stems that never return to their lattice, and described them by their statistics: a mean divergence near 185° and a spread of fifty to eighty degrees. Those are the numbers a disordered sequence would give, and reporting them and stopping would have been a mistake, because the sequences are not disordered at all — a mean and a spread are the two statistics that cannot see it.

They repeat, exactly, and they go on repeating for as long as the run is continued.

Eight angles, exactly #

Take the stem whose organ eight places back was removed. Three hundred organs later its divergence sequence is

46.6° · 96.1° · 136.9° · 272.6° · 137.6° · 270.9° · 229.9° · 272.1°

and then those eight again, and again, to the resolution of the azimuth grid, for as long as the run is continued. Not approximately: the check looks for an exact repeat, with a tolerance of half a degree against a grid whose step is a quarter of one, and refuses to report a period that only nearly repeats.

The cut four places back gives a different cycle of the same length —

138.5° · 136.6° · 137.8° · 138.3° · 138.1° · 270.7° · 231.3° · 271.2°

— and the cut six places back gives a cycle of four:

230.4° · 271.9° · 138.3° · 270.7°

Five of the eight angles in the first cycle and five in the second are within a degree or two of 137.8° or of 270.9°, which is 137.8° doubled and wrapped. The cycles are not arbitrary sequences; they are made mostly of the settled divergence and of two of it — so a stem that reads as disordered by any summary is built almost entirely out of the one angle it was supposed to have lost, and its double.

Every one of them is the same arrangement #

Five stems never healed, with cycles of eight, eight, eight, eight and four angles and five different-looking motifs. They are one object.

Add each cycle up. The eight angles of the stem cut eight places back sum to 1462.7°, which is four full turns and 22.7° over. The cut four places back sums to 1462.5°, and the four-angle cycle of the cut six places back sums to 911.25°, which is two and a half turns — so over two of its cycles, eight organs again, it also advances 22.5°.

Measured directly off the finished stems rather than off the motifs, the azimuth advance from an organ to the one eight places later is 22.54°, 22.73°, 22.97°, 22.73° and 22.54° on the five arrangements. Every stem that fails to heal ends up as a block of eight organs that rotates by about 22.7° per block, and the one with a four-angle cycle is that same structure whose motif happens to repeat after half a block.

The effective divergence follows: 1462.7° over eight organs is 182.84° per organ. Which is to say each organ is placed nearly opposite the one before it, and the pair of ranks so formed precesses slowly — a twenty-second of a turn every eight organs.

The effective divergence is the control’s, plus a turn over eight #

Subtract, and the number that comes out is not an arbitrary one.

The control settles at 137.8°. The wrecked stems run at 182.84°. The difference is 45.04°, and 360 divided by eight is 45.

So the wrecked stem has not drifted to some new angle. It is running at the arrangement it was cut from plus exactly one whole turn per eight organs, to within four hundredths of a degree — which is a fifth of the azimuth grid’s own step and is as close as this measurement can come to exact.

Eight is the block’s length, and it is also the lag whose hop the wrecked stem holds unchanged. So the relation is one turn per period of the family that stayed rigid, which is the law a later thread derives from the rigidity itself — arrived at here, two rounds earlier, from adding up a motif.

That also disposes of the precession. Twenty-two point seven degrees per block is eight times the 2.84° by which 182.84 exceeds a half turn, so the slow rotation of the two ranks is not a separate phenomenon: it is the same 45° slip seen against a two-ranked arrangement rather than against the golden one.

And the conjugates all move the same way #

The counter’s readings are worth one more sentence than they get, because the four conjugate numbers are not scattered.

They are 14, 16, 19 and 26 against the control’s 13. Every one of them is larger. The family that stayed rigid keeps its number and the family that was rebuilt comes back coarser, on all five stems and in only one direction.

One of them is worth flagging separately. 8/16 is not a pair a settled arrangement carries, because a pair’s two numbers are coprime — that is what makes the two families between them touch every organ once. A counter returning a number and its double is reporting a rotational symmetry rather than a lattice, which is exactly what a stem placing organs nearly opposite one another has, and it appears on two of the five.

What a counter makes of it #

A sequence of angles is one description. This site’s discipline is that a pattern’s claims are settled by machinery shown the positions and told nothing else, so the tops of the wrecked stems were handed to the counter.

It reads 8/16 off two of the five, and 8/14, 8/19 and 8/26 off the others. The 8 is unanimous and it is the block. The second number is what the counter makes of the precession: a block advancing 22.7° goes round in 360/22.7 = 15.9 blocks, so 16 is the arrangement’s own answer and 14, 19 and 26 are a counter struggling with a lattice whose two hop lengths are wildly unequal. That is the counter behaving as it should — this collection has spent a long time establishing that a counter which never refuses and never wobbles is a counter that is not measuring anything — and it is a reason to lean on the direct measurement of the block advance rather than on the reading.

A pair of the form (m, 2m) is the signature of a two-jugate lattice: a pattern built of two identical spirals half a turn apart rather than one. Two of these read that way, and the arrangement they are reading is genuinely two-ranked — 182.8° between consecutive organs is a hair off exactly opposite.

Why this is a statement about the rule and not about a run #

The site’s central figure is a bifurcation diagram: sweep the one parameter of the placement rule and watch which divergence it settles on. It gives a broad golden branch, a transition, and a two-whorl regime at exactly half a turn.

That diagram is built by running the model to convergence at each parameter from an arbitrary start. It is therefore a picture of which attractor a run happens to find, and by construction it cannot show a second one at the same parameter, because nothing in the procedure ever tries to reach one.

The ablation does try. Here the growth parameter never changes: the rise is 0.005 from the first organ to the last, the exponent is inverse cube throughout, and the only difference between the stem that settles at 137.84° and the stem that runs an eight-cycle is that one organ was deleted from a shared history. So the rule has at least two attractors at this parameter, and the diagram has been showing one of them since it was first drawn.

The second one is not an exotic arrangement, which is the part worth pausing on. A divergence of 182.8° is two-ranked phyllotaxis with a slow precession, and two-ranked — distichous — arrangements are, after spiral ones, the commonest thing a plant does. The bifurcation diagram already contains a regime at half a turn; it puts it at a different value of the growth parameter, reached by making the meristem large relative to the organs. What the ablation shows is that the half-turn arrangement is available at the golden parameter too, as a second stable state of the same rule, and that the way to it is not through the parameter at all.

That is a different kind of claim about the model from any this site has made before. Everything up to here has been of the form the rule produces X at parameter G, tested by sweeping G. This is of the form the rule produces X or Y at one G, and which one depends on the history — bistability rather than a branch — and bistability is not visible to any procedure that reports where a run from an arbitrary start ends up.

Why eight, and why 22.7° #

Neither number is put in by hand, and it is worth asking where they come from, even though the honest answer is that only half of it is understood.

The eight is the smaller parastichy number of the lattice that was cut. That is suggestive and it is not yet an explanation: the cut stems’ own smaller parastichy number is also eight, so the block could be inherited from the pattern that was destroyed or could be a property of the arrangement that replaced it, and nothing here separates those. The test is obvious and is not run here — ablate a stem on the 5/8 rung, where the smaller number is five, and see whether the block is five or eight.

The 22.7° is the residue. Eight organs at a divergence near 180° advance nearly four full turns, and what is left over after those four turns is the precession. Nothing chooses it directly; it is what the rule’s own minimisation leaves when it settles into the near-opposite arrangement, and a change of a hundredth of a degree in the effective divergence would change it by a tenth. That it comes out within half a degree of the same value on five independent cuts is the interesting part, and it is why the five arrangements can be called one object rather than five.

How it was found, which was by accident #

The orbit was not looked for. It turned up while checking whether the azimuth grid decides the ablation’s answer.

The rule takes its minimum over a finite number of sampled azimuths, and every flat run in this collection uses 384 of them. At 384, with no noise at all, a stem at this rise does not settle on the golden angle. It settles into a cycle, at a mean divergence of 185.8° — the same neighbourhood the cut stems end up in — before anything has been removed.

Every flat run on this site is a noisy one, and the noise is what keeps them off the cycle: a jostle of a quarter of a degree at 384 azimuths gives a clean 8/13 at 137.8°, which is why nobody had seen this. The quantisation is acting as a disturbance of about a quarter of a degree in its own right, and evidently a disturbance of the wrong colour.

That is worth recording as a hazard rather than as a result. A deterministic model on a coarse grid can find a periodic orbit that is a property of the grid; the reason this essay’s orbits are not that is that they survive refinement. At 1,536 azimuths and at 4,608 the same cuts give the same verdicts, and the cycles are still exact.

Noise does not undo it either #

The other way an orbit can be an artefact is by being a knife edge — a state the arithmetic balances on and that any real system would fall off. So the cut was repeated with a jostle in the run.

At 0.1° the wrecked stem’s tail is 186° ± 83°, which is the noiseless run to within the reporting precision. At 0.4°, close to the amplitude at which the lattice fails on its own, it is 149° ± 92° — still nowhere near 137.8°, with the disturbance now doing most of the spreading.

The orbit has a basin, and the basin is wide enough to hold a stem that is being jostled hard enough to be close to losing its pattern altogether.

What this is worth, and what it is not #

It is not a claim about plants. No plant has been ablated here, and the model’s response to a deletion is exactly the thing an experiment would be testing. If a real apex reorganises to a two-jugate arrangement after a mid-front ablation, that is a striking confirmation; if it heals every time, that is evidence about how a real apex differs from this rule, and both are more interesting than what was available before the intervention was computed.

It is a claim about the model, and a fairly strong one. The rule was introduced on this site as a mechanism whose attractor is the golden angle, and that is right and has been tested from a dozen directions. What is added here is that the golden angle is not its only attractor at a fixed parameter, that the second one is periodic rather than fixed, and that a single deletion is enough to move between them. None of the site’s earlier machinery could have found that, because all of it varies a parameter and reports where a run settles.

It changes what this collection’s earlier negative results mean, slightly. The essays on noise and on the cut-off both concluded that noise cannot move a grown pattern off its branch except by destroying it: placement noise never changes which minimum is chosen, a jostle changes it once or twice in a thousand placements, and field noise changes it only once the lattice is already coming apart. Those findings stand — they are about noise. What they were sometimes taken to mean, here included, is that the pattern’s state is effectively unique at fixed parameter, and that reading is now wrong. The state is not unique; noise is simply not the way to the other one.

And it makes the previous essay’s map more interesting than a table. Which removals a stem can undo is now a question about basins: five of the thirteen offsets inside the front put the pattern over a boundary it cannot come back across, and eight do not. Work that wanted to draw that boundary would be mapping a basin in a space nobody has yet written down.

There is also a specific thing a plant could be looked at for, which is worth stating because so much of this collection’s output is a specification for an experiment nobody has run. If a mid-front ablation on a real apex sends the shoot two-ranked, the shoot does not need to be measured to a tenth of a degree or counted at all: two ranks of leaves is a thing that can be seen from across a greenhouse. It is the most visible prediction this site has produced, and it comes from the least measurable-looking of its results.

Not always an orbit #

At the arrangements measured here a wrecked stem settles into a repeating block of angles. One arrangement coarser it does something else: it settles onto a single divergence, and that divergence is 360° minus the one it was cut from — the same lattice, wound the other way.

The distinction is worth making carefully, because a routine looking for exact repeats reports a period for both. What separates them is the span of the motif against the resolution of the azimuth grid: hundreds of grid steps for a real orbit, three or four for a constant the grid cannot write down.