A cut of two organs

The intervention this collection keeps returning to is one sentence long. Grow a stem until its pattern has settled, take a single organ out of its recent history, and let the rule place what comes next against what is left. Everything it reports is a difference between two runs that share a history and differ in one organ, which is why it can say things a photograph of a plant cannot.

It has produced three results and refused a fourth. The removal is felt if and only if the organ removed is one of the most recent n, where n is the larger parastichy number — a spiral count obtained without counting anything. A cut in the middle of that band is never undone, and the stem settles instead into a repeating block of angles. And what fixes the band is two edges: three organs at the tip repair, three or four at the far edge repair, and whatever is left in between does not.

The refusal is the coarsest arrangement. On a stem carrying three spirals one way and five the other, the front is five organs deep, the two edges account for all five, and every single-organ cut heals. There is no middle. That was reported as a condition on the experiment — a plant patterned too coarsely will not go wrong at all — and it left the block prediction with no test at the one place where the answer would have been a small number — the 2/3 rung, which is coarse enough that its own settling is the thing in question.

This essay is about the experiment that reaches it, and about the thing that makes it worth doing beyond stubbornness: it is not the same experiment with more damage. It has a second parameter, and the second parameter is a control the first version could not have.

Two organs, and the offset between them #

The intervention is named by two numbers instead of one. The first is how many places back the nearer of the two removed organs sits, counted from the tip. The second is the gap — how many further places back the second one sits. A cut at an offset of two with a gap of one takes out the organs two and three places back; the same offset with a gap of five takes out two and seven.

Nothing else changes. The control and the cut run share their history to the last digit; the heights are prescribed by the rise, so a removal leaves a vacancy rather than shifting anything above it; only the azimuths of organs placed afterwards are free. The quantity reported is what it was: how far the next organ ends up from where it would have been.

The reason for wanting the coarse arrangement in particular is that the numbers are small enough to be complete. A front of five has five offsets in it; a table over both parameters is nine rows by nine columns, and a claim about all eighty-one cells is a claim with nowhere to hide.

The first thing to check is that the front has not moved #

The obvious worry about removing two organs is that it is a bigger intervention and therefore a different one. If the front widened with the size of the damage, the count it returns would be a property of the experiment rather than of the pattern, and the whole reason for doing it — a spiral count out of a yes-or-no answer — would go.

It has not moved. The shaded rows run out at five, which is the larger parastichy number, and every row past it is quiet in every column: the largest displacement anywhere below the boundary is 2.34°, against displacements of eight to one hundred and seventy-eight above it. Whatever the second organ is doing, it cannot make the pattern notice an organ it was not going to notice.

The same holds one arrangement finer, where the pair is 5/8. The table is twelve by twelve, the shaded rows end at eight, and the largest displacement past the boundary is 1.41°.

So the nearer of the two organs decides whether the removal is felt at all. That is worth stating as a rule because it is not obvious: a vacancy nine places back sits well inside the neighbourhood the rule actually sums over, and it changes the energy at every candidate azimuth. It just does not change which azimuth is least.

And the second thing is that the gap does something #

A control that is idle is not a control. If the gap changed nothing, the two-organ cut would be the one-organ cut with extra steps, and the table would be a set of identical columns.

Inside the front it does a great deal. With the nearer organ one place back, moving the second one from one place behind it to four swings the answer from 80.9° to −60.7° to 177.7° to 9.4°. There is no trend in that and none should be expected: the second vacancy is being moved through a profile the first one has already deformed, and the two are not independent contributions to be added.

Why the interior of the table is not additive #

It is tempting to read the two-organ cut as two one-organ cuts happening at once, and to expect the displacement to be something like the sum of the two separate answers. It is not, and the table says so at a glance: an offset of one with a gap of four gives 9.4°, while the single cuts at offsets one and five give 139.7° and 8.7°, whose sum is nowhere near it.

The reason is that the rule does not add displacements. It places the next organ at the least of a profile summed over the whole neighbourhood, and the position of a minimum is not a linear function of the terms that make it. Removing one organ lowers the profile near where that organ was; removing a second lowers it somewhere else; whether the least point moves a degree or half a turn depends on how close the two lowest candidate positions were to begin with, which is a property of the arrangement and not of either removal.

That is worth saying because it is the same non-additivity that makes the front sharp. If displacements added, a vacancy far behind the front would contribute a small but non-zero amount and the boundary would be a slope rather than a step. It is a step — 2.34° on one side and tens of degrees on the other — precisely because what is being reported is which of several candidate positions wins, and a loser that is lifted slightly is still a loser.

The limit that makes it a control #

Here is the part that earns the extra parameter. Once the second organ is behind the front, the two-organ cut has to reproduce the one-organ cut — and it does.

With the nearer organ one place back and the second one two or more places behind the front, the displacement averages −139.8° over the gaps available, with a spread of 2.6°; the single-organ cut at the same offset, computed by the earlier intervention rather than by this one, gives −139.7°. At an offset of two the two numbers are 78.6° and 79.0°; at three they are −42.6° and −42.9°.

That is a limit rather than an agreement of convenience. It says the two-organ experiment contains the one-organ experiment, so a reading that comes out of the larger table can be checked against a number that was measured before the larger table existed. An experiment whose extra knob has a known setting where it must reproduce the simpler experiment is a much safer instrument than one that does not.

The convergence is not instant, and the residual is worth naming. At a gap that puts the second organ exactly one place behind the front, the answer is still about eight degrees off — 131° where the single cut gives 140°. Two places behind and it is inside the resolution of the azimuth grid. That is why the comparison is defined with a margin rather than at the boundary.

What it buys: the coarsest arrangement stops being immune #

With the instrument checked, the question it was built for. Does a stem carrying 3/5 recover from a cut of two?

Not always. Four of the sixty-four pairs of offsets tried never return to the divergence they were cut from: an offset of two with gaps of one and two, and an offset of three with gaps of one and three. The other sixty do, within a few dozen organs.

Which four they are is the part that connects this table to the account it was built to test. Written as pairs of organs rather than as offset-and-gap, the four are {2, 3}, {2, 4}, {3, 4} and {3, 6}. The front is five organs deep; the organ one place back is the tip, which repairs by taking the vacancy; the organ five places back is the far edge, whose vacancy the arrangement closes over. What is left in between is 2, 3 and 4 — a middle of three organs on a front that was said to have no middle for a single cut.

There are exactly three pairs to be made out of those three offsets, and all three of them fail. That is not a rate and it is not a selection: it is every available way of putting two vacancies into the interior of this front, and every one of them defeats the repair. The edge account, which was derived from single-organ cuts at two finer arrangements, therefore predicts the coarse arrangement’s two-organ failures exactly — not their number, which would be arithmetic, but which cells they are.

The fourth cell is the informative exception. {3, 6} puts one vacancy in the middle and one past the front, at an offset whose removal on its own moves the next organ by less than a degree and a half. So an organ the pattern cannot feel can still stop the pattern healing. That is the clearest case on this site of the displacement and the fate being two different measurements: the first is decided by the nearer organ alone, and the second is not.

Four in sixty-four is a small number and it is meant to be. One arrangement finer the same table has twenty-two unrepaired cells in thirty-six, and the contrast is the useful part: the same intervention that wrecks two thirds of a 5/8 stem’s offsets wrecks one sixteenth of a 3/5 stem’s. Coarse arrangements are not immune, but they are robust, and the robustness is quantitative rather than categorical.

Why the coarse arrangement resists #

The reason is the one the single-organ work already found, applied twice. The front is repaired from its two ends: a vacancy within about three organs of the tip is filled by the organs that come immediately after it, and a vacancy within three or four of the far edge is outside the region the next few placements are sensitive to. A front of five is edges all the way across.

Two vacancies can defeat that only if they are placed so that the repair of one interferes with the repair of the other, which is why the four failing cells are the ones with small gaps at middling offsets, and why an offset of one never fails whatever the second organ does. The organ one place back is the tip, and the tip is repaired by definition — the next organ simply takes the vacancy.

What it costs an experiment #

The single-cut version of this proposal had four conditions attached, and one of them was that the plant must not be too coarsely patterned. That condition is now about the intervention rather than about the plant, and it is cheaper to lift than it looked: one more organ.

What the two-organ version adds is a second thing to record. An experimenter who removes two primordia has to say which two, because the answer depends on the gap between them wherever both are inside the front — and the values inside the front are the ones with the large displacements, so a study that reported “two primordia removed” without the offsets would be reporting a number drawn from a distribution spanning two hundred and forty degrees.

Outside the front the gap does not have to be recorded, because the answer does not depend on it. That is the same statement as the limit above, read as advice.

What this does not say #

It does not say two organs are the minimum. Nothing here tried removing one organ and displacing another, or removing one and adding one, and there is no reason to think the two-organ cut is the smallest intervention that reaches a coarse arrangement. It is the smallest one that keeps the whole experiment inside the same description — a set of organs present or absent — which is worth something for an experiment somebody would have to actually perform.

It does not say the boundary is exactly the larger parastichy number at every gap. It says the run of felt offsets ends there, as it does when one organ is removed in the middle of a rung, and that the largest displacement past it is under two and a half degrees at the coarse arrangement and under one and a half at the finer one. Those are small against the displacements inside the front and they are not zero.

It does not say four in sixty-four is a rate. The four failing cells are four particular pairs of offsets, found by trying all of them, and they are what they are for reasons about the repair of the front rather than by chance. A different cut point on the same stem gives the same four cells to a tenth of a degree, which is checked.

And it does not say anything directly about a plant. These are stems grown by a rule that places each organ where the repulsion from the ones already there is least. What the measurement licenses is a prediction about what an ablation experiment would find if that rule is what a meristem does, which is the only reason to run one.

The check that would refuse it #

Three claims here can each fail on its own, and each is asserted while the figures are drawn.

The first is that the run of felt offsets ends at the larger parastichy number at both arrangements, at every gap, with nothing past it felt. A single filled cell below the boundary in any column would stop the collection being built.

The second is the limit: with the second organ clear of the front, the displacement has to agree with the single-organ intervention’s own answer to within eight degrees, at three offsets. That check compares two functions written for different purposes, and it is the one most likely to catch an indexing mistake — removing the further organ first is what keeps the nearer one’s offset meaning what it says, and doing it the other way round would silently shift every row of the table by one.

The third is that the gap matters inside the front — by far more than the azimuth grid could account for, which is how it is stated, because the number of gaps that keep the second organ inside the front falls as the offset grows and a fixed threshold in degrees would be a threshold on the offset as well. It is the assertion that would fail if the second cut were somehow being ignored, and it is the one that makes the limit above worth having: a parameter that changes nothing anywhere would satisfy the limit trivially.