The stem that changed hands

Every arrangement of spirals has a hand. The organs of a stem step round the axis by a fixed angle, and that step is either clockwise or anticlockwise; a divergence of 139.6875° and a divergence of 220.3125° are the same lattice seen in a mirror. Every count is the same, every distance between organs is the same, every spiral is as tight — and a photograph of one cannot be superimposed on a photograph of the other.

Plants have hands too, and they are famously not fixed. A sunflower head is roughly as likely to run one way as the other, and both heads on one plant may differ. That is usually explained as a symmetry breaking that happens once, very early, and is then locked in. This essay is about a way of unlocking it after the fact — and the way is not subtle. Take two organs out of a settled stem and the rule, left to itself, sometimes rebuilds the whole pattern the other way round.

What was expected instead #

The intervention has a well-established answer at finer arrangements. A stem cut in the middle of its front never returns to the divergence it had; it settles into an exactly repeating block of angles — five of them at the 5/8 arrangement, eight at 8/13 — precessing slowly around the axis, at an effective divergence far from anything the undisturbed rule produces.

Those orbits are unmistakable. Their motifs span a hundred and forty-eight and two hundred and twenty-six degrees; nobody looking at the sequence could think the stem had settled on a single angle. The prediction, extended downward, was that the coarsest arrangement would give a block of three.

What the coarsest arrangement does #

It gives no block at all.

The undisturbed stem settles at 139.6875°. The cut stem is thrown, wanders for a few dozen organs, and settles at 220.3125° — which is 360° − 139.6875° to the last digit that the arithmetic carries. It holds it for the remaining three hundred organs, with a scatter of a third of a degree at the azimuth grid used and a sixteenth of a degree at a grid three times finer.

Three of the four cells that fail to repair do this, at two different offsets and two different gaps. The agreement between them is exact rather than close: the same four decimal places, from cuts at different places in the stem’s history.

The counts come back with it #

A divergence that happens to sum to 360° with another one would be a curiosity if nothing else agreed. Everything else agrees.

A counter shown the wrecked stem’s positions, with no access to the angle it was built from, returns 3 and 5 — the pair the stem was cut from. The three shortest hops are five, three and eight organs, in that order, exactly as before. The spacing between neighbours is the same, because it is a function of the same lattice.

So the stem has not been damaged into some third thing. It has been moved from one member of a pair of equivalent lattices to the other, and there was never a sense in which it was on the first one rather than the second except that it had been started there.

The arithmetic of why a mirror is the same lattice #

The identity is worth writing out, because it is what makes the result a statement about handedness rather than about a coincidence between two numbers.

An organ i on a cylindrical lattice sits at azimuth iδ and at height ih, with the azimuth read around a circle of circumference one. The distance from organ i to organ i + k depends on δ only through the wrapped quantity kδ modulo a whole turn, and it depends on it only through the magnitude of that wrap, since a step of +x and a step of −x are the same distance either way round the cylinder. Replace δ by 360° − δ and every wrap changes sign and keeps its size. So every hop length is unchanged, for every k, exactly.

Everything a counter uses is a hop length. The parastichy numbers are the two offsets with the shortest hops; the incoming number is the shortest hop longer than those; the whole ladder of transitions is solved from the same table. None of them can distinguish a lattice from its mirror, and none of them should: the two point sets are congruent, and a rule that placed organs by repulsion alone cannot prefer either.

What does distinguish them is the sign of the step, which is not a distance and is not in the table. It is in the sequence of azimuths and nowhere else — which is to say, it is in the order the organs were made in. That is the sense in which handedness is a fact about growth rather than about form, and it is why a measurement made on a photograph reports it and a measurement made on a neighbour graph cannot.

What the transient looks like #

The stem does not jump. It is thrown by the cut — the organ placed immediately afterwards is sixty-two degrees from where it would have been — and then spends somewhere between thirty and sixty organs producing divergences that belong to neither lattice, before the sequence tightens onto the mirrored value and stays there.

That transient is not noise and it is not a slow drift. Watched closely it is the rule doing what it always does: each organ placed at the least of a profile, with the profile still deformed by the two vacancies and by every organ placed since. The vacancies pass out of the neighbourhood after a few dozen placements, and from then on the rule is being asked its ordinary question about an arrangement that is already nearly the mirror.

The length of the transient is the reason the recovery test needs a hold. A test that asked only whether the last few angles are close to a settled value would call a wandering stem settled somewhere in the middle of this stretch. The test used here requires every divergence from some organ onward to stay within a degree and a half of the value being tested, with at least sixty of them to check, which is why a wrecked stem is reported as wrecked rather than as slowly recovering.

A single organ cannot do it #

This is the part that makes the result an experiment rather than an observation about symmetry. At this arrangement, every single-organ removal repairs itself — all five offsets of the front, at three cut points four dozen organs apart, back on 139.6875° within thirty-nine organs.

Two organs can do it and one cannot, at four of sixty-four pairs of offsets. That gives the pair of lattices a quantitative separation: they are further apart than one removal throws a stem and closer than two removals at the right offsets. It is a statement about the size of a basin, made with an intervention that has a size.

The fourth cell is not a mirror, and it is worth keeping #

Of the four cells that fail, three settle on the mirror and one does not. A cut at offsets two and four settles at 258.73°, whose three shortest hops are four, three and seven — a 3/4 lattice, which is neither the original nor its mirror.

It is reported rather than smoothed away because a result that held in four cells of four would be a weaker claim than one that holds in three: the fourth cell is what says the mirror is a destination the rule can reach rather than the only thing available once a coarse stem is disturbed. There are at least two places to go, and most of the failures go to the same one.

The 3/4 lattice is also interesting on its own account. Four is not a Fibonacci number and 3/4 is not a rung of the ladder a golden-angle stem walks; it is a lattice the rule can sit on at this rise without being on the ladder at all. Its divergence, 258.73°, has a mirror at 101.27°, which nothing here reaches — so the cell that does not go to a mirror does not go to a mirror in both senses. What it shares with the other three is that it is a lattice: a single divergence held for three hundred organs, not a block of angles, and a counter shown its positions returns a pair.

Taken together the four cells say the coarse arrangement has at least three resting places within reach of a two-organ disturbance, and that all of them are lattices. That is a different situation from the finer arrangements, where every failure lands on an orbit, and it is the shape of the difference this essay is about.

What it would mean on a plant #

Handedness is one of the four things this collection’s survey specification asks a real census to record, and it is the one nobody reports. Spiral counts are published constantly; the counting radius, the element count, the rotational symmetry and the hand are the fields that turn a count into a measurement, and they are usually treated as context.

The prediction here gives the hand something to do. On a coarsely patterned shoot — three and five, which is common on young stems and on many rosettes — removing two nearby primordia should, at a minority of offsets, produce a shoot whose subsequent phyllotaxis runs the other way, with the same counts as before. That is a large, obvious, photographable outcome from a small intervention, and it is one that no account in which the hand is fixed early and locked in can produce.

It is also a test with an unusually good null. If the hand were fixed by something upstream of the placement rule, an ablation could not change it at any offset; if the hand is nothing but which of two equivalent lattices the neighbourhood currently sits on, it should be changeable by a disturbance of about the size measured here. Those two predictions differ in kind rather than in degree.

Why the coarse arrangement and not the fine ones #

The finer arrangements do not do this, and the reason is visible in what they do instead. At 5/8, twenty-two of thirty-six pairs of offsets fail to repair, and every one of them lands on a repeating block rather than on a lattice. The mirrored divergence, 223.22°, is not among the values reached.

The natural reading is about how much room there is. A coarse lattice has five organs in its front and two vacancies in it are a large fraction of the neighbourhood; the rule is thrown a long way, and the two nearest places to land are the two hands of the same lattice. A fine lattice has thirteen organs in its front, two vacancies are a small fraction of it, and what the rule finds nearby is a rearrangement rather than a reflection.

That is a reading rather than a measurement, and the measurement that would test it is available: the same table at 5/8 with three organs removed, which is a larger disturbance on a bigger neighbourhood. Nothing here has run it.

What this does not say #

It does not say the rule prefers one hand. It has no preference: the rule is exactly symmetric under reflection, and a stem seeded at 220.3125° stays there as firmly as one seeded at 139.6875°. What the measurement shows is that the barrier between them is finite and has a size that an intervention can be compared with.

It does not say a plant’s handedness is unfixed. It says that in a model where the hand is nothing but a property of the current arrangement, a two-organ ablation flips it at some offsets. Whether a meristem carries anything else that remembers the hand is exactly what the experiment would find out, and the model cannot answer it.

It does not say the mirror is reached from every failure. Three of four cells at one arrangement, and none at all one arrangement finer. This is a property of coarse patterns and it is stated as one.

And it does not say the sequence is noiseless. The wrecked stem’s divergence scatters by a third of a degree at the grid the table is computed on. That scatter is the grid — at three times the resolution it falls to a sixteenth of a degree while the mean does not move — which is the same discretisation that produced a spurious repeating block in an earlier reading of this run, and which has its own essay.

The check that would refuse it #

The claim is three things at once, and all three are asserted while the figures are drawn.

The settled divergence of the wrecked stem has to be within a twentieth of a degree of 360° minus the divergence it was cut from. That tolerance is a fifth of the azimuth grid’s own step, and the three cells that satisfy it satisfy it to 0.0000°, so nothing is near the threshold. A run that landed near the mirror rather than on it would be reported as near.

A counter shown the wrecked stem’s positions has to return the pair the stem was cut from. This is the strong half: two angles summing to 360° is arithmetic, and a counter agreeing about both parastichy numbers is a statement about the point set.

And not every failing cell may be the mirror. If all four went to the same place the figure would be consistent with the rule having exactly one destination after a coarse stem is disturbed, which is a different and less interesting claim; the check requires at least one cell that does something else, and the 3/4 lattice at 258.73° is what supplies it.