What a cut costs a whorl

A bijugate pattern puts two organs at every node instead of one, half a turn apart, and steps between nodes by a divergence of its own. Its parastichy numbers are both even; a stem counted at four and six is a stem whose two families come in pairs. The arithmetic of it is settled here already: a k-jugate lattice is the ordinary lattice at k times the divergence and k times the rise, seen k times over, with every count multiplied by k, and the bijugate limit divergence is exactly half the golden angle.

That makes it the sharpest available test of one particular account of what a stem does after an ablation it cannot repair. The account says the repeating block it settles into is a repeat unit — the count of the ordinary lattice underneath a jugate arrangement. On a bijugate stem there really is an ordinary lattice underneath, and its counts are half the bijugate ones. So the prediction is unambiguous, and it is different from every other account’s.

The rule has to be extended, and only in one place #

The placement rule this collection uses puts one organ at a time at the least of a repulsion profile. It cannot make a whorled pattern: nothing in it produces two organs at one height.

So the rule is extended in the smallest way that does. Organs arrive k at a time at one height, and each one is placed at the least of the same profile against everything already there — its own whorl-mates included. The exponent, the neighbourhood, the window and the argmin are untouched, and the k-fold symmetry is deliberately not imposed: the second organ of a whorl is placed by the same argmin as the first, and where it lands is a result.

Grown that way at three rises, the rule produces lattices with rotational symmetry of order 2, measured on the positions afterwards by rotating the whole point set and requiring every rotated organ to land on an existing one. Their pairs are 4/6 at rises of 0.026 and 0.013 and 6/10 at 0.0065 — which are rungs of the bijugate ladder this collection derived from the arithmetic, 2/4, 4/6, 6/10, 10/16.

The first thing the experiment changes is the measurement #

Removing one organ from a whorled stem breaks the symmetry, and the displacement of the next organ has to be read differently because of it.

The two organs of a whorl are interchangeable. An intervention that swaps which one the rule calls “next” reports a displacement of half a turn while nothing has moved. Taken at face value the response never ends: at the 6/10 arrangement, offsets eleven and twelve report displacements of 0.5° and 179.5°, and both stems demonstrably recover within a handful of organs.

Read modulo 180°, the response is a clean run that ends at ten, which is the larger parastichy number, with everything past it under half a degree. The boundary result therefore survives multijugacy — and it needed to be checked, because a rule that gives the count on ordinary stems and half a turn of noise on whorled ones would be a rule with a hidden assumption in it.

And the response comes in pairs #

The clean part. Removing the organ one place back and the organ two places back displaces the next organ by the same amount to the last digit — 67.97° at both — then three and four by 42.89° at both, five and six by 20.63°, seven and eight by 76.64°.

That is the pattern’s own symmetry appearing in an experiment rather than in a picture. The organ one place back and the organ two places back are the two members of one whorl, and they are equivalent under the rotation that maps the lattice onto itself, so the rule cannot distinguish them. It is the cheapest test on this page that a stem really is bijugate, and it costs one extra ablation.

It is also a test a photograph cannot do. A pattern can look whorled and be an ordinary lattice whose two families happen to share a factor; this collection has built exactly that case, an ordinary lattice at 180.5° counting two and four with rotational symmetry of order one. Counts cannot separate the two. A response constant on whorls can.

How a whorled stem’s divergence is even defined #

There is a measurement question here that the ordinary case does not raise, and getting it wrong produces a result rather than an error, so it is worth setting out.

On an ordinary stem the divergence is the angle from each organ to the next, and a settled stem gives the same number over and over. On a bijugate stem the angle from each organ to the next alternates: half a turn to its whorl-mate, then whatever is left to the next whorl’s first organ. A sequence measured that way has no settled value at all, and a first version of this work duly reported an undisturbed bijugate stem as a period-two orbit — a perfect lattice reported as already wrecked.

The quantity that behaves is the whorl divergence: the angle from organ i to organ i + k, which is the same rank one whorl up. On a settled bijugate stem at the finest rise here it is 68.98° with a scatter of 0.11°, which is within a quarter of a degree of half the golden angle. It is reduced modulo one k-th of a turn for the same reason the displacement is, and for a reason this collection already had: a k-jugate divergence is only defined modulo 360/k, because adding that much leaves the point set identical.

So a whorled stem needs two conventions an ordinary stem does not, and both of them are forced rather than chosen. Every recovery time and every block below is measured on the whorl divergence, folded.

What the block is, and what it is not #

At the 6/10 arrangement, eight of thirteen offsets fail to repair, and their orbits are wide — motifs spanning a hundred and seven to a hundred and fourteen degrees, so they are orbits and not the azimuth grid rounding a constant. The blocks are six and ten.

Six and ten are the two parastichy numbers of the bijugate lattice. The ordinary lattice it is two copies of has the pair 3/5, and neither three nor five appears at any offset. The repeat-unit account predicted exactly those two numbers, on the strongest ground it had, and they are absent.

The two coarser bijugate rises do not produce orbits at all, and that is reported rather than folded in. At 0.026 the wrecked stem takes a constant whorl divergence — 23.44° at two offsets and 156.56° at two others, with a motif spanning 0.00° — and at 0.013 the motifs span two to five degrees, which is ten to twenty steps of the azimuth grid and not an orbit either. A period read off either would be the grid’s period rather than the pattern’s, which is a mistake this collection has already made once and now tests for.

The symmetry does not come back #

The result that has nothing to do with blocks, and is the one an experimenter could see fastest.

Every wrecked bijugate stem here has rotational symmetry of order 1 where it had 2, at every rise, and none of them recovers it in three hundred organs. The pattern goes on producing a lattice — a counter shown the positions returns a first parastichy number that makes sense — and the property that made it bijugate is gone permanently.

That is worth putting beside what this collection already says about jugacy: the counts cannot decide it. An ordinary lattice and a genuine bijugate one can carry the same pair, and only the symmetry of the point set separates them. So jugacy is the one property of a whorled pattern that a count cannot see, and it is the property a single missing organ destroys.

An experimenter looking for the cheapest observable in this whole thread should probably look here. Counting parastichies on a shoot is work; noticing that a shoot which used to have two organs per node no longer does is not.

Why the underlying lattice does not surface #

A reading, offered as one. The bijugate stem’s organs are placed against a neighbourhood, and the neighbourhood does not know it is two interleaved copies of anything. What it contains is ten short chains one way and six the other, and the rule’s profile is dominated by the organs at those two offsets. When the symmetry is broken the two copies stop being copies, and there is no longer any sense in which a smaller lattice is present to be fallen back onto.

Put the other way: the ordinary lattice underneath a bijugate one is a fact about how the pattern can be described, not about what any organ is placed against. A description does not have to survive an intervention.

What the experiment would look like on a plant #

The translation is unusually direct here, which is the argument for doing it.

Find a shoot with two organs at every node — teasel and valerian are the textbook cases, and both are common. Count its parastichies; they will be even. Remove one primordium at a stated offset and watch what the shoot does next.

Three things are predicted, in decreasing order of how easy they are to see. The shoot should stop having two organs per node, permanently. The organs placed immediately after the cut should move if and only if the primordium removed was one of the most recent n, where n is the larger of the two counts. And removing either member of one node should give the same answer, which is a control the ordinary experiment cannot offer at all — two interventions that must agree, on the same shoot, decided by the pattern’s own symmetry.

The first of those needs no protractor and no counting after the fact. It is a yes-or-no question about whether a shoot still looks whorled, asked a few weeks later, and every account in which a whorled pattern is maintained by something other than the local arrangement predicts that it should recover.

What this does not say #

It does not say the whorled rule is the rule plants use. It is the smallest extension of this collection’s own placement rule that can produce organs arriving in pairs, and what recommends it is that it produces the bijugate ladder the arithmetic predicts without being told to. Whether a meristem does anything like it is a different question.

It does not say bijugate stems are more fragile. The front is the larger count, as on an ordinary stem; the offsets that fail are in the middle of it, as on an ordinary stem. What is different is which property is lost, not how easily.

It does not say the repeat-unit account is finished. It says the account’s one sharp prediction — a block belonging to the underlying ordinary lattice — fails where it should have been strongest. An account that predicted the bijugate numbers for a different reason would be untouched by this, and nobody has written one.

And it does not say the symmetry could not return over a longer run. Three hundred organs after the cut it has not, at any offset, at any of the three rises. Longer runs are cheap and have not been done.

The check that would refuse it #

Four assertions run when these figures are drawn.

The whorled rule has to produce a lattice with the rotational symmetry of its own jugacy, with both parastichy numbers multiples of it, and with a settled whorl divergence. If the extension quietly produced an ordinary lattice with a doubled count, everything after it would be about the wrong object.

The response has to be a run ending at the larger parastichy number when read modulo one k-th of a turn, with nothing past it — and the unfolded displacement has to reach past a quarter turn somewhere it has not moved. The second half is what makes the folding a correction rather than a convenience: if the raw numbers never misbehaved there would be nothing to fold.

Removing either organ of one whorl has to displace the next organ by the same amount, to within a degree. That is the symmetry showing up in the experiment, and a stem that failed it would not be bijugate whatever the counter said.

And every block reported has to belong to the bijugate pair and none of them to the ordinary lattice underneath. The check is made at the arrangement where the orbits are wide enough to be orbits, because at the two coarser rises there is no block to check — and reporting their periods as blocks would have put a two into the table at exactly the rung where the account under test predicts a two.