The symmetry that is not there

Three of the six places a many-organ cut can send a stem are counted at a pair whose numbers share a factor: 2/6, 4/6 and 3/6. A shared factor is the signature of a whorled pattern — one that arrives k organs at a time and therefore carries k copies of every family — and if a cut could produce one, then the property counting is least able to decide would be reachable by damage.

The test takes one measurement, and it is not a count.

Jugacy is a property of the positions #

Multijugacy is not a fact about numbers; it is a fact about where the organs are. A pattern is k-jugate when rotating the whole of it by a k-th of a turn maps it onto itself — every organ landing on another organ, to within a small fraction of the local spacing.

That is a different statement from anything a spiral count makes, and this collection has drawn the difference before, on patterns written down from formulas rather than grown: a bijugate lattice and an ordinary lattice at a divergence near half a turn are counted identically and have different symmetry.

The measurement is direct. Take the positions, take the median nearest-neighbour distance as the scale, rotate the whole set by 1/k of a turn for every k from two to eight, and ask whether every rotated organ lands within a fraction of that scale of an organ that is already there. The largest k that passes is the order.

The controls, first #

A measurement that never returns anything but one is not a measurement, so the instrument is run on two patterns whose answers are known before it is run on the three in question.

The positive control is a stem the whorled rule actually grew. It is important that it was grown rather than written down: the rule places each organ of a whorl at the minimum of the same profile the ordinary rule uses, with its whorl-mates already in the neighbourhood, so the k-fold symmetry that comes out is a result and not an imposition. A control whose symmetry was built in could not fail.

Grown at two organs per node, at the same rise the cuts are made at, it comes out counted 2/6 with rotational symmetry of order 2. Grown at three, it comes out counted 6/9 with order 3. The instrument sees what is there.

The negative control is the undisturbed stem the cuts are made in. It is counted 5/8 and its rotational symmetry is order 1 — no rotation maps it onto itself, which is what one organ per height means.

How the measurement is made #

Worth setting out in full, because a null result stands or falls on whether the instrument could have reported anything else.

The pattern is the top two hundred and twenty organs of a run, taken as points on an unrolled cylinder of circumference one. The scale is the median nearest-neighbour distance, computed from the pattern itself rather than stated, so that a coarse arrangement and a fine one are judged on the same terms.

For each order from two to eight, every point is rotated by that fraction of a turn and asked whether an existing point sits within a small fraction of the scale of where it landed. The largest order at which every point succeeds is the answer.

Two details are load-bearing and both were settled when the whorled rule was written, against patterns whose symmetry was known.

The test is made on the interior of the pattern. The top and bottom rows of a finite point set have nothing to rotate onto, so a test applied to all of it reports order one for everything — including a genuinely bijugate stem — and is a measurement that cannot fail.

And the lookup that finds the nearest existing point lives on the cylinder rather than on a rectangle. A grid keyed on the raw azimuth puts a point at 0.998 of a turn and one at 0.002 in buckets a whole turn apart, so every query near the seam misses — and nine per cent of a spiral pattern lies within a cell of the seam, which is exactly where its neighbours are. That bug once reported a bijugate lattice with eight parastichies in a family as having thirty-one.

The three destinations have order one #

All three.

The strongest form of the comparison pairs them off rather than reporting the number alone. The stem that settles at 175.01° is counted 2/6. The stem the whorled rule grew at two organs per node is counted 2/6. A counter shown the positions of both returns the same pair and has nothing further to say. Rotated by half a turn, the second maps onto itself and the first does not — not marginally, not at a looser tolerance, not at any order from two to eight.

So a cut does not convert a spiral stem into a whorled one. The rule kept placing one organ at a time throughout, which is what it always does, and the pattern it produced is a one-organ-at-a-time spiral whose two shortest families happen to share a factor.

The inference that fails, stated as an inference #

It is worth writing out the step that breaks, because it is a step almost nobody writes down and almost everybody takes.

A pattern with k organs per node has k-fold rotational symmetry. True, by construction.

A pattern with k-fold rotational symmetry has every parastichy family in k copies. True: the rotation carries each family to another family of the same count.

Therefore a pattern whose counted pair shares a factor of k has k organs per node. False. It reverses an implication. Sharing a factor is a consequence of the symmetry and is not equivalent to it, and the three destinations here are exactly the counterexample.

The reason the reversal usually works is empirical rather than logical. Ordinary spiral patterns sit near the golden angle, where no small number of consecutive steps comes back near the start, so an ordinary pattern’s shortest two families are consecutive Fibonacci or Lucas numbers and are coprime. A pair with a factor in it therefore usually does mean a whorl — because the other way of getting one requires a divergence nothing normally arrives at.

A cut is a way of arriving at one.

Why the counter is not wrong #

It is worth being careful here, because “the counter says 2/6 and the pattern is not bijugate” reads like an instrument failure and is not one.

The counter’s job is to report the two families of nearest chains in the positions. At a divergence of 175.01° that is exactly what it does: consecutive organs sit almost opposite each other, so the pattern really does have two near-vertical rows of organs and really does have six chains in the other direction. Two and six are the right answer to the question the counter is answering.

What is wrong is the inference. “A pair with a shared factor implies k organs per node” is a rule of thumb that holds across the patterns botanists usually meet, because those patterns sit near the golden angle where no small number of consecutive steps comes back to the start. It fails for a lattice that has drifted somewhere else, and a wrecked stem has drifted somewhere else by construction.

What would have had to be true #

Suppose the answer had gone the other way. It is worth being explicit about what that would have meant, because a test whose alternative outcome has no consequences is a test nobody needed to run.

A rotation of half a turn mapping a wrecked stem onto itself would mean that organ i and some organ at nearly the same height sit nearly opposite each other, for every organ in the run. The rule places one organ per height, so the partner would have to be an organ at a different height that happened to land there — and it would have to happen for all two hundred and twenty of them at once, to within a fraction of the local spacing.

There is a rise at which that is nearly true, and it is worth naming because it is the case that makes the test non-trivial. If the rise were small enough that consecutive organs were separated vertically by much less than the local spacing, a stem near half a turn would approach a two-ranked arrangement in which organs really do pair off. The rises here are not small enough — at 0.013 the vertical step is more than a tenth of the spacing — but the failure is quantitative rather than categorical, and a test with a tolerance stated relative to the spacing is what turns that into a number.