A count with a factor in it

Remove several organs from a settled stem and let the rule place what comes next. Most arrangements of the cut never repair, and the stems that do not repair land on a strikingly short list of places: at one lattice, over more than three hundred wrecked runs made by cuts of one to five organs, exactly six settled divergences, with the largest cuts reaching nowhere the smallest had not.

Three of the six are understood. They are the lattice the stem was cut from with one family left standing and a whole number of turns threaded through it, and they sit at 136.99°, 208.80° and 280.43° — steps of 71.81° and 71.63° against 360° divided by five, which is 72°. Each keeps the five-family. Three rungs, one ladder, one hop.

The other three are not that. They keep no lag of the old lattice at any period up to twenty-four, and a counter shown their positions returns a pair the original lattice does not contain. This essay is about what those pairs are, and what they normally mean — and what has to be true of the divergence for them to mean it.

The three, and their counts #

The destinations are 175.01°, 189.96° and 235.00°. A counter shown the top two hundred organs of each returns:

Every one of those pairs has a factor in common. The undisturbed stem is counted 5/8, and five and eight are coprime — as are 3/5, 8/13, 4/7, 7/11 and every other pair this collection has ever measured on a stem the rule grew one organ at a time.

Coprimality is a theorem, not a regularity #

The comparison in the census above — every undisturbed pair coprime, every one of these three not — is stronger than a run of observations, and it is worth saying why, because it decides how much the three counts are allowed to weigh.

A stem grown one organ at a time is generated by a single displacement: organ i sits at i divergences round and i rises up. Its contact families are the denominators of successive convergents of the divergence, which is the constraint that makes a ladder a ladder rather than a list of numbers. And successive convergents of any real number satisfy the determinant relation — the cross-product of the two fractions is exactly one — from which their denominators are necessarily coprime.

So the census could not have come out any other way. Coprimality of the counted pair is not a property this collection happened to observe on every rise it swept; it is forced by the arrangement being generated one organ at a time, and a counter returning 2 and 6 on such a stem would be reporting an error in itself rather than a fact about the plant.

That is what makes the three destinations worth an essay. A regularity with three hundred supporting cases and no reason behind it can be broken by a four-hundredth; a consequence of how the lattice is generated can only be broken by the lattice not being generated that way. The shared factor is therefore evidence about the generating rule, which is the claim the rest of this thread is about.

It also sharpens what the alternative reading has to be. If the three stems are not whorled, then they are one-at-a-time spirals, and a one-at-a-time spiral cannot have a counted pair with a factor in it — so the counter’s answer would have to be wrong. That is a real possibility and it is the failure this collection has already had, but it is a much more specific charge than “the counts are misleading”, and it is checkable on the positions.

Why six is the other number in all three #

One thing about the table is not explained by any of the above: the larger count is six at all three destinations, though the smaller runs 2, 4 and 3 and the divergences are forty and sixty degrees apart.

The reading is that six belongs to the rise rather than to the cut. A family is counted when the hop to the organ q places above is short across the surface, and its height component is q times the rise. At a rise of 0.013 the offsets that give a hop comparable to the spacing between organs cluster near nine, and the whole counted band sits between about four and thirteen — which is where the undisturbed 5 and 8 come from as well. A destination with a q-step file has q forced into its pair by the file, and the second family then has to be a multiple of q lying in that band. Multiples of two in it are six and eight; multiples of three are six and nine. Six is the one available to both.

There is a second consequence, and it is the one that would make the number useful. If the band is set by the rise, then the pair of numbers a wrecked stem returns carries the rise as well as the divergence — the file count says where the divergence landed and the larger count says how coarse the stem is. A single count would then be doing what a count is elsewhere unable to do, which is reason enough to distrust it until the sweep below has been run.

Stated as a test rather than a story, because as it stands it is a story: repeat the census at a different rise. If six belongs to the rise, the recurring second number moves — to the multiple of the file count nearest the new band’s centre — while the file counts themselves, which come from where the divergence landed, do not. If six comes back unchanged at a rise two or three times coarser, the reading is wrong and the number is saying something about the cut that this essay has not found.

Where the six come from #

The sweep behind the list is worth describing, because “six destinations” is a number that depends on how hard anybody looked.

A cut is named by the offsets of the organs removed: [3, 5] takes out the organs three and five places back, furthest first so that each offset means what it says. For each size from one organ to five, every arrangement described by a starting offset and a set of gaps is tried — a hundred and eighty-eight arrangements in all — and each one is continued for three hundred organs against a control that shares its history exactly.

The share that never repairs climbs with the size: 0.25 at one organ, then 0.56, 0.71, 0.83 and 0.98 at five. So the intervention is doing what an intervention of increasing size should. What it is not doing is inventing new places to land — the destinations reached by the five-organ cuts are the ones the single cuts already reached, and the total across every size is six.

Two destinations are within a degree of each other and are counted as one place; the tolerance for that is stated and nothing in the table is near it. The six are separated by twenty-six degrees at the closest.

Why the counts are worth taking seriously #

A reflex here would be to say the counter has been shown something it was not built for and its output can be ignored. That reflex is wrong, and it is worth saying why before the counts are questioned.

The counting machinery on this site is deliberately blind. It is given coordinates — nothing else — and it finds chains of near neighbours, follows them, and reports how many run in each direction. It has never been told a divergence angle, and it does not know whether the pattern it is looking at was grown, cut, forged or written down from a formula. That is the whole reason a count here is evidence rather than a restatement of what the pattern was built from.

So when it returns 2/6 it has found two chains one way and six the other, and those chains are in the positions. The question is not whether the count is right. It is what a count of that shape licenses somebody to conclude, and the answer turns out to be less than the textbooks assume.

What a shared factor normally means #

This is not an obscure signature. It is the single most-cited structural fact about phyllotaxis after the Fibonacci sequence itself.

A whorled stem puts k organs on every node at once, spaced a k-th of a turn apart. The result has k-fold rotational symmetry by construction; every parastichy family comes in k copies; and the pair a botanist writes down is k times the pair of the single-organ lattice underneath. A bijugate specimen with a 5/8 lattice hidden inside it is counted 10/16. A trijugate one is counted 15/24.

So a stem counted 2/6, 4/6 or 3/6 is, on the face of it, a stem with two or three files running up it — an arrangement that arrives several organs at a time rather than one. And that would be a remarkable thing for a cut to produce, because the number of organs per node is the one property a spiral count is normally understood to be about, and it is a property of how the pattern is generated rather than of where it ended up.

The other three, for contrast #

The half of the list that is understood is worth laying out beside the half that is not, because the contrast is what makes the three counts strange rather than merely unfamiliar.

The three slips settle at 136.99°, 208.80° and 280.43°. Each keeps the five-family of the old lattice rigid — the angle from an organ to the one five places above it is unchanged from the control, organ by organ, to a fraction of a degree — and each sits at a different whole number of turns of that family from the divergence the stem was cut from.

A counter shown those three returns 5/11, 5/12 and 5/9. Every one of those pairs is coprime, every one carries the five of the family that survived, and none of them looks like anything unusual: they are ordinary counts of ordinary lattices that happen not to be the lattice this stem started on.

So the six divide cleanly on two independent measurements at once. Three keep a lag and are counted coprime; three keep no lag and are counted with a factor. The two halves of that split were found by different machinery — one by comparing hops against a control, one by counting positions — and they agree on which three are which.

Why a cut converting a spiral stem into a whorled one would matter #

Set out plainly, because the claim is worth taking seriously before it is tested.

The rule this collection uses places one organ at a time, at the minimum of a sum over what is already there. There is a separate rule for whorls, and it places k at a time. They are different rules. A stem grown by the first has never once been observed to become a stem of the kind the second produces, and there is no mechanism in the first by which it could — the organs are still arriving one at a time, at prescribed heights, after the cut exactly as before.

If a large enough cut could nevertheless produce a genuinely whorled pattern, the consequence would be sharp and would reach outside this collection. It would mean that jugacy — the property counting is least able to decide and morphologists have argued about longest — is reachable by damage, and that a whorled specimen in a herbarium is not necessarily a specimen of a whorled kind.

What the three are not #

Three readings can be disposed of before the measurement, because each is checked by something already in the runs.

They are not transients. Each destination is the mean over a whole number of periods of the motif the stem repeats, taken from the last hundred and twenty organs of a three-hundred-organ continuation, and the recovery test has already refused to call these stems recovered. A stem still moving would not produce the same four decimal places from arrangements that differ in which organs were taken.

They are not artefacts of the azimuth grid. The grid is 1,536 steps of 0.234°, and the motifs at these destinations span hundreds of steps. A motif narrower than a few steps is a constant the grid could not write down rather than an orbit, and this collection has been caught by that once; the check has been in the machinery since and these cells pass it comfortably.

And they are not on the ladder. Every slip’s divergence is the original plus a whole number of turns of the family it kept; 175.01°, 189.96° and 235.00° are not at any multiple of 72° from 136.78°, and none of them has a rigid lag at any period up to twenty-four. Whatever they are, they are not the lattice the stem was cut from, displaced.

One more thing is worth fixing before the test. The three destinations are not rare corners of the sweep: the one at 175.01° is reached by a cut of two organs, which is the smallest intervention that reaches anywhere other than a slip, and the two at 189.96° and 235.00° by cuts of four. So whatever they are, they are ordinary outcomes of an ordinary intervention rather than the tail of something.

Two candidate readings, and how they differ #

There are two ways the three counts could have come about, and they make different predictions about a measurement that has nothing to do with counting.

The first is that the cut really has produced a whorled arrangement. Then the positions should have k-fold rotational symmetry: rotate the whole pattern by a half turn, or a third, and every organ should land on another organ.

The second is that the stem is an ordinary one-organ-at-a-time spiral whose two shortest families happen to share a factor. Then rotating it should land nothing anywhere, because a spiral lattice with one organ per height has no rotational symmetry at any order.

The two readings are distinguished by one measurement on the positions, made without reference to counts, and it has been in this collection’s machinery since the whorled rule was first written — for exactly this reason. The next essay makes it.

One more thing is worth fixing before the measurement, because it is the sort of detail that decides an answer quietly. The symmetry is measured on the interior of the pattern rather than on all of it. The top and bottom rows of a finite point set have no partner to rotate onto, so a test applied to every organ reports order one for every pattern including a genuinely bijugate one — which is a measurement that cannot fail and therefore says nothing.

The tolerance is a fraction of 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. Both of those choices were made when the whorled rule was written, against patterns whose answers were known, and neither is being made here for this measurement.

Stated in advance, so that the answer cannot be arranged afterwards: if the rotational symmetry comes back as two for the stem counted 2/6 and three for the one counted 3/6, then a cut can change the jugacy of a pattern, and a great deal of the literature on whorled arrangements has a new failure mode to worry about. If it comes back as one, then the shared factor is a fact about where the stem’s divergence landed, the counter is doing exactly what it should and being misread, and the question becomes what those three divergences have in common.