A period that is not a count

A count is a good instrument and this collection has leaned on it heavily. Hand a counter a set of points and nothing else — no divergence angle, no order of arrival, no knowledge of what produced them — and it returns two numbers, and those two numbers pin down the lattice tightly enough to recover the angle to hundredths of a degree. The whole method of this collection depends on that separation being real.

It has a limit, and there is now a case that shows exactly where it is.

Nineteen stems in this collection never repair after an organ is removed, and each settles into an exactly repeating block of divergence angles. The period of that block turns out to be a lattice step that the removal left standing: the angle from an organ to the one p places above it is unchanged, organ by organ, while the divergence swings through eighty degrees. At eighteen of the nineteen, p is one of the two spiral counts the stem was cut from.

At the nineteenth it is four, on a lattice counted 8 and 13.

Four is not a spiral count of this lattice #

The lattice in question has parastichy numbers 8 and 13. Four is neither, and it is not a near miss for either — the numbers a counter shown these positions returns are 8 and 19, so the wrecked stem is not being counted as 4-and-something by any reading.

What four is is a lag. The four-hop of this lattice is the step from an organ to the one four places above it, and its length across the surface is 6.8 times the length of a contact hop. It is a stride right across the pattern, through the middle of several other organs’ neighbourhoods, joining points that are in no sense adjacent.

And it is rigid, on the same measurement that identifies the survivor everywhere else: a spread of 0.117° over a hundred and twenty organs, and a displacement of 0.006° from where the undisturbed control put it. That is the steadiest survivor in the whole census by the second measure. There is nothing marginal about it.

Why the counter still finds eight #

The apparent contradiction dissolves in one line of arithmetic. If the four-hop is rigid then so is the eight-hop, because the eight-hop is two four-hops. So is the twelve, and the sixteen.

A counter, shown the positions, looks for the shortest steps that link the points into families, because that is what counting spirals means. Among the rigid lags — 4, 8, 12, 16, 20, 24 — the eight-hop is far shorter than the four-hop, since four steps of the divergence take an organ most of the way round the stem and eight take it nearly back again. So the counter returns eight, and it is right to.

Both answers are correct and they are answers to different questions. The counter is asked which steps are short. The block is set by which step is conserved. Nothing requires those to be the same step, and here they are not.

What the wrecked stem looks like from the inside #

It helps to have a picture of the object rather than only its statistics.

Start from the undisturbed lattice at this rise. Its divergence is 137.84°, its two shortest steps are the thirteen-hop and the eight-hop, and every organ has six near neighbours arranged the way a lattice arranges them. The four-hop of that lattice takes an organ four divergences round, which is 551.4°, or 191.4° after a turn is taken off — very nearly half way round the stem, at a height four rises up. That is a long way, and there is nothing between those two organs that would make a rule notice the relation.

Now remove the organ six places back from the tip and let the rule continue. The divergences that follow never settle: they run a repeating cycle of four values spanning tens of degrees, so no protractor reading of the divergence of this stem would return anything stable. The mean of those four sits 90.00° from the angle the stem was cut from, which is one turn spread over four organs.

And the four-hop is where it always was. Organ i and organ i + 4 sit at the same relative azimuth, to six hundredths of a degree, for every i in the last hundred and twenty organs — which means that long stride across the pattern has been carried through the whole disturbance untouched while everything shorter than it was rearranged.

That is a strange object and it is worth pausing on. The relations a placement rule is sensitive to are the short ones; the relation this stem conserved is a long one. The rule did not hold the four-hop because it cared about it. It held it because holding the four-hop was the arrangement that satisfied the rule’s local demands, and the long relation is a consequence rather than a cause.

The eighteen that agree, explained rather than confirmed #

Once the exception is understood, the rule that holds eighteen times stops being a fact and becomes a consequence.

A placement rule puts each organ where the repulsion from the ones already there is least, and the sum is dominated by the nearest organs. A short hop is therefore a step the rule has a strong grip on: displacing one of its members costs a great deal, so a family of short steps is the kind of thing that can survive a disturbance intact. A long hop is not held by the rule at all in that direct sense.

So the usual outcome is that the survivor is a contact family, which is to say a parastichy number, which is to say a spiral count. The exception is what happens when the arrangement conspires to leave a long step untouched — and the fact that it can happen is the evidence that the quantity being conserved was never the count.

What a count cannot see, stated carefully #

It is easy to overstate this, so it is worth putting the limit precisely.

A blind count on a settled lattice is complete: two counts fix the divergence and the rise, and everything else about the lattice follows. Nothing is lost. That is the result this collection’s method rests on and it is not touched here.

A blind count on a disturbed stem is not complete, and this is the case that shows why. The pattern of what a disturbance conserves lives in the lags, and a count reports only the two shortest of them. Any structure carried by a longer lag is invisible — not obscured, not noisy, invisible — because the instrument selects on shortness before it reports anything.

That is a limit with a shape. It says a census of wrecked plants would recover the block period correctly whenever the block is a contact family and would report a different number, its own smallest rigid multiple, whenever it is not. And it says the two would be indistinguishable without measuring the divergences, because the point set alone does not carry the distinction.

How often would it matter #

One row in nineteen is not nothing and is not a rate. The offsets in this census were chosen to cover the front at four rises of the golden branch and two of the Lucas branch, which is a survey of one thing rather than a sample of stems.

What can be said is where to look. The exception here is a stem whose surviving lag is half one of its counted numbers — four of an eight — so the signature to look for is a block whose period divides a parastichy number rather than equalling one. That is a cheap thing to check on any future census and it had not occurred to anybody to check it, because the framing was about counts.

There is also a negative worth recording. Nothing about the displacement at that offset is unusual — it sits in the middle of the range the other wrecked offsets produce — so there is no cheap early warning. The only way to know which lag a stem kept is to measure the lags.

A note on what made this findable #

The measurement that separates the two readings is not clever, and the reason it was not made earlier is worth naming: the quantity everybody plots is the divergence, and the divergence is lag one. A stem’s whole lag spectrum is available from the same positions at no extra cost, and asking for it turns a puzzle about periods into a one-line description.

That generalises past this thread. A summary statistic selects, and what it selects on is usually invisible in its output — a count selects on shortness, a mean side count selects away everything Euler’s relation does not fix, an equivalent width selects on the window it was integrated over. The habit worth keeping is to ask what the instrument had to throw away in order to return one number, and then to go and look at the thing it threw away.

The same lesson, twice before #

This is not the first time an instrument in this collection has been found to select before it reports, and the earlier cases are worth putting beside it because together they make a pattern rather than an anecdote.

The first was the count itself, in its original form. A blind counter returned the two smallest lattice offsets rather than the two shortest steps — 21 and 34 for a head whose real neighbours were 34 and 55. The drawing looked right: twenty-one spirals were drawn and there were twenty-one of them. Only the angle recovery caught it, by refusing, because no divergence makes 21 and 34 the closest pair at that radius. An instrument that selects on the wrong quantity produces output that is internally consistent and wrong.

The second was a period read off a grid. A block of three angles was reported here, precessing by an amount that matched a standing prediction in the right units — and the three numbers turned out to be three consecutive samples of the azimuth grid the stem was computed on. Their mean was a grid point. At a finer grid the same run reported a block of two. The repair was to return the span of a motif beside its period, so that the scale of the thing being summarised travels with the summary.

The case in this essay is the third, and it is the mildest of the three, because nothing here is wrong. The counter’s answer is correct, the block’s period is correct, and the only error available is in the reader who assumes they must be the same number. What the three share is the shape: a summary returned without the thing it selected on can be read as a claim it does not support.

Two questions this leaves open, both cheap #

The exception is a single row and it points at two measurements that would cost almost nothing and have not been made.

Which lags can be rigid at all. The census reports the lag that survived at nineteen offsets, which is a sample of outcomes rather than a map of what is available. The direct question is whether every lag of a lattice can be the survivor at some offset, or whether only a few can — and it is answerable by sweeping the offset at one lattice and tabulating the survivors, which is a run this collection already makes for other reasons. If only a few lags are ever rigid, the next question is what distinguishes them, and the first candidate is the length of the step.

Whether a long survivor is less stable. The rule holds a short step by holding its members’ positions, and a long step has no such support. That suggests a wrecked stem whose survivor is long should be more easily knocked off it by noise than one whose survivor is short — which is a prediction with an amplitude in it, on machinery this collection already has. If it is right, a plant carrying a long-lag defect would be a rarer thing than one carrying a short-lag defect, whatever the offsets did.

Neither is attempted here. Both are named because a single anomalous row is worth more as a pointer than as a curiosity, and because the version of this essay that only reported the row would have been shorter and less useful.

What this does not say #

It does not say a count is a poor instrument. On a settled lattice it is complete and it is the cheapest complete instrument available, which is why it is used throughout this collection. The limit is specific: it is blind to structure carried by a lag longer than the two it reports.

It does not say four is a special lag. It is the smallest rigid lag of one stem in one census. What makes the case useful is that four is not a count of that lattice, not that four is anything in particular.

It does not say the counter is reporting an error. Eight and nineteen are correct as counts of that point set; the wrecked stem really does have eight short chains through it. The two instruments disagree about the period because they are measuring different things, and neither is malfunctioning.

And it does not say anything about a plant. These are stems grown by a rule that places each organ against its neighbours, and the transferable part is the warning about instruments rather than a claim about meristems.

The check that would refuse it #

Two assertions carry this and they are written to fail in opposite directions.

The first requires that the census contain at least one offset whose surviving lag is not one of the lattice’s counted numbers, and that its hop be more than three times the length of a contact hop rather than a near miss. That assertion fails if the awkward case ever disappears — from a change to the rule, to the range of rises, or to how a survivor is identified — and it is written that way deliberately, because the eighteen agreeing rows would otherwise leave the wrong statement standing.

The second requires the opposite: that the surviving lag is one of the counted numbers at all but a couple of offsets. A census in which survivors were long lags half the time would mean the argument about why short steps are held has failed, and the pair of assertions together is what makes “usually a count, and here is why, and here is one that is not” a claim rather than a hedge.

Both are checked against the same nineteen rows on every build, and both are checked with the tolerance that identifies a survivor stated separately from them — a spread under half a degree and a displacement under three — so that neither assertion can be satisfied by loosening the definition of what survived.