A counter that sees no positions

There are now three instruments on this site that report a parastichy number, and the useful thing about having three is not that they agree. It is that they are given different things.

The first is parastichy, which counts a disc. It receives an array of points and a radius band. The second is cylCount, which counts a stem. It receives an array of points and a height band. The third arrived with the previous essay and it receives a list of angles.

What follows is what that difference buys, what it costs, and the one property a blind instrument must have that the other two never needed.

What each one is not told #

The discipline is the same in all three cases and it is the reason any of the numbers mean anything.

None of the three knows the divergence angle. None knows the rise, or the growth rate, or which numbers a reader is hoping for. Each is a function from data to a number, and the data does not contain the answer in any form that could be read off without doing the work.

That is easy to state and it is the thing that is most often quietly violated. A counter that took the divergence angle and returned the pair the ladder predicts would agree with all of these on every case in this collection, and would be a restatement rather than a measurement. the founding essays turned on exactly this: the recovery step refused a count, and it could only refuse because it had never been told what the count was supposed to be.

The third counter takes the discipline further than the other two, because a list of angles does not contain a picture. There is no arrangement in it to look at. The information is there — the previous essay is about why — but nothing in the data resembles the answer.

What each one returns #

Here they differ, and the third comes off worst.

The disc counter returns a pair — the two shortest offsets — along with the whole curve it picked them from, so a reader can see how close the third was. The cylinder counter returns a pair and the same curve. Both, on a good band, are unambiguous.

The angle counter returns one number. It is one member of the counted pair, it is the nearer family away from a transition, and there is no second number available: the sequence’s spectrum contains the other families, as the previous essay shows, but nothing in it distinguishes a second family from a harmonic of the first.

So the third instrument is strictly weaker in what it produces. Its interest is entirely in what it consumes.

That last claim did not survive, and the correction belongs here rather than only in the essay that made it. The significant lags are not a list but two arithmetic progressions: one at multiples of the smaller counted number and one at the same spacing displaced by the difference of the pair. Sorting the lags by their remainder rather than by their height recovers both numbers from the same spectrum this essay calls unreadable.

So the third counter returns a pair after all — the spacing exactly, and the second number modulo the first, with one convention to close the gap. Nothing new was measured to get there; the peaks were already drawn and the harmonics were already labelled. What was missing was the observation that the leftovers are not leftovers.

Which leaves the comparison between the three where the essay wanted it to be and for a better reason. All three return a pair, none is told the angle, and the third is handed the least information of any of them — a list of numbers with no arrangement in it. Its interest is still in what it consumes, and it is no longer weaker in what it produces.

The round trip, run three ways #

The site’s habit is to close a circuit: build a pattern from stated parameters, throw the parameters away, recover them from the output, and report the error.

Run on the same stems, the three counters agree. At a rise of 0.032 the positions say 3 and 5 and the angles say 3; at 0.013 the positions say 5 and 8 and the angles say 5; at 0.005 the positions say 8 and 13 and the angles say 8.

That agreement is worth something specific and it is easy to overstate. It does not show that either instrument is correct — two instruments implementing the same mistake would agree too. What it shows is that they do not share a mistake, because they do not share a code path, a data type, or a definition. One measures distances between points. The other measures products of angle differences. They have the integer in common and nothing else.

What a counter got wrong once, and how it was caught #

The site’s first counter was wrong, and the way the error surfaced is the reason this essay treats refusal as a feature rather than as a nuisance.

parastichy computes, for every offset, the median distance between nodes that many places apart, and returns the offsets at which that distance is smallest. The first version returned the two smallest offsets among the local minima rather than the two shortest hops. On a head whose real neighbours are 34 and 55 it returned 21 and 34.

Nothing looked wrong. Twenty-one spirals were drawn and there were twenty-one of them; the figure was correct, the assertions inside it passed, and a reader checking the picture against the caption would have found them in agreement. The pattern genuinely has a 21 family. It is simply not one of the two nearest.

What caught it was the recovery step refusing. Given the pair 21 and 34 and the band they were counted in, the recovery asks which divergence angles make those two the shortest offsets — and there is no such angle. Not a large error: no solution at all. The instrument downstream could not be satisfied, and its complaint was specific enough to locate the fault.

That is the shape every check on this site is built to have, and it is why the third counter’s threshold is not an afterthought. An instrument that can only return answers can only be wrong. An instrument that can refuse has a second output, and the second output is where errors show up first.

The same story has repeated twice since. A cylinder counter applied a local-minimum rule that is right on a disc and wrong on a stem, drew five families where there were five, and was caught by a recovery that refused. A figure asserted a tolerance tighter than its own library’s and failed on a run the library passed. In all three cases the picture was fine.

The Lucas control, which is the one that matters #

Agreement on Fibonacci numbers is weak evidence, because Fibonacci numbers are what a careless instrument would produce.

Seeded at 99.502° the pattern walks the Lucas ladder and the angle counter follows it: 3, then 4, then 7. Every one of those agrees with the position counter on the same stem, and the last two are numbers that no Fibonacci-flavoured bug could produce.

This is the same control the site’s disc counter was given in the founding essays and it is worth applying to every counting instrument as a matter of course. An instrument that only ever returns 8, 13, 21 and 34 has not been shown to be counting.

A blind instrument must be able to refuse #

Here is the property the other two counters never needed, and it is the reason this essay exists rather than being a paragraph in the last one.

The position counters can fail visibly. A band with too few points, a curve with no local minimum, a pair sharing a factor — each of those is a recognisable state and each throws. More to the point, their output is checkable by eye: draw the polylines joining every 34th point, and if there are thirty-four of them the counter was right.

The angle counter has no such recourse. It is an argmax over thirty lags, and an argmax always returns an index. Handed a sequence with no period in it at all it returns a number, promptly and with no sign of difficulty. There is no picture to check it against, because the whole point of the instrument is that it never saw one.

So it needs a threshold, and the threshold has to be set from outside the data it is judging. It is set from sequences with no structure whatever: white noise of the same length gives a largest-of-thirty-lags value of about 0.10, worst case 0.15, and the threshold sits at three sampling bands, which is 0.34. Anything under that is refused.

Where it refuses, it would have been wrong #

A threshold that never fires is decoration. This one fires, and the case is instructive.

At a rise of 0.05 the pattern is a 2/3 lattice. The position counter reports 2 and 3 without hesitation, and the pattern is intact by every measure the site has — the scatter is four tenths of a degree, every run is coherent, nothing is wrong with it.

The angle counter reports 4, 23, 12, 2 and 9 across five runs.

Four of those five are not in the counted pair. The fifth, 2, is correct by accident. And the clearance test refuses all five, because the peaks are 0.10 to 0.22 and the line is at 0.34.

That is the whole argument for the gate in one row of a table: five wrong answers, five refusals. Without it, this essay would be reporting that a 2/3 lattice sometimes has a period of twenty-three.

Why it fails there, and what that says about the instrument #

The failure is at the coarse end, and it is not arbitrary.

A period of two or three has few cycles inside a thirty-lag window, and the family carrying the correlation has few members to carry it. Going finer, the peak grows: 0.32–0.46 at the 2/3 rung, 0.72 at 3/5, 0.59 at 5/8, 0.78 at 8/13.

So the three instruments have different blind spots, which is the useful property. The position counters are weakest near a transition, where two pairs are equally short. The angle counter is weakest at the coarse end, where there is barely a period to find. A pattern that defeats one is not usually the pattern that defeats another.

What each one costs to use on a plant #

The three instruments are not interchangeable in the field, and the differences are larger than the differences in what they return.

The disc counter needs a head, photographed square on, with the primordia resolvable and the centre locatable. That is an afternoon with a camera and a sunflower, and it is the reason this is the count the literature is full of. Its weakness is that it answers about a band and is usually reported as though it answered about a head.

The cylinder counter needs a stem with its leaves or scales still attached and their positions measurable in two dimensions. Harder — the organs are on a curved surface and the far side is hidden — but it has the compensating property that the answer does not depend on where it is taken, which on a disc it always does.

The angle counter needs sixty consecutive internodes, a steady rung, and the angles measured to better than a quarter of a degree. The first two are cheap and checkable in advance. The third is not: it is a demand for photogrammetry rather than a protractor, and the essay two along is about exactly how the requirement arises and why it cannot be relaxed by averaging.

So the order of difficulty is not the order of sophistication. The blind instrument is the most demanding of the three, and it is demanding in precision rather than in quantity — sixty internodes is one good stem, and a quarter of a degree is an instrument most fieldwork does not carry.

What the third counter is for #

Not for counting spirals. Given a photograph, the spirals should be counted.

It is for the case of a sequence and no photograph — which is not a contrivance, because a botanist walking up a stem with a protractor produces exactly that, and because the angles are the quantity every published claim about divergence is stated in. It is also for the case where the count and the angles are both available and disagree, which is a state no single instrument can detect.

And it is for the argument the next essays make, which is that the order of the angles carries several things nobody had looked for. The count is the first, and it is the one that could be checked against an instrument the site already trusted.

What a fourth counter would have to use #

It is worth asking what is left, because the answer is short and it says something about how much of a pattern a count uses.

A stem hands over three kinds of information: where the organs are, what order they were made in, and how big they are. The disc and cylinder counters use the first two — positions, indexed by placement order, which is why they can speak about an offset at all. The angle counter uses the second and a projection of the first.

Nothing on this site uses the third. Organ size varies systematically up a stem, it is measurable with far less precision than an angle needs, and it is not obviously independent of anything: a primordium’s size and the space available to it are the same quantity seen twice. Whether a size sequence carries a parastichy number the way an angle sequence does is a question with a definite answer and this collection does not know it.

What is not left is a fourth reading of the positions. The disc and cylinder counters differ in their metric and not in their data, and a third metric on the same coordinates would be a third way of saying the same thing — which is worth having for robustness and is not worth calling an independent instrument.

The honest summary #

Three instruments, three kinds of data, one integer. The third is weaker in every respect except independence, and independence is what a round trip is for.

Its distinguishing feature is not accuracy. It is that it has a threshold with something behind it — a measured noise floor, a case where it fires, and a demonstration that the case where it fires is the case where the answer would have been wrong. That is a smaller claim than the other two counters make and it is supported by a kind of evidence they were never asked for.

And a third counter, on the same principle #

The instrument in this essay reads one number out of a sequence. The work after found the second number in the same spectrum, at lags nobody had sorted.

What a cut costs the symmetry #

The index-free counter exists because a whorled pattern has no genetic spiral to count along, and this essay’s closing point is that only the symmetry of the point set separates a genuine k-jugate lattice from an ordinary one whose families share a factor.

That property turns out to be the one a single missing organ destroys. Removing one organ from a bijugate stem grown by a rule that places two at a time drops the rotational order from two to one, at every rise tried, and it does not come back in three hundred organs.