The order carries the count

Every count in this collection has been made from coordinates. The disc counter is handed the positions of nine hundred primordia; the cylinder counter is handed the positions of three hundred nodes; both are careful to be told nothing else, and that carefulness is the reason a returned 34 is evidence rather than a restatement.

This essay counts the spirals without any positions at all.

What it is given is the list of divergence angles — one number per internode, in the order they were made, which is what a botanist walking up a stem with a protractor writes down. There is no picture in it. There is nothing in it that could be called a spiral. And the smaller of the two parastichy numbers is in it, plainly, at a place nobody had looked.

The measurement #

Hold the rise fixed.

That is the one departure from everything else in this thread, and it is not a convenience. A growing stem climbs the ladder — its rise falls, and the counted pair changes with it — so a band over which the pair is constant is forty to eighty nodes long at the rate the rest of this site uses, and forty nodes cannot show a period of eight. Fixing the rise puts the pattern on one rung and leaves it there for four hundred nodes.

Then disturb it a little, drop the first hundred and forty nodes as a transient, and compute the correlation between each divergence and the one k internodes later, for k from one to thirty.

At a rise of 0.032 the positions count 3 and 5, and the angles peak at 3 — and at 6, 9, 12, 15, 18. At 0.013 the positions count 5 and 8 and the angles peak at 5, 10, 15. At 0.005 the positions count 8 and 13 and the angles peak at 8, 16, 24.

The peaks are not small. At the finest of the three the correlation at lag eight is 0.78 against a sampling band of ±0.11, which is seven bands.

Why there is anything there #

A lattice is an arrangement in which node i’s near neighbours are i ± m and i ± n. That is what a parastichy number is: the offset in placement order that lands on a neighbour.

The placement rule is a sum over neighbours. So a disturbance to node i is felt most strongly not by node i + 1, which is half a turn away and far, but by the nodes that have i as a near neighbour — which are m and n places later. The sequence of divergences is therefore a feedback loop whose delay is m, and a loop with a delay of m rings at m.

That explanation makes a prediction beyond the number itself, and it is the one that turns a tall bar into a mechanism: a ringing loop produces harmonics. If the delay is eight, the correlation should reappear at sixteen and at twenty-four, decaying, and it does — 0.78, 0.51, 0.32 on the finest stem, all of them above the band.

A single peak at lag eight would be consistent with a great many things. Peaks at eight, sixteen and twenty-four are consistent with almost none of them.

Which of the two numbers comes out #

One, not both, and it is worth being exact about which and why.

The correlation is carried by the nearest family — the offset whose two nodes are closest, because that is the offset along which a disturbance is transmitted most strongly. At the three rises above, that is the smaller number of the pair, and the readout returns the smaller number every time.

It is not a law that it must be. The two families’ hop lengths cross as the pattern moves along a rung, and near a transition they are within a per cent or two of each other; there the readout is ambiguous in exactly the way every other counter on this site is ambiguous there. What the readout gives, stated honestly, is one member of the counted pair, reliably away from a transition and unreliably at one.

That is less than the position counter gives, which returns a pair. It is also obtained from a different kind of data, and the next essay is about what the two instruments together are worth.

The other families are in there too #

The readout reports one number. That is not because one number is all the sequence holds.

On the 8/13 rung, the correlation at lag 8 is 0.78. At lag 13 it is 0.59, and at lag 21 — which is 8 + 13, the third parastichy family — it is 0.47. All three are above the band of ±0.11, and the pattern continues downward into the small offsets: 5 and 3, which are the differences.

So the sequence carries the whole family structure, not merely its strongest member. The lattice’s entire set of near neighbours is written into the order of the angles, and the peak heights rank them by how near.

What the sequence does not carry is any way to tell which peak is which without already knowing the answer. Presented with peaks at 8, 13, 16, 21 and 24, a reader who did not know the pattern could not say which are families and which are harmonics — 16 and 24 are multiples of 8, 21 is a sum, and nothing in the spectrum distinguishes those cases. That ambiguity is why the readout reports the tallest peak and stops, and it is a real limit rather than a conservative choice.

It also connects this thread to one that looks unrelated. The set of families a head has is closed under addition — an essay two fields away establishes that every family but the two smallest is the sum of two others — and the reason the spectrum is hard to read past its first peak is exactly that closure. A set closed under addition cannot be told from the set of multiples of its smallest member by looking at which numbers are present.

It is not a fact about Fibonacci #

The obvious worry about any result on this subject is that Fibonacci numbers have been smuggled in somewhere. Here the worry is sharp, because the numbers coming out — 3, 5, 8 — are the ones a reader expects, and a readout that had them built in would look exactly like this.

So run it on a lattice whose numbers are different.

Seeded at 99.502° the pattern walks a different ladder, and the readout follows it: 3 where the positions count 3 and 4, 4 where they count 4 and 7, 7 where they count 7 and 11. Seven and eleven are not Fibonacci numbers and the instrument has never heard of them.

It reads the lattice.

The bottom row of that table #

The last row is a rise of 0.05, and it is the most important line in the essay.

At that rise the pattern is an unarguable 2/3 lattice. The position counter says so without difficulty. The angles say 4, 23, 12, 2, 9 across five runs — five different answers, four of them not in the counted pair.

And the readout refuses all five.

The peak has to clear three sampling bands before a number is reported, and the threshold was set from white sequences with no pattern in them at all rather than from these ones: a sequence with nothing in it produces a largest-of-thirty-lags value of about 0.10, worst case 0.15, and the threshold sits at 0.34. At a rise of 0.05 the peaks are 0.10 to 0.22. Under the line, every time.

This matters more than the successes do, and the reason is structural. The readout is an argmax, and an argmax always returns an index. It can never say nothing. A number reported without the clearance test in front of it is the largest of thirty noisy values wearing a measurement’s clothes, and at this rise it would be a confident 23.

Five wrong, five refused. That is the gate doing the only job a gate has.

Where it works and where it gives out #

The coarse end is where it fails, and the reason is the same one that makes it work.

A short period has few cycles inside a thirty-lag window, and the family carrying the correlation has few members. At the 2/3 rung the peak is 0.32–0.46 — right, and clearing the threshold on about three runs in five. One rung coarser it is gone altogether.

At the fine end it gets better: 0.72 at 3/5, 0.59 at 5/8, 0.78 at 8/13.

That is worth setting beside the other statistic this thread has produced, because the two run opposite ways. The lag-one correlation that separates one kind of noise from another needs a quiet plant and gets harder as the pattern gets finer, because the tolerance tightens up the ladder. This one needs a disturbed plant and gets easier as the pattern gets finer.

Two statistics on the same list of numbers, with opposite requirements, is a better position than one. It is also the beginning of an answer to the mixture problem the previous essay ended on, which is that one number cannot separate two unknowns.

The instrument needs a disturbed plant, which is not a figure of speech #

At a fixed rise with no noise at all, the rule converges exactly. Every divergence is the same number to the resolution of the sample grid, the variance is zero, and there is nothing to autocorrelate.

The readout throws.

That is worth stating plainly because it inverts an assumption that runs through the rest of this collection. A noiseless pattern is the clean case everywhere else here — it is what the round trips are run on, it is what the ladder is computed from. For this measurement it is the useless case, and a perfectly regular plant would hand over a list of identical angles from which nothing could be read.

Between the two ends there is a wide band. The readout is right on every run from about a fifth of a degree of scatter to about 1.3°, which reaches most of the way to the degree and a half at which a lattice fails. The instrument stops roughly where the thing it measures stops, which is the honest place for it to stop.

What a real stem would have to do #

The fixed rise is the artificial part, and it is worth pricing rather than apologising for.

A real shoot climbs the ladder. What the readout needs is a stretch over which the counted pair does not change — one rung — long enough to carry the measurement. The first of those is set by the plant and the second by the statistic.

A rung lasts $2T\ln\varphi$ nodes, where $T$ is the rate at which the rise declines, so a shoot at the rate the rest of this thread uses holds a rung for about sixty-seven nodes. The readout needs about sixty. The two numbers are the same size, which is a coincidence of this site’s parameter choices and not a law, but it says the right thing about what to look for: a stem on which one rung lasts long enough is a stem on which the counted pair is visibly steady over five or six dozen internodes, and that is checkable before any angles are measured.

Slower is better and the gain is direct — at three times the rate a rung holds two hundred nodes and every band on the stem can be read separately. Faster fails, and it fails in the way that is hardest to notice: a band that spans a transition contains two lattices, its spectrum contains both periods, and the readout returns whichever is taller.

So the eligibility test has two clauses and the first is free. Count the spirals in two places sixty internodes apart; if the pair is the same, the band is a rung. Only then is it worth measuring angles.

What this is and is not evidence for #

It is evidence that the divergence sequence contains the parastichy number. That is a fact about the model, checked on four ladders and two seed angles, and it is not a fact about plants until somebody reads a sequence off one.

It is not evidence that the model is right. Everything here would follow from any placement rule in which a node’s near neighbours are m and n places back, which is to say from the lattice rather than from the mechanism that built it. That is a strength for the measurement and a weakness for anybody hoping to use it to choose between mechanisms: the readout would return the same number for a rule this site has never written down.

And it is not a replacement for counting the spirals. It gives one number where a photograph gives two, it needs the rise to be steady over sixty internodes, and — as the essay after next sets out at length — it needs the angles measured far more precisely than a protractor measures anything.

What it is, is a second route to a number this site has spent four phases counting one way. The two routes share no code, take different data, and agree.

A note on what was in the picture all along #

The autocorrelation function that opens the previous phase’s essays runs to six lags. It was drawn to six because the argument it served was about lag one, and six seemed like enough to show that nothing else was happening.

At six lags a period of eight is invisible. The negative dip at lag five that the previous phase noticed and set aside as a curiosity is the trough before the peak at eight, on a stem whose bands were too short to resolve it.

The measurement did not need new machinery. It needed the same curve drawn further along, on a stem held still enough to have one answer.