Two windows on one stem

A real shoot does not sit at one rise. Its internodes shorten as it grows, the pattern climbs the ladder, and the parastichy pair changes underneath any measurement made on it. The pair can still be read from the angles — the earlier work established that — but only through a window, and only if the window fits inside one rung of the ladder.

That left a problem the earlier work named and could not solve. A stem that returns no reading is reporting that one of four things is true: too quiet, too disturbed, too fast, or a window in the wrong place. The sequence distinguishes none of them, and a refusal that could mean four things is close to a refusal that means nothing.

The proposal was a second overlapping window. This essay implements it, and the first surprise is that the decision about what agreement means is harder than the measurement that follows.

Four outcomes, and each is given a name #

Two windows, each of which either reports a pair or refuses. Four cases:

Agree — both report and report the same pair. Disagree — both report and report different pairs. One-sided — exactly one reports. Silent — neither reports.

There is nothing clever about that enumeration, and the temptation is to skip past it to a rule like take the reading if either window gives one. The enumeration is the point: each of the four is a different piece of evidence, and collapsing them into report-or-not throws away most of what the second window bought.

Agreement certifies the rate #

The numbers are stark enough not to need statistics. Over twenty-five stems at each rate: at 130 nodes per rung, 0 agreements, and 1.92 windows to a rung; at 250, 1 agreement and 1.00; at 400, 14 agreements and 0.63; at 700, 14 agreements and 0.36.

The transition is not gradual and it sits exactly where the arithmetic puts it: when a rung is shorter than the window, at least one of the two windows contains a transition, and a window containing a transition has no single pair in it to find.

That makes agreement a certificate. A botanist who reads two overlapping windows and gets the same pair from both has established — from the angles alone, without measuring internode lengths or counting transitions or knowing anything about how fast the shoot grew — that the shoot is slow enough for the reading to be about one rung.

That is the cause the earlier work most wanted removed, and it is the one a botanist can least easily check by other means. The others have external evidence available: a protractor gives the scatter, and a ruler down the stem gives some idea of the rate. But the rate that matters is not the rate in centimetres per week — it is nodes per rung, which needs the ladder, which needs the pair, which is the thing being measured. The certificate breaks that circle.

The controlling ratio is the span, not the window #

The four rates are reported as windows to a rung — 1.92, 1.00, 0.63 and 0.36 — and that is the natural quantity to reach for, because it is the one the single-window instrument was specified against. It is the wrong quantity here, and the four numbers say so.

If agreement required both windows to sit inside one rung, the threshold would be at 0.63 windows to a rung — a window and a half of stem inside one rung, since the lower window is shifted by half a window — and the agreement rate at 0.63 would be a few per cent rather than the fourteen in twenty-five observed. The naive placement arithmetic gives 6% there and 46% at the slowest rate; the measurements are 56% and 56%. It is right at one rate and wrong by a factor of nine at another, which is not a model with noise in it.

The quantity that does order the four rows is the total span of the two windows against the rung. Two 250-internode windows offset by 125 span 375 internodes between them, so the spans are 2.88, 1.50, 0.94 and 0.54 rungs — and the agreement counts are 0, 1, 14 and 14. The break sits at a span of about one rung, and it is sharp: nothing above 1.5 agrees at all, and everything below 1.0 agrees about half the time.

Two things follow, and the second is a design change rather than a remark.

Agreement is a weaker certificate than the enumeration implies. At a span of 0.94 rungs the two windows cannot both be inside one rung more than a fraction of the time, yet they agree in over half of stems — so the readout is surviving a window that crosses a rung boundary, presumably because the pair that dominates most of a window is the pair reported. An agreement therefore certifies that the stem is slow enough for the pair to persist across the span, which is close to but not the same as the claim that both windows sat on one rung.

And the offset is the parameter to tune, not the window. The span is the window plus the offset, and the window is fixed by how many internodes a reading needs. Halving the offset from 125 to 62 shortens the span from 375 to 312 and moves the certifiable rate about a fifth faster, at no cost in reading quality — the cost is paid in independence instead, which is the trade the section below is about. That is a knob nobody turned, and it is the one that buys rate.

What a survey should expect to keep #

The grid also answers a question a survey designer asks before anything else, and the arithmetic is one sum over the tallies.

Pooling the five disturbances gives 29 agreements over the hundred stems in the grid. Excluding the largest disturbance, where the pattern has come apart and the ruler in the specification would have excluded the specimen before it reached the readout, it is 29 in eighty — a 36% yield.

So a survey wanting thirty certified specimens needs to collect and read something like ninety, and wanting a hundred needs three hundred. That is a real number to plan against and it is not a discouraging one, because the discarded specimens are discarded for a stated reason rather than by judgement: a one-sided read is a reading without a certificate, and throwing it out is what makes the kept ones mean something.

It also says where effort is best spent. Two thirds of the loss is one-sided verdicts, which are a matter of window placement rather than of specimen quality — so the offset change above is worth more to a survey’s yield than any improvement to the readout itself.

Why the enumeration is worth keeping #

A reader who has followed the survey thread will notice that this essay produces no new statistic. It runs the same readout twice on the same stem and looks at the pattern of outcomes, and that is all.

That is the point, and it is worth defending because the alternative was tried. The obvious way to use two windows is to combine them into one better estimate — average the two correlograms, or concatenate the internodes and read four hundred instead of two hundred and fifty. Both make the reading better and both destroy the thing the second window is for. A combined reading is a single number again, and a single number cannot say whether the pattern changed underneath it; that information lives in the disagreement between two readings, which averaging is designed to remove.

So the second window is not more data. It is a second measurement of the same thing under a slightly different condition — a lower stretch of stem — and the value is entirely in whether the two agree, and what a refusal is worth follows from the same enumeration. This is the standing pattern of this site’s strongest results: the round trip, the two instruments that share no code path, the angle readout against the position counter. Each is two ways of getting one number, and each is worth more than either way alone by exactly the amount that a disagreement would have cost.

What is different here is that the two readings are not independent, which is what the next section is about.

What the overlap costs #

The lower window is shifted by half a window, so the two share half their internodes. That is a choice and it is the one parameter here worth arguing about.

Disjoint windows would be better evidence. Two readings from the same internodes are not independent: a wrong reading driven by a peculiarity of the data can appear in both, and agreement between them is then agreement of a measurement with itself.

And disjoint windows would be unusable. Two 250-internode windows laid end to end span 500 internodes. At every rate this instrument works at, 500 internodes is longer than a rung — 500 against 400, or against 700 with the pair sitting near a boundary — so the lower window would be almost guaranteed to straddle a transition, and the verdict would be one-sided whatever the shoot was doing.

So the overlap is a trade between independence and placement, and half is neither end of it. What that costs is measured in the next essay: agreement is not correctness, and at the quiet end of the disturbance range there are stems where both windows agree on a pair the position counter does not report.

Reading the grid the other way #

The rows of the grid are rates and they carry the result. The columns are disturbances and they carry a second one that is worth extracting, because it says which of the four verdicts a survey should expect to spend its time on.

Over the twenty stems at each disturbance, pooled across all four rates, the tallies of agree, disagree, one-sided and silent run: 8, 1, 9, 2 at a disturbance of 0.05; 6, 5, 7, 2 at 0.15; 9, 1, 6, 4 at 0.25; 6, 0, 12, 2 at 0.5; and 0, 0, 2, 18 at 0.9.

Three things fall out of that.

One-sided is the commonest verdict over most of the range, and it is the least informative of the four. It happens when one window is on a rung and the other is not, which is what a stem at an awkward rate looks like from below, and it happens at every disturbance from the quietest to the strongest. A survey should expect to discard roughly a third of its specimens for this reason alone, and discarding them is correct: a one-sided read is a reading with no certificate attached.

Silence is concentrated at one end. Eighteen of the twenty-two silences in the whole grid are at the largest disturbance, where the pattern has come apart. That is the reason the ruler is in the specification, and it is why silence is much less ambiguous in practice than the four-cause enumeration makes it sound: most silences are accompanied by a scatter that says immediately which end of the range the stem is at.

And disagreements cluster in the middle. Five of the eight are at a disturbance of 0.15, which is not where the pattern is fragile and not where it is quietest. That is the disturbance at which the second comb is weakest relative to the first — strong enough to clear, not strong enough to be unambiguous — so it is where a window is most likely to name the wrong partner. A disagreement that is not between adjacent rungs is therefore a signature of a particular disturbance range rather than of anything about the shoot, which is useful to know and would have been read as noise without the grid.

A disagreement can name a transition #

The rarest of the four verdicts is the most informative when it happens.

Across the grid there are eight disagreements. Six of them are the two windows agreeing on the smaller parastichy number and differing on the larger — 8/13 against 8/11, or 8/13 against 8/10 — which is one window misreading the second comb and is not a fact about the plant.

Two of them are different. At 250 nodes per rung, one stem returns 8/13 above and 5/8 below: two rungs that the ladder puts next to each other, in the right order, on a shoot whose rung length is about the window length. That is a transition caught inside the read.

The distinction is mechanical rather than a matter of judgement: check whether the two readings are consecutive pairs on the Fibonacci ladder. If they are, the windows are on opposite sides of a transition and the shoot’s rate can be estimated from where the boundary must lie. If they are not, one of the two readings is wrong and the stem should be discarded.

Two of eight is not many, and the reason there are so few is instructive rather than disappointing. A genuine transition inside the read requires the rate to be in a narrow band: fast enough that a boundary falls between the two windows, slow enough that each window is mostly on one side of it. Faster than that and both windows straddle boundaries and neither reads; slower and no boundary falls in the pair of them at all. So the disagreement-names-a-transition case is a diagnosis that is available exactly at the rate where it is needed, and is absent — correctly — everywhere else.

It also means the count of adjacent-rung disagreements across a survey is itself a measurement: it estimates how many of the sampled shoots are climbing at around one rung per window, which is the population’s rate distribution, which is a quantity no survey of phyllotaxis has ever reported.

The certificate, stated as a claim a plant could refute #

It is worth writing the result in the form the site uses for anything it expects to be held to, which is a statement with a stated way of failing.

Claim. If two windows of 250 internodes, overlapping by half, both return a pair and return the same one, then the shoot’s rung is longer than the window.

How it could fail. A shoot fast enough that both windows straddle transitions and both nevertheless return the same wrong pair. That is not impossible in principle — a pattern spends part of a rung near each boundary, and two windows sharing half their internodes share whatever the overlap contains.

What the measurement says about that. Over fifty stems at the two fast rates it happened once, and the once was at 250 nodes per rung with the window exactly one rung long, which is the boundary case. At 130 nodes per rung — two windows per rung — it did not happen at all in twenty-five.

So the claim is not a theorem and is not offered as one. It is a rate of false certificates measured at one in fifty at the worst rate tried, which is the form a claim about an instrument should take.

What this does not fix #

The second window removes one cause of refusal and identifies a second. It does not remove the other two, and the reason is that they are properties of the whole stem rather than of where the window sits: a plant that is too quiet is too quiet in both windows, and one that is too disturbed is too disturbed in both.

The full accounting, and what is left after it, is the next essay. The short version is that the four causes go to three separated and one remaining, and the one remaining is the one this instrument was built to remove.