Phyllotaxis
the spirals are counted, not admired
The claims, measured

What a refusal does not say

The readout can decline for four different reasons — too quiet, too disturbed, too fast, or a window in the wrong place — and a stem that returns nothing does not say which. That is the third time this thread has failed to close the mixture problem, and the first time the failure has a shape.
18 min read 9 figures State the claim, state the testForm is not mechanismCounted, not admired

Worth reading first: The survey this site cannot do · The sequence has a memory · A pattern with a rate.

An instrument that refuses is better than one that guesses. This site has said so four times and built the refusals to prove it: the angle recovery that declines because no divergence makes that pair the closest at that radius; the readout that returns nothing five times out of five at the rise where it would have been wrong; the fitter that has no solution when a parent is thinner than a branch it carries.

But a refusal is only a measurement if it says something about the specimen. And this one does not, because it has four causes.

The four

The two-comb readout declines when any of its tests fails, and the tests fail for different reasons.

The plant is too quiet. Below a disturbance of about 0.1 the sequence has too little variation to be a sample of anything, and in the model it locks onto the sampling grid. The cap on the correlation catches most of that, and what it catches it reports as a refusal.

The plant is too disturbed. Above about 0.5 the lattice is gone: divergences wander over the whole circle and the coherence test declines.

The shoot is too fast. A window of two hundred and fifty internodes spanning more than a rung has two combs in it at different spacings, neither of which wins by a sampling band, so the margin test declines.

Or the window was in the wrong place. Even on a slow shoot, a window that happens to straddle a transition has the same problem as a fast one.

Both statistics, on the same stems, at a rise of 0.005Five seeded stems at each disturbance, held at a fixed rise. Bars are how many returned the pair the position counter finds; open portions are refusals. The pair comes out from 0.1 to 0.25, and across that whole range the lag-one correlation of the *same* sequences is -0.33, -0.58, -0.59 — decisive, negative and flat. There is no trade between the two: one stem supplies both. Below the window the sequence has locked onto the sampling grid and is a cycle rather than a sample; above it there is no lattice left, at 117° of scatter.012345-1.70-1.30-1-0.824-0.602-0.398-0.222disturbance amplitude, degrees of azimuth per nodestems out of five returning the counted pair0.020.050.10.150.250.40.6lag-one correlation = 0lag one, on the same sequencesrise 0.005 · 5 stems per point · bars are the pair, line is lag onefilled where the pair agrees with the position counter
Fig. 1 Two of the four causes, on one axis: the disturbance sweep, whose left-hand end is a plant too quiet to have a sequence and whose right-hand end is a plant with no lattice. Both ends refuse and the readout’s output is identical at the two — no pair, no number, nothing to distinguish a specimen at one end from a specimen at the other.
A window that fits inside a rungStems that climb the ladder at four rates, read over a window at the fine end. The condition is a ratio: the window has to be shorter than a rung. 250 internodes at 130 per rung is 1.92 rungs and agrees on 0 of 3; 400 internodes at 130 per rung is 3.08 rungs and agrees on 0 of 3; 250 internodes at 260 per rung is 0.96 rungs and agrees on 3 of 3; 400 internodes at 260 per rung is 1.54 rungs and agrees on 1 of 3; 250 internodes at 520 per rung is 0.48 rungs and agrees on 2 of 3; 400 internodes at 520 per rung is 0.77 rungs and agrees on 3 of 3; 250 internodes at 1040 per rung is 0.24 rungs and agrees on 3 of 3; 400 internodes at 1040 per rung is 0.38 rungs and agrees on 3 of 3. Read over the whole stem instead, every rate returns nothing — 0 of 3, 0 of 3, 0 of 3, 0 of 3 — because the quantity the comb is periodic in changes as the pattern climbs.nodes per rung250-node window400-node windowwhole stem1301.92 rungs0/3 · 1 wrong3.08 rungs0/30/32600.96 rungs3/31.54 rungs1/3 · 1 wrong0/35200.48 rungs2/30.77 rungs3/30/310400.24 rungs3/30.38 rungs3/30/33 stems per cell · rise falls from 0.4 to 0.004 on every onefilled where the angles and the positions agree
Fig. 2 The other two, on another: the rate grid, where a window longer than a rung refuses for reasons that have nothing to do with the disturbance. A stem in the failing cells and a stem at the quiet end of the sweep above return the same thing, which is nothing.

Why that is worse than it sounds

A survey is a set of specimens, and what a survey does with a refusal decides what its numbers mean.

Suppose fifty stems are measured and thirty return a pair. The twenty that do not are either plants whose organs were not placed one at a time — which would be the result of the century — or plants that were quiet, or noisy, or growing quickly, or measured across a transition. A rate of refusal is not an estimate of anything unless the causes can be separated, and reporting “sixty per cent returned a pair” as though the other forty per cent were evidence about mechanism would be the worst possible use of the instrument.

That is the shape of the mixture problem this thread keeps failing to close, and it is worth naming precisely. It is not that the statistics are weak. Both are strong where they work: the comb is 0.64 against a band of 0.11, and the lag-one correlation is −0.6 against the same band. It is that the null has more than one cause, and no single sequence distinguishes them.

Both vary; only one of them varies enough to findEach organ's step exponents, divided by its own mean so the two are comparable. The ogive's run over 15 per cent of their mean across 5 rings. The convex head's run over 1.15 per cent across 5 — inside the band a 2 per cent error on each ring position leaves, so no ruler separates it from a flat disc.0.9000.95011.0501234which step of the ladderexponent ÷ its meanan ogive — 15%a convex head — 1.15%what 2% per ring allows5 rings on the ogive · 5 on the head15% against 1.15%
Fig. 3 The same shape from the cone thread: a quantity is measurable when the difference it makes exceeds what the error allows, and the boundary belongs to the measurement rather than to the object. A refusal here is the region below that boundary, and the region has four walls.

What a ruler removes

Two of the four are visible without the sequence at all.

The recorded scatter of the divergences — a mean and a standard deviation, which a ruler gives — is small at the quiet end and enormous at the loud one. In the model, the quiet end has a scatter under 0.4° and the loud end over a hundred degrees, and there is no ambiguity between them.

So the first line of the specification is: report the scatter alongside the readout. A refusal at 0.2° of scatter is a plant too quiet; at 40° it is a plant with no lattice; at 0.7° it is one of the other two.

That is not a subtle instrument and it is not free either — it needs the same azimuths measured to the same quarter of a degree — but it comes at no additional cost, because the sequence has already been recorded.

A lattice or a wreck, with nothing in betweenEvery run in the sweep, at both kinds of noise. The line at 6° is where a run stops being counted as a lattice, and nothing lands near it: the intact runs reach 1.97° and the destroyed ones start at 8.06°, a factor of 4.1 away.00.50011.5001234567amplitude, by stepscatter, log₁₀ degrees6°intactno latticeplacementfield48 runs · both kindsan empty factor of 4.1 at the cut
Fig. 4 What a ruler reads against what was put in. The scatter is monotone in the disturbance over the whole working range, which is what makes it able to separate the two ends of the window — and the reason a survey that records only the readout is throwing away the cheapest thing it measured.

What a second window would remove

The third is separable too, and the method is one this site has used twice before.

The foundation phase caught a counting bug by counting in three bands of one head and requiring one answer. The expansion phase established that a cylinder’s pair is constant up the stem by counting in three disjoint bands and getting one answer — with the disc as the control, where three bands give three different pairs.

The same trick applies here. Read the window, then read a second window shifted by half its length. On a stem where the pair is not changing the two agree. On a stem where the window straddles a transition, or on a shoot too fast for a window to fit in a rung, they do not — because the two windows contain different mixtures of the two rungs.

That would separate “too fast or badly placed” from the other causes, and it would cost half as much stem again: three hundred and seventy-five internodes rather than two hundred and fifty.

It is specified rather than implemented, and the reason is worth being honest about: implementing it means deciding what “agree” means when one window refuses and the other does not, and that decision is not obvious. Two refusals could be one long stretch of unreadable stem or two different failures. Rather than write a rule and assert it, this is left as the next measurement.

The same counter, on a stem and on a discThe stem returns 2 and 3 in all three bands. The disc returns 13/21, 34/55, 55/89 — three answers to one question, which is why a published count needs to say where it was taken.stem, lower third2 and 3stem, middle third2 and 3stem, upper third2 and 3disc, r = 0.18–0.3213 and 21disc, r = 0.45–0.6234 and 55disc, r = 0.82–0.9955 and 89stem at rise 0.060, disc of 900 points, both at 137.51°counted from coordinates onlyone answer against three
Fig. 5 The three-band control in its original form: the same blind counter in three disjoint bands of a cylinder gives one answer, and in three bands of a disc gives three. The two-window check proposed above is this design applied to a sequence, and it inherits the same logic — one answer means the object has one, and three answers mean the question was about an annulus.

The fourth, which is not separable

That leaves a window that fits inside a rung on a slow shoot, with a scatter in the working range, that still returns nothing.

There are two things that could be: the sequence has no comb because the organs were not placed one at a time in the presence of the ones already there, or the sequence has a comb too weak to clear at this length. And those are not separable by more of the same measurement, because the second is cured by a longer stem and the first is not — so a null at two hundred and fifty internodes has to be followed by a null at a thousand before it means anything at all.

Which puts a number on what the mechanism claim would cost. A positive result is two hundred and fifty internodes; a negative result is a thousand, and the asymmetry is the ordinary asymmetry of a null result, arriving here in a form that can be priced.

Every open question here needs under 28 specimensThe sample size at which each comparison reaches 80 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives.plants show consecutive Fibonacci pairs far more…4and more often even than a coin weighted to a half10a conifer cone's rings are spaced as a cone rather…1multijugate patterns are a real minority rather than…28against 14.7%, if the truth is 90%needs: the pair, at a stated rungagainst 14.7%, if the truth is 50%needs: the pair, at a stated rungagainst φ² = 2.62, if the truth is φ^(2/1.88) = 1.67needs: three ring positions, to ±3%against 2%, if the truth is 15%needs: the pair; the whorl's symmetryspecimens neededexact binomial · α = 0.05 · power 0.81 to 28 specimens
Fig. 6 The population half of the same arithmetic, from the measurement phase: how many specimens a claim about a frequency needs. Multiply by the internode cost of one specimen and the survey this site has specified six times over becomes a quantity of fieldwork rather than an aspiration.

The specification, as it now stands

Every phase since the measurement phase has ended by restating what a survey would have to be, and each time the specification has got longer and more specific. Here it is with the two lines this phase adds:

  • 250 consecutive internodes on an unbranched stem, azimuths to a quarter of a degree per organ. Half a degree costs sixty per cent more stem; one degree does not work at all.
  • A shoot slower than 250 nodes per rung, or an established stem whose parastichy pair does not change across the window — checkable by counting the spirals at both ends of the window.
  • Report the recorded scatter with every readout, refusal or not, so that two of the four causes of a null are separable.
  • Two overlapping windows where the stem allows, so that a third is.
  • And report attempted and returned, not returned alone.

The last line is borrowed from the branching thread, which found the same shape in a different subject: a sample of junctions whose impossible members are silently dropped looks clean and gives an answer that is wrong by more than the distance between the hypotheses being tested. The diagnostic there is one integer — attempted and retained. It is one integer here too.

The counting radius is worth about 10 per centFor each reported pair, the divergence angles consistent with the pair alone and with the pair plus the rise its counting radius implies. The gap between the two series is a factor of 1.10 to 1.10. The radius is still the measurement for where the transitions sit along an axis; it is not what recovers the angle.-1-0.50000.500the reported pairangles left open, log₁₀ °5/88/1313/2121/3434/55the pair aloneand with its radius5 pairs · rise from the rung each pair occupies×1.10 on average
Fig. 7 What the survey specification asks for by the other route: a counted pair pins the divergence angle to a band, and adding the radius the count was made at buys about a tenth more. Every line of a specification is a cost, and the honest summary of six phases is that the costs have gone up each time a new instrument was built, because each new instrument asks for something the previous ones did not.

The asymmetry between the two answers

There is a structural feature of this instrument that is easy to miss and worth stating on its own, because it decides how a survey should be designed.

A pair returned is nearly unambiguous. Three tests have been passed, the readout has named two integers, and those integers can be checked against a spiral count on the same stem. There is one way to get a wrong pair past all three tests — a window straddling a transition with one rung dominant — and it is detectable with a second window.

A refusal is highly ambiguous. Four causes, of which the sequence distinguishes none.

So the instrument is asymmetric in a way that most measurements are not. A thermometer reading twenty degrees and a thermometer reading nothing are both informative, the second saying the thermometer is broken. This readout’s silence says the plant might be quiet, or noisy, or fast, or badly sampled — or that the result of the century is sitting on the bench.

That asymmetry should shape the sampling. A survey that wants to establish a positive — that plants place organs one at a time, that this species is on the Lucas branch — can take specimens as they come and use the ones that return something, provided it reports the refusal rate and does not interpret it. A survey that wants to establish a negative has to control every one of the four causes on every specimen, which is a different and much more expensive design.

A count of m and n pins the divergence to 223°/mnEach dot is one reported pair, and its height is the total width of the divergence angles that could have produced it at some rise. 3/4 leaves 20.7° open; 18/29 leaves 0.426°. The line is 223°/mn, taken from the three highest pairs and drawn back through the rest.-0.50000.50011.5022.50product of the two counts, log₁₀angles left open, log₁₀ °3/44/77/1111/1818/295 pairs · edges found by bisectionwidth × mn = 223°
Fig. 8 The positive side of the same instrument on the Lucas branch, where a returned pair pins the divergence angle to a band whose width falls as the product of the two numbers. A refusal pins nothing, which is the asymmetry stated in the units the survey thread uses.

What this thread has and has not closed

Four phases have worked on the same question — what can be told from a finished plant about the process that made it — and it is worth setting out the ledger.

Closed. The recorded scatter cannot distinguish three kinds of noise; the lag-one correlation distinguishes the one that arrives after the placement from the two that arrive before it; the sign of that correlation reports whether the stem’s rise is falling; the spectrum’s main comb gives the smaller parastichy number and the second comb gives the larger; a lattice with no rule behind it has no comb at all; and both statistics come off one stem over a wide window of disturbance.

Not closed. Which of the two noises that arrive before the choice produced a given pattern — the field and the jostle are still not separated by anything measured. Whether a real plant’s disturbance is inside the readable window, which no model can answer. And the causes of a refusal, which is this essay.

Not attempted. The correlated jostle, carried over untouched from two phases ago: every disturbance in this thread is independent from node to node, and a real apex’s perturbations almost certainly are not. What a correlated disturbance does to the comb is unknown, and it is the kind of unknown that could remove the result rather than qualify it — a disturbance correlated at the parastichy number would manufacture the comb, and nothing here would notice.

That last one is the most important item in the phase’s leavings and it belongs here rather than in a summary, because it is the assumption every essay in this thread rests on and none of them tests.

Three kinds of noise, matched at 0.75° of divergence scatterThe amplitudes differ — field 0.0056 (fraction of the barrier), jostle 0.15 (degrees of azimuth), placement 0.18 (degrees of azimuth) — and are in different units, so they cannot be compared directly. What can be compared is what they produce, and matched here they are within 27% of one another. Everything a finished pattern records about its noise is shared between the three.field — before the choice0.92°amplitude 0.0056jostle — before the choice0.70°amplitude 0.15placement — after it0.79°amplitude 0.183 runs each, at the amplitude that reaches 0.75°27% apart on the ruler
Fig. 9 The pair the earlier thread could not separate: two kinds of noise that arrive before the rule’s choice, matched at the same recorded scatter, producing patterns a ruler cannot tell apart. Neither statistic in this phase separates them either, and the correlated jostle would be a third.

Why this is progress

Three phases have failed to close this problem and it would be easy to read a fourth failure as a thread that should be dropped.

What has changed is that the failure now has a stated cause. The measurement phase recorded that the scatter alone cannot say which of three noises produced a pattern. The phase after found a statistic that separates one of the three from the other two. The phase after that found a second statistic and worried that the two wanted different plants. This phase shows they do not, and finds that the remaining obstacle is not about the statistics at all — it is that a null has four causes, three of which are properties of the specimen and the measurement rather than of the plant.

That is a problem with a design solution rather than an inferential one, and the design is written above. Which is a better place to leave a thread than where it was: a worry about signs, restated three times, never measured.

The line for the specification file

Three phases have added lines to the survey specification and this one adds the line that changes how it is read: a null is not a datum unless its cause is controlled.

Everything else in the specification is a cost — internodes, degrees of precision, nodes per rung. This is a design constraint, and it is the difference between a survey that can report “sixty per cent of stems returned a pair” as a fact about stems and one that can report it only as a fact about the survey.

The four, as a table

For the record, since the essay is mostly about them:

  • too quiet — no pair, and a main comb near 0.98 — separated by the recorded scatter, and caught by the cap
  • too disturbed — no pair and no comb at all — separated by the scatter
  • the shoot too fast — no pair, no spacing winning by a band — separated by a second window
  • the window badly placed — the same reading, and not separable from the one above

Two of the four are separated by a number a ruler gives. The other two are separated by measuring twice, and are not separated from each other at all — which does not matter, because the response to either is the same: move the window, or find a slower plant.

What is not in the table is the fifth possibility, that the plant’s organs were not placed one at a time. It is not in the table because nothing separates it from the others on a single stem, and that is the whole content of the essay.

Where this goes next

nodes per rung250-node window400-node windowwhole stem1301.92 rungs0/3 · 1 wrong3.08 rungs0/30/32600.96 rungs3/31.54 rungs1/3 · 1 wrong0/35200.48 rungs2/30.77 rungs3/30/310400.24 rungs3/30.38 rungs3/30/33 stems per cell · rise falls from 0.4 to 0.004 on every onefilled where the angles and the positions agree A window inside a rung A stem that climbs the ladder has no comb in it at any rate, because the quantity the comb is periodic in changes as it goes. Read a window instead and it comes back, on one condition: the window has to be shorter than a rung — which makes the shoot's rate the thing that decides whether a plant can be asked. plants show consecutive Fibonacci pairs far more…4and more often even than a coin weighted to a half14a conifer cone's rings are spaced as a cone rather…1multijugate patterns are a real minority rather than…34against 14.7%, if the truth is 90%needs: the pair, at a stated rungagainst 14.7%, if the truth is 50%needs: the pair, at a stated rungagainst φ² = 2.62, if the truth is φ^(2/1.88) = 1.67needs: three ring positions, to ±3%against 2%, if the truth is 15%needs: the pair; the whorl's symmetryspecimens neededexact binomial · α = 0.05 · power 0.91 to 34 specimens The survey this site cannot do Four phases of asking for a dataset, and it is still not here. What this phase can do instead is specify it — the fields, the sampling, the sizes, and which of this collection's claims each one would settle. Two of the four fields asked for turn out to be worth less than the asking implied, and one was never asked for at all. -0.20000.2000.4000.6000.5000.75011.251.50divergence scatter, in degrees — the one quantity a plant offerscorrelation between one divergence and the nextplacement noisejostle noisefield noise4 runs per point · band ±0.13every point is a lattice What a quiet plant is worth Almost every measurement gets easier as the effect gets larger. This one gets harder — a stem's divergence sequence stops carrying information about its noise at precisely the scatter where the noise becomes obvious. The specimens worth measuring are the ones that look least interesting.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

  • The test a plant could settle — both name autocorrelation, discrimination, divergence angle, evidence, falsifiability, identifiability, measurement, noise, sample size, specimen, survey
  • What the pair costs — both name autocorrelation, discrimination, divergence angle, honest limits, identifiability, measurement, sample size, specimen, survey
  • A comb is evidence of a rule — both name autocorrelation, discrimination, divergence angle, evidence, falsifiability, measurement, noise
  • The band decides the answer — both name discrimination, evidence, honest limits, identifiability, measurement, sample size, specimen
  • Two readings from one stem — both name autocorrelation, divergence angle, identifiability, measurement, noise, specimen, tolerance
  • What one angle says about the next — both name autocorrelation, discrimination, divergence angle, identifiability, measurement, noise, tolerance

Named objects

A flat tag is an object no other essay names yet.

AutocorrelationDiscriminationDivergence angleEvidenceFalsifiabilityHonest limitsIdentifiabilityMeasurementNoiseRateSample sizeSpecimenSurveyToleranceUnderdetermination