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

The survey loses its second outcome

The specification written one phase ago names three results the survey could return, and the second — a ratio near or above 1.30, read as evidence against the placement rule — is the one that would have been worth publishing. It does not survive this phase. The ratio moves with where the plant sits between two transitions, and it moves again with the colour of the plant's own disturbance.
20 min read 11 figures State the claim, state the testForm is not mechanismCounted, not admired

Worth reading first: The survey this site cannot do · Errors that pass between organs · A disturbance with a memory.

One phase ago this collection wrote down a survey specification with six requirements and three outcomes. The outcomes were the important half. A specification that does not say in advance what would count as which result is not a test, and the three were:

Pairs agreeing across six plants, ratio near 0.65. Consistent with a placement rule and with any transport whose coupling leans about three to one towards the smaller parastichy number.

Pairs agreeing, ratio near or above 1.30. Evidence against the placement rule as this site implements it.

Pairs disagreeing across plants. Either the shoots are not all on the same rung, or the comb is being manufactured by something that varies from plant to plant.

The middle one is gone. This essay says why, adds the two requirements that follow, and gives the specification as it now stands.

The first problem: the number is not one number

The ratio was measured at one rise. Swept across two rungs it is a U — a floor of about 0.79 two thirds of the way up a rung, climbing to 2.8 at the fine end of the coarser rung measured.

The ratio is a U across every rung, and its floor is the number that was reportedThe ratio of the second comb to the main comb, on five stems at each of 15 rises spanning two rungs, against the ladder's own coordinate for where each rise sits inside its rung. Both rungs give the same shape: a floor of 0.71 and 0.79 about two thirds of the way up, climbing towards the transition at either end. The dashed line is a transported disturbance with no rule in it at 1.29, which does not vary with the rise at all — a kinematic lattice's angle sequence has no rise in it. Where the rule's curve crosses that line the two accounts are indistinguishable.1200.2500.5000.7501where the rise sits in its rung — 0 at the next transition down, 1 at the last onesecond comb ÷ main comb5/8 rung8/13 runga transported disturbance, no rulethe floor, 0.795 stems a point · 1152 azimuthsgenerated from a stated rule, not drawn to look right
Fig. 1 The ratio against where the rise sits in its rung. A plant near the middle of a rung reads near 0.8; the same plant, same species, same disturbance, caught a fifth of a rung from a transition, reads above 1.2.

So a reading of 1.3 is not evidence against a placement rule. It is what a placement rule gives on a shoot that is approaching a transition, and a shoot approaching a transition is not a rare or pathological object — the ladder puts one transition every factor of 2.6 in the rise, and a growing shoot passes through them all.

The rival’s curve, meanwhile, is flat. A kinematic lattice’s angle sequence does not contain the rise at all, so a transported disturbance gives the same number at every rise on a rung — 1.24 at equal coupling and 1.29 weighted by distance. The two curves cross, and near a transition the quantity distinguishes nothing.

The two combs, in the proportions the rule gives themThe ratio of the second comb to the main one, for a kinematic lattice whose errors are inherited from its two contact neighbours, against how unevenly that inheritance is split. The horizontal line is where the placement rule's own stems sit, at 0.65. Weighted by distance — the coupling a d⁻³ interaction would give, which at this rise favours the 13-neighbour by 1.26 to one because the 13-hop is the shorter — the forgery sits at 1.46, well above the rule. It reaches the rule's value only at about 3 to one the other way, which is a factor of 4 against what distance supplies and in the opposite direction.0.4000.6000.80011.201.40-0.30100.1760.3010.4770.699how much more strongly the error is inherited from the 8-neighbour than from the 13-neighbourthe second comb's strength as a fraction of the main comb'sthe placement rule: 0.65equal combs1:21:11.5:12:13:15:1at 3:1 the ratio is 0.75kinematic lattice · 3 seeds a pointgenerated from a stated rule, not drawn to look right
Fig. 2 The comparison as the specification assumed it: two numbers a factor of two apart. That factor exists in the middle of a rung and nowhere else.

The second problem: the number follows the disturbance

Worse, and it is the reason the outcome is retracted rather than qualified.

Drive the same placement rule with disturbances of different correlation structure — all at the same displacement per organ, all leaving a lattice standing — and the ratio moves across most of the range the discriminator was supposed to occupy. Independent errors give 0.80. Errors that remember the last one give 0.76 to 0.81. A disturbance that repeats every eight organs gives 0.45. And errors inherited from the two contact neighbours give 1.02 and 1.09.

The ratio follows the disturbance, not the ruleThe ratio of the second comb to the main comb on stems grown by the placement rule and jostled by seven different disturbances, all at 0.25° of displacement per organ and all on the same rule. Independent errors and errors with a memory return 0.76–0.81, which is the value this site measured for the rule. A periodicity at the smaller parastichy number takes it down to 0.45; errors inherited from the contact neighbours take it up to 1.09, most of the way to the 1.24 a transported disturbance gives with no rule in it at all. So the quantity separates arrangements by how their errors are related, not by whether anything computed the positions.second comb ÷ main comb, at 0.25° of displacementthe rule, 0.79no rule at all, 1.24independent0.80a memory, ρ = 0.50.78a memory, ρ = 0.90.76a memory, ρ = 0.970.81repeating every 80.45inherited, a = 0.51.02inherited, a = 0.71.095 stems a row · rise 0.005generated from a stated rule, not drawn to look right
Fig. 3 Seven disturbances, one rule, one rise, one displacement per organ. The quantity that was supposed to identify the process spans 0.45 to 1.09 without the process changing at all.

A plant that both computes its organ positions and passes a fraction of each organ’s displacement to the organs it touches — which is what a plant most plausibly is — reads at about 1.05. That is inside the band the specification called evidence against the rule.

So the second outcome does not merely need a wider tolerance. It has to be withdrawn: a high ratio is consistent with a placement rule whose disturbance is transmitted between contacts, and there is no reading of the ratio that counts against a placement rule.

The third problem, which is smaller and was found first

The number itself was wrong by a fifth. The rule takes its minimum over 384 sampled azimuths in every flat run on this site, which is a step of 0.94° against a disturbance of a quarter of a degree; the quantisation is white, it dilutes the second comb more than the first, and refining the grid moves the ratio from 0.62 to 0.79.

At 384 azimuths the ratio is 0.62; converged it is 0.82The comb ratio and the recorded divergence scatter at a rise of 0.005, against how finely the rule samples the circle when it takes its minimum. At 384 azimuths — the grid every flat run on this site uses, and the grid the previous phase's 0.65 was measured on — the step is 0.94°, which is larger than the 0.25° disturbance the stems carry. The quantisation is white noise, it dilutes both combs, and it does not dilute them equally. The ratio settles at 0.82 from 1152 azimuths up, and the scatter loses 0.19° that belonged to the grid rather than to the stem.0.4000.6000.8001azimuths the rule samples the circle at (logarithmic)ratio, and the scatter a protractor would record, in degrees384768115215362304ratioscatterthis phase works hererise 0.005 · 5 stems a pointgenerated from a stated rule, not drawn to look right
Fig. 4 The correction that arrived first and matters least. It narrows the discriminator’s margin from a factor of two to a factor of 1.6 — which would have been the headline had the other two problems not made the margin irrelevant.

It is worth putting the two problems side by side, because they are different in kind and only one of them was foreseeable.

The rung dependence is a confound: a second variable the reading depends on, which can be measured and controlled for. Requirement seven does exactly that, and once the rise is recorded the confound is gone. Nothing about it says the ratio is the wrong quantity.

The disturbance dependence is a failure of the quantity. There is no variable to record, because the thing the ratio turns out to depend on — how a plant’s errors are correlated from organ to organ — is exactly what nobody can measure and what the ratio was going to be used to infer. A survey cannot control for the autocorrelation of a plant’s disturbance by recording it; recording it is the harder measurement.

The previous phase came within one step of seeing this. It removed a requirement from the specification on the grounds that a disturbance with a memory manufactures no comb — so a survey need not control for correlation length. That was right about a memory and it was generalised, here included, into a belief that the comb machinery was insensitive to the disturbance’s structure. It is insensitive to a correlation at lag one and highly sensitive to a correlation at the contact offsets, and the contact offsets are the physically obvious place for a plant to have one.

A memory manufactures nothingThe largest comb mean found in a kinematic lattice whose azimuth errors are an AR(1) process, against the coefficient of that process, over eight seeds at each point. The dashed line is where the rule's own stems sit, at 0.64; the shaded strip is three sampling bands. Every point is inside the strip — 0.028, 0.026, 0.022, 0.014, 0.015 at ρ = 0.3, 0.5, 0.7, 0.9, 0.97 — and the readout returns nothing on 40 runs out of 40. A correlated error is not a periodic one.00.2000.4000.6000.3000.5000.7000.9000.970how strongly each error remembers the last, ρthe largest comb mean anywhere in the thirty lagsthe rule's own stems: 0.64three sampling bands0.0280.0260.0220.0140.015kinematic lattice · AR(1) errorgenerated from a stated rule, not drawn to look right
Fig. 5 The control the previous phase removed, correctly. What it does not license is the more general belief that the readout does not care what shape the disturbance has, and this phase is the correction to that belief rather than to this figure.

The specification as it now stands

Requirements one to six are unchanged and are not repeated here. Two are added.

Seven. The rise, measured on the same stretch of shoot as the angles. Height gained per organ, divided by the shoot’s circumference, which is two lengths and a division. It is needed to place the plant on the ladder, and therefore to know whether the reading is near a transition. A shoot whose rise puts it within a fifth of a rung of a transition should be excluded, and the exclusion has to be decided from the rise rather than from the reading, because the reading is what is being protected.

Eight. A statement of what the ratio is being used for. This sounds like bookkeeping and it is the substantive one. After this phase the ratio supports exactly one inference — the errors on this shoot are correlated at the contact offsets in a particular proportion — and does not support any inference about whether a rule chose the positions. A survey may still report it, and it should, because it is a number about a plant that nobody has ever measured. What it may not do is call a high value evidence against a mechanism.

Agreement between two windows happens only on a slow enough shootFive stems at each of four rates and five disturbances, each read through two overlapping windows of 250 internodes. A filled mark is agreement — both windows reported the same pair; a half mark is a disagreement; a small mark is one window reporting and one refusing; an open mark is silence. Agreement appears 0 times in 25, 1 times in 25, 14 times in 25, 14 times in 25 at 130, 250, 400, 700 nodes per rung, and the two rates it is almost absent from are the two at which a rung is no longer than the window.nodes per rung0.050.150.250.50.9disturbance1301.92 rungs0 of 25 agree2501.00 rungs1 of 25 agree4000.63 rungs14 of 25 agree7000.36 rungs14 of 25 agreeagreedisagreeone-sidedsilentfive stems a cellgenerated from a stated rule, not drawn to look right
Fig. 6 Requirement five, from the previous phase: two overlapping windows, and only agreement reported. It survives untouched — the windows certify the rate, which is a verdict rather than a continuous number, and verdicts are the part of this instrument that has held up.

What it cost to find out, which is worth recording

The retraction did not come from re-reading the specification. It came from two sweeps run for other reasons.

The rung dependence turned up because the previous phase’s plan asked for a curve and predicted the wrong reason for expecting one — it said the hop asymmetry would change with the rise and take the ratio with it. The asymmetry does change; the ratio does not follow it; and the curve that does exist has a different shape and a different cause. So the plan was right that a sweep was needed and wrong about everything the sweep would show, which is the usual outcome when a prediction is made from an argument rather than from a measurement.

The disturbance dependence turned up because the same plan asked for a coloured jostle inside the rule, with a prediction attached: that a self-correcting rule would shorten whatever correlation length it was given, so the previous phase’s negative result should hold a fortiori. The rule does damp a correlated disturbance — a stem survives three times the displacement when the organs share it — and the ratio moves anyway. Both halves of that prediction were tested and one of them failed.

Two predictions written into a plan file, both tested in the phase that followed, both wrong in informative ways. That is the arrangement working: the value of writing a prediction down is not that it turns out right.

What the survey can still establish

Stripping an outcome out of a specification is only worth doing if what is left is stated as carefully as what went.

The parastichy pair, from the angles alone. Solid, unaffected by everything in this phase, and still the thing no botanist does: a count obtained from a list of divergence angles with no photograph and no positions.

That the plant’s errors are not independent. A comb rules out an arrangement whose organ positions are independent draws around an ideal lattice. That is a real statement about a plant and it is the null model a botanist would otherwise be arguing against.

The correlation structure of the errors, weakly. A ratio near 0.45 says something repeats; a ratio above one says the errors are transmitted between contacts more strongly than the geometry alone would give. Both are statements about the disturbance, and both are new.

And the shoot’s rate, from the two windows agreeing. Also a verdict, also unaffected.

It is worth noticing what those four have in common. Three of them are verdicts — a pair, a presence, an agreement — and the fourth is a constraint rather than an identification. Every quantity in this collection that has survived contact with a forgery, a grid refinement or a change of disturbance has been of that kind, and every one that has not survived has been a continuous number asked to carry an inference about mechanism.

That is now a pattern with four instances and it is worth stating as a working rule for the phases after this one: a continuous statistic read off a finished pattern is a description of the pattern, and turning it into a claim about the process is the step that keeps failing. The comb was such a step. The ratio was such a step. What has replaced them is an experiment whose output is a yes or a no.

A count of m and n pins the divergence to 221°/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. 2/3 leaves 38.8° open; 34/55 leaves 0.118°. The line is 221°/mn, taken from the three highest pairs and drawn back through the rest.-10111.5022.503product of the two counts, log₁₀angles left open, log₁₀ °2/33/55/88/1313/2121/3434/557 pairs · edges found by bisectionwidth × mn = 221°
Fig. 7 What a count is worth on its own, from the phase that priced it. Everything in the list above is of this kind: a fact about the arrangement or about its errors, rather than about what made it.

What has replaced the lost outcome

Not nothing, which is the reason this essay is not a retreat.

The same phase that took the second outcome away produced an experiment that does what the second outcome was supposed to do. Remove one organ from a settled apex and the placement rule predicts that the next organ moves — by between 2.6° and 168°, depending on which organ was removed — and that the number of organs whose removal matters is the larger parastichy number. The transported-error account predicts no displacement at any offset, because in it no organ’s position was ever a function of which organs exist.

The next organ moves for the last 13, and for no othersOne row per organ removed, counted back from the tip of a stem at a rise of 0.005 whose counted pair is 8 and 13. Removing any of the last 13 moves the next organ by 2.6° to 167.6°; removing an older one moves it by at most 0.47°, which is under the azimuth grid. The boundary is at 13, and 13 is the larger parastichy number — so the experiment counts the spirals without measuring an angle.organ removed, counted back from the tiphow far the next organ moves, in degrees1138.0°284.4°353.4°4167.6°529.3°6101.7°7120.7°816.4°9165.2°1056.7°1181.1°12140.6°132.6°— the front ends here140.0°150.0°160.5°rise 0.005 · pair 8/13generated from a stated rule, not drawn to look right
Fig. 8 The replacement. Where the ratio offered a factor of 1.6 between two accounts on a quantity that turns out to depend on two things nobody would have controlled for, the intervention offers tens of degrees against nothing.

That is a better test on every axis that matters. The effect is thirty times the plant’s own noise rather than a fifth of it. It needs one apex rather than six shoots of 375 internodes. It does not need a protractor good to a quarter of a degree. It does not care where the plant sits in its rung — indeed the boundary it measures is the rung. And it has a control built into it: the offsets past the boundary, where nothing is predicted to happen.

One thing the intervention does not replace, and it should be said here rather than in an essay about the intervention. The survey measures plants; the ablation measures one apex under a needle. A shoot that has been operated on is a shoot that has been operated on, and a mechanism established on surgically disturbed apices is established on surgically disturbed apices. The two experiments answer to different objections and neither retires the other.

Every open question here needs under 34 specimensThe sample size at which each comparison reaches 90 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 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
Fig. 9 What the observational survey costs in plants, from the phase that computed it. The intervention’s costs are in apices and in surgery, which is a different budget and a smaller one.

What a survey would have to see to move anything

With the second outcome gone, it is fair to ask what result a survey could now return that would change anybody’s mind about a mechanism. Three, and they are weaker than the one that was lost.

No comb at all, on a shoot that meets every requirement. That excludes both accounts as this site implements them: a placement rule makes a comb, and a contact-transported disturbance makes a comb. An arrangement with neither would be one whose organ positions are independent draws about an ideal lattice, which is the null model, and finding it would be a large negative result about the subject.

A ratio well outside the range any of the arrangements here produce. The seven disturbances measured span 0.45 to 1.09 through the rule, and the kinematic forgery gives 1.24 to 1.29. A plant reading 2.5 in the middle of its rung, or 0.2, would be doing something none of the machinery in this collection does.

A ratio that varies systematically between species in a way the rise does not explain. That would be a measurement of how differently different apices transmit, which is a comparative question and the kind a survey is actually good at.

None of those is the clean discriminator the previous phase thought it had, and all three are honest.

Why the specification is kept anyway

It would be tidier to withdraw the survey and put everything into the intervention. That would be a mistake for two reasons.

The survey is the only thing that can say what plants actually do. An ablation on one species reports on that species’ apex. A survey across a genus reports how much of the variation the subject’s claims are made about is real. Those are different questions and this collection has never confused them.

And the survey’s surviving outcomes are the ones nobody has data on. No one has published the autocorrelation of a plant’s divergence errors. No one has reported a parastichy pair recovered from angles. The measurements that remain are the measurements the specification was originally for, before a discriminator was hung on them.

What is being retired is one inference, not one experiment.

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 3 per cent error on each ring position leaves, so no ruler separates it from a flat disc.0.9000.95011.050123which step of the ladderexponent ÷ its meanan ogive — 15%a convex head — 1.15%what 3% per ring allows5 rings on the ogive · 5 on the head15% against 1.15%
Fig. 10 What a real specimen supports, from the thread that has been building the specification for eight phases. Losing an outcome from it is the first change in three phases that made it smaller.
Both ends of the window are silent, and a ruler tells them apartThe scatter recorded on stems at 400 nodes per rung, against the disturbance that produced it, with the stems that returned no reading at all marked as open. Silence at the quiet end comes with a scatter of 0.38 and 0.44°, which any botanist would call an orderly plant; silence at the disturbed end comes with 56°, which nobody would call a pattern. The two refusals look identical in the instrument's output and are three orders of magnitude apart in a quantity measured with a protractor.-0.50000.50011.50-2.30-2-1.50-1-0.500disturbance, in degrees of azimuth (logarithmic)the scatter a protractor would record, in degrees (logarithmic)no lattice left above here3 of 5 silent3 of 5 silent400 nodes a rung · five stems a pointgenerated from a stated rule, not drawn to look right
Fig. 11 And the standing lesson it belongs to: an instrument that reports has to be asked what its report excludes, and one that refuses has to be asked why. The ratio reported, and for a phase nobody asked what its reporting ruled out.

Where this goes next

stemwhat the angles say18/10not the lattice's pair28/12not the lattice's pair3refused48/10not the lattice's pair58/12not the lattice's pair68/12not the lattice's pair78/12not the lattice's pair88/11not the lattice's pairthe positions say 8/13kinematic lattice · error of period 8generated from a stated rule, not drawn to look right The control a survey would need A comb no longer shows that a plant computes its pattern, so the survey this site has been specifying for five phases has to change. What it loses is its headline; what it gains is a measurement a botanist can actually make — six requirements, four of them already in the specification, and a quantity nobody has ever reported. 1200.2500.5000.7501where the rise sits in its rung — 0 at the next transition down, 1 at the last onesecond comb ÷ main comb5/8 rung8/13 runga transported disturbance, no rulethe floor, 0.795 stems a point · 1152 azimuthsgenerated from a stated rule, not drawn to look right The ratio was the floor of a curve The last phase left one number standing between a placement rule and a transported disturbance, measured at one rise, with the explanation that the geometry there happens to favour the larger parastichy number. Swept across two rungs the number is a U — a floor of about 0.79 two thirds of the way up a rung, climbing past 2.8 as a transition approaches — and the geometry is flat exactly where the curve is steepest. organ removed, counted back from the tiphow far the next organ moves, in degrees1138.0°284.4°353.4°4167.6°529.3°6101.7°7120.7°816.4°9165.2°1056.7°1181.1°12140.6°132.6°— the front ends here140.0°150.0°160.5°rise 0.005 · pair 8/13generated from a stated rule, not drawn to look right The experiment this site can specify For eight phases the outstanding item has been a survey — photographs, a protractor, hundreds of specimens — and it has not been done. The intervention is a different kind of ask, and a cheaper one: a needle, one apex, and a yes-or-no per ablation. Here is what it would cost, what it would settle, and the four ways it could come out.

Shares its objects with

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

  • A refusal with a reason — both name autocorrelation, discrimination, honest limits, measurement, measurement error, rung, sample size, specimen, survey
  • The test a plant could settle — both name autocorrelation, discrimination, evidence, falsifiability, measurement, measurement error, sample size, specimen, survey
  • What a quiet plant is worth — both name autocorrelation, discrimination, evidence, honest limits, measurement, measurement error, sample size, specimen, survey
  • What a refusal does not say — both name autocorrelation, discrimination, evidence, falsifiability, honest limits, measurement, sample size, specimen, survey
  • The band decides the answer — both name artefact, discrimination, evidence, honest limits, measurement, measurement error, sample size, specimen
  • What the pair costs — both name autocorrelation, discrimination, honest limits, measurement, measurement error, sample size, specimen, survey

Named objects

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

ArtefactAutocorrelationDiscriminationEvidenceFalsifiabilityHonest limitsMeasurementMeasurement errorNull modelRiseRungSample sizeSpecimenSurveyTransitions