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

The second statistic was the first

The experiment this collection has been specifying was priced as two readings off one sequence, the second of them free. The two readings turn out to be one function looked at twice, so the specification loses a statistic — and gains a cheaper one, a warning about how observables get priced, and a question it could not previously ask.
21 min read 12 figures State the claim, state the testCounted, not admiredOne parameter, one family

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

This collection has been assembling a specification for an experiment nobody has run: what to measure on a real shoot, how much of it, and what each measurement would settle. The specification is now long enough to be worth auditing, and one of its lines has just failed.

The line was a second statistic. It promised that a divergence sequence long enough to read a comb from is long enough to read a slow wander from as well, and that the two disagree about whether a plant’s errors are inherited between touching organs. It was the cheapest thing in the specification, because it required no additional measurements at all.

It is not a second statistic. Both readings are the autocorrelation function of the same disturbance, one evaluated at a lag and one at a block size, and the identity between them holds across every disturbance this collection has built.

What the specification said, line by line

Worth setting out before amending, because most of it is unaffected.

The counts. Record the parastichy pair at a stated radius or height. This is the field everybody reports and the qualifier is the field nobody does.

The radius. A count without the place it was taken is a statement about an annulus with the annulus missing, and this collection’s own measurements put four distinct pairs in one head.

The symmetry and the hand. Two fields that separate a genuine whorled pattern from an ordinary lattice whose families share a factor, and that separate a lattice from its mirror. Neither is expensive; both are absent from the literature.

The sequence. Divergence angles organ by organ, about nine hundred of them at half a degree of precision, which is what it takes to read a comb.

The intervention. Remove one primordium at a stated offset and record whether the next one moves. Repeated across offsets it returns the larger parastichy number with no protractor anywhere.

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. 1 What a single published spiral count settles and what it does not. The specification exists because the answer to the second is most of it.

The line that fails

The sequence line carried a rider: two statistics, one sequence, and no extra cost. A comb at the contact offsets says the errors are shared between touching organs; a slow wander in the block means says they are also passed on and on.

The wander is real and it is in the disturbance. A divergence is a difference of two organs’ errors, and differencing removes low-frequency power, so the sequence a plant hands over is the one place the wander cannot appear. The inherited disturbance’s own block means carry forty-nine times an independent stream’s variance and the divergences it produces carry 0.83.

The wander is in the disturbance and not in what a plant lets you measureEach disturbance measured twice, in the same statistic. On the left, the variance of the block means of the disturbance's own deviates, over blocks of 100, as a multiple of what independent draws would give; on the right, the same quantity for the divergence sequence those deviates produce, over blocks of 128. The left column is what this site measured when it proposed a slow wander as a second observable. The right column is what a botanist would have: a divergence is the difference of two organs' errors, and differencing is exactly the operation that removes power at low frequencies. The disturbance inherited between touching organs goes from ×49.1 — the largest here — to 0.83, which is what independent errors give. The one with a memory in time keeps most of its own.in the disturbanceblocks of 100in the divergencesblocks of 128the horizontal rule is what independent errors giveindependent×0.930.96a memory, ρ = 0.9×16.2410.25inherited, a = 0.7×49.090.83shared once, a = 0.7×2.550.956000 organs · 6 runs eachgenerated from a stated rule, not drawn to look right
Fig. 2 Each disturbance measured twice in the same statistic, on its own deviates and on the divergences those deviates produce. The two columns reorder the disturbances almost completely.

So the specification loses a discriminating statistic, and the sequence’s price does not fall: it was nine hundred organs for the comb and it is nine hundred organs for the comb.

What the pair costs, at a rise of 0.005Five seeded stems at each length, read at four protractor errors. With no reading error the pair needs 250 internodes — against the sixty the single parastichy number costs. At 0.25° per organ it needs 250; At 0.5° per organ it needs 400; At 0.75° per organ it needs 1100. The pattern's own scatter here is 0.70°, so the last of those is a reading error larger than the signal being read.0123451502504007601.1e+3internodes measured on one stemstems out of five returning the counted pairno reading error0.25° per organ0.5° per organ0.75° per organrise 0.005 · disturbance 0.25 · pattern scatter 0.70°generated from a stated rule, not drawn to look right
Fig. 3 What reading a sequence costs in organs and in protractor precision. Nothing in this accounting changes; what changes is how many independent answers the sequence yields once it is paid for.

What replaces it, and it is cheaper

An inherited disturbance does leave something in the block-mean statistic, and it is not a wander. It leaves a hole: at block sizes equal to the offsets it couples at, the statistic falls well below one — 0.22 at eight and 0.21 at thirteen against a null of one, and 0.06 and 0.12 on stems a placement rule grew.

That is cheaper than the comb it comes from, and the difference is worth spelling out for somebody who would have to compute it. A comb is a set of scores across many lags, judged against a sampling band, and it needs the parastichy pair either known or recovered from the sequence itself. The hole is a single ratio at a stated block size, and the block sizes to use are the two parastichy numbers, which a count of the shoot supplies before the sequence is even taken.

The disturbance with the largest wander leaves none in the sequenceHow much of a divergence sequence's variance survives being averaged over blocks, on kinematic lattices. The vertical quantity is B² times the variance of the block means divided by the variance of the sequence, which is one at every block size for independent errors — the arithmetic is normalised for a *differenced* stream, since a divergence is the difference of two organs' errors. A line that climbs is a sequence with power at frequencies below one per block. The disturbances that remember the last error climb to 32 at a block of 128. The ones inherited between touching organs do not climb at all — 1.06 and 0.83 at the same block — although their own deviates carry ×5 and ×49 an independent stream's variance in exactly this statistic. What they do instead is dig a hole: at block sizes of 8 and 13, which are the offsets they couple at, the statistic falls to 0.22 and 0.21.10.11024813163264128block size, in organsvariance of the block means, against independent errorsindependenta memory, ρ = 0.5a memory, ρ = 0.9a memory, ρ = 0.97inherited, a = 0.5inherited, a = 0.7shared once, a = 0.76000 organs · 6 runs eachgenerated from a stated rule, not drawn to look right
Fig. 4 The block-mean statistic across every disturbance shape. The inherited ones are flat at one except at the two offsets they couple at, where they dig a hole; the drifting ones climb.

The null is not one, and that has to go into the specification too. A stem placed by a rule with independent errors still digs a shallow hole at its own contact offsets — 0.20 at a block of eight — because the rule’s placements are correlated where its neighbours are. A test calibrated against a null of one would have an inflated false-positive rate, and the calibration is available.

Through the rule, the drift survives and the inheritance still does notHow much of a divergence sequence's variance survives being averaged over blocks, on stems the rule grew. The vertical quantity is B² times the variance of the block means divided by the variance of the sequence, which is one at every block size for independent errors — the arithmetic is normalised for a *differenced* stream, since a divergence is the difference of two organs' errors. A line that climbs is a sequence with power at frequencies below one per block. The disturbances that remember the last error climb to 46 at a block of 64. The ones inherited between touching organs do not climb at all — 1.51 and 1.90 at the same block — although their own deviates carry ×— and ×— an independent stream's variance in exactly this statistic. What they do instead is dig a hole: at block sizes of 8 and 13, which are the offsets they couple at, the statistic falls to 0.06 and 0.12.10.11024813163264block size, in organsvariance of the block means, against independent errorsindependenta memory, ρ = 0.9a memory, ρ = 0.97inherited, a = 0.5inherited, a = 0.7900 organs · jostled at 0.25° · 3 stems eachgenerated from a stated rule, not drawn to look right
Fig. 5 The same statistic on stems the rule grew rather than on lattices with errors added. The holes are deeper and so is the shallow one under independent noise, which is the null a real test would have to be read against.

And a question the specification could not previously ask

The withdrawal comes with an addition, and the addition is not a consolation prize.

A wander in a divergence sequence says the disturbance has a memory in time: that the conditions under which organ five hundred was placed resemble those of organ four hundred more than those of organ one. That is a question about a shoot’s environment and its own development, and this collection had no observable for it at all.

It is also readable quantitatively rather than as a yes or no. For a disturbance that remembers the last error with coefficient ρ, the block-mean statistic climbs to one over one minus ρ and stops — measured at 1.38, 1.90, 3.15, 9.51 and 30.94 against closed-form values of 1.43, 2.00, 3.33, 10.00 and 33.33. So the height of the plateau names the memory’s length, and a sequence long enough to see the plateau is a sequence that measures it.

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 116° of scatter.012345-1.70-1.30-1-0.824-0.602-0.398-0.2220disturbance amplitude, degrees of azimuth per nodestems out of five returning the counted pair0.020.050.10.150.250.40.61lag-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. 6 Two statistics of one sequence as this collection has used the phrase elsewhere. The lesson from this round applies to every such pair: two numbers off one series are two measurements only if they are not functions of each other.

Why “free” was the word that went wrong

The rider’s actual failure was not in the statistics. It was in the word “free”, and the way that word is used about measurements is worth taking apart because it recurs.

A statistic computed from data already collected is free in one sense: no additional specimen, no additional protractor work, no additional season. It is not free in the sense that matters for an argument, which is whether it carries information the other statistic does not. Those two senses of free were run together in a single sentence, and the sentence read as though the second implied the first.

They come apart in both directions, which is why the confusion is not simply carelessness. A statistic can cost nothing extra and add a great deal — the symmetry of a point set is read off a photograph already taken and settles a question the counts cannot touch. And a statistic can cost nothing extra and add nothing at all, which is this case.

The way to tell them apart is not intuition about the statistics. It is to compute both on the same simulated data and look at whether one predicts the other. Here that took a few minutes and produced an exact identity; had it produced a scatter plot with structure in it, the rider would have stood.

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. 7 The same distinction elsewhere in this collection: which quantities an instrument can measure and which it merely returns numbers for. A number that is a function of another number is on the wrong side of that line, however cheaply it was obtained.

The specification as it now stands

Amended in three places and unchanged everywhere else.

The sequence line loses the wander as a test of inheritance and keeps the comb. It gains the hole at the contact offsets, with a stated null of about 0.2 rather than 1, and it gains the plateau as a measurement of environmental drift.

The intervention line gains the two-organ variant, which reaches coarse arrangements a single removal cannot disturb, and gains the whorled variant, where removing either organ of one node must give the same answer — a control internal to a single shoot.

And a new line: report the scatter and the block-mean curve together. A small organ-to-organ scatter reads as a well-behaved shoot and can equally be a rule correcting hard against a large slow disturbance. The two numbers separate those, and neither does it alone.

The measurement is limited by the protractor, not by the plantThe peak falls as the reading error grows, and it falls by an arithmetic factor with nothing fitted: a position error enters two consecutive divergences with opposite signs, adding variance at every lag while the pattern's signal sits at one. At a quarter of a degree the readout is right on all 5 runs; at half a degree on 2; at a degree on 1. Below the dashed floor the peak is the largest of thirty noisy numbers rather than a measurement.00.2000.4000.6000.80000.50011.502reading error on each organ's position, in degreesheight of the peak at the parastichy numberwhat noise alone givesthe threshold a reading must clear5/5 right5/5 right2/5 right1/5 right1/5 rightpredictedrise 0.008 · 5 runs · pattern scatter 0.75°peak × σ²/(σ² + 2ε²), nothing fitted
Fig. 8 How much of what this collection wants to measure is decided by protractor precision rather than by specimen count. The sequence statistics live at the demanding end of it, which is why removing one of them matters and adding a cheap one helps.
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 The sample-size arithmetic for the claims this specification carries. A per-stem false-positive rate on a null arrangement is the number that decides how many shoots a claim about combs would need.

What the specification is for

It is worth restating, because a specification that keeps being amended can start to look like an end in itself.

Every frequency, every threshold and every angle in this collection is about the geometry or about a model, and says so. The census of divergences is a census of what the arithmetic hands over, not of what grows in a field. The rate at which a pattern climbs the ladder is a property of a rule. The boundary at which a lattice stops being a lattice is a boundary in a simulation. None of that is a defect — the whole discipline here is to state a claim and give it a test it could fail — but it means the collection has exactly one way of being wrong that it cannot detect from the inside, which is that the rule is not what a plant does.

A specification is the list of things a botanist could record that would put a number from a field beside a number from a model. It is not a wish list: every line on it is a field that is cheap to record, absent from the literature, and attached to a specific claim that would be settled by it.

That is why an amendment is worth an essay. Removing a line means one claim has lost its route to a measurement. Adding one means a claim has gained one. And this round does both, which is the ordinary shape of finding out that an instrument measures something slightly different from what it was said to.

The list is now five fields and two interventions, none of them requiring equipment a plant physiology laboratory does not have, and the whole of it would fit on a page. Its cost is dominated by the sequence, which is a season of patient measurement on one shoot; everything else is a photograph and a count.

At 3.0 per cent on each ring, no number of rings shows the driftThe gaps between consecutive rings are 1.626, 1.631, 1.657, 1.763. Two rings give one gap and no way to disagree with itself; three give two gaps and a fit with nothing left over. The question is how many gaps it takes for their spread to exceed what the measuring error can explain, and the answer depends on the error as much as on the organ.error per ringrings neededspread vs bound1%4 of 5 rings1.9% > 1.4%1%5 of 5 rings8.2% > 2.8%2%5 of 5 rings8.2% > 5.7%3%not on this oneunder 8.5%5%not on this oneunder 14.1%8%not on this oneunder 22.6%an ogive · 5 rings availableno rings at 3%
Fig. 10 How many rings of a head are needed before a count means anything, from this collection’s own machinery. The specification’s expensive line is the sequence, and its cheap lines are the ones this kind of arithmetic settles.

How an observable gets mispriced

The general lesson is short and this collection is going to keep needing it.

An observable is a function of what can be measured. Between a hypothesis about a plant and a number a botanist can write down there is always at least one operator, and every operator here destroys something. Positions are differenced to get divergences. Divergences are wrapped modulo a turn, and modulo a fraction of a turn on a whorled shoot. Cell areas are read against a background. Counts are taken in an annulus.

The wander was proposed by measuring it on the hidden side of a difference operator and asserting it on the visible side. Nothing about the model was wrong and nothing about the measurement was wrong; the sentence between them did not survive the operator, and nobody had computed the statistic on the side a plant supplies.

The protection costs minutes: compute the proposed statistic on the visible side before writing the sentence. That is now a rule here, and it is the third time a version of it would have saved a claim.

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. 11 The same principle in the specification’s oldest line. Recording where a count was taken costs about a tenth more effort than recording the count, and without it the count is a statement about an unspecified annulus — an operator applied and not reported.
What the experiment costs, in internodesThe combined sampling band of two autocorrelations falls as one over the root of the sequence length. The difference to be resolved is 0.76 — between noise that arrives before the primordium is placed and noise that arrives after — so the count needed is 56 internodes on a single stem. Every other open question in this collection is priced in tens of specimens.00.2500.5000.750100200300internodes counted on one stemsmallest difference in correlation the count can resolvethe difference to resolve — 0.7656 internodesmatched at 0.75° of scatterone stem, counted once
Fig. 12 How much stem a sequence measurement needs, in internodes. The cost of the experiment is dominated by this rather than by the number of statistics computed afterwards, which is exactly why a statistic that is free sounds like such a good bargain.

What this does not say

It does not say the specification is weaker overall. It loses one test of one hypothesis and gains a cheaper test of the same hypothesis plus a new test of a different one. The intervention half has gained two variants in the same round.

It does not say the comb is in doubt. The comb is a separate statistic, measured separately, and it is untouched — including this collection’s earlier concession that it is evidence of a re-transmitted disturbance rather than of a placement rule.

It does not say the identity makes the block-mean statistic useless. A quantity that equals the autocorrelation at one lag is still worth computing if it is easier to compute or easier to interpret, and at large block sizes it is both. What it is not is independent information.

It does not say the amendment is the last one. The specification has been amended in every round it has existed, and the amendments have so far been about what a measurement can settle rather than about what to measure. That is the shape of a specification that is being tested rather than written.

And it does not say a real sequence will behave like these. Everything here is measured on stems grown by a rule and on lattices with stated errors added. What the specification does is say what to record so that a real sequence could be compared with them.

The three amendments in one place

For anybody reading the specification rather than this essay, the changes are:

Delete the rider on the sequence line that promises a slow wander as a second test of inheritance. It is not a second test and the sequence cannot show it.

Add to the sequence line: the block-mean statistic at block sizes equal to the two parastichy numbers, with a null of about 0.2 rather than 1, as a cheap confirmation of the comb. And, separately, the shape of the same statistic across block sizes as a measurement of environmental drift, with the plateau naming the memory’s length.

Add to the reporting line: the per-organ scatter and the block-mean curve are to be given together, because a small scatter is ambiguous between a quiet environment and hard correction against a loud one.

The check that would refuse it

The claims in this essay are about what a measurement would cost and what it would settle, and two of them are checkable arithmetic.

The identity between the block-mean statistic and the autocorrelation has to hold at every disturbance shape and every block size, within the sampling error of the noisier estimator. If it failed anywhere the two statistics would be independent after all and the line would not need amending.

The hole at the contact offsets has to be well below one at those offsets and near one away from them. Both halves are required: a statistic that was below one everywhere would be a statistic with a scale error rather than a signature.

And the null has to be measured rather than assumed. The shallow hole a rule-grown stem digs under independent noise is what a real test would be read against, and asserting that it is not one is what keeps the specification honest about its own false-positive rate.

Where this goes next

in the disturbanceblocks of 100in the divergencesblocks of 128the horizontal rule is what independent errors giveindependent×0.930.96a memory, ρ = 0.9×16.2410.25inherited, a = 0.7×49.090.83shared once, a = 0.7×2.550.956000 organs · 6 runs eachgenerated from a stated rule, not drawn to look right A difference forgets a drift This collection proposed a second observable and priced it as free: if a plant's errors are inherited between touching organs, the divergence sequence should carry a slow wander as well as a comb. The disturbance with the largest wander of any built here leaves none at all in the sequence, because a divergence is a difference and differencing is what removes a drift. riseorgans back from the tip →run · isolated24681012140.023/55 · 70.0135/880.0095/88 · 120.0085/88 · 120.0068/13120.0058/13136 rises · 1536 azimuthsgenerated from a stated rule, not drawn to look right The ablation a plant would survive The intervention this site proposed returns a spiral count from a yes-or-no answer, needs no protractor, and was specified at one rise. Measured across the ladder it acquires three conditions a real experiment would have to meet — and one of them is that the plant must not be too coarsely patterned, or nothing will go wrong at all. 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 rounds of asking for a dataset, and it is still not here. What the work here 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.

Shares its objects with

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

  • The control a survey would need — both name autocorrelation, discrimination, evidence, falsifiability, honest limits, measurement, measurement error, sample size, specimen, survey
  • The survey loses its second outcome — both name autocorrelation, discrimination, evidence, falsifiability, honest limits, measurement, measurement error, sample size, specimen, survey
  • The test a plant could settle — both name autocorrelation, discrimination, evidence, falsifiability, measurement, measurement error, sample size, specimen, summary statistic, survey
  • The experiment this site can specify — both name discrimination, evidence, falsifiability, honest limits, 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

Named objects

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

AutocorrelationClaim testingCounting radiusDiscriminationEvidenceFalsifiabilityHonest limitsMeasurementMeasurement errorNegative resultSample sizeSpecimenSummary statisticSurveyUntested claim