Field

The claims, measured

The nautilus, the sunflower and the golden angle arrive with more confident wrong statements attached than any other subject on this fleet. Each one gets a test and a number — including the one that turns out to be right.

The spiral counts four different divergence angles produce. Fibonacci counts come from one angle. The Lucas angle — which the same dynamical model reaches on a different branch — gives 47 and 76, and neither number is a Fibonacci number.

Fibonacci is a branch, not a law

Fibonacci counts come from one branch of the model. The Lucas branch — which the same rule reaches at a different growth rate — gives 47 and 76, and neither is a Fibonacci number. The sequence is a consequence of an angle rather than a property of plants.

The version of the claim that does survive measurement. The golden angle scores 0.4377, against 0.3462 for the best of 938 other angles sampled. The dashed line is Hurwitz's 1/√5, which no number can exceed.

The claim that survives

Of the three famous assertions about this subject, one is out by a factor of two, one is true of a branch rather than of plants, and one is right — in a sharper form than the version usually told, and about arithmetic rather than about packing.

What the "whorled" bucket contains, at a rise of 0.008. 7 pairs, sharing 7 different factors, and every one of them is k and 2k: the second-shortest hop is the shortest taken twice. What the two-hop rule files as whorled is one family read twice, and no copies of anything.

What "whorled" was hiding

The earlier work's census put 35% of divergences in a bucket labelled whorled and moved on. Opened, every pair in it is k and 2k — the coarsest pair of the sequence, repeated k times — and reading the census up to jugacy takes the Fibonacci share from 14.7% to 50.1% without describing a single extra plant. By the lattice's own two families it moves nothing, and the share is 3.2% either way.

How often is it Fibonacci

The claim that plant spirals come in consecutive Fibonacci numbers is stated as a near-universal. Asked of the geometry, the answer collapses with scale — at a coarse rise 67% of divergences give Fibonacci pairs, and at a fine one 15%, with whorled and unnamed pairs taking the rest.

The plane of stems: divergence across, rise up. Each shade is one parastichy pair. The marked points are the lattices where three families are equally short — the forks — and the Fibonacci ones run up the middle towards 137.51°.

The angle is not the object

Every popular account of phyllotaxis is organised around a number. After a round of work spent on stems, forks and frequencies, the number looks like the wrong thing to organise an account around — it is the limit of one path through a branching structure, it is at no fork, and a plant that has it got there by not jumping.

A count of m and n pins the divergence to 221°/mn. Each 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.

What a count is worth

A reported parastichy pair pins the divergence angle to a band 221°/mn wide — so 2 and 3 says almost nothing and 34 and 55 fixes it to a tenth of a degree. Each step up the Fibonacci sequence is worth a factor of φ², and recording the radius a pair was counted at adds only ten per cent.

14 specimens separate 14.7% from 50%. The exact binomial power against sample size, for a one-sided test at 5 per cent. It is a staircase rather than a curve because the decision rule is a whole number of specimens: at 14 the cut sits at 5 and the power is 91.0 per cent. A normal approximation smooths that staircase away and reports a different answer.

How many plants would it take

Read by the two shortest hops, fourteen specimens separate the geometry's Fibonacci share from a coin weighted to a half, and four separate it from what a grown history gives. Read by the lattice's own two families, it takes seven and three. One fir cone measured at three rings settles whether its transitions are spaced as a cone's or an ogive's. The sample sizes are small, and that is the uncomfortable part.

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.

What the experiment costs, in internodes. The 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.

The test a plant could settle

Every other open question in this collection is priced in tens of specimens, and one of them in a hundred and sixty. This one is priced in internodes on a single stem, and the number is fifty-six — because it is a statistic of one sequence rather than a share of a population.

What the sequence sees that the scatter cannot. Each point is an ensemble at one amplitude, placed at the scatter it produces. A stem at three quarters of a degree of scatter has a lag-one correlation near zero if its noise arrived after the primordium was placed, and near 0.7 if it arrived before — and no measurement of the scatter can tell those apart. The separation closes above about a degree, because what the other two kinds preserve is the correlation of 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.

The measurement is limited by the protractor, not by the plant. The 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.

What the protractor has to be

The readout that names the parastichy number costs sixty internodes, which is cheap. It also needs every organ's position measured to better than a quarter of a degree, which is not — and the requirement follows from arithmetic rather than from care, so no amount of averaging relaxes it.

The statistic everybody reports is the one that cannot vary. Six arrangements of 900 points, from a whorled lattice to a set with no rule in it. The mean number of sides per cell is 5.97–6.04 on all six, because Euler's formula forces it. The mean squared departure from six runs from 0.023 to 1.83 — a factor of 79 — and the most hexagonal tissue in the set is the whorled one, at a rational angle.

What a summary throws away

Four statistics this collection has relied on turn out to be incapable of varying with the thing they describe — one is invariant to shuffling, one is fixed by a theorem, one is a parameter that stopped mattering, one is a fitted number selected into being wrong. In each case the second statistic was free and nobody had taken it.

What the pair costs

The single parastichy number cost sixty internodes. The pair costs two hundred and fifty, and a protractor error of three quarters of a degree takes it to eleven hundred. The arithmetic that predicts the second of those is right about the shape and wrong about the scale by a consistent factor, which is recorded rather than fitted away.

The disorder of a head against its divergence angle, 900 points. μ₂ is the mean squared departure of a Voronoi cell's side count from six, measured on 900 points inside 86% of the radius. Swept across 1.32° it is a staircase: flat over stretches of a few hundredths of a degree, with sharp steps between them and narrow deep dips wherever a rational falls. At 137.143° — which is 360 × 8/21 — it is 0.025; At 137.882° — which is 360 × 18/47 — it is 0.125; At 138.002° — which is 360 × 23/60 — it is 0.089. The golden angle is marked and sits at 0.253, in the middle of a flat stretch and nowhere near the largest value on the range.

The most irrational is not the most disordered

If rational angles make ordered tissue, the most badly approximable angle should make the most disordered — which would at last give the golden angle a criterion it wins. Swept across the interval it is the most irrational point of, μ₂ peaks at 138.42° and the golden angle sits unremarkably in the middle.

Both statistics, on the same stems, at a rise of 0.005. Five 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.

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.

Both ends of the window are silent, and a ruler tells them apart. The 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.

A refusal with a reason

Three note left with the work running have recorded that a refusal has four causes and the sequence separates none of them. With a second window and a protractor, three are separated: silence at 0.38° of scatter is a quiet plant, silence at 56° is a disorderly one, and agreement certifies the rate. The fourth survives, and so does a worse discovery — agreement is not correctness.

A periodicity reports a different partner every time. eight kinematic lattices, differing only in the seed of their disturbance, each read by the same instrument. The disturbance repeats every 8 organs at a weight of 0.9: it puts a strong comb at spacing 8 — 0.75 against a band of 0.07 — and the partner it names is 8/10, 8/12, 8/11 across the 8 stems and never 8/13, which is what the position counter finds in every one of them. There is no placement rule in any of these arrangements.

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 a long time 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.

The next organ moves for the last 13, and for no others. One 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.

An experiment a needle could run

For eight instalments 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.

The ratio is a U across every rung, and its floor is the number that was reported. The ratio of the second comb to the main comb, on five stems at each of 9 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 1.28, 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.

The survey loses its second outcome

The survey specification written earlier here 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 the measurements here. The ratio moves with where the plant sits between two transitions, and it moves again with the colour of the plant's own disturbance.

$The order follows the window, so it was never the fractions'. The four fractions with a denominator of 34, ordered four ways. The left column puts them in order of how close the nearest other rational is — the crowding — with the most crowded at the top. The other three order them by the area of their dip, at windows of 50, 100, 200 scaled units. The residual claim this thread carried was that the most crowded fraction gives the widest dip, which would make all four columns the same order. They are not: the order changes between the first two windows and settles, from a window of 100 outwards, into 11/34 > 15/34 > 13/34 > 9/34 — which is not the crowding order either. A quantity that reverses when the measurement is stopped somewhere else is a property of the stopping.$

The order belonged to the method

A residual was left over after the two width laws, and it looked ordered: the most crowded fraction gave the widest dip, in all three families, in the direction a measurement artefact would take. Measured again with an instrument that has no level in it, the order changes with the window, disagrees between families, and in one of them comes out backwards.

On a rung the response is a run of offsets; near a transition it has a hole. One row per rise, from 0.02 at the top to 0.005 at the bottom, and one column per offset: the organ one place back at the left, 14 places back at the right. A cell is filled where removing that organ moves the next organ by more than 2.5°, and empty where it does not. On a rung the filled cells are a run from one to the larger parastichy number — 8, 12, 13 at the pairs shown on the left. Between a rise of 0.02 and 0.008 the run ends at 5 and one more cell is filled at 7, with the offsets between them quiet to under a degree. That isolated column is one place inside the larger number of the pair the stem is climbing towards.

The ablation a plant would survive

The intervention proposed earlier 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.

Both edges of the front heal; the middle of it does not. The same removals, followed for 300 organs each. A cut one to three places back is undone within fifty organs and a cut ten to thirteen places back within sixty. A cut in between is never undone: the divergence sequence settles into an exactly repeating cycle of angles and holds it for the rest of the run. The rule corrects a displacement and cannot correct a deletion.

A period the grid invented

A wrecked stem was reported as settling into a repeating block of three angles — 219.84°, 220.31°, 220.78° — which is the smaller of its two spiral counts and would have confirmed a standing prediction. Those three numbers are three consecutive samples of the azimuth grid. There is no block; there is a constant the grid cannot write down, and the routine that found the block was working perfectly.

The disturbance with the largest wander leaves none in the sequence. How 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.

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.

$Hold the neighbourhood and the denominator stops mattering. The equivalent width of the disorder dip — the area of the deficit divided by its own depth — for seven fractions whose nearest neighbours sit at the same distance and whose denominators run from 19 to 47. Each is measured at a head size chosen so that all of them share one scaled unit, which makes a window in scaled units the same window in degrees and the same fraction of the way to the neighbour for every member. At a window of 25 the seven widths are 38.8, 38.8, 39.0, 39.0, 38.7, 39.0, 39.0 — a spread of ×1.010 across a factor of 2.47 in denominator. The lines separate as the window widens, to ×1.147 at 200, and when they do they order by denominator rather than by crowding. So the residual this thread carried was the window: hold it and there is nothing left that belongs to the fraction.$

The residual was the window

After the depth and the q over n squared scale are taken out of a disorder dip, something looked left over and looked ordered by how crowded the fraction's neighbourhood is. Measured on fractions whose neighbourhoods are identical by construction, seven widths across a factor of two and a half in denominator agree to one per cent. There is no residual; there was a comparison made at different effective windows.

$Five fractions with one neighbour distance and every denominator. Each member of a matched set drawn on its own stretch of the divergence axis, 0.5° either side of itself, with the nearest other rational marked. The distances are 0.3651°, 0.3529°, 0.3692°, 0.3640°, 0.3557° — a spread of 4.6% — while the denominators run 17, 20, 25, 43, 44, a factor of 2.59. That is the construction this thread needed. Every instrument for the width of a disorder dip has a free parameter set by how close the neighbour is, so a hypothesis about the neighbourhood cannot be tested by varying the neighbourhood; on this set the neighbourhood is held fixed and the arithmetic of the fraction is what varies.$

Matching instead of correcting

Two rounds of work failed on one question because every instrument's free parameter was set by the thing under test. The repair was not a better instrument or a model of the bias: it was choosing what to compare so that the confound could not vary. That move is available in four other places here, and three of them have already used it without anybody naming it.

For a high pair a miscount does not blur the report: it moves it. The divergence axis from 50° to 145°. For 21/34 and 34/55, the dark band is what the report allows if both counts are right and the pale bands what it allows if either is wrong by one. 21/34 allows 0.308° exactly and 0.931° in all, in 3 pieces: 20/33 at 54.37° and 22/35 at 82.13° beside the true band at 137.48°; 34/55 allows 0.118° exactly and 0.354° in all, in 3 pieces: 33/56 at 109.23° and 35/54 at 113.27° beside the true band at 137.52°. Each wrong band is as narrow as the right one and tens of degrees from it, so the reading does not widen; it becomes a short list of sharp candidates, and neither wrong candidate contains the golden angle.

A count that can be wrong by one

A reported parastichy pair pins the divergence angle to a band 221°/mn wide only if both counts are right. Allowing either to be off by one adds the bands of every neighbouring pair whose counts share no factor, and those bands sit where their own lattices live: for 2/3 they swallow the whole range, and for 34/55 they are two bands as narrow as the true one at 109° and 113°, twenty-five degrees away. So a high count that may be wrong is not a blurred reading but a short list of sharp candidates, costing log₂ 3 bits. And on the Fibonacci pairs, two in every six — 21/34 and 34/55 among them — cannot be miscounted silently by one count at all, because every such miscount shares a factor.

Every angle whose counts from 34 to 144 resist approximation within a per cent of the golden angle's. Over the counts a head shows from 34 to 144, an angle scores like the golden angle when those counts add up, each the sum of the two before, from a first pair near the golden ratio. 46 angles between 20° and 180° come within one per cent of its score of 0.44718, each drawn as a stem at its angle. The five nearest are 137.51° with counts 34, 55, 89, 144 at 100.000 per cent; 99.50° with counts 47, 76, 123 at 99.989 per cent; 106.45° with counts 44, 71, 115 at 99.931 per cent; 151.14° with counts 50, 81, 131 at 99.919 per cent; 132.18° with counts 49, 79, 128 at 99.907 per cent. The golden angle is the highest, and the Lucas angle at 99.50° is a ten-thousandth of the score behind it.

What a head can mean by most irrational

Hurwitz's bound, the one famous claim about this subject that survives, is a limit over every denominator, and a head shows only the counts between its innermost spirals and its rim. Over those counts an angle resists approximation like the golden angle exactly when the counts it shows add up, each the sum of the two before, from a pair near the golden ratio — and every such pair has an angle of its own. The golden angle still scores highest over every window measured, by a ten-thousandth: over counts from 34 to 144 the Lucas angle is 99.989 per cent of it and forty-six angles are within one per cent. What separates the golden angle from them is below the counts they share, at the centre of the head.

A count that drifts by two

A reported pair of 34 and 55 that may be wrong by one allows three sharp bands; allowed to drift by two it allows thirteen, and by three, twenty-nine — and the information lost is exactly the logarithm of that count, because every band is as narrow as the true one. The nearest wrong band stays twenty-four degrees away until a drift of three brings one to eleven. What does not survive is the protection: 21/34 and 34/55 could not be miscounted silently by one, but every Fibonacci pair can be by two, so a counter who drifts by two as readily as by one reports 34/55 silently wrong 9.5 per cent of the time rather than 0.13.

The first organs of a golden head, a Lucas head and a head at 104.67°, with each one's closest pair. The organs within a radius of 7.2 of three Vogel heads — the first 51 organs after the one at the centre — at the golden angle, the Lucas angle and 104.67°, three of the forty-six angles whose counts from 34 to 144 resist approximation within a per cent of the golden angle's. The joined pair in each is its closest: organs 1 and 4, 1.602 apart, at the golden angle; 1 and 5, 1.574 apart, at the Lucas angle; 12 and 19, 1.241 apart, at 104.67°. Out here the three are already different drawings; by a radius of seventeen they are not.

The first three hundred organs

Over the counts a head shows, forty-five angles resist approximation within a per cent as well as the golden angle, and what separates them is at the centre. Grown as heads and measured there, the golden angle has the widest closest pair of all forty-six — by organs 1 and 4, the count its arithmetic names — and keeps first place only while the centre is in the reading. Its rivals stay a per cent apart from it out to a radius that tracks where their spiral counts start to add up, and every one of them is within a per cent by the 289th organ. By the largest hole it is never the best.

The divergence a reading of 34/55 reports when the counter closes the circle early or late. A counter traces each family round 1 + ε of a turn, so both counts are multiplied by 1 + ε and rounded; the horizontal axis is ε in degrees of the circle, out to 54.0° either way. Where the rounded pair shares no factor it is drawn at the centre of the band it would put the divergence in; where it shares one it is a grey tick in the lane below. 34/55 is read right for closing errors under 3.27°; the first silent readings are 33/53 at −9.82° and 35/57 at +9.82°, reporting 54.4° and 82.2°. The silent band nearest the truth is 31/49's, 1.86° from it, at a closing error of 36.0°. Marked: a closing error of 9.8°, which reads 35/57 · reports 82.2°, silently wrong.

Two counts that slip together

A counter who closes the circle a few degrees late counts a sliver of the head twice, in both families at once, so the two counts of a reported pair drift together rather than apart. Coupled that way the count is safer than it was: fourteen Fibonacci pairs in twenty-three admit no silent equal shift of one, against seven that admit no silent single miscount, and 34/55 announces every closing error short of 9.82°. The check is what breaks. Two annuli closed at the same wrong mark pass 17.6 per cent of wrong readings of 34/55 and 76.8 per cent of 13/21's, because a linear relation survives multiplication — and what catches them instead is a protractor good to twelve degrees.

The census wants a low count

Four specimens separate the geometry's Fibonacci share of 14.7 per cent from the ninety per cent a grown history gives — if every count is right. Counted with a closing error spread over 7.2°, the same census needs six specimens counted at 13/21, fifty-four at 34/55 and 449 at 55/89, because the geometry's own pairs are all small enough that no closing error under 11° moves them, while a grown plant counted high loses its Fibonacci reading first. Counted at 55/89 with a spread of 9.83° the census reads plants as less Fibonacci than random angles, though only against that share: against the 3.2 per cent the lattice's own two families give, those sizes fall to four, eleven and twenty-six, and the reversal goes. The count that pins the divergence best is the one a census should avoid.

What a reading of 34/55 becomes when each family is closed at a mark of its own. Each axis is one family's closing error in degrees of the circle, out to 24° either way; every rectangle is the set of closing errors that give one rounded pair, keyed as read right, sharing a factor (so announcing itself), or sharing none (so passing silently). The diagonal is one mark shared by both families, where the first silent readings are 33/53 and its mirror at 9.82°; off it the nearest are 33/56 and its mirror at 5.29°, with the two marks erring on opposite sides. The ellipses are one and two spreads of the marks' joint distribution at 7.2° a mark and a correlation of 0.50: 21.0% of readings right, 58.6% announced and 20.4% silent.

Two marks chosen by one eye

A counter traces each family of spirals from a starting organ of its own, so a reported pair carries two closing errors, correlated because one eye chose both. Letting them differ costs 34/55 its ten-degree margin — 33/56 and 35/54 share no factor, and marks that err 5.3° in opposite directions reach them — while 21/34 keeps its margin whatever the marks do. And it decides the second annulus. At a spread of 7.2° the relation passes right readings 2.8 times as readily as silent ones when the marks are independent, 1.25 times at a correlation of 0.9, and stops telling them apart at 0.98; where it does work it keeps one reading in forty-six.

Counting it again

A reading whose two counts share a factor says the count went wrong, and the specimen is still there to be counted again. Counted afresh, the reading kept is exactly one reading conditioned on not announcing itself — the second chance a silent error gets is matched by the second chance a right reading gets — so a recount changes which specimens a census keeps, not what a kept reading says. At 34/55 with closing errors spread over 7.2° it takes the census from fifteen kept specimens to ten and from about thirty counts to twenty-one, and against scoring every reading it turns 449 counts at 55/89 into 52. It never makes a high count as cheap as counting 13/21 once. Against the lattice's own two families it is six to five, and the share set aside runs the other way: strong at one count, and spent by each recount.

What one recount reads after 34/55 was first read as 34/54, for each belief the counter aims from. A head whose true pair is 34/55, first read as 34/54 with closing errors spread over 7.2°, recounted once; the two closings of the head are correlated by 0.9. Each bar is one counter, split into the recount reading 34/55, announcing itself again, and reading another pair silently: unaimed, aiming +0.0°, 41.8% right, 55.2% announced, 3.0% silent; expects Fibonacci, aiming +3.8°, 69.6% right, 30.2% announced, 0.2% silent; expects Fibonacci, capped at 2σ, aiming +3.8°, 69.6% right, 30.2% announced, 0.2% silent; expects any spiral, aiming −4.5°, 5.5% right, 62.7% announced, 31.8% silent; expects nothing, aiming +0.3°, 44.7% right, 52.8% announced, 2.5% silent.

The recount aims where the counter expects

A counter who recounts an announced reading knows it went wrong, and if the same habit spoils both counts of a head, the first error says where to aim the second. But the reading alone does not say which way the first erred: a reading of 34/54 is as well explained by a whorled 34/54 read right, or by 34/53 read long, as by 34/55 read short. The direction comes from what the counter expects. Expecting Fibonacci, an aimed recount at 7.2° and a habit correlated at 0.9 reads 34/55 69.6 per cent of the time where an unaimed one reads it 41.8, and the census needs seven specimens rather than fifteen. Expecting only a spiral, it aims the wrong way and reads 34/55 5.5 per cent of the time. The belief that helps is the hypothesis the census is testing: uncapped, it reads the geometry's whorled 3/6 heads as 3/5 and a census of a hundred rejects a true null 40 per cent of the time; capped, it still reads a silent 33/53 as Fibonacci twice as often. And no aimed recount spends fewer counts than 13/21 counted once.

The high heads the geometry rarely makes

A Fibonacci census draws its null from the geometry's heads, and at the rise it was specified at none of them counts past sixteen — which is why a counter who aims a recount at the Fibonacci pair they expect, capped at twice their own spread, left the null alone. Drawn at finer rises, where the geometry's heads count into the forties, the null gets easier to beat rather than harder: its Fibonacci share falls from 22.5 to 7.5 per cent and the census needs four kept specimens instead of seven. The capped aim still adds almost nothing to it, because the high heads the geometry makes are rarely the ones it moves: the geometry makes a 33/53 only at a rise of 0.0003, near a divergence of 54°, and it is a twentieth of a per cent of the null. What the census does lose at a fine rise is its five per cent — to the counter's habit, not to the aim, and every recount policy loses it alike.

A disc reads almost no Fibonacci heads by chance

A Fibonacci census draws its null from the geometry, and every null so far was a cylinder's: one rise, one dominant pair per divergence angle. Read the same angles as discs, counted in the band a surveyor uses, and the null's Fibonacci share falls by a factor of ten to forty — because nearly all of the cylinder's Fibonacci heads were the pair 1/2 at the smallest angles, a single tight arm that no band reads as 1 and 2. The census then needs three kept specimens. Half of a large head's bands do straddle a transition, but a band that straddles reads one edge's pair or its own middle pair, and the handful of true mixtures are nowhere near a Fibonacci pair.

Displacing a disc's organs makes no new near misses

A Fibonacci census's null, drawn from Vogel discs counted in a surveyor's band, holds a few tenths of a per cent of heads that a capped recount could move. Every disc was exact, and a real head's organs sit a fraction of a spacing off the lattice, where a band can read a family the lattice holds only marginally. Displaced by a twentieth, a tenth and a quarter of a spacing, the discs' readings do change — at up to one angle in six — and on large heads mostly to a pair the exact band already reads at one of its edges. But near the golden angle they change no more often than anywhere else, almost never into a Fibonacci pair, and the heads within a capped aim's reach move by under six hundredths of a percentage point. The census still needs three kept specimens.

The same head displaced organ by organ and by a drift along its spirals, the displacements drawn four times their size. A Vogel head of 300 organs at 137.6°, every organ in the band from 0.55 to 0.95 of its radius drawn with a line from where the lattice puts it, four times the length of its displacement. On the left each organ is moved by an independent gaussian of a quarter of a spacing; on the right by a drift of the same size, each organ inheriting 0.9 of the mean error of its two nearest earlier organs. The drift's lines run in the same direction over whole stretches of the band, so neighbours move together: at this correlation an organ moves against its four nearest neighbours by 0.55 of what the independent displacement moves it. The band reads 21/34 independently displaced and 21/34 drifted.

A drift along the spirals changes fewer counts

A grown head's errors are not independent: an organ placed against the organs already there inherits their errors, so whole stretches of a spiral move together. Laid down that way over the census's Vogel discs, at the same size as the independent displacement measured before, the drift changes fewer of the band's readings, not more — a sixth to a quarter fewer at a quarter of a spacing — never more often near the golden angle, and at most two into a Fibonacci pair. A count reads an organ against its neighbours, and a drift moves neighbours together: at a strong correlation an organ moves against its four nearest by just over half what an independent displacement of the same size moves it, and the readings changed are about what an independent displacement of that relative size changes, plus a little for the coherent part.

Where a chosen mark closes: the organ nearest one place, and where its two spirals cross the counting circle. A stretch of a golden-angle head around one place on a circle inside the band of radii where 34 and 55 are the two shortest families of spirals, 75 per cent of the way out. The ringed organ is the one nearest the place, and both counts start on spirals through it. Its 34-spiral crosses the circle 0.48 of that family's spacing to one side of the place and its 55-spiral 0.42 of a spacing to the other, because the organ sits outside the circle and the two families wind in opposite senses. A count that closes on the place, rather than on the organ, is off by those amounts, which round to nought here.

A chosen mark errs by a third of a spiral

A counter who starts every family of spirals at the organ nearest one place on the head, and stops when the count comes back round to that place, makes a closing error the head decides: the gap between the place and where that organ's spiral crosses the counting circle. It is three tenths of a spiral spacing in every family of every pair, so in degrees it falls as one over the count — 3.2° on a 34-family, not the 7.2° assumed — and the two families' errors are correlated by where on the head the place is, from +0.35 at the inner edge of a pair's band of radii to −0.5 at the outer. From the geometry alone a reading is wrong one time in eleven and almost never in both counts, so 34/55 is silently wrong once in six thousand. An eye that misplaces the mark by a tenth of a spacing is enough to turn the correlation positive.

The same head drifted in patches and in streaks, the displacements drawn four times their size. A Vogel head of 300 organs at 137.6°, every organ in the band from 0.55 to 0.95 of its radius drawn with a line from where the lattice puts it, four times the length of its displacement, both fields a quarter of a spacing in size on the same seed. On the left each organ inherits 0.9 of the mean error of its two nearest earlier organs, which carries an error along both families crossing there and makes round patches. On the right it inherits 0.9 of its single nearest earlier organ's error, which carries it along one family and makes streaks. Both move an organ against its four nearest neighbours by about the same amount — 0.55 and 0.55 of an independent displacement — but the streak moves it against the neighbours on its own streak by 0.34 and against the rest by 0.70. The band reads 21/34 in patches and 21/34 in streaks.

A streak is a shear the count can see

Let each organ of a census disc inherit its error from its nearest earlier neighbour alone, rather than from the two it sits against, and the error travels along one family of spirals: streaks instead of patches. Measured against its four nearest neighbours a streak moves an organ exactly as much as a patch does — just over half what an independent displacement of the same size moves it — and yet it changes more of the band's readings at every size and in every draw, and on heads of three and five hundred organs as many as the independent displacement itself, or more. Its movement is nearly all across its own streak and hardly any along it, so it shears one family's chains against the other's, and the relative size that predicted a patch's effect underestimates a streak's by up to three points. Near the golden angle it changes no more readings than anywhere else, and the census still needs three kept specimens.

What a pinned count of 34/55 reads for an eye that slips 1 in 500 crossings at an organ's spacing, beside a placed mark. Shares of readings exact, wrong and announced by a shared factor, and wrong and silent: pin, outer edge, 94.8% exact, 5.1% announced, 0.07% silent, both counts wrong on 0.04%, a count two or more out on 0.05%; pin, a third in, 91.0% exact, 8.8% announced, 0.21% silent, both counts wrong on 0.13%, a count two or more out on 0.15%; place, a third in, 98.9% exact, 1.1% announced, 0.00% silent, both counts wrong on 0.00%, a count two or more out on 0.00%. The placed mark is read from 2000 places on its circle and is never two out.

A pinned count errs where the spirals crowd

A counter who closes each family on a chosen place is wrong one time in eleven from the geometry alone. Pin the starting organ instead and the closing costs nothing; the only error left is a slip on the way round, and a slip is likeliest where neighbouring spirals cross the counting circle closest. They cross evenly round any one circle and much closer at the inner edge of a pair's band, and the larger family's always cross closer by the ratio of the counts — so it slips about four times as often per count. A pin is best at the outer edge, where a place is worst, and beats a place there for an eye that slips less than once in 156 crossings at an organ's spacing; to beat a place a third of the way in it must slip less than once in 2,400. And a pin makes an error a place never does: a count two out, silent on 34/55.

Every report within three of 21 and 34, and what a census would call each one. Rows are the first count from three below to three above 21, columns the second about 34. One count away, 21/33 would be read as 3 × 7/11; One count away, 21/35 would be read as 7 × 3/5. A report in an unshaded cell shares no factor and belongs to no family.

A shared factor has two readings

A report whose two counts share a factor has been read two ways. One account says it is a miscount announcing itself, since a true ordinary pair cannot share one. The other says it is a plant grown k organs at a time, and that reading took a census's Fibonacci share from a seventh to a half. It is one piece of evidence, and nobody has compared the rates. Above the smallest pair, at least two of the four reports one count from a Fibonacci pair share a factor and carry a family's name: 21/35 reads as seven whorls of 3/5, and 33/55 as eleven. At a chance of 5 per cent that a count is off by one, with 5 per cent of plants truly whorled, a report with a shared factor is a true whorl less than half the time.

A count weighted by direction closes a third of the gap

A streak moves an organ against its neighbours about as far as a patch does and changes far more of a count's readings, so the size of an error, averaged over directions, was the wrong number. Measured the way the count reads it, along each of the two families it is counting, the larger of the two closes between a quarter and under half of the gap, on thirty combinations of three errors at once: independent, patch and streak. It still leaves a strong streak a fifth short. The wobble of each family's spirals, which can be measured from a photograph with no knowledge of where anything belonged, does as well. And the two families are not lopsided on the heads where the gap is widest, so direction was not the whole of what the average missed.

A slip that remembers is caught by a second pass

A pinned count errs only by slipping between spirals on the way round, and on 34/55 only a count two out is silent, which independent slips make rare. An eye that has lost its place once is likelier to lose it nearby, and runs of slips ought to make that silent error common. Holding the mean rate of slipping fixed, runs do raise it, by more than five times when each slip makes the next hundred times likelier. It still never reaches the whole error of a count taken at a chosen place at the rim, which is fifteen in a hundred against under half of one. Bunching the same slips together leaves fewer counts wrong in all. And two passes that must agree take the wrong readings kept from 5 per cent to a tenth of one, with runs or without.

How often a random divergence reads Fibonacci, by the two shortest hops and by the lattice's two families. Thirty-six hundred divergences at each rise. At 0.12: 67.5 per cent by hops, 10.9 per cent without the pair 1/2, and 32.4 per cent by the lattice's families; At 0.03: 31.1 per cent by hops, 7.3 per cent without the pair 1/2, and 14.3 per cent by the lattice's families; At 0.008: 14.7 per cent by hops, 1.9 per cent without the pair 1/2, and 3.2 per cent by the lattice's families; At 0.001: 4.7 per cent by hops, 0.2 per cent without the pair 1/2, and 0.3 per cent by the lattice's families.

Most of the Fibonacci share was one family read twice

The census behind How often is it Fibonacci takes the two shortest hops at each divergence and calls them the pair. A pair of spiral families has to be two families, and the second-shortest hop is often the first taken twice in the same direction. That is the whole of the bucket once marked whorled, and it is most of the Fibonacci share too, since the pair 1/2 is one family read twice. Read by the lattice's own two families, the share at a rise of 0.008 is 3.2 per cent, not 14.7; at 0.12 it is 32 per cent, not 67.5; and Fibonacci's tenfold lead over Lucas at the finer rises is gone, 3.2 against 3.0.

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