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

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.
21 min read 11 figures State the claim, state the testCounted, not admiredForm is not mechanism

Worth reading first: What a summary throws away.

The disorder dip at a rational divergence has two laws. Its depth is a property of the fraction, and its scale in degrees goes as q/n² — the denominator over the square of the head size. Both are established here, both hold over factors of two and a half in denominator and four in head size.

After them, something looked left over. Within a single denominator the widths were not identical, and the differences appeared ordered: the fraction whose nearest neighbour is closest gave the widest dip, in all three families measured, which is the direction a hypothesis about crowding would predict.

That ordering was withdrawn a round ago, on the ground that every instrument available for the width has a free parameter set by the crowding, so the measurement could not distinguish the hypothesis from its own bias. This essay makes the measurement the withdrawal said was needed, on fractions whose neighbourhoods are identical by construction. The answer is short.

The measurement

Seven fractions — 6/19, 7/20, 8/21, 9/23, 16/33, 19/45 and 15/47 — whose nearest other rationals sit within 3.1% of the same distance and whose denominators span a factor of 2.47. Each measured at a head size chosen so that all seven share one scaled unit, which makes one window the same window in degrees and the same fraction of the way to the neighbour for every member.

At a window of 25 scaled units the equivalent 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, on an axis where the two width laws move the answer by exactly that factor.

Hold the neighbourhood and the denominator stops matteringThe 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.25×1.01050×1.022100×1.075200×1.147window, in scaled units of δ·n²/qequivalentwidth6/197/208/219/2316/3319/4515/47the neighbourhood at 0.31° · heads 487–766 organsgenerated from a stated rule, not drawn to look right
Fig. 1 The equivalent width of each member’s dip against the window it was integrated over. At the narrow end the seven lines are indistinguishable; at the wide end they separate, and the separation orders by denominator.

So there is no residual. What was left over after the two laws was not a property of the fraction at all.

The spread that does appear

The lines do separate, and where they separate is the whole explanation.

At windows of 25, 50, 100 and 200 scaled units the spreads are ×1.010, ×1.022, ×1.075 and ×1.147. The disagreement grows monotonically with the window, and at the widest window it comes out very nearly in order of denominator: the rank correlation between the width and the denominator is 0.857 in this set and 1.000 in the second one.

That is the same window-dependence this collection already measured and reported as the reason the equivalent width has no value: the dip has a depth and a scale and no outer edge, so an integral of it reports where it was stopped. A comparison between fractions at what looked like a common window was a comparison at different effective windows, because the effective window is set by the crowding.

Hold the crowding and the window together, and the difference goes.

The area never settles, so the number reported is the windowThe equivalent width of the disorder dip — the area of the deficit divided by the dip's own depth — for four fractions with a denominator of 21, against the window it was integrated over. The window is in scaled units, u = δ·n²/q, so a head of any size is sampled at the same places on its own dip; 200 scaled units is about 0.0176° at these head sizes. A width would show as a flat stretch. There is none: every line climbs to 417 scaled units or so and then turns over, because past that the window has reached far enough towards the next rational that it is integrating the neighbour's dip instead of this one's background. The dip has a depth and a scale; it does not have an outer edge, and an integral of something with no outer edge is a statement about where it was stopped.50100200400800window, in scaled units of δ·n²/qequivalentwidth8/215/2110/214/21q = 21 · two head sizes agree to 10% at a window of 200generated from a stated rule, not drawn to look right
Fig. 2 The same behaviour on the unmatched family the residual was read off. Every line climbs and none settles, so any comparison between them is a comparison at a stopping point rather than at a width.

And it orders the wrong way for the hypothesis

There is a second, sharper point in the wide-window numbers, and it is worth having because it goes beyond “the effect vanishes”.

The hypothesis said the most crowded fraction gives the widest dip. Matched, the rank correlation between the width and the distance to the nearest neighbour is −0.673 in the first set and −0.200 in the second — negative, meaning the least crowded fraction is widest. The residual, where it appears at all, points the opposite way from the claim it was invented for.

That is what a bias produces when the confound it rides on has been removed. The crowding is now the same for everybody, so what is left correlating with width is the denominator, and the denominators happen to run slightly the other way against the small residual differences in crowding within the set.

The order follows the window, so it was never the fractions'The four fractions with a denominator of 55, 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 23/55 > 17/55 > 12/55 > 21/55 — which is not the crowding order either. A quantity that reverses when the measurement is stopped somewhere else is a property of the stopping.crowdednearest neighbour firstwindow 50widest firstwindow 100widest firstwindow 200widest first23/5517/5521/5512/5523/5517/5521/5512/5523/5517/5521/5512/5523/5517/5521/5512/55crowding: 21/55 0.193° · 12/55 0.205° · 23/55 0.152° · 17/55 0.156°q = 55 · 1279 organsgenerated from a stated rule, not drawn to look right
Fig. 3 The withdrawal as it was originally made, on one unmatched family. The order changes between windows and settles into something that is not the crowding order — which was enough to withdraw the claim and not enough to settle it.

What a common window buys, in numbers

The claim that the earlier comparison was made at different effective windows is worth putting in figures rather than leaving as an explanation, because the figures are what the matching changes.

In the unmatched design — four fractions of one denominator — a window stated in scaled units reaches a different fraction of the way to each fraction’s nearest neighbour, because the neighbours are at different distances. Measured on the family the residual was read off, one common window reached 6.1% of the way to the neighbour for the most crowded member and 3.8% for the loneliest. That is a sixty per cent difference in the quantity the integral is most sensitive to, and it runs in the same direction as the hypothesis: the most crowded fraction was being integrated further into its own dip.

In the matched design the same window reaches 5.1% to 5.2% for every member — a two per cent spread, which is the residual disagreement in the crowdings passed through unchanged.

So the correction that would have been needed is about thirty times larger than the effect that remains, and it would have had to be modelled with the same machinery whose behaviour was in question. That is the argument for matching rather than correcting, stated as a ratio.

A window that fits inside a rungStems that climb the ladder at four rates, read over a window at the fine end. The condition is a ratio: the window has to be shorter than a rung. 250 internodes at 130 per rung is 1.92 rungs and agrees on 0 of 3; 400 internodes at 130 per rung is 3.08 rungs and agrees on 0 of 3; 250 internodes at 260 per rung is 0.96 rungs and agrees on 3 of 3; 400 internodes at 260 per rung is 1.54 rungs and agrees on 1 of 3; 250 internodes at 520 per rung is 0.48 rungs and agrees on 2 of 3; 400 internodes at 520 per rung is 0.77 rungs and agrees on 3 of 3; 250 internodes at 1040 per rung is 0.24 rungs and agrees on 3 of 3; 400 internodes at 1040 per rung is 0.38 rungs and agrees on 3 of 3. Read over the whole stem instead, every rate returns nothing — 0 of 3, 0 of 3, 0 of 3, 0 of 3 — because the quantity the comb is periodic in changes as the pattern climbs.nodes per rung250-node window400-node windowwhole stem1301.92 rungs0/3 · 1 wrong3.08 rungs0/30/32600.96 rungs3/31.54 rungs1/3 · 1 wrong0/35200.48 rungs2/30.77 rungs3/30/310400.24 rungs3/30.38 rungs3/30/33 stems per cell · rise falls from 0.4 to 0.004 on every onefilled where the angles and the positions agree
Fig. 4 The same kind of reasoning where this collection uses it elsewhere: a window has to be stated in units the thing being measured supplies, or the comparison is between two windows rather than between two objects.

Both halves reproduce

An agreement at one window on one set would be a thin result. Both halves repeat.

At a second head size. Every member re-measured with its head a quarter larger, so that every dip is a quarter narrower in degrees: the spreads at the same four windows are ×1.008, ×1.020, ×1.067 and ×1.119, and each member’s own width agrees between the two scales to within a few per cent.

On a second set. Five fractions — 5/17, 9/20, 9/25, 15/43 and 21/44 — at a crowding a fifth larger, sharing no members with the first. Spreads of ×1.001, ×1.028, ×1.091 and ×1.112 at the same windows.

Four independent versions of the same measurement, agreeing on both halves: one per cent or so at the narrow window, ten to fifteen at the wide one.

Five fractions with one neighbour distance and every denominatorEach 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.5/17q = 1717/589/20q = 2023/519/25q = 2514/3915/43q = 438/2321/44q = 4411/23the fractionits neighbourneighbour distances 0.3529° to 0.3692° · denominators 17 to 44neighbours looked for among denominators up to 60generated from a stated rule, not drawn to look right
Fig. 5 The second set, drawn. Five fractions with a common neighbour distance and denominators from seventeen to forty-four, which is where the replication comes from.
Hold the neighbourhood and the denominator stops matteringThe equivalent width of the disorder dip — the area of the deficit divided by its own depth — for five fractions whose nearest neighbours sit at the same distance and whose denominators run from 17 to 44. 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 five widths are 39.0, 39.0, 39.0, 39.0, 39.0 — a spread of ×1.001 across a factor of 2.59 in denominator. The lines separate as the window widens, to ×1.112 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.25×1.00150×1.028100×1.091200×1.112window, in scaled units of δ·n²/qequivalentwidth5/179/209/2515/4321/44the neighbourhood at 0.36° · heads 461–742 organsgenerated from a stated rule, not drawn to look right
Fig. 6 The second set measured. The same shape: indistinguishable at the narrow window, separating at the wide one, ordering by denominator when it does.

The agreement is not an insensitive instrument

The obvious worry about a null result is that the measurement cannot tell anything apart. It can, and the same figure shows it: the spread reaches ×1.147 at the widest window, which is fourteen times the width of the agreement at the narrow one.

So the instrument distinguishes these seven fractions perfectly well when it is allowed to reach far enough. What it distinguishes them by is how far it reached, and at a window where the integral is inside the dip it finds nothing to distinguish.

That is the shape of a null result worth trusting: an instrument with demonstrated sensitivity, reporting no difference where the hypothesis predicted one.

The dip at 8/21 — 137.1429° — at three head sizesWalking the divergence angle off an exact rational, at 300, 600, 1200 points. The floor falls as the head grows — 0.078 at 300, 0.038 at 600, 0.019 at 1200, halving for each doubling — and the dip narrows faster: half-widths of 0.0195°, 0.0088°, 0.0023°, a factor of four for each doubling rather than two. Half-width times the square of the head size is 1754, 3151, 3333, which is what makes the width a property of the sample rather than of the angle.00.2000.4000.600-3-2.50-2-1.50-1-0.700offset from the rational, log₁₀ of degreesμ₂300 points600 points1200 points8/21 of a turn · 12 offsets · vertical rules are the half-widthsgenerated from a stated rule, not drawn to look right
Fig. 7 One member of the set at three head sizes, for scale. The depth is a property of the fraction and the width in degrees is not, which is why the head size has to be chosen per member rather than fixed.

What is settled, and what the thread was

This closes a question that has been open since the work that established the width laws, and it is worth setting out the whole arc because the arc is the result.

A summary statistic was found to throw away most of what a head’s cell areas contain. A second statistic was built. It turned out to be a staircase in the divergence angle rather than a smooth curve. The staircase’s steps were dips at rationals. The dips were given a depth law and a scale law. Something looked left over. The something was measured with a level crossing, which had a resolution problem; then with an integral, which reproduced beautifully and had no limit; then the ordering was withdrawn because both instruments were calibrated by the thing under test; and now it is measured on a construction where they are not.

After the depth and the q/n² scale, there is nothing left that belongs to the fraction — to within one per cent, over a range of denominators where the laws themselves move the answer by a factor of two and a half.

The statistic everybody reports is the one that cannot varySix 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.mean sides per cell(forced to six)mean squared departure from six(not forced)whorled, 144°5.9860.023golden, 137.508°5.9900.253rational, 137.5°5.9900.255Lucas, 99.502°6.0360.255137.0°5.9900.291Poisson5.9691.830six433–637 interior cells each, inside 86% of the radiussame cells, same cut, two statistics
Fig. 8 Where the thread started: the summary statistic that throws most of a head’s information away, and the second statistic built to keep it. Everything since has been about what that second statistic can and cannot measure.
The coefficient is not one numbern²·w, measured at the two larger of each denominator's three head sizes and averaged, for six Fibonacci fractions. Undivided it runs 1188, 1285, 2999, 7005, 9125, 14297 — a spread of 12.0. Divided by the denominator it runs 149, 99, 143, 206, 166, 161 — a spread of 2.1. So the law is w ≈ 150·q/n² to within a factor of two, and that earlier work's constant of 1,400 to 3,300 was three measurements of the small-denominator end of it.33.504divergence angle, as a fraction of a turnn²·w, the coefficient of the width law (logarithmic)11881285299970059125142973/85/138/2113/3421/5534/89spread 12.0 across the sixsix denominators · n²·wgenerated from a stated rule, not drawn to look right
Fig. 9 The law that does hold, drawn: the width carries the denominator, which is the scaling every comparison in this essay is made through. What this essay settles is that nothing survives that scaling.

What closes with it

This is the thirteenth essay against one idea, and the idea is finished — not in the sense that nothing more could be said about a disorder statistic, but in the sense that the specific question the thread was built around now has an answer and the instruments have been characterised to the point where a further essay would be repeating one of them.

The question was: what does a summary statistic throw away, and what does a better one find? The answers, in order: a mean side count throws away nearly everything, because a head at almost any divergence has a mean of six by Euler’s formula and the departures are what differ. The second moment keeps it. The second moment across the divergence axis is a staircase rather than a curve. The staircase’s steps are dips at rationals. The dips have a depth that belongs to the fraction and a scale that goes as the denominator over the square of the head. They have no outer edge, so they have no width, and an integral of one reports its own window. And after the depth and the scale there is nothing left.

Six statements, each with a test it could have failed, two of them retractions of earlier statements in the same list. What ends the thread is that the last question in it has a construction that answers it rather than an instrument that cannot.

How many sides the cells actually haveThe mean is 5.908, which Euler's formula forces. The spread around it is not noise — a tiling of only hexagons cannot close up on a finite patch.5 sides13420%6 sides45869%7 sides7311%665 bounded cellsmean 5.908 sidessix is forced, not chosen
Fig. 10 Where the thread began: the side-count distribution of a head, and the fact that its mean is fixed by a topological identity rather than by the pattern. Everything since has been about the shape the mean discards.
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. 11 And what the thread mostly turned out to be about: which quantities the instrument can measure and which it merely returns numbers for. Three of the thirteen essays are about that boundary rather than about heads.

Why a null is the right outcome here

It is worth saying plainly that this is a satisfying answer rather than a disappointing one.

The two width laws are simple and they came out of the geometry: a dip’s scale is the distance over which a head at a nearby angle stops resembling a head at the rational one, and that distance is q/n² by an argument with nothing fitted in it. A residual would have meant the geometry was incomplete — that some further property of the fraction leaks into the width by a route nobody had identified.

There is no such route, at the precision available. The dip is a two-parameter object: a depth and a scale. Everything else about how it looks is the window it was measured through, which is a fact about the observer.

What this does not say

It does not say the crowding is irrelevant to a measurement. It sets both instruments’ free parameters and it decides whether a dip can be measured at all. What it does not do is change the dip.

It does not say the earlier ordering was fabricated. It was measured, in three families, with the direction the hypothesis predicted. It was a real property of measurements made at different effective windows, and it was withdrawn before this essay existed.

It does not say one per cent is zero. It says the spread across a factor of 2.47 in denominator is a per cent, at a window where the measurement is well-posed, at two head sizes, in two sets. A residual an order of magnitude smaller than that could be there.

It does not settle whether a residual exists at some other window. At a window of 200 the seven fractions disagree by fifteen per cent and the ordering is by denominator; a hypothesis that the residual is a real property of the fraction which only appears when enough of the dip’s tail is included cannot be distinguished here from the window’s own reach. What can be said is that the denominator ordering appears exactly where the instrument’s known failure mode appears, and disappears exactly where it does not.

And it does not say the equivalent width is a width. It is not — that is the finding it inherits, and it is why every number here is quoted with the window it was taken at. What the matching buys is that at a common window every matched fraction gives a common answer, which is a statement about fractions rather than about the quantity.

What a reader should take from a null of this kind

A negative result is easy to file and hard to use, so it is worth saying what this one is for.

It says the disorder dip is a two-parameter object. Anyone who wants to predict how ordered a head at a rational divergence will be needs the denominator and the head size and nothing else; anyone who observes a departure from that prediction has found something the geometry does not currently account for, and the threshold for “found something” is now about a per cent rather than about the fifteen per cent an unmatched comparison would have allowed.

And it says something about the design of the next question. The matched-set construction cost a search over five hundred and twenty-six fractions and a choice of head size per member — an afternoon — and it answered in one measurement a question two instruments had failed at over two rounds. Where a confound can be held fixed by choosing what to compare, that is worth more than any amount of care with the instrument.

The check that would refuse it

Four assertions run whenever this measurement is drawn.

At the narrow window every member has to give the same width, to within three per cent. That is the result.

At the widest window they must not, by more than three times the narrow window’s disagreement. That is the sensitivity check, and it is compared as an excess over one rather than as a ratio of ratios — ×1.147 is not three times ×1.010, and a check written that way would pass only if the wide window disagreed by a factor of three, which no window here does.

The wide-window widths have to follow the denominator with a rank correlation of at least 0.7, and to follow the crowding with a correlation that is smaller and negative. Both halves are the essay’s explanation of where the spread comes from, and either could fail on its own.

And the whole thing has to hold at a second head size, member by member, to within five per cent. Without that the result would be consistent with the disorder statistic having a head-size dependence beyond the scaling law, which would put it back inside the instrument.

Where this goes next

6/19q = 1919/607/20q = 2020/578/21q = 2121/559/23q = 2320/5116/33q = 3317/3519/45q = 4511/2615/47q = 478/25the fractionits neighbourneighbour distances 0.3064° to 0.3158° · denominators 19 to 47neighbours looked for among denominators up to 60generated from a stated rule, not drawn to look right Fractions with the same neighbours Every instrument this collection has for the width of a disorder dip has a free parameter set by how close the next rational sits — which makes a hypothesis about the neighbourhood untestable with any of them. The repair is not a better instrument. It is a set of fractions whose neighbourhoods are identical and whose denominators are not, and the arithmetic supplies twenty-four of them. crowdednearest neighbour firstwindow 50widest firstwindow 100widest firstwindow 200widest first15/3413/349/3411/3415/3413/349/3411/3415/3413/349/3411/3415/3413/349/3411/34crowding: 13/34 0.193° · 9/34 0.200° · 15/34 0.179° · 11/34 0.286°q = 34 · 791 organsgenerated from a stated rule, not drawn to look right 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. 50100200400800window, in scaled units of δ·n²/qequivalentwidth13/349/3415/3411/34q = 34 · two head sizes agree to 8% at a window of 200generated from a stated rule, not drawn to look right A dip with no outer edge The disorder of a head dips at every rational divergence, and how wide that dip is has carried a long argument. Reading the width as a level crossing has a resolution problem, and the obvious repair is to integrate instead. The integral reproduces beautifully across head sizes and never settles on a value, because there is nothing out there for it to settle against.

Shares its objects with

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

  • Matching instead of correcting — both name artefact, bias, claim testing, honest limits, measurement, negative result, residual, summary statistic
  • A difference forgets a drift — both name artefact, claim testing, honest limits, measurement, negative result, summary statistic, untested claim
  • A dip belongs to the head — both name artefact, disorder, honest limits, measurement, order and disorder, rational approximation, summary statistic
  • A period the grid invented — both name artefact, bias, claim testing, honest limits, measurement, negative result, summary statistic
  • The background is not one sample — both name artefact, disorder, honest limits, measurement, order and disorder, rational approximation, summary statistic
  • The width carries the denominator — both name artefact, disorder, honest limits, measurement, order and disorder, rational approximation, summary statistic

Named objects

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

ArtefactBiasClaim testingDisorderHonest limitsMeasurementNearest neighbourNegative resultOrder and disorderRational approximationRational divergenceReproducibilityResidualSummary statisticUntested claim