Phyllotaxis
the spirals are counted, not admired
Packing and tiling

The window is the neighbour

An integral needs a limit, and this one has two conditions on it that pull opposite ways. It has to scale with the dip, so that two head sizes are comparable, and it has to stay clear of the next rational, which is a fixed distance in degrees. Between them there is no stretch where the answer holds still — and the limit that decides it is the crowding.
21 min read 12 figures State the claim, state the testOne parameter, one familyCounted, not admired

Worth reading first: What a summary throws away · Two laws that want opposite tissue.

The area under a dip is a well-defined number once somebody says where to stop integrating. This essay is about where that is, and the answer turns out to have a name already: it is the distance to the nearest other rational — the crowding — which is the quantity this thread has spent its whole length trying to measure the effect of.

Two conditions, pulling opposite ways

The window has to satisfy two requirements and they are not compatible.

It has to scale with the dip. A head of 1,224 organs has a dip a quarter the width of a head of 612’s, so a window fixed in degrees contains the whole of one dip and a sixteenth of the other. Measuring both at the same angular limit reports mostly-dip for one head and mostly-shoulder for the other, and the comparison between head sizes — the one thing the width law is about — becomes a comparison of two different measurements.

The dip at 21/55 — 137.4545° — at three head sizesWalking the divergence angle off an exact rational, at 825, 1279, 1980 points. The floor falls as the head grows — 0.148 at 825, 0.094 at 1279, 0.060 at 1980, halving for each doubling — and the dip narrows faster: half-widths of -0.0160°, 0.0052°, 0.0028°, a factor of four for each doubling rather than two. Half-width times the square of the head size is -10861, 8532, 11100, which is what makes the width a property of the sample rather than of the angle.00.1000.2000.300-3-2.50-2-1.50-1-0.700offset from the rational, log₁₀ of degreesμ₂825 points1279 points1980 points21/55 of a turn · 12 offsets · vertical rules are the half-widthsgenerated from a stated rule, not drawn to look right
Fig. 1 The problem stated in one figure. One rational at three head sizes: the dip narrows as the head fills, so any limit fixed in degrees means something different at each of them.

Measured: with a window fixed at a tenth of the way to the neighbour — a natural choice, and fixed in degrees — the equivalent widths of the four fractions of denominator 21 disagree between two head sizes by factors of 2.0, 2.3, 2.6 and 2.1, and at wider settings the larger head returns negative widths while the smaller one is still positive.

It has to stay clear of the neighbour. Every rational has a dip; the nearest other fraction to 23/55 is 18/43, a seventh of a degree away. Integrate far enough and the profile being subtracted from is somebody else’s shoulder, the deficit turns negative, and the integral eats the dip it just measured.

The gaps close faster than the dips narrowFor each Fibonacci fraction, the distance to the nearest other rational with a denominator of 60 or less, and the half-width of its own dip at the smallest head that resolves it. The gaps fall from 0.462° at 5/13 to 0.1925° at 21/55; the dips stay between 0.0103° and 0.0155°. The dips never touch — the closest they come is a factor of 15 — so what stops the measurement is not the dips overlapping but the background between them ceasing to be flat. The clear offsets available fall from 70 to 54.-2-1.50-1-0.500divergence angle, as a fraction of a turndegrees (logarithmic)5/138/2113/3421/55to the nearest other rationalthe dip's own half-widthsix denominatorsgenerated from a stated rule, not drawn to look right
Fig. 2 How far it is to the nearest neighbour, for the fractions this thread measures. Between 0.15° and 0.29°, against dips whose own scale is a hundredth of that — so there is room, but the room is not the same for everybody, and that turns out to be the difficulty.

The first condition wants a limit proportional to q/n². The second wants a limit proportional to the crowding, which is a property of arithmetic and does not know how many organs the head has. A single window cannot satisfy both across a range of head sizes, and the whole of what follows is what happens when one of them is chosen.

What a fixed-degree window does, measured

The abstract statement above — that a window fixed in degrees is not comparable between head sizes — is worth putting numbers on, because the size of the failure decides whether it is a nuisance or a wall.

Take the window at a fixed fraction of the distance to each fraction’s nearest neighbour, which is a defensible choice: it is the same relative position in every neighbourhood, and it is well clear of the neighbour’s own dip. Then measure each fraction’s equivalent width at two head sizes.

A fifth of the way to the neighbour is already a wall. At that setting every one of the eight fractions of denominators 21 and 34 gives a positive equivalent width at the smaller head and a negative one at the larger: 437 against −202, 383 against −75, 318 against −578, and so on through the table. The two heads are not disagreeing about a number there; they are disagreeing about whether there is a dip.

A tenth of the way is unreliable rather than impossible. Three of the four fractions of denominator 21 agree between heads to within 22%, and the fourth disagrees by 50%. In the denominator-34 family two agree within 20% and two disagree by factors of 4.1 and 9.4.

The reason is the same in both cases. At the larger head the dip is 2.4 times narrower in degrees, so a window fixed in degrees puts most of its length on the shoulder — where the profile is crossing its own background and the deficit is small and signed — and the integral is dominated by the part of the range that has nothing to do with the dip.

That is what makes the scaled axis compulsory rather than elegant. Without it the head-size comparison, which is the whole of the 1/n² law, cannot be made with an area at all.

The dip at 8/21 — 137.1429° — at three head sizesWalking the divergence angle off an exact rational, at 315, 488, 756 points. The floor falls as the head grows — 0.074 at 315, 0.047 at 488, 0.030 at 756, halving for each doubling — and the dip narrows faster: half-widths of 0.0194°, 0.0125°, 0.0059°, a factor of four for each doubling rather than two. Half-width times the square of the head size is 1925, 2975, 3389, which is what makes the width a property of the sample rather than of the angle.00.2000.400-3-2.50-2-1.50-1-0.700offset from the rational, log₁₀ of degreesμ₂315 points488 points756 points8/21 of a turn · 12 offsets · vertical rules are the half-widthsgenerated from a stated rule, not drawn to look right
Fig. 3 The same statement for the coarser denominator: three heads, three dips, each a good deal narrower than the last. A limit drawn in degrees on this picture cuts all three in different places, and the cut lands further out on the shoulder each time.

Choosing the scaled window, which is the right choice

The scaled axis is the one that makes the measurement reproducible, so it is what the area is measured on: u = δ·n²/q, with the window stated in those units.

It works. Across the eight fractions of denominators 34 and 55, at a window of 200 scaled units, the two head sizes agree to between 0.3% and 8%.

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 55, 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 0.0067° at these head sizes. A width would show as a flat stretch. There is none: every line climbs to 595 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.50100200400window, in scaled units of δ·n²/qequivalentwidth21/5512/5523/5517/55q = 55 · two head sizes agree to 3% at a window of 200generated from a stated rule, not drawn to look right
Fig. 4 The equivalent widths of the four fractions of denominator 55, against the window. The lines are reproducible between head sizes and none of them is flat — which is the pair of facts this thread now has to work with.

But now look at what a common window means in each fraction’s own neighbourhood. At 200 scaled units, on a head of 791 organs, the window reaches:

fraction how far the window reaches the neighbour
15/34 6.1% of the way 0.1795° away
13/34 5.6% 0.1925°
9/34 5.4% 0.1998°
11/34 3.8% 0.2862°

The most crowded fraction’s window reaches 1.6 times further into its clear space than the loneliest one’s. The window is common in units of the dip and uncommon in units of the neighbourhood — and it is the neighbourhood that decides how much of the neighbour’s shoulder has been integrated.

The neighbourhood of 15/34, and where its background was taken fromμ₂ across nine tenths of a degree either side of 15/34, on a head of 791 organs — 23 in each of its 34 rows. The deep notch at the centre is the dip. The shaded columns are the offsets this site now takes a background from: those at least 0.03° from every rational with a denominator of 60 or less, of which there are 48 here. The marked sample at 0.2° is the one the earlier work used, and it lands 0.379° from 26/59. It reads 0.288 against a floor of 0.053. The clear offsets give 0.287.0.1000.2000.3000.400-0.400-0.300-0.200-0.10000.1000.2000.3000.400degrees from 15/34, at 158.8235°μ₂, the second moment of the side-count distribution26/59the background from the clear offsets: 0.287the old single sample: 0.28815/34 · head of 791generated from a stated rule, not drawn to look right
Fig. 5 Where the background is taken for the most crowded member of the family, and how little clear space there is either side of it. The band has to be far enough out to be background and near enough in to be this fraction’s background, and the two constraints are tightest exactly where the residual claim was measured.

Which is why the residual cannot be tested this way

The residual claim was this: after the 1/n² law and the denominator’s coefficient are taken out, what is left is ordered by crowding, with the most crowded fraction giving the widest dip.

An area measured at a common scaled window integrates more of the neighbour’s shoulder for a crowded fraction than for a lonely one. The neighbour’s shoulder lowers the profile, which reduces the deficit, which makes a crowded fraction’s area come out smaller — the opposite direction to the claim, by construction, before any measurement.

So there is a systematic effect running the other way, of unknown size, built into the instrument. Which means neither an agreement with the claim nor a disagreement with it settles anything: an agreement would be the residual winning against the instrument, and a disagreement could be either the instrument or the absence of a residual.

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.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
Fig. 6 And what the ordering does in practice, in the family the claim was first measured in. Three windows, three orderings, none of them the crowding order. The instrument’s own bias is one reason to expect that and it is not the only one.

Why this is not a complaint about precision

A reader who has followed the thread this far might reasonably say: every measurement has a free parameter somewhere, and choosing one and stating it is what a method is. That is right, and it is worth being exact about why this case is different.

A well-posed integral has a range of limits over which the answer is stable. Choosing inside that range is bookkeeping: the number does not move, so the choice is not a measurement. That is what makes an integrated line strength in spectroscopy, or a total under a peak in a diffraction pattern, a quantity rather than a convention — the peak returns to a background, and any limit past the return gives the same total.

Here there is no such range. The equivalent width at windows of 100, 200 and 400 scaled units is 152, 325 and 445 for one fraction, and every one of those windows satisfies both conditions. The answer is roughly proportional to the limit over the whole safe range, which means the limit is not a tolerance — it is the measurement’s dominant term.

The distinction has a practical form. If somebody publishes a dip width for a rational divergence, the question to ask is not “at what precision” but “out to what offset”, and the answer will be doing most of the work. This site has published widths, and this essay and the one before it are the correction: the numbers were level crossings, they were compared with each other consistently, and none of them means anything on its own.

Four fractions of 21, one widthThe half-width of the second-moment dip at four divergence angles with the same denominator, on heads of 488 organs — 23 in each of 21 rows. 8/21 is the Fibonacci convergent, the fraction every earlier measurement of this law was made at and the one whose neighbourhood is emptiest; 5/21, 10/21, 4/21 are not. The four agree within a factor of 1.15, well inside the factor of two the law is stated to, so the width is a function of the denominator and not of how well the fraction approximates its neighbours. What is left over is ordered by the note beside each bar: the more crowded the neighbourhood, the wider the dip comes out, which is the direction a neighbour's own shoulder would push it.half-width of the dip, in degrees, at 488 organsnote: how far the nearest other rational sits8/211.14e-221/55 at 0.312° · convergent5/211.06e-214/59 at 0.291°10/219.91e-321/44 at 0.390°4/211.11e-211/58 at 0.296°q = 21 · 488 organsgenerated from a stated rule, not drawn to look right
Fig. 7 The comparison as the thread has been making it: fractions of one denominator measured the same way and set against each other. Comparisons of that kind survive the correction — what does not survive is quoting any single one of these as a width.

The measurement that would settle it, and what it costs

There is a version of this that would work, and it is worth setting out because it says why nobody has run it.

Hold the window at a fixed fraction of each fraction’s own crowding — so that every dip is integrated over the same share of its own clear space — and then restore comparability across head sizes by measuring each fraction at the head size that makes its dip the same size relative to that window. That means a different head size for every fraction, chosen so that q/n² times the window equals a fixed share of the crowding.

The heads that come out are not free. For the four fractions of denominator 34, matching the loneliest member’s ratio would require the most crowded one to be measured at a head about 25% larger, and cell areas on a head that size cost what they cost. The whole table is six denominators times four fractions times a head size that is now a function of the fraction rather than of the denominator — and every one of them needs its own background band re-chosen, because the band is defined relative to the neighbours.

Four fractions of 55, one widthThe half-width of the second-moment dip at four divergence angles with the same denominator, on heads of 1279 organs — 23 in each of 55 rows. 21/55 is the Fibonacci convergent, the fraction every earlier measurement of this law was made at and the one whose neighbourhood is emptiest; 12/55, 23/55, 17/55 are not. The four agree within a factor of 1.28, well inside the factor of two the law is stated to, so the width is a function of the denominator and not of how well the fraction approximates its neighbours. What is left over is ordered by the note beside each bar: the more crowded the neighbourhood, the wider the dip comes out, which is the direction a neighbour's own shoulder would push it.half-width of the dip, in degrees, at 1279 organsnote: how far the nearest other rational sits21/555.54e-313/34 at 0.193° · convergent12/555.24e-37/32 at 0.205°23/556.67e-318/43 at 0.152°17/556.71e-313/42 at 0.156°q = 55 · 1279 organsgenerated from a stated rule, not drawn to look right
Fig. 8 The comparison as it stands: one denominator, four fractions, measured at the head sizes the denominator sets. Making the comparison the way the paragraph above describes means giving each of these four a different head, which is a different table from the one the thread has been building.

That is not an unreasonable experiment. It is a bigger one than the repair was supposed to be, and its result would be a residual measured against a control that is itself a function of the quantity being tested — which is the shape that has gone wrong twice already in this thread.

The same difficulty, one level up

It is worth noticing that this is the third time this thread has met the same shape, because that is what says it is a property of the object rather than a run of bad luck.

The first was a background taken as a single sample. At 21/55 the sample sat seven thousandths of a degree from 13/34 — inside the neighbour’s dip — and reported a background below the floor, which inverts the verdict and makes a dip look like a bump. The repair was to take a band and choose the offsets in it that are clear of every other rational.

The second was the width read as a level crossing. The level lands on a tread of a staircase and the number reported is the tread’s edge; two fractions of twelve had to be refused.

The third is this one: a window whose outer bound is the distance to a neighbour.

Each repair fixed the previous instrument and inherited the same difficulty in a new place, and the difficulty is that the object has no clear space. A sample can land in a neighbour, a level can land on a neighbour’s shoulder, and a window can reach a neighbour. The dips at the rationals are not features on a ground; they are the ground.

The neighbourhood of 21/55, and where its background was taken fromμ₂ across nine tenths of a degree either side of 21/55, on a head of 1279 organs — 23 in each of its 55 rows. The deep notch at the centre is the dip. The shaded columns are the offsets this site now takes a background from: those at least 0.03° from every rational with a denominator of 60 or less, of which there are 54 here. The marked sample at 0.2° is the one the earlier work used, and it lands 0.007° from 13/34. It reads 0.152 against a floor of 0.094. The clear offsets give 0.258.0.1000.2000.3000.400-0.400-0.300-0.200-0.10000.1000.2000.3000.400degrees from 21/55, at 137.4545°μ₂, the second moment of the side-count distribution13/34the background from the clear offsets: 0.258the old single sample: 0.15221/55 · head of 1279generated from a stated rule, not drawn to look right
Fig. 9 The fraction the first of those three repairs was found on, drawn with its band. Its nearest neighbour is 0.19° away, and the single sample the old method used was inside that neighbour’s dip — the same relationship, at a different scale, as a window that reaches too far.

What the thread is left with

Three statements, in decreasing order of confidence.

The dip’s depth and scale are measurable and stand. The second moment at a rational, and the way the whole profile shrinks as q/n², are properties of the head and not of any window.

The dip’s width is not a quantity. Whatever number is reported for it is a joint statement about the profile and about where the measurement stopped, and the two cannot be separated because the profile has no outer edge.

The residual ordering is not testable with the instruments here. Not “is false” — untestable, which is a weaker and more useful thing to say. The level crossing has a resolution set by the treads of a staircase; the area has a limit set by the crowding. Both instruments’ free parameters are entangled with the quantity the residual is about.

Both vary; only one of them varies enough to findEach organ's step exponents, divided by its own mean so the two are comparable. The ogive's run over 15 per cent of their mean across 5 rings. The convex head's run over 1.15 per cent across 5 — inside the band a 3 per cent error on each ring position leaves, so no ruler separates it from a flat disc.0.9000.95011.050123which step of the ladderexponent ÷ its meanan ogive — 15%a convex head — 1.15%what 3% per ring allows5 rings on the ogive · 5 on the head15% against 1.15%
Fig. 10 The distinction this site keeps having to draw. A quantity that moves with a knob nobody can set on principle is not a small measurement problem; it is a statement that the quantity was not defined. Saying so is more use than reporting the number the knob happened to be at.

What this does not say

It does not say crowding has no effect. It plainly does have one — it is why a background sample can land inside a neighbouring dip and invert a verdict, which is a defect this thread found and repaired. What is untestable is the residual claim: crowding as an ordering of dip widths after the main laws are removed.

It does not say the area was a wasted repair. It measures two fractions the level crossing could not, it is reproducible where the level crossing was a lottery, and it makes the shape of the difficulty visible. A knob one can turn is better than a knob nobody knew was there.

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. 11 The result that does survive both instruments, drawn without the denominator divided out. Whatever the width is measured with, the coefficient is a function of q — a claim about the arithmetic of the fraction rather than about the geometry of its neighbourhood.

And it does not say the dips are shallow or unimportant. A rational divergence really does make a tidier head, and the tidiness is large: at 5/13 the second moment at the rational is under an eighth of its level nearby. Nothing in this essay is about the size of the effect. It is about the shape of the thing whose size is being asked for.

The disorder of a head against its divergence angle, 800 pointsμ₂ is the mean squared departure of a Voronoi cell's side count from six, measured on 800 points inside 86% of the radius. Swept across 0.60° 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 137.646° — which is 360 × 13/34 — it is 0.071; At 137.874° — which is 360 × 18/47 — it is 0.096; At 138.000° — which is 360 × 23/60 — it is 0.104. The ticks along the top are the fractions p/q, placed from arithmetic rather than from the curve.00.1000.2000.3000.400137138138138divergence angle, degreesμ₂, the mean squared departure of a cell's side count from six13/34201 angles · 0.0030° apart · 800 points eachmarks are the fractions, placed from arithmetic
Fig. 12 The landscape once more, narrowly. Every visible feature is a rational’s dip, and the ground between any two of them is the shoulders of both. The picture is the argument: there is no ground.

The check

The window’s outer condition is asserted rather than described, in two parts.

A window wider than the profile that was sampled is refused outright, so an extrapolated area cannot be reported by accident. And the window every comparison here uses is required to sit inside a third of the way to the nearest other rational — checked against the crowding computed from the arithmetic, for the most crowded fraction in the set, so that the constraint is tested where it binds.

The reproducibility claim is asserted too: at the stated window the equivalent width must agree between two head sizes to within a fifth, over a whole family. That one exists to keep the essay honest in the other direction — the area’s failure is specific, and it is not a failure to measure anything.

Where this goes next

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. 00.2000.400-0.400-0.300-0.200-0.10000.1000.2000.3000.400degrees from 21/55, at 137.4545°μ₂, the second moment of the side-count distribution13/34the background from the clear offsets: 0.339the old single sample: 0.11521/55 · head of 825generated from a stated rule, not drawn to look right The background is not one sample The dip in disorder at a rational angle is a comparison against a background, and the background was one measurement taken two tenths of a degree away. At 21/55 that lands seven thousandths of a degree from 13/34 — inside another rational's dip — and the comparison inverts. Fixed, the dip survives to a denominator of 89. half-width of the dip, in degrees, at 1279 organsnote: how far the nearest other rational sits21/555.54e-313/34 at 0.193° · convergent12/555.24e-37/32 at 0.205°23/556.67e-318/43 at 0.152°17/556.71e-313/42 at 0.156°q = 55 · 1279 organsgenerated from a stated rule, not drawn to look right Four fractions with one denominator The dip in a head's side-count disorder is as wide as 150·q/n², measured over six fractions — every one of them a Fibonacci convergent, which is the emptiest neighbourhood a denominator ever gets. So the law could be about the denominator or about how well the fraction approximates its neighbours. Four fractions of 55 at one head size settle it in one figure.

Shares its objects with

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

  • The order belonged to the method — both name artefact, convergents, disorder, honest limits, identifiability, measurement, measurement error, null model, rational angle, rational approximation, sampling, summary statistic
  • A width read off a staircase — both name artefact, convergents, discretisation, honest limits, identifiability, measurement, measurement error, rational angle, sampling, summary statistic
  • A dip belongs to the head — both name artefact, disorder, divergence angle, honest limits, measurement, rational angle, rational approximation, sampling, summary statistic
  • The width carries the denominator — both name artefact, convergents, disorder, honest limits, measurement, rational angle, rational approximation, sampling, summary statistic
  • The disorder is a staircase — both name artefact, disorder, divergence angle, measurement, rational angle, rational approximation, sampling, summary statistic
  • The grid was in the number — both name artefact, discretisation, divergence angle, honest limits, measurement, measurement error, sampling

Named objects

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

ArtefactConvergentsDiscretisationDisorderDivergence angleHonest limitsIdentifiabilityMeasurementMeasurement errorNull modelRational angleRational approximationSamplingSummary statisticVoronoi cell area