How wide a band should be

Six bands, six widths, and a prediction that looks like it cannot fail. A band ends where the settled divergence has moved a twentieth of a degree from its value at the handover. If the divergence slides at some rate across a rung, then a band should be about two twentieths of a degree divided by that rate wide, and a steep rung should give a narrow band.

It is wrong by factors of five, in both directions, and the reason it is wrong is the reason a band exists at all.

The quantity being predicted #

A width in the rise is a ratio, not a difference, because the ladder is geometric and everything measured on it has to be. So a band’s width is quoted as the factor between its coarse end and its fine one — 1.148, 1.248, 1.284, 1.030, 1.185 and 1.279 for the six — and the arithmetic is done on the logarithm of that.

In those units the six measure 0.138, 0.221, 0.250, 0.030, 0.170 and 0.246. The Lucas 3/4 band is an order of magnitude narrower than the rest and everything below is about why.

The straight-line prediction #

Take the divergence at a rung’s coarse end and at its fine end, divide the difference by the rung’s width in log rise, and call that the slope. Across the six rungs it measures −4.298, 1.321, −0.338, −0.654, −2.913 and 0.516 degrees per unit of log rise, so the steepest rung is thirteen times the shallowest.

A band ends where the divergence has moved 0.05° either way. On a straight line of slope s that takes 0.1/|s| in log rise, which for the six is 0.023, 0.076, 0.296, 0.153, 0.034 and 0.194.

Those are the predictions. The measurements are 0.138, 0.221, 0.250, 0.030, 0.170 and 0.246.

How badly it fails #

Measured over predicted: 5.92, 2.92, 0.84, 0.20, 4.94 and 1.27.

Two of the six are within a third of right. Two are out by factors of three and five in one direction and one is out by a factor of five in the other. A prediction that is right to a third on some rows and out by five on others, in both directions, is not a prediction with a missing constant in it; there is no constant that would help, and no reordering of the rows that would turn the failures into a trend.

It fails hardest on the two rungs whose end-to-end slope is steepest — the golden 3/5 at −4.298 and the Lucas 4/7 at −2.913 — which is where the straight line promises the narrowest bands and the measurement gives some of the widest.

Why it fails #

Because a band is exactly the stretch of a rung where the divergence is not sliding.

The reason a band exists is that near a handover the settled divergence has a shallow floor — that is what the design rests on — so the curve is flat there and steep elsewhere. A rung-average slope is a number computed over the whole rung, most of which is the steep part, and it is being used to predict the width of the flat part.

Put the other way round: if the divergence really did slide at a constant rate down a rung, there would be no flat stretch, and there would be no band. The prediction assumes away the thing it is trying to predict the size of.

The second prediction #

At a stationary point the linear term vanishes and the leading behaviour is quadratic. If the divergence near a handover goes as a curvature c times the square of the distance in log rise, then it moves 0.05° at a distance of √(0.05/|c|) either side, and a band is twice that: 2·√(0.1/|c|).

The curvature is read by fitting a quadratic to the band’s own rises. Across the six it measures −9.97, 9.45, −5.29, −75.98, −6.80 and 5.81, so the Lucas 3/4 rung is seven to fourteen times more curved than the others.

The predictions are 0.200, 0.206, 0.275, 0.073, 0.243 and 0.263, against measurements of 0.138, 0.221, 0.250, 0.030, 0.170 and 0.246.

How well it does #

Measured over predicted: 0.69, 1.08, 0.91, 0.41, 0.70 and 0.94.

Five of the six are between 0.69 and 1.08. The sixth is 0.41, and it is the Lucas 3/4 band — the one whose divergence is not stationary at its handover, and therefore the one this arithmetic does not apply to.

So the prediction that assumes a stationary point works on the five rungs that have one and fails on the one that does not. That is the right pattern for an account to have, and it is worth saying that it is also the pattern a lucky guess would have if the outlier had been chosen after the fact. It was not: the stationarity is measured independently, as the linear term of the same fit.

The linear terms, which are the check #

The same quadratic fit that gives the curvature gives a slope at the handover. Across the six it measures 0.075, −0.117, 0.005, −2.193, −0.131 and −0.134 degrees per unit of log rise.

Five of the six are under a seventh of a degree, which on a curve whose rung-wide slope is 0.3 to 4.3 is stationary to any resolution available here. The sixth is 2.19, a factor of seventeen above the next.

That is the measurement that makes the outlier an outlier for a stated reason rather than by exclusion. The Lucas 3/4 band is not “the one that does not fit”; it is the one whose fitted linear term is not small, and the width prediction that assumes a small linear term therefore has no business being applied to it.

What kind of test this is #

Weaker than the first one, and the weakness has to be stated because it is easy to miss.

The straight-line prediction uses a slope taken over the whole rung, which is independent of the band. The curvature prediction uses a fit to the band’s own rises, which are the rises whose extent is being predicted. That is not circular — the fit uses the shape of the curve inside the band and the prediction is about where the curve leaves a bound — but it is a consistency check rather than a forecast.

An honest forecast would fit the curvature on rises outside the band and predict the band from it. That is available and is not done here, because the rises outside a band are on the steep part of the rung where a quadratic about the handover is not the right description either.

The residual is not random #

The five ratios that work are 0.69, 0.70, 0.91, 0.94 and 1.08 — four of them under one. A quadratic under-predicts nothing and over-predicts a little on most of these bands, which means the real curve flattens faster away from the handover than a parabola does on four rungs and slower on one.

That is a fourth-order effect and there is not enough here to fit one. Six bands, one free shape, and a spread of thirty per cent; anything fitted to that would be fitted to five points and a plausible story.

How consistent is worth a number, since “consistent enough” is the kind of phrase that hardens into a claim on re-reading. Taken as logarithms — the right scale for a ratio — the five have a mean of −0.16 and a spread of 0.20 about it, so the quadratic over-predicts by about 15 per cent on average with a scatter of twenty. Against a null of no bias that is a t of 1.8 on four degrees of freedom, which is roughly one chance in seven.

So the bias and the scatter are the same size at five points, and one in seven is not a result by any standard this collection uses. What makes it worth recording anyway is the direction: four of the five under one is the shape a real fourth-order term would produce, and if a seventh band ever arrives it will either move that number decisively or leave it where it is. A leaving with a t attached is a leaving somebody can act on; one without is a hunch that gets quoted.

What can be said is the sign, and it is worth recording as a leaving rather than a result.

Why a width matters at all #

Because it decides whether the design is usable. A band’s job is to put two opposite orderings at its two ends with the divergence held between them, and whether the two contact steps have separated by the ends is a question about how far the rise has moved.

The band that moves nothing is the case where the answer is no. Its ends differ by 0.03 and 0.43 per cent in step length, against 3.4 to 8.3 per cent for the other five, and the whole reason is that its band is three per cent of its rung rather than fifteen to seventy.

So a prediction of the width is a prediction of whether a rung can carry the design at all — which is what would make it worth having on a ladder whose finer rungs nobody has swept.

What it would predict below the ladder #

Nothing usable, and that is worth saying rather than extrapolating. The ladder stops at a rise of 0.0040 because below it a stem does not settle onto a lattice at any run length. There is no 13/21 rung to predict a band on.

Above the ladder there are two rungs with no handover at all, so there is nothing to predict there either. Six bands is what this ladder holds, and a prediction tested on six is tested on all of them rather than on a sample.

That is an unusual and slightly uncomfortable position: the test has no holdout, because there is nothing held out.

The two arithmetics compared #

The first is wrong and cheap; the second is right and slightly circular. Between them they say something the measurement alone does not.

A band’s width is not set by how fast the divergence changes across its rung. It is set by how sharply the divergence turns at the handover — a second derivative rather than a first — and the two are unrelated: the golden 3/5 rung has the steepest average slope and a middling curvature, and the Lucas 3/4 rung has a middling slope and by far the largest curvature.

A reader who had only the first arithmetic would predict the widths in the wrong order. The rank correlation between the straight-line predictions and the measurements is negative.

What was expected #

The prediction was written down before the widths were measured, and it was expected to work. It has the shape of arithmetic that usually does: one tolerance, one rate, one division, no free parameter.

The reason to record that it failed rather than quietly replacing it is that the failure is the informative part. Being out by a factor of five in both directions says the divergence curve has structure a rung-average cannot see, and that is a statement about the curve that the successful prediction, on its own, would not have produced.

What is being measured when a band is measured #

A width in the rise, and it is worth separating from the other number each band carries.

A band occupies some fraction of its rung — 3.0, 14.6, 18.7, 23.2, 48.4 and 72.1 per cent — and that fraction is not what either prediction is about. It depends on where the rung’s transitions happen to sit, which is a fact about the ladder rather than about the divergence near a handover.

The width in the rise is the quantity the arithmetic is stated over, and the two readings order the six bands differently: the golden 8/13 band is the widest by fraction of rung and only just the widest by factor in the rise, while the Lucas 7/11 band is second on both. Quoting one and predicting the other would be a straightforward error and it is an easy one to make.

Where the tolerance enters #

Both predictions divide by the same twentieth of a degree, so both scale the same way with it: doubling the tolerance doubles the straight-line prediction and multiplies the curvature one by the square root of two.

That gives a check neither of them was designed for. If the tolerance were halved, the measured widths should shrink by a factor of two on the straight-line account and by 1.41 on the curvature account — and the measured widths are what they are, so the two accounts predict different responses to a choice nobody has to make again.

It is not run here. It costs one re-sweep of six bands and it would separate the two accounts on a quantity that is not the widths themselves, which is a better test than either of the ones above.

Two other things the widths track #

Neither is an account and both are worth recording.

The two widest bands are the two on the finest rungs of their branches — 8/13 and 7/11 — and the narrowest is on a coarse one. That could be a fact about curvature at the fine end or it could be a fact about rung width, since the fine rungs are narrow and a band bounded by a transition is bounded by the rung.

And the two bands whose coarse end is set by a transition rather than by the divergence are among the three widest, which is what a lower bound looks like when it is quoted as a measurement. Their true widths are larger than the numbers here.

Why the failure is the useful half #

A prediction that works tells a reader that an account is consistent with six numbers. A prediction that fails by factors of five in both directions tells a reader something about the object.

Here it says the divergence curve is not straight down a rung — that it has a flat stretch and a steep one, and that the flat stretch is where the handover is. That is a statement about the shape of the curve, and nothing else in this collection measures it. The slide across a rung has been drawn several times and read for its total, never for its shape.

So the straight-line arithmetic, run and refuted, is a measurement of curvature by a route that never mentions curvature.

What this predicts about a rung nobody has swept #

Nothing on this ladder, since there is no seventh handover. But the arithmetic is not about this ladder.

Any rule with a geometric ladder of rungs, a settled divergence that slides along each of them, and a crossing where two contact steps swap places, will have a band around that crossing whose width is set by the curvature there. The prediction is 2·√(2·flat/|c|) and it does not mention phyllotaxis.

Whether it transfers is not testable here and is worth stating as the shape of the claim rather than as a claim: the arithmetic is general, the six numbers it is tested on are not, and six is the population of this ladder rather than a sample of anything wider.

The one line #

A band ends where the divergence has slid a twentieth of a degree. Predicted from the rung’s average slope, the six widths come out wrong by factors of 0.20 to 5.92, in both directions and worst where the rung is steepest. Predicted from a curvature about a stationary point, they come out at 0.41 to 1.08 — five inside a third, and the sixth being the one rung whose divergence is not stationary at its handover.