Phyllotaxis
the spirals are counted, not admired
Stems and cones

How wide a band should be

A band ends where the divergence has slid a twentieth of a degree, so its width should follow from how fast the divergence slides. Predicted from the rung's slope that is wrong by factors of 0.20 to 5.92; predicted from a stationary point it is 0.41 to 1.08, and the outlier is the rung that has no stationary point.
13 min read 20 figures State the claim, state the testTwo routes, one number

Worth reading first: Where a handover sits · Counting the spirals · The angle the ladder returns to.

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.

Two ways of predicting how wide a band is. A band ends where the settled divergence has moved 0.05° from its value at the handover, so the width should follow from how fast the divergence changes there. Reading that rate as the rung's average slope predicts widths that are wrong by factors of 0.20 to 5.92 — wrong in both directions, so no constant rescues it. Reading it as a curvature about a stationary point gives 0.41 to 1.08, with five of the six inside a third. The difference between the two is the difference between a curve and its average, and a band is exactly where the two are least alike.
Fig. 1 Each band’s measured width divided by the width two different arithmetics predict for it.

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.

Where the two contact steps change places, on every rung. One row per rung of the two branches, drawn from its coarse end to its fine one on a logarithmic axis. The mark on each row is the rise at which the two contact steps change places — the rung's handover — and the shaded stretch is the band around it over which the settled divergence holds still. six of the eight rungs have a handover and every one of those sits between 6 and 40 per cent of the way down its rung, never past the middle. The two rungs without one are the coarsest on each branch, whose crossing is above the range this ladder reaches.
Fig. 2 The six bands as stretches of their own rungs, which is what the widths are widths of.

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.

The divergence slides along the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not.
Fig. 3 The divergence across one whole rung, whose end-to-end slope is what the first prediction uses.

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.

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.

Two ways of predicting how wide a band is. A band ends where the settled divergence has moved 0.05° from its value at the handover, so the width should follow from how fast the divergence changes there. Reading that rate as the rung's average slope predicts widths that are wrong by factors of 0.20 to 5.92 — wrong in both directions, so no constant rescues it. Reading it as a curvature about a stationary point gives 0.41 to 1.08, with five of the six inside a third. The difference between the two is the difference between a curve and its average, and a band is exactly where the two are least alike.
Fig. 4 The straight-line prediction alone, on which the ratios run from a fifth to nearly six.

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.

Two lines across the 5/8 rung, crossing once. The divergence the rule settles on, against the divergence at which the two contact steps would be exactly the same length. The second is arithmetic on the lattice and no stem is grown for it. Across this rung the balanced line moves 2.281 degrees and the rule's own line moves 1.262, so the shallower line crosses the steeper one, and it does so exactly once at a rise of 0.0154 — 16 per cent of the way down from the coarse end. That crossing is the handover: above it one family has the shorter step and below it the other does. So a rung has one handover, its position is fixed by the arithmetic rather than by any experiment, and a sweep of the rise carries a stem across it at a place nobody chose.
Fig. 5 The two curves whose crossing is a handover, and near which the rule’s own divergence flattens.

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.

The 3/5 rung at a tenth of the ladder's step. Every rise of one rung, sampled ten times as finely as the ladder that found the locked band. All 23 are counted at 3 and 5 spirals and all 23 settle: the largest wander is 0.221 degrees, against the 0.5 degree threshold and against the 0.79 to 1.60 degrees the band on the coarse rung wobbles by. The divergence slides smoothly from 139.0625 to 136.7344 degrees with no rise stuck on a rational and none stuck on anything else. Whatever the band is, it is not something a coarser sampling was hiding here.
Fig. 6 The divergence near a transition, resolved finely enough for a curvature to be fitted rather than guessed.

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.

Two ways of predicting how wide a band is. A band ends where the settled divergence has moved 0.05° from its value at the handover, so the width should follow from how fast the divergence changes there. Reading that rate as the rung's average slope predicts widths that are wrong by factors of 0.20 to 5.92 — wrong in both directions, so no constant rescues it. Reading it as a curvature about a stationary point gives 0.41 to 1.08, with five of the six inside a third. The difference between the two is the difference between a curve and its average, and a band is exactly where the two are least alike.
Fig. 7 The curvature prediction alone, with five bands inside a third and the sixth being the rung with no stationary point.

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.

The two contact steps change places inside the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not.
Fig. 8 The other quantity read across a rung, whose crossing is what the divergence is stationary near.

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.

What each band holds, and what it moves. One row per band. The counted pair is held by construction and the settled divergence is held to a twentieth of a degree; the quantity a band exists to move is which of the two contact steps is the shorter. five of the six do move it — the ordering changes hands exactly once inside, and at both ends the two steps differ by enough for an ordering to mean anything. The remaining one changes hands three times and its two steps are never more than 0.0 per cent apart, so it holds all three quantities and is a control rather than an experiment.
Fig. 9 The six bands, whose own rises are what the second prediction’s curvature is fitted to.

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.

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

Two ways of predicting how wide a band is. A band ends where the settled divergence has moved 0.05° from its value at the handover, so the width should follow from how fast the divergence changes there. Reading that rate as the rung's average slope predicts widths that are wrong by factors of 0.20 to 5.92 — wrong in both directions, so no constant rescues it. Reading it as a curvature about a stationary point gives 0.41 to 1.08, with five of the six inside a third. The difference between the two is the difference between a curve and its average, and a band is exactly where the two are least alike.
Fig. 10 Both predictions on one axis, on which the second one’s residuals sit mostly to one side.

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.

The families left standing, on both sides of every handover. One row per band, with every offset cut at rises spread across it and always at both ends and at the handover itself. The last column is every family left standing anywhere on that band. On three of the four bands that wreck at all it is a single family, unchanged across a rise at which the two contact steps swap places — so the step ordering is not what decides which family survives, and the result now rests on four counted pairs rather than on two. The shortest-hop reading scores 44 of 114 across the whole set, which is what a reading looks like when the quantity it is stated over is not in the mechanism.
Fig. 11 The cuts each band carries, which is what a usable width buys.

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.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.381, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 12 The whole ladder, on which six rungs carry a handover and nothing outside it does.

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.

Where a rung's handover sits, over six rungs. The rise at which the two contact steps change places, as a fraction of the way down each rung from its coarse end, over every rung on both branches that has one. All six fall in the coarse half and none in the middle: the positions measure 6, 12, 15, 33, 35, 40 per cent. That is a fact about the arithmetic of the lattice rather than about any experiment run on it, because the crossing is where the two step lengths are equal and both are functions of the rise and the divergence alone.
Fig. 13 Where the handovers sit, which is where both arithmetics are anchored.

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.

The settled divergence down the golden branch. Every rise from 0.07 down to 0.00482, plotted against the divergence the rule settles on, with each rung drawn in its own stroke and the branch's limit angle marked. The curve does not slide: it turns three times in four rungs, climbing across one and falling across the next, so a value it takes on one rung it takes again on another. That is what makes a matched pair possible — two rises, different counted pairs, one angle — and it is the whole reason the design exists on this branch. The widest excursions from the limit angle, coarse rung first, are 3.195°, 3.352°, 0.961°, 0.422°.
Fig. 14 The whole curve, on which a rung-average slope and a local curvature are very different quantities.

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 two contact steps change places, on every rung. One row per rung of the two branches, drawn from its coarse end to its fine one on a logarithmic axis. The mark on each row is the rise at which the two contact steps change places — the rung's handover — and the shaded stretch is the band around it over which the settled divergence holds still. six of the eight rungs have a handover and every one of those sits between 6 and 40 per cent of the way down its rung, never past the middle. The two rungs without one are the coarsest on each branch, whose crossing is above the range this ladder reaches.
Fig. 15 The bands as stretches of their own rungs, which is the reading the predictions are not about.

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.

What each band holds, and what it moves. One row per band. The counted pair is held by construction and the settled divergence is held to a twentieth of a degree; the quantity a band exists to move is which of the two contact steps is the shorter. five of the six do move it — the ordering changes hands exactly once inside, and at both ends the two steps differ by enough for an ordering to mean anything. The remaining one changes hands three times and its two steps are never more than 0.0 per cent apart, so it holds all three quantities and is a control rather than an experiment.
Fig. 16 The six bands at the tolerance they were grown at, which is the one number both predictions divide by.

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.

The 3/5 rung at a tenth of the ladder's step. Every rise of one rung, sampled ten times as finely as the ladder that found the locked band. All 23 are counted at 3 and 5 spirals and all 23 settle: the largest wander is 0.221 degrees, against the 0.5 degree threshold and against the 0.79 to 1.60 degrees the band on the coarse rung wobbles by. The divergence slides smoothly from 139.0625 to 136.7344 degrees with no rise stuck on a rational and none stuck on anything else. Whatever the band is, it is not something a coarser sampling was hiding here.
Fig. 17 The fine end of the ladder, where the two widest bands sit and where the rungs are narrowest.

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.

The divergence slides along the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not.
Fig. 18 The slide across a rung, whose shape the failed prediction is a measurement of.

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.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13 → 13/21. Consecutive transitions are 0.382, 0.382, 0.383, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 19 The ladder the six bands sit on, which is the whole of what the prediction has been tested against.

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.

What each band holds, and what it moves. One row per band. The counted pair is held by construction and the settled divergence is held to a twentieth of a degree; the quantity a band exists to move is which of the two contact steps is the shorter. five of the six do move it — the ordering changes hands exactly once inside, and at both ends the two steps differ by enough for an ordering to mean anything. The remaining one changes hands three times and its two steps are never more than 0.0 per cent apart, so it holds all three quantities and is a control rather than an experiment.
Fig. 20 The six bands with their widths and their ends, which is the table both predictions are scored against.

Where this goes next

Where the two contact steps change places, on every rung. One row per rung of the two branches, drawn from its coarse end to its fine one on a logarithmic axis. The mark on each row is the rise at which the two contact steps change places — the rung's handover — and the shaded stretch is the band around it over which the settled divergence holds still. six of the eight rungs have a handover and every one of those sits between 6 and 40 per cent of the way down its rung, never past the middle. The two rungs without one are the coarsest on each branch, whose crossing is above the range this ladder reaches. Four crossings nobody visited Six rungs of the ladder carry a handover and two of them had a band built on them. The other four are here: 70, 126, 16 and 124 rises wide, found by sweeping at a ratio rather than at a fixed step in the rise, which is why the fine ones had been stepped over. What each band holds, and what it moves. One row per band. The counted pair is held by construction and the settled divergence is held to a twentieth of a degree; the quantity a band exists to move is which of the two contact steps is the shorter. five of the six do move it — the ordering changes hands exactly once inside, and at both ends the two steps differ by enough for an ordering to mean anything. The remaining one changes hands three times and its two steps are never more than 0.0 per cent apart, so it holds all three quantities and is a control rather than an experiment. A band that moves nothing One of the six bands holds the counted pair, holds the divergence, and does not move the ordering: its two contact steps stay within four parts in a thousand of each other across the whole of it, so the ordering changes hands three times and neither end has one worth the name. Two lines across the 5/8 rung, crossing once. The divergence the rule settles on, against the divergence at which the two contact steps would be exactly the same length. The second is arithmetic on the lattice and no stem is grown for it. Across this rung the balanced line moves 2.281 degrees and the rule's own line moves 1.262, so the shallower line crosses the steeper one, and it does so exactly once at a rise of 0.0154 — 16 per cent of the way down from the coarse end. That crossing is the handover: above it one family has the shorter step and below it the other does. So a rung has one handover, its position is fixed by the arithmetic rather than by any experiment, and a sweep of the rise carries a stem across it at a place nobody chose. Two lines that cross once The divergence at which a lattice's two contact steps are exactly equal is a curve across each rung, computable from the geometry with nothing grown. The rule's own settled divergence is a second, shallower curve, and where they cross is where the step ordering changes hands.

What links here

Computed from the collection, not written here: the essays that point at this one.

Shares its objects with

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

Named objects

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

Claim testingDivergence angleFalsifiabilityFittingGeometric ladderHandoverMeasurementNegative resultPredictionResidualResolutionRiseRungTolerance