Phyllotaxis
the spirals are counted, not admired
Stems and cones

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.
12 min read 16 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.

A band is supposed to move exactly one thing. The counted pair is held because a rise returning a different pair ends the band; the settled divergence is held to a twentieth of a degree because that is what the band’s extent is defined by; and the step ordering reverses, because a band is grown around the rise where the two contact steps change places.

Five of the six do that. The sixth holds all three, and this essay is about why it is in the table rather than out of it.

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. 1 The six bands, with how many times the ordering changes hands inside each and how far apart the two steps are at the ends.

The band

The Lucas 3/4 rung runs from a rise of 0.0640 down to 0.02390 and its handover — the rise at which its 3-step and its 4-step change places — sits at 0.04299, which is 40.4 per cent of the way down it. That is the deepest handover on the ladder and still in the coarse half.

Grown outwards at two parts in a thousand — the ratio step the four new bands needed — the band around it carries 16 rises and spans a factor of 1.030 in the rise. That is three per cent of the rung, against 14.6, 23.2, 72.1, 18.7 and 48.4 per cent for the other five. It is an order of magnitude narrower than anything else here.

Sixteen rises is not too few to sweep. It is too few for the thing being swept: the band’s whole purpose is to put two orderings at its two ends, and sixteen steps of two parts in a thousand do not carry the rise far enough from the crossing for the two steps to come apart.

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 Every rung with its band shaded, on which this one is a mark rather than a stretch.

What is wrong with it

Two things, and they are the same thing.

The ordering changes hands three times inside those sixteen rises rather than once. And at the two ends the two contact steps differ in length by 0.03 per cent and 0.43 per cent — against 3.4 to 8.3 per cent at the ends of the other five bands, and against a threshold of four parts in a thousand below which an ordering between two steps is not an ordering.

A quantity that flips three times over sixteen samples, while the difference it is the sign of is a few parts in a thousand, is a quantity that is not doing anything. It is the sign of a number that is zero to the resolution available.

Two lines across the 4/7 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 3.537 degrees and the rule's own line moves 2.602, so the shallower line crosses the steeper one, and it does so exactly once at a rise of 0.0222 — 7 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. 3 The two length curves whose crossing is a handover, on the Lucas branch, where the crossing here is very shallow.

Why the two steps stay together

Because the band is narrow, and the band is narrow because the divergence is not flat there.

Fitted a quadratic across the band’s own rises, five of the six bands have a slope at their handover of between 0.005 and 0.13 degrees per unit of log rise — which is stationary to the resolution available. The Lucas 3/4 band has 2.19, a factor of seventeen larger than the next.

That is a measurement rather than an explanation. Why the divergence should have a turning point at a handover on five rungs and not on the sixth is not answered anywhere here; the two curves whose crossing makes a handover are the balanced divergence and the rule’s own, and nothing says the second has to be stationary where they meet.

So on five rungs the divergence has a genuine turning point at the handover and the curve stays inside a twentieth of a degree for a long way either side. On this one the divergence is running downhill through the crossing, hits the bound after eight steps, and the band ends before the two steps have had room to separate.

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 widths against a prediction from curvature, with the outlier being the band whose divergence is not stationary.

The two facts are one fact

The two contact steps are equal at the handover — that is what a handover is — and they separate as the rise moves away from it. How far the rise can move is what the band’s width is. So a narrow band is a band whose ends are close to the crossing, and ends close to a crossing are ends where the two steps are nearly equal.

Stated that way it stops being a coincidence and becomes arithmetic. A band is wide when the divergence is flat; the ordering is meaningful at a band’s ends when the band is wide; so the same property of the divergence curve decides both.

The Lucas 3/4 rung fails on both counts because it fails on one.

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. 5 The ratio between the two contact steps across a rung, which is the quantity that is small at a band’s centre and has to be large at its ends.

What it would have taken to hide this

Very little, and that is the reason to write it down.

The first version of the check over the six bands required every one of them to change its ordering exactly once. That version fails, and the obvious repair is to loosen it to “most of them” — at which point a table of six reports five matched pairs and one unremarked row, and nothing in the output says a band exists that holds all three quantities.

The second obvious repair is worse: drop the row. Six handovers, five bands, and a sentence saying the sixth was unsuitable. That is a design reporting five for five.

What the check does instead is require that exactly one band fail, and that the one that fails do so because its steps are never separated rather than because its divergence moved. Both halves are asserted, so a future change that broke a different band in a different way would not pass.

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. 6 The table as it is written, with the failing row in it and the reason for its failure in its own column.

It is a control

Having one is useful. Five bands hold two quantities and move a third, and the answer does not change; one band holds all three, and the answer does not change either. The second is what a null control looks like.

Without it, “the surviving family does not change across a band” is a statement about bands that move the ordering, and the natural worry is that a band is simply a stretch of rises too narrow for anything to change across. The Lucas 3/4 band is narrower than any of the others and its cuts behave the same way, which is a small piece of evidence that the constancy is not about width.

Small, because that band wrecks at no offset at all — so what it controls for is the geometry rather than the cut.

There is a second reason to keep a row that measures nothing. The census this thread’s readings were scored on turned out to have been very nearly holding the quantity it was thought to be varying, and that was found by computing a column nobody had computed. A table that lists the cases where a design does not apply is a table someone can compute such a column from later; a table that lists only the cases where it worked is not.

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. 7 The cuts made at every band, including the two that wreck at no offset.

It wrecks at nothing

The Lucas 3/4 band sits at rises around 0.043, which is the coarse end of the Lucas branch. Cut a stem there at every offset the front reaches and every cut repairs.

That is the coarse end behaving as it has been measured: a front of four organs is its own two edges with no middle, and a single removal has nothing to sever. The golden 3/5 band, also coarse, does the same thing — two of the six bands wreck at no offset, and both are the coarse ones.

So the Lucas 3/4 band is doubly not an experiment: it does not move the ordering, and it produces no cut to see the effect of moving it.

What a two-organ cut does at each rise of the 2/3 rung. At every rise the coarse rung is a lattice on, all 36 arrangements of two organs removed, with the ones that never repair counted and split by where they end up. 22 of 324 cuts across the rung reverse the stem's handedness onto the mirror of the divergence they were cut from. 40 fall instead into a cycle whose mean is half a turn, which the lattice they came from has no number for. 4 rises give only the first, 4 give only the second, and at a rise of 0.075 both happen in the same table, which is what says the fate belongs to the cut and not to the rise.
Fig. 8 What happens to a coarse stem when an organ is removed, which is nothing that lasts.

Two coarse bands, two different failures

The golden 3/5 band and the Lucas 3/4 band both wreck at nothing, and they are not the same case.

The golden 3/5 band carries 70 rises, spans 1.148 in the rise, reverses its ordering exactly once, and has ends separated by 3.9 and 4.7 per cent. It is a perfectly good matched pair with nothing to measure on it. If a way were found to wreck a coarse stem — a removal of two organs, say — that band would be ready.

The Lucas 3/4 band would not. Its ends are not a matched pair whatever is done to them, because its two steps are the same length at both.

That is worth separating because the two rows look identical in a column headed “wrecked cuts: 0”.

The distinction has a use. Removing two organs wrecks a coarse stem where removing one cannot, so the golden 3/5 band is a design waiting for an experiment that now exists. The Lucas 3/4 band is not: a two-organ cut across it would still be a cut across a band whose two ends carry the same ordering.

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. 9 The handovers by position in their rungs, on which the two coarse bands sit at 11.6 and 40.4 per cent.

What the grid can and cannot resolve

The two steps at this band’s ends differ by 0.03 and 0.43 per cent in length. The azimuths those lengths are computed from are read on a grid of 1,536 candidate positions, so a divergence is quantised at 0.234° — and a difference of three parts in ten thousand in a step length is well below what that grid can settle.

Which means the sign of the difference at this band’s ends is not a measurement. It has a value, the arithmetic produces it, and it would change if the grid were finer. Calling one step shorter there is reporting the last digit of a number whose earlier digits are all zero.

The other five bands are not in that position: 3.4 per cent is more than a hundred times the resolution.

This site has been caught by exactly this before, in the other direction. A period the grid invented was a structure read off a quantity the azimuth grid had quantised, and the check that found it was re-measuring on a finer grid. The same check is available here and is not worth running: a difference of three parts in ten thousand would move on a finer grid, and the point is that it is not a difference worth having a sign for.

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. 10 How much the divergence itself scatters at this resolution, which is the floor a difference in step length has to clear.

Where the width was checked

The band’s extent is found by stepping outwards until the divergence leaves the bound, and the crossing is located to within one step — two parts in a thousand. On a band spanning a factor of 1.030 that is about a tenth of the whole width.

So the width quoted for this band, unlike the others, is good to roughly ten per cent rather than one. That does not change any of the conclusions — three flips in sixteen rises is three flips whatever the ends are, and 0.03 per cent is 0.03 per cent — but it is the one number here worth a caveat, and it is the number the essay leans on least.

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. 11 The width predictions again, with the band whose width is least well measured being the outlier of both.

What this says about the other five

Nothing, directly, and something indirectly. The five that work do so because their divergence has a stationary point at their handover, and this one does not. Nobody chose which rungs had one.

That is a small warning about the design. A band is available where the divergence happens to be flat, and the divergence is flat at five of six handovers for reasons nothing here explains. If it were flat at two of six, the ordering result would rest on two bands and the same argument would be made about it — and the number of bands available would be a fact about the curve rather than about the effort.

The settled divergence down the Lucas branch. Every rise from 0.07 down to 0.0057, 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 one 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 5.264°, 2.920°, 2.295°, 0.427°.
Fig. 12 The Lucas divergence curve, on which this rung’s descent is steep through its own handover.

The rung it belongs to

Worth a paragraph, because the Lucas 3/4 rung is the widest on the ladder and its band is the narrowest.

It runs from 0.0640 to 0.02390, a factor of 2.68 in the rise, carrying a hundred rises at the ladder’s one per cent step — more than any other rung on either branch. Its divergence slides only 0.645° across the whole of that, from 102.422° to 101.777°, which is the smallest end-to-end slide of any rung except the fine 8/13.

So it is a wide, flat rung with a narrow band on it. That combination is what the width prediction gets most wrong, and it is the reason: a rung-average slope of −0.654 says the band should be wide, and the curve is doing something local that the average cannot see.

The Lucas ladder, rung by rung. Each bar is one rung — a run of rises over which a counter returns one pair — drawn from its coarse end on the left to its fine end on the right, with the whole bar scaled to the same width so that positions inside different rungs can be compared. The mark on each bar is the rise at which the two contact steps change places, and it falls between 6 and 40 per cent of the way down on every rung that has one. The dots are the lattices this collection's ablation census was grown at, dropped onto the rungs they belong to. They are not spread across the bars; three of them sit past the mark.
Fig. 13 The Lucas rungs, of which this one is the widest and carries the narrowest band.

What it does to the count of bands

Six handovers, six bands, five usable. That is the number to quote, and quoting “six bands” without the qualifier would be the ordinary way for a table to overstate itself.

It also changes the arithmetic of the ordering result. That claim is now carried by the four bands that wreck, of which this is not one, so the Lucas 3/4 band contributes nothing to it at all — neither support nor a counter-example. A row that contributes nothing is worth listing precisely because a reader counting rows will otherwise count it.

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. 14 The cuts each band carries, on which two rows contribute no cuts.

What a reader should take from a row like this

That a design’s coverage and a design’s applicability are different numbers.

Six handovers exist and six bands were built, so the coverage is complete: nothing was skipped, nothing was chosen. The applicability is five, because one of the six turns out to hold the quantity it was built to move. Both numbers are true and only the second belongs in a claim.

The same distinction runs through the other design this round built. Five matched pairs exist and four produce an experiment, because the fifth sits at the coarse end and neither of its stems wrecks. In both cases the unusable cell is unusable for a reason the ladder supplies rather than for a reason the experimenter chose, which is why both are worth listing.

Which rungs of the golden branch share a divergence. One row per pair of rungs. A pair whose divergence ranges overlap has a rise on each rung where the rule settles on the same angle; a pair whose ranges do not overlap has none, whatever the search. On this branch four of six pairs match, and three of those match to 0.0000° — the same value of a quantity read on a grid of 1,536 azimuths. The rises differ by factors of 1.48 to 5.09, so the design holds one angle while changing everything the rise controls.
Fig. 15 The other design’s cells, of which one is likewise built and unusable.

The one line

The Lucas 3/4 band carries sixteen rises, spans three per cent of its rung, holds its counted pair and its divergence, and does not move the ordering: the two contact steps differ by 0.03 and 0.43 per cent at its ends and change places three times inside it. It is the one rung whose divergence is not stationary at its handover, it wrecks at no offset, and it is in the table because five for five is not what happened.

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 with the one that moves nothing named as such rather than left out.

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. Which lag survives, at every lattice and every offset. A row for each of twelve lattices and a column for each offset an organ is removed at, with the lag whose hop survived written in the cell. 30 cells never repair. The lag left standing is one of the two contact families at 29 of them, and at 17 it is the second shortest step on the lattice rather than the shortest, which is what refuses the reading that a rule holds its nearest neighbours. The ringed cells are the five the offset rule gets wrong. Two lattices carry no cells at all because no single removal wrecks them. Not the shorter of the two If a damaged stem keeps one contact family standing, the obvious guess is that it keeps the nearer one. Across thirty wrecked offsets that is true twelve times and false seventeen, and on one lattice the two steps differ by a quarter of a per cent — where the words shorter and longer are doing no work 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. 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.

What links here

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

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

Named objects

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ControlDiscretisationDivergence angleHandoverMatched designMeasurementNegative resultParastichy pairRefusalResolutionRiseRungSelection effectTolerance