Phyllotaxis
the spirals are counted, not admired
Stems and cones

When nine rises are enough

A coarse design was shown to be misleading on one band and it has been criticised on that ground ever since. On the second band it is exactly right, and the difference between the two cases is a property of the band rather than of the design — which is the awkward part.
13 min read 23 figures State the claim, state the testCounted, not admired

Worth reading first: Where a handover sits · The organ that was taken away.

The design under examination cuts a band at nine rises, evenly spaced in the logarithm of the rise, always including both ends and the handover itself. It was chosen for a stated reason: sixteen cut stems a rise at a hundred rises a band is hours of work for a quantity the design predicts to be constant, and a constant is tested at the extremes and at the crossing.

Then the quantity turned out not to be constant, and the sweep at full resolution said so. The design has been treated as the villain of that story since. It is not, and the second band is what shows it.

What nine rises find on each band, against what all of them find. Two bars per band: the changes of surviving family a nine-rise design finds, and the changes the full sweep finds. On the Lucas band the two agree exactly, at none and none. On the golden band they agree that something changes and disagree about how much — 5 against 19 — because one step of that design is 16 rises and the band carries features one to three rises wide. A sample was never wrong about whether; it was wrong about how many.
Fig. 1 What nine rises find on each band against what all of them find.

Two questions, and they come apart

There are two things a sampling design can be asked. Does anything change? and how much, and where? The nine-rise design answers the first correctly on both bands and the second correctly on only one.

On the golden 8/13 band it says yes and there are nineteen changes; it finds five. On the Lucas 7/11 band it says no and there are none; it finds none. So it has never been wrong about whether, and it was wrong about how many by a factor of nearly four.

That split matters because the two questions were asked in different essays. The first was asked by the design that built the six bands and the second only arose once the first had been answered.

The two widest bands on the ladder, each cut at every rise. One block per band, one row per offset that wrecks anywhere on it, one column per rise, coarse on the left. A filled cell is a cut that wrecks, and its tone is the family left standing; a pale cell is a cut that recovers. The golden 8/13 band above changes its answer at three of its six offsets, 19 times in all. The Lucas 7/11 band below changes it nowhere: every cut that wrecks on it keeps the 7 family, at every offset and every one of its 124 rises.
Fig. 2 Both bands at full resolution, which is what the nine-rise design is being scored against.

What was actually claimed at nine rises

It is worth going back to the wording, because the criticism has to be aimed precisely. The nine-rise sweep reported that the survivor at a fixed offset is not constant across a band, naming three offsets on the golden band that change.

That claim is true. What it also did was quote the changes’ positions in its own steps — two to four sweep steps below the crossing — and that reads as a measurement and is not one, because one of its steps is sixteen rises.

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. 3 The changes of answer the nine-rise design found across all six bands.

So the failure was a units problem

A position quoted in sampled steps is a position quoted in the sample. At full resolution the changes sit 8 to 58 rises below the crossing rather than two to four of anything, and the three offsets that change cross at three different rises.

That correction is worth more than the numbers on either side of it, and it is not a criticism of nine rises. It is a criticism of quoting a distance in units the instrument does not resolve.

What nine rises could see of 126. Above, offset 7's answer at every rise of the band. Below, the same row with only the rises a 9-cut design visits, which is one every 16. The design was built for a quantity expected to be constant and it reports the ends and the crossing correctly; what it cannot report is where inside the band the answer changes, or that it changes back. Every island here is 2 rises wide, against a step of 16, so the sample can only land on one by accident.
Fig. 4 One offset’s answer at every rise of the golden band, with only the rises a nine-cut design visits marked.

And a resolution problem

The other thing nine rises could not do is see a feature narrower than its own step. The golden band’s changes are thirteen islands one to three rises wide with gaps of 1 to 48 rises, and one step of the nine-rise design is sixteen rises.

A sample whose step is sixteen rises hits an island about one time in eight. It hit two of thirteen, which is what the arithmetic predicts, and hitting two of them with a run of the other family between them produced the alternating reading that made the whole thing look like a period.

A period fitted to the speckle, at every period it could have. Each mark is one candidate period, drawn at the share of rises it gets right when it is given its best phase and its best family in each residue class — the most generous reading of periodic there is. The flat rule is what saying nothing gets: name the commonest family and stop. The best period scores 76 per cent against 76 for no period at all, a gain of 0 points over 123 rises, so the alternation the coarse design reported is not a period being sampled badly.
Fig. 5 Every candidate period scored against the golden band’s speckle, with the score for no period drawn as a rule.

What the second band adds

On the Lucas 7/11 band there are no islands. There is nothing narrower than the sample’s step because there is nothing at all, and the sample and the sweep agree exactly: none and none.

That is a case the criticism could not have anticipated and it changes what the criticism can say. The design is not unreliable; it is reliable exactly when the thing it samples has no structure below its step, and it has no way of knowing whether that is the case.

The band it is right about is uniform in every cell that holds an answer, which is a strong statement precisely because it was made at full resolution and could have been made at nine rises and meant almost nothing.

Which offsets wreck across the Lucas 7/11 band. One row per offset and one column per rise, with a mark where a single removal at that offset wrecks the stem. The set is not the same at every rise: on this band one offset wrecks at only 27 of its 124 rises, in several separate stretches, while others wreck at all of them. So a census taken at one rise of a band and a census taken at another are censuses of different sizes, and every claim of the form "at every offset that wrecks" is quantified over a set the rise decides.
Fig. 6 The band where the sample is right, and where the picture is one tone throughout.

Which is the general form of the problem

That last sentence is the uncomfortable part and it is not special to this design. A sample cannot report on structure finer than its own resolution, and it cannot report that it has failed to, because failing to see something and there being nothing to see produce the same output.

So was this sample enough? is not a question a sample can answer. It is a question answered by running the finer measurement, which is what makes the finer measurement worth its cost even when it agrees.

Offset 6 across the 8/13 band, rise by rise. The family this one offset keeps at each of the band's 126 rises, coarse on the left. It wrecks at 110 of them and keeps the 4-family and the 8-family at different rises. The ticks below mark two islands — runs of 2 and 1 rises where the coarse family comes back inside the fine one. The handover is the taller rule and the change of answer is nowhere near it.
Fig. 7 An offset whose changes sit inside stretches the coarse design steps over.

The cost of finding out

Cutting a band whole is 1,612 to 1,890 cut stems and thirty-eight to sixty-seven minutes. Cutting it at nine rises is about a tenth of that. So the finer measurement costs roughly ten times the coarse one, and it was run twice.

Both were worth running and for different reasons. The first changed several numbers the collection had been quoting; the second changed none and bounded a claim, which is the return a null result buys and is usually undersold.

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. 8 The six bands with their extents, two of which have now been cut at every rise.

Where the sample was right about the ends

One more thing the nine-rise design got right on both bands, and it is the thing it was built for. The coarse end of the golden band keeps 8 at every offset and the fine end keeps 4 at the three offsets that ever change.

Both are confirmed at full resolution. The design was built to test a constant at its extremes and at the crossing, and at the extremes and the crossing it is right on both bands.

The ordering changes and the survivor does not. Every offset that wrecks, at every rise of the band, with the family left standing written in the cell. The counted pair is 5 and 8 at all 18 rises and the settled divergence is held to a twentieth of a degree, so the one quantity moving across the columns is which of the two contact steps is the shorter — and it changes hands at the marked rise. The cells do not: the 5 family survives at all 24 wrecked cuts, on both sides. Scored on this band, the reading that a wrecked stem keeps its shortest hop is right 14 times out of 24, for an answer that never changed.
Fig. 9 The families a band’s cuts keep at its two ends, which is what the coarse design was built to compare.

And about the handover

The claim the thread rests on is that no change of surviving family sits at the rise where the two contact steps change places. The nine-rise design says so on all six bands and the full sweeps say so on the two they cut.

That is a claim about a single rise, which is a rise every version of the design visits. A sample that always includes the point of interest can be trusted about the point of interest, and that is not a coincidence — it is why the design includes 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. 10 How far apart the two contact steps get across each band, which is the quantity a handover is the crossing of.

Designing a sample for what it will be asked

The lesson that generalises is about the second question rather than the first. A design built to test is this constant? answers that well and answers where does it change? badly, because the two need different resolutions.

Nine rises were chosen against the first question and the answer to it produced the second question immediately. That is the ordinary sequence and the ordinary mistake is to answer the second with the instrument built for the first.

Every rise of the 8/13 band, cut at every offset. One column per rise of the band, coarse on the left and fine on the right, and one row per offset that wrecks anywhere on it. A filled cell is the family the cut stem keeps; a pale cell is an offset that recovers at that rise and has no survivor to report. The band holds 126 rises and 1890 cut stems. three of the six offsets change their answer somewhere inside, three never do, and the vertical rule is the handover — the rise where the two contact steps change places. Not one of the 19 changes is at it.
Fig. 11 The full-resolution table, which is what the second question needed and the first did not.

What a step of sixteen rises is

It is worth making the step concrete because nine rises sounds like a lot. The golden band spans a factor of 1.28 in the rise across 126 steps of two parts in a thousand each; nine cuts across that is a step of about three per cent in the rise.

Three per cent is enough to step clean over a band on some rungs — that is the argument for sweeping a band at a ratio rather than an absolute amount in the first place. Inside a band it is enough to step over an island thirteen times out of fifteen.

The flat band, re-measured on a finer grid. A quantity that comes out constant is the first thing an azimuth grid should be suspected of, so the whole band is grown again on a grid of 6144 steps against the 1536 the site uses. The finer grid does resolve structure the coarse one flattened: a shallow minimum 0.0820 degrees deep, with its floor at a rise of 0.0158. What it does not do is separate the ends, which still agree to 0.0000 degrees while carrying opposite step orderings. The matched pair the band is for survives the check that would have broken it.
Fig. 12 The same band on an absolute grid and on a proportional one, which is the correction both sweeps inherit.

The narrow bands are still sampled

Four of the six bands have only ever been cut at nine rises, and two of those are too narrow for the question to arise: the Lucas 3/4 band is sixteen rises wide, so its nine cuts are a step of under two rises and are nearly a full sweep already.

The other two — the 3/5 band at 70 rises and the Lucas 4/7 at 86 — are sampled at steps of eight and ten rises. Those are the cases where the same undercounting could be happening and nothing has looked.

Neither wrecks at its coarse end, because a coarse enough stem cannot be wrecked by a single removal, so a full sweep of either would be measuring over a shorter stretch than its width suggests.

The two steps changing places inside the 4/7 band. Measured at every rise of a band on the Lucas branch, where a counter returns 4 and 7 spirals throughout. The settled divergence moves by 0.0195 degrees across the whole band, which is a fraction of the azimuth grid step and a fortieth of the slide across the rung it sits in. The ratio of the two contact steps does move: it falls to 1.0010 and the ordering changes hands at a rise of 0.0225, so above that rise the shorter step belongs to the 4 family and below it to the 7 family. Two of the three quantities that vary along a rung are therefore held here and the third is not, which is what makes the ends of this band a matched pair.
Fig. 13 One of the two bands that is wide enough for a step of ten rises to hide something.

What it would cost to close that

The 3/5 band is 70 rises and the Lucas 4/7 is 86, so cutting both whole is about 2,300 cut stems and roughly fifty minutes. That would take the sample of fully cut bands from two to four.

It would also change what can be said. Two cases support one band has a transition region and one does not; four would support a rate, and a rate over four bands is still small but is a different kind of statement.

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. 14 Where the handovers sit inside their own rungs, which is what any band-sampling design assumes something about.

A sample that includes its own ends

There is one design feature worth defending explicitly. The nine rises always include both ends of the band and the handover, which means three of the nine are placed rather than spaced.

That is why the ends are right on both bands and why the handover claim is safe. A design that spaced nine rises evenly and let them fall where they fell would have been worse at the questions it was built for and no better at the ones it was not.

A window that fits inside a rung. Stems that climb the ladder at four rates, read over a window at the fine end. The condition is a ratio: the window has to be shorter than a rung. 250 internodes at 130 per rung is 1.92 rungs and agrees on 0 of 3; 400 internodes at 130 per rung is 3.08 rungs and agrees on 0 of 3; 250 internodes at 260 per rung is 0.96 rungs and agrees on 3 of 3; 400 internodes at 260 per rung is 1.54 rungs and agrees on 1 of 3; 250 internodes at 520 per rung is 0.48 rungs and agrees on 2 of 3; 400 internodes at 520 per rung is 0.77 rungs and agrees on 3 of 3; 250 internodes at 1040 per rung is 0.24 rungs and agrees on 3 of 3; 400 internodes at 1040 per rung is 0.38 rungs and agrees on 3 of 3. Read over the whole stem instead, every rate returns nothing — 0 of 3, 0 of 3, 0 of 3, 0 of 3 — because the quantity the comb is periodic in changes as the pattern climbs.
Fig. 15 Where a sampled window sits inside a rung, which is the placement question a design has to answer once.

What the two bands say about the design’s future

The design is still the right one for a band that has not been cut whole, because it is a tenth of the cost and it answers the question it was built for. What has changed is what may be quoted from it.

Positions are out: a change located between two sampled rises is located to within one step and should be quoted that way. Counts are out: a count of changes from a sample is a lower bound. Ends and crossings are in.

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. 16 What the nine-rise design reports about each band’s ends, which is the part that survives.

The comparison that makes it a measurement

None of this would be sayable from the sweeps alone. It is sayable because the nine-rise design was re-run on both bands from the same table, so its output and the full output are the same code reading the same stems.

That is the same discipline as keeping the six original lattices inside the larger table when the slot design was extended: a comparison between two designs is worth nothing if the two designs are also two implementations.

What nine rises could see of 126. Above, offset 8's answer at every rise of the band. Below, the same row with only the rises a 9-cut design visits, which is one every 16. The design was built for a quantity expected to be constant and it reports the ends and the crossing correctly; what it cannot report is where inside the band the answer changes, or that it changes back. Every island here is 1 or 1 or 2 or 1 or 2 or 1 or 2 or 1 or 3 or 1 rises wide, against a step of 16, so the sample can only land on one by accident.
Fig. 17 The offset that carries most of the golden band’s changes, with the sampled rises marked on it.

What the design cost in the other direction

There is a symmetric error the design did not make and it is worth crediting. It did not report a change it had not seen. The five changes it found are five of the nineteen that are there; none of them is spurious.

That is not automatic. A design that reads a noisy quantity at nine points and calls each difference a change would manufacture changes out of a stable row, and this one does not, because the quantity it reads is a family label rather than a number with an error on it.

So the design’s errors run one way. It undercounts and it misplaces; it does not invent.

Offset 8 across the 8/13 band, rise by rise. The family this one offset keeps at each of the band's 126 rises, coarse on the left. It wrecks at 123 of them and keeps the 4-family and the 8-family at different rises. The ticks below mark ten islands — runs of 1 and 1 and 2 and 1 and 2 and 1 and 2 and 1 and 3 and 1 rises where the coarse family comes back inside the fine one. The handover is the taller rule and the change of answer is nowhere near it.
Fig. 18 The offset the sample found four of its five changes on, drawn at every rise.

A label is not a measurement with error

That deserves a sentence of its own, because it is what makes a nine-rise design tolerable at all. Each cell of these sweeps is the answer to which family did this cut leave standing? and the answer is one of a small set of lags rather than a number.

A label read at nine points is nine correct labels. A number read at nine points is nine numbers with error bars and a temptation to draw a line through them. The first is the safer thing to sample and the sweeps here are of the first kind.

The order of the angles carries the count. Three stems, each held at a fixed rise so the pattern sits on one rung of the ladder. At a rise of 0.032 the positions count 3 and 5 spirals and the angles peak at 3; At a rise of 0.013 the positions count 5 and 8 spirals and the angles peak at 5; At a rise of 0.005 the positions count 8 and 13 spirals and the angles peak at 8. Each panel marks the peak and its multiples; the pale strip is what an uncorrelated sequence of this length gives.
Fig. 19 How a family is read off a cut stem, which is why the answer is a label rather than a number.

Where the same argument has already bitten

This is the third instrument in the collection to be caught quoting a result in its own units. A residual was read against the window it was measured over; an onset was reported at the last organ of a run, which a run length produces whether or not the condition holds; and now a position was quoted in sampled steps.

All three have the same shape: a quantity that looks like a property of the object and is partly a property of the instrument. None of the three was found by a check; all three were found by moving the instrument setting and looking again.

Two windows on a shoot at 130 nodes per rung. A stem grown at 130 nodes to the rung with a disturbance of 0.25, its rise falling from 0.4 to 0.004 over 623 nodes. The upper window is the last 250 internodes — where a count would be made on a real plant — and the lower is the same length shifted down 125. The upper reads nothing; the lower reads nothing. The verdict is silent, and the window holds 1.92 of a rung.
Fig. 20 The same quantity read through two settings of one instrument, which is how each of the three was caught.

The cheapest habit that would have caught it

Vary the setting once. Not a sweep of settings, not a sensitivity analysis — one alternative value, run once, compared.

Every one of the three would have been caught by that. It costs whatever the measurement costs, once, and it is the difference between reporting a number and reporting a number with a claim about the instrument attached to it.

How steady each class is, over 300 organs and over 600. Each mark is one wrecked cut, placed across at the widest spread found inside any one of its residue classes over the shorter run and up at the same reading over the longer one. A mark on the diagonal is a row the two lengths agree about. The rules are the 10 degrees that separates a profile called periodic from one that is not: three rows fall in different quadrants at the two lengths, two of them becoming periodic and one ceasing to be. The gap between the two groups narrows from 1.69 times to 1.27.
Fig. 21 A quantity measured at two settings of one instrument, which is the cheapest check there is.

What a reader should carry

That the coarse design was right about everything it was built to be right about, and that its one bad number was a position quoted in its own steps.

And that whether nine rises are enough is not a property of nine rises. It is a property of what is being sampled, it differs between two bands on the same ladder, and the only way to find out is to stop sampling.

The two widest bands on the ladder, each cut at every rise. One block per band, one row per offset that wrecks anywhere on it, one column per rise, coarse on the left. A filled cell is a cut that wrecks, and its tone is the family left standing; a pale cell is a cut that recovers. The golden 8/13 band above changes its answer at three of its six offsets, 19 times in all. The Lucas 7/11 band below changes it nowhere: every cut that wrecks on it keeps the 7 family, at every offset and every one of its 124 rises.
Fig. 22 The two bands stripped to their content: one with structure below the sample’s step, one without.

What the picture at the top shows

Two pairs of bars, one pair per band. In each pair the upper bar is how many changes of surviving family the nine-rise design finds and the lower is how many all the rises find.

On the Lucas band both bars are nothing. On the golden band the upper is five and the lower is nineteen, and the gap between them is one step of sixteen rises meeting thirteen islands one to three rises wide.

What nine rises find on each band, against what all of them find. Two bars per band: the changes of surviving family a nine-rise design finds, and the changes the full sweep finds. On the Lucas band the two agree exactly, at none and none. On the golden band they agree that something changes and disagree about how much — 5 against 19 — because one step of that design is 16 rises and the band carries features one to three rises wide. A sample was never wrong about whether; it was wrong about how many.
Fig. 23 The two designs’ counts on the two bands, which is the whole comparison in four numbers.

The one line

The nine-rise design finds none of none on the Lucas band and five of nineteen on the golden one: it has never been wrong about whether a band’s answer changes, and it undercounts by a factor of nearly four where the changes are narrower than its own step.

Which case a band is in cannot be read from the sample, so the finer sweep is what tells anybody whether the coarser one was enough — and on one of the two bands it was.

Where this goes next

The two widest bands on the ladder, each cut at every rise. One block per band, one row per offset that wrecks anywhere on it, one column per rise, coarse on the left. A filled cell is a cut that wrecks, and its tone is the family left standing; a pale cell is a cut that recovers. The golden 8/13 band above changes its answer at three of its six offsets, 19 times in all. The Lucas 7/11 band below changes it nowhere: every cut that wrecks on it keeps the 7 family, at every offset and every one of its 124 rises. The second band, cut whole One band was cut at every one of its rises and came back with a transition region — a stretch where three offsets change their answer, in short islands with uneven gaps. The obvious question is whether that is a picture of bands or a picture of that band. The other wide band answers it. Every rise of the 8/13 band, cut at every offset. One column per rise of the band, coarse on the left and fine on the right, and one row per offset that wrecks anywhere on it. A filled cell is the family the cut stem keeps; a pale cell is an offset that recovers at that rise and has no survivor to report. The band holds 126 rises and 1890 cut stems. three of the six offsets change their answer somewhere inside, three never do, and the vertical rule is the handover — the rise where the two contact steps change places. Not one of the 19 changes is at it. Every rise of a band A band is cut at nine rises because the quantity it was built to test is a constant, and a constant is checked at the ends and at the crossing. On the widest band that quantity turned out not to be constant, which makes nine the wrong number. This is all hundred and twenty-six. Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it. One rise per rung is a sample Every census on this site takes one rise from each rung, because the question was always which pair. Any rule later scored on those rows inherits a variable that was never varied — and two of this collection's results turn out to be about the sampling as much as about the rule.

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.

AblationClaim testingHandoverHonest limitsMatched designMeasurement errorNegative resultResolutionRiseRungSample sizeSampling