Phyllotaxis
the spirals are counted, not admired
Stems and cones

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.
12 min read 20 figures State the claim, state the testCounted, not admired

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

A band is the stretch of rise around a handover on which a stem’s counted pair holds and its settled divergence stays flat, so that the only thing moving is which of the two contact steps is shorter. It is the design that isolates the step ordering from everything else the rise controls.

Six of them were grown and each was cut at nine rises, evenly spaced in the logarithm of the rise, always including both ends and the handover itself. Nine, because sixteen cut stems a rise at a hundred rises a band is hours for a quantity the design predicts to be constant — and a constant is tested at the extremes and at the crossing.

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. 1 Every rise of the widest band, cut at every offset. One column per rise, one row per offset, and the family the cut stem keeps in each cell.

Where the design broke

On the golden 8/13 band it is not constant. Three offsets change their surviving family somewhere inside, and at one of them the nine sampled rises came back 8, 4, 8, 4.

An alternation is the one thing a nine-rise design cannot interpret. It could be a period, it could be two changes that happen to be sampled either side of, or it could be a single rise doing something unusual. The design has no way to tell those apart and it was not built to.

The previous round said so, in the sense that it reported the alternation as an open question rather than as a finding. What it also did was quote the changes’ positions in its own steps, which reads as a measurement and is not one — and that is the part this sweep corrects rather than confirms.

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. 2 The changes of answer the nine-rise design found across all six bands, which is the reading this sweep was built to check.

The sweep

Cut every rise of that band. It holds 126 rises at the step the geometry is swept at, and each rise is a control and fifteen or sixteen cut stems, so the sweep is 1,890 cut stems and it took sixty-seven minutes.

That is not cheap and it is not the several hours the nine-rise design was chosen to avoid. The estimate that produced nine was made when the bands were narrower and before the sweep’s underlying stems were kept on disk between runs.

Take away the organ eight places back, and the next one goes into the hole. The last 30 organs of a stem at a rise of 0.005, unrolled. The open circle is the organ removed — eight places before the tip. The ring at the top is where the rule puts the next organ with every organ present; the filled mark beside it is where the rule puts it with that one missing. The two are 16.4° apart, against a local spacing of 25°, and the vacancy itself is 22.7° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.
Fig. 3 One cut: the removed organ, the organs placed after it, and a control that shares the history below the hole.

Which band, and why that one

The golden 8/13. It is the widest band on the ladder — 126 rises against 16 for the narrowest — and it covers seventy-two per cent of its own rung, spanning a factor of 1.28 in the rise.

Width is the whole reason. A band sixteen rises wide has no room for a feature two rises across, so the question this sweep is about could not be asked of it. The narrow bands could be cut whole for a tenth of the cost and would answer a smaller question.

The narrowest is the Lucas 3/4, and it is narrow because its two contact steps never separate — so it is both the band with least room for an interior and the band whose interior would say least.

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. 4 Every band on the ladder with its extent. The one cut here is the long one.

The step is a ratio

Two parts in a thousand between consecutive rises, which is the step the band’s geometry is swept at. It is a ratio rather than a fixed amount because the ladder is geometric: an absolute step is three different instruments on it, one per cent at one handover and half a per cent at another.

So the 126 rises are 126 multiplications rather than 126 additions, and a feature two rises wide is two parts in a thousand of the rise wherever on the band it sits.

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. 5 The same band on an absolute grid and on a proportional one, which is the correction this sweep inherits.

What nine rises got right

Both of the things the design was for.

The ends. The coarse end of the band keeps the 8 family at every offset that wrecks there, and the fine end keeps the 4 family at the three offsets that ever change. Both are confirmed at full resolution.

And the handover. Not one of the nineteen changes at full resolution sits at the rise where the two contact steps change places. That is the claim the whole thread rests on and the finer sweep does not touch 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. 6 How far apart the two contact steps get across each band, which is the quantity the handover is the crossing of.

What it got wrong: how many

Five changes in the sample; nineteen at full resolution. Thirteen of the nineteen are at one offset.

That is not a failure of the sample so much as a description of what a sample is. One step of the nine-rise design is sixteen rises of the sweep, and a change that happens and reverses inside sixteen rises is invisible to it.

Nineteen is also not a stable number. It counts transitions between consecutive wrecked rises, so an offset that recovers for a stretch and comes back with a different answer contributes one — and whether that is one change or two depends on a convention nobody has had to state before, because at nine rises the question does not arise.

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. 7 One offset’s answer at every rise, and the same row with only the rises the nine-cut design visits.

And where

The sample bracketed the changes two to four of its own steps below the crossing, and the previous round wrote that down as a distance. At full resolution they sit 8 to 58 rises below it, and the three offsets that change cross at three different rises.

A position quoted in sampled steps is a position quoted in the sample. That sentence is the correction and it is worth more than the numbers on either side of it.

Offset 7 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 126 of them and keeps the 4-family and the 8-family at different rises. The ticks below mark one islands — runs of 2 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. 8 The offset that crosses closest to the handover, at every rise. Its change is eight rises below the crossing, not two.

The three results

They get essays of their own and it is worth naming them here. The alternation is not a period: it is thirteen short islands with uneven gaps, and a period fitted at its best phase buys exactly nothing over saying the commonest family and stopping.

The three offsets that change cross at three different rises, none of them the handover. And three of the six offsets that wreck never change at all, while which offsets wreck is itself a function of the rise.

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. 9 Every candidate period scored against the speckle at its best phase, with the score for no period at all drawn as a rule.

The offsets that wreck, rise by rise

One thing the sweep produces that the sample could not is a picture of which offsets wreck at all, across a whole band. It is not the same set at every rise: offset 7 wrecks at all 126, offset 8 at 123, offset 6 at 110, offset 4 at 98, offset 9 at 81 and offset 5 at only 22 of them.

So a census taken at one rise of a band and a census taken at another are censuses of different sizes. Every claim in this thread of the form “at every offset that wrecks” is quantified over a set that the rise decides, and nothing had drawn that set moving.

Offset 5 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 22 of them and keeps the 8-family throughout. The ticks below mark no island: the answer changes once and stays changed. The handover is the taller rule and the change of answer is nowhere near it.
Fig. 10 The offset that wrecks least often across the band. It has no answer at 104 of the 126 rises and a clean one at the other 22.

What a sample is for

Not this. A nine-rise design answers “is the quantity the same at both ends and at the crossing”, and that is a real question with a real answer, and it got it right.

What it cannot answer is anything about the interior. The mistake was not in choosing nine; it was in reading the sample’s output as though it were the band’s. The 8, 4, 8, 4 was a genuine reading of four sampled rises and it was never a reading of a period.

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 What each band keeps, which is the reading the nine-rise design supports and which this sweep confirms.

The refusal that now sits on it

Asking a nine-rise design for the islands is refused rather than answered. The islands are one to three rises wide and the design steps sixteen rises at a time, so it can only land on one by accident.

That is a piece of machinery rather than a caution in prose, and it is the piece that would have stopped the 8, 4, 8, 4 from being reported as an alternation in the first place. A design that cannot resolve a feature should decline to describe it.

What nine rises could see of 126. Above, offset 6'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 or 1 rises wide, against a step of 16, so the sample can only land on one by accident.
Fig. 12 A different offset at both resolutions. The island in the middle of it is one rise wide against a step of sixteen.

What the sweep cannot do either

It has a step of its own. Two parts in a thousand is fine enough to resolve islands of one rise and it is not infinitely fine: a feature at four parts in ten thousand would be invisible here exactly as these islands were invisible to nine rises.

Nothing in this sweep says the speckle has a floor. The honest form of the finding is that at a step of two parts in a thousand the transition region is speckled, and a finer step might show the speckle to be structure.

A divergence that does not move across the 5/8 band. Measured at every rise of a band on the golden branch, where a counter returns 5 and 8 spirals throughout. The settled divergence moves by 0.0469 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 1.0013 and the ordering changes hands at a rise of 0.0156, so above that rise the shorter step belongs to the 5 family and below it to the 8 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 A band on its own grid, which is where the sweep’s own step is visible as a limit.

What was already on disk

Most of the cost. The stems the sweep grows are shared with the census and with the band geometry, and the ones that had been grown before were read from disk rather than regrown.

That is why sixty-seven minutes rather than several hours, and it is the reason the “nine is enough” estimate was out of date. It was made against a cost that has since fallen by more than an order of magnitude, and nobody revisited it.

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. 14 The survivors on a narrower band, from the design this one extends. Most of what it computed is shared with what this sweep needed.

What the other five bands would cost

Between a tenth and three quarters of this one each, and about two hours for all five. The narrow ones are cheap and would answer little; the Lucas 7/11 is 124 rises and would be the real second test, since it is nearly as wide as this one and on the other branch.

Not done. The reason is the ordinary one: the sweep was built to check one alternation on one band and it answered that, and extending it is a separate decision that this round did not take. It is recorded as an outstanding check rather than as a sentence about intending to.

Both bands, on two branches and two pairs. One row per band. Each runs from its coarse end on the left to its fine end on the right, with the rise at which the two contact steps change places marked, and the family that survives every wrecked cut written at the end. The 5/8 band on the golden branch keeps the 5 at all 24 of them and the 4/7 band on the Lucas branch keeps the 4 at all 31. Two branches, two counted pairs, one result: the quantity the band varies is not the quantity that decides the answer.
Fig. 15 Two bands drawn together. Whether the speckle found on the widest one occurs on the others is the obvious next question.

The one thing the sample could not have been fixed to see

Suppose the nine-rise design had been given sixteen rises, or thirty-two. The islands are one to three rises wide and the gaps between them run from one rise to forty-eight, so a design with a step of four would still be landing on islands by accident and reporting a sequence of coincidences.

What resolves speckle is not a denser sample; it is every rise. That is an unusual conclusion for this collection, which nearly always finds that a moderately denser sweep suffices, and it follows from the feature being one step wide at the sweep’s own step. There is nothing between “sample it” and “do all of it”.

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. 16 Another band’s own grid. A feature one grid step wide is a feature no sampling of that grid can characterise.

What the sweep measures at each rise

The same thing the census does. Every offset out to the front and two past it is cut in turn; each cut stem is grown and compared with a control that shares its history to the last digit; the surviving lag is the one whose hop is unchanged.

Nothing new is computed. The sweep is the existing measurement made 126 times instead of ten, which is what makes its results comparable with everything the thread already has — the same discipline the wider slot design follows on the other side of the site, where thirty lattices are a superset of six rather than a replacement for them.

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.
Fig. 17 The census’s own reading of which offsets wreck and what they keep, which is the measurement this sweep repeats at every rise of one band.

Which is the argument for doing it

A finer sweep that measures something new is two changes at once and its results are hard to attribute. This one changes the sampling and nothing else, so every difference between the nine-rise reading and the 126-rise reading is a difference the sampling made.

That is worth the sixty-seven minutes on its own, independently of what turned up. Most of the corrections in this collection have come from varying one instrument setting and leaving everything else alone — doubling a run length in the ablation thread is the same move made this round on a different instrument, and it produced the same kind of result.

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. 18 The same picture without the handover marked, so the transition region can be read on its own terms.

What a reader should carry

That a band’s ends and its crossing are what nine rises measure, and that everything this thread has said about a band’s interior was a statement about ten sampled rises out of a hundred and twenty-six.

And that the correction runs in one direction. Nothing the sample said turned out to be false; several things it said turned out to be about the sample.

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. 19 Where the handovers sit inside their rungs, which is the distribution any band-sampling design is making an assumption about.

What the picture at the top shows

Six rows and a hundred and twenty-six columns, coarse on the left. The dark cells are the 8 family kept, the mid cells the 4 family, and the pale ones are offsets that recover at that rise and have no survivor to report.

Read across the top three rows and the band is nearly uniform. Read the bottom three and there is a clean left half, a speckled middle and a clean right half — and the speckle is in a different place on each of the three. That is the whole of what this sweep found, and it is legible in one picture because there is nothing between the sampling and the band.

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. 20 The band once more, with the handover marked. The vertical rule is nowhere near the middle of any row’s speckled stretch.

The one line

The widest band on the ladder, cut at all 126 of its rises and 1,890 cut stems: the nine-rise design’s two claims — that the band’s ends differ and that no change of answer sits at the handover — both survive, and everything it said about where inside the band the changes are was a statement about a step of sixteen rises.

Nineteen changes rather than five, eight to fifty-eight rises below the crossing rather than two to four, and three offsets crossing at three different rises.

Where this goes next

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. The ordering on six bands A hundred and fourteen wrecked cuts across four bands, and at every offset of every one of them the family left standing is the same immediately above the handover and immediately below it. Where the answer does change — on the widest band, at three offsets — it changes somewhere else. 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. The alternation is not a period Nine sampled rises gave 8, 4, 8, 4 at one offset of one band, and a period was the obvious thing to look for. At full resolution it is thirteen islands one to three rises wide, with gaps of 1, 2, 3, 6, 7, 8, 9, 16, 31, 44 and 48 — and a fitted period buys exactly nothing. 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.

Shares its objects with

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

  • One offset, two answers — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rigid hop, rise, rung
  • The front deepens down a rung — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rigid hop, rise, rung
  • The shortest hop was a coin flip — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rigid hop, rise, rung
  • The side the census sat on — both name ablation, claim testing, control, handover, honest limits, lattice offset, negative result, parastichy pair, rise, rung, sampling
  • When the second wall is free — both name ablation, claim testing, control, honest limits, lattice offset, matched design, measurement, negative result, parastichy pair, rise, rung
  • One rung, two answers — both name ablation, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rigid hop, rise, rung

Named objects

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

AblationClaim testingControlHandoverHonest limitsLattice offsetMatched designMeasurementNegative resultParastichy pairResolutionRigid hopRiseRungSample sizeSampling