Phyllotaxis
the spirals are counted, not admired
Stems and cones

Three offsets, three crossings

The claim the band design rests on is that the survivor does not change where the two contact steps change places. It holds at full resolution: nineteen changes and not one at the handover. Where they are is three different rises, eight, nineteen and twenty-nine below it.
12 min read 20 figures State the claim, state the testForm is not mechanism

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

The band design exists to separate one thing from everything else. Across a band the counted pair holds and the settled divergence stays flat, and the quantity that moves is which of the two contact steps is shorter — the step ordering. If the survivor of a cut changed where the ordering does, the ordering would be deciding it.

At nine rises a band it does not, and at every rise of the widest band it still does not. Nineteen changes of survivor and not one at the handover.

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 band with the handover marked. No row changes its shade at the vertical rule.

Which is the claim, strengthened

The nine-rise reading had five changes to make this statement about. This one has nineteen, over a hundred and twenty-six rises rather than ten, and the nearest change to the handover is eight rises away.

Eight rises at two parts in a thousand is 1.6 per cent in the rise. That is not adjacent and it is not far: it is comfortably outside anything the sweep’s own resolution could confuse.

The strengthening matters because the earlier version of the claim rested on a design that could not have seen a change at the handover if one had been hiding between two sampled rises. The finer sweep removes that: there are twelve rises either side of the crossing and all twenty-four of them give the same answer at every offset.

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. 2 How far apart the two contact steps get across each band, which is the quantity whose crossing the handover is.

And three crossings, not one

The three offsets that change do not change together. Offset 7 crosses from the 8 family to the 4 family at a rise of 0.00595, offset 6 at 0.00582 and offset 8 at 0.00571.

Those are eight, nineteen and twenty-nine rises below the handover at 0.00605. The band does not have a boundary; it has three, one per offset, spread over twenty-one rises.

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. 3 Offset 7, which crosses closest to the handover. Its change is eight rises below it.

Which the sample could not have said

The nine-rise design bracketed all three between the same pair of sampled rises or the pair next to it, and reported them as “two to four sweep steps below the crossing”. One of its steps is sixteen rises, so the three crossings — twenty-one rises apart — fit inside two of its steps.

A design whose step is sixteen cannot report three positions twenty-one apart as three positions. It reports them as one region.

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 Offset 7 at both resolutions. The sample’s nearest pair of rises brackets a stretch seventeen rises wide.

What a change of survivor means

A stem with an organ removed either recovers — returning to its control’s divergence and counted pair — or it does not, and a stem that does not keeps exactly one lag rigid: the angle from an organ to the one that many places above it is unchanged from the control, organ by organ, while every other lag moves. That lag is the survivor, and on this band it is 8 or 4 and never anything else.

So a crossing is a rise at which cutting the same organ from a stem grown at a slightly smaller rise leaves a different family standing. Nothing about the cut changes, nothing about the counted pair changes, and the divergence moves by less than the width of the grid the azimuths sit on.

One wrecked stem, lag by lag — golden, rise 0.013, organ 4 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 65 degrees. The lag-5 hop swings by 0.11 degrees and sits 0.09 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 5, which is the surviving lag and not a coincidence.
Fig. 5 The lag spectrum that identifies a surviving hop, which is the measurement behind every cell of the band picture.

The order they cross in

Coarse to fine: 7, then 6, then 8. That is not the numerical order of the offsets and it is not their distance from the front.

Nothing here accounts for it. It is three numbers and any ordering of three numbers is one of six, so it would be foolish to make anything of the sequence — but the fact that they are separated at all is the finding, and it was not visible before. The census’s own account of which family survives turns on the offset and not on the lattice, so three offsets behaving separately is what that account would predict and one boundary would have been the awkward result.

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. 6 Offset 6, which crosses nineteen rises below the handover — eleven rises after offset 7 does.

What a crossing is here

The first rise, reading from coarse to fine, at which the offset’s answer changes and does not change back for the rest of the band. Islands after it are counted as islands rather than as further crossings, which is a decision and is stated because it has to be.

Under the other convention — every transition is a crossing — there are nineteen rather than three, thirteen of them at offset 8, and no useful sentence can be written. The three are the transitions that separate the band’s two halves at each offset.

A convention that has to be stated is a convention doing work, and this one is doing a lot: it turns a ragged transition into a number. The picture at the head of this essay is the version with no convention applied, and a reader who prefers it should take it as the primary reading and these three rises as a summary of it.

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. 7 The scoring that establishes the islands are not a rhythm, which is what licenses treating them as islands rather than as crossings.

Twenty-one rises is five per cent of the rise

Worth converting, because rises are the sweep’s unit and not the band’s. The three crossings span 0.00595 to 0.00571, which is 4.2 per cent in the rise, on a band spanning 28 per cent.

So the crossings occupy about a seventh of the band, sitting a fifth of the way down it. They are not spread through it and they are not at a point.

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. 8 A band on its own grid, where the fraction of a band a feature occupies can be read directly.

Which is a shape nobody predicted

The design’s prediction was a constant: the survivor is the same everywhere on the band. Its refutation at nine rises was a scatter of changes with no structure. At full resolution it is a band with a clean coarse half, a clean fine half, and a transition occupying about a seventh of it in which three offsets cross at three rises and the answer is speckled.

That is more structure than “the survivor sometimes changes” and less than “the band has a boundary”. It is the shape the sampling could not have produced.

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. 9 The band with nothing marked, so the clean halves and the transition between them can be read without a rule drawn through them.

What the handover does

Nothing to the survivor, which is the point. Nineteen changes, none at it, and the nearest eight rises away.

What it does do is sit above all three crossings. Every crossing is on the fine side of the handover, which is a weaker statement than “at” and is not nothing: if the crossings were unrelated to the handover they would be as likely above it as below. Three of three below is not evidence of much on its own — one in eight under a coin, and the three are not independent, since they are three offsets of one band and could easily be moved together by one thing.

The honest reading is therefore that the crossings sit below the handover on this band, and that a second band would say whether that is a fact about bands or about this one.

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. 10 Every handover on the ladder with its band. The one this band is grown around sits above every crossing in it.

What else moves across the band

Everything the band does not hold. The rise moves by twenty-eight per cent, the two contact steps lengthen and shorten at different rates, the shorter of them changes at the handover, and the number of organs in a turn moves with the rise.

That is the honest position: a band holds two quantities still and lets a dozen move, so “something correlated with the ordering” is a list of a dozen candidates rather than a hypothesis. The band design was never a controlled experiment in the strong sense; it is a design that holds the two quantities the thread’s rival accounts turn on.

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. 11 What moves inside a rung. A band holds two of these still and the rest are free.

Which is a hypothesis worth stating

That the ordering does not decide the survivor, and something correlated with the ordering shifts the rise at which the survivor changes. The band was designed to test the first and it has no instrument for the second.

An instrument would be a second band on the same rung — grown around a different feature — or the same measurement on a band whose handover sits at the other end. Both are available and neither has been run.

The second is the cheaper and the sharper. Handovers sit at different fractions of their rungs, from five per cent to forty, so a band whose handover is near its rung’s fine end would put the crossings above it rather than below — or would not, which would be the more interesting outcome.

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. 12 Where each handover sits inside its own rung, which is the variation a second band on a different rung would exploit.

What is unchanged from the earlier reading

The claim the thread rests on: the two contact steps changing places does not change which family a cut leaves standing. That was established on a hundred and fourteen cuts across four bands and it is established here on 1,890 cut stems on one.

The finer sweep is not an independent confirmation — it is the same band, more finely — but it removes the one way the earlier reading could have been wrong by luck: that a change sat between two sampled rises straddling the handover and was invisible.

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. 13 What every band leaves standing, from the design this sweep refines on one of them.

What is changed

Every sentence about where inside a band something happens. The previous round’s “two to four sweep steps below the crossing” was a position quoted in a sample’s own steps, and the corrected figure is eight to fifty-eight rises across the nineteen changes and eight, nineteen and twenty-nine for the three crossings.

That correction is small in consequence and general in kind. A sample can locate a feature no better than its own step, and a step reported as a unit reads like a measurement.

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. 14 Offset 8 at both resolutions. Its crossing is twenty-nine rises below the handover and the sample puts it inside a stretch of sixteen.

The three offsets are consecutive

6, 7 and 8, with 4, 5 and 9 never changing at all. The three that move are adjacent and they sit around the band’s larger counted number, which is 13, at a distance of five, six and seven organs below it.

Whether that is meaningful is not decidable here. Three consecutive offsets out of six is not a strong pattern and the six are themselves consecutive, so any three of them that move together would be adjacent about half the time. The census has a standing result that the offset mostly decides the survivor, and three adjacent offsets behaving alike is consistent with it and adds nothing to it.

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. 15 Which offsets wreck at each lattice of the census and what each keeps, which is where the offsets that change here sit in the wider picture.

The check that would sharpen it

The Lucas 7/11 band, at 124 rises, is nearly as wide as this one and is on the other branch. If its changing offsets also cross at separate rises below its handover, the shape is a property of bands. If they cross together, or at the handover, this band is unusual.

Two hours of runs, and it is the single most informative thing that could be added to this thread. It is not in this round, and it goes into the record as an outstanding check with its cost attached.

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. 16 Two bands drawn together. The comparison this section asks for is this picture with both cut at every rise.

Why three crossings is more useful than one

A single boundary would have been a cleaner result and a less informative one. If all three offsets crossed at one rise, the natural reading would be that something about the lattice changes there and every cut feels it — and the next question would be what, with no way to narrow it.

Three separate crossings say the change is not a property of the lattice alone. It depends on which organ was removed as well, and that halves the space of candidate accounts at a stroke: whatever moves the crossing has to be something a cut’s offset can interact with, which rules out every quantity that is a function of the rise alone.

The next organ moves for the last 13, and for no others. One row per organ removed, counted back from the tip of a stem at a rise of 0.005 whose counted pair is 8 and 13. Removing any of the last 13 moves the next organ by 2.6° to 167.6°; removing an older one moves it by at most 0.47°, which is under the azimuth grid. The boundary is at 13, and 13 is the larger parastichy number — so the experiment counts the spirals without measuring an angle.
Fig. 17 How a removal’s cost depends on which organ was taken, which is the kind of quantity a crossing that moves with the offset would have to involve.

What a reader should carry

That the band’s central claim survives at fourteen times the resolution, and that everything else the thread has said about a band’s interior was said at a resolution of one rise in sixteen.

And that “the survivor changes somewhere below the handover” is now three numbers rather than one region, which is the difference between a description and a measurement — the same difference a count taken at one radius has to be careful about at the other end of this site.

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 Offset 8, the last of the three to cross and the one whose transition region is longest.

Three crossings and thirteen islands

The two readings belong together. Below each offset’s crossing there is a stretch in which the coarse family comes back — thirteen such returns across the three offsets, one to three rises wide each, with no period in them.

So a crossing is the point after which the fine family wins on balance, not the point after which it wins outright. Stating the crossings as three sharp rises is a simplification the essay makes deliberately and the picture does not: read the band image and the three transitions are visibly ragged rather than clean.

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. 19 The band once more. The three transitions in the lower rows are staggered and none of them is a clean edge.

What the design would look like rebuilt

Grow a band, cut every rise of it, and report three things per offset: whether it wrecks, which family it keeps, and where its crossing is. That is what this sweep does and it is not what the band design does, which reports one family per band and a list of exceptions.

Rebuilding it that way costs sixty-seven minutes a band and would replace six numbers with about twenty. Whether that is worth two hours across the remaining five bands is a judgement, and the argument for it is that four of the six currently report no change at all — a reading that a sixteen-rise step could produce from a band with a transition region narrower than 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. 20 The full reading the band design currently produces, which this sweep suggests is under-resolved on the wide bands.

The one line

Cut at every one of its 126 rises, the widest band on the ladder changes its surviving family nineteen times and not once at the handover, with the nearest change eight rises away.

The three offsets that change cross at three different rises — eight, nineteen and twenty-nine rises below the handover — which a design stepping sixteen rises at a time reported as one region two to four steps below the crossing.

Where this goes next

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. 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. The same angle, two rungs, two answers. Each block is one matched pair: two rises whose stems settle on the same divergence and whose counters return different pairs. Under each is the family every wrecked cut leaves standing. On three of the four pairs both rises wreck at some offset, and on every one of those the two stems keep different families — so the divergence, which is held, is not what decides the survivor. The two stems keep exactly the counted numbers their two pairs share, including the pair that shares none and keeps none. The angle is not the actor Cut an organ out of two stems that settled on the same divergence and return different counted pairs, and the family left standing is different at every one of the four pairs where both stems wreck. The angle is held to a hundredth of a degree underneath.

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.

  • Six lattices were not enough — both name ablation, claim testing, control, honest limits, lattice offset, matched design, measurement, negative result, parastichy pair, rise, rung, sampling, underdetermination
  • One offset, two answers — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rigid hop, rise, rung, underdetermination
  • The exception was already labelled — both name ablation, claim testing, control, handover, honest limits, lattice offset, matched design, measurement, negative result, parastichy pair, 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, underdetermination
  • One rung, two answers — both name ablation, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rigid hop, rise, rung, underdetermination
  • The family that lost a member — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rigid hop, rung, underdetermination

Named objects

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

AblationClaim testingControlHandoverHonest limitsLattice offsetMatched designMeasurementNegative resultParastichy pairResolutionRigid hopRiseRungSamplingUnderdetermination