Phyllotaxis
the spirals are counted, not admired
Stems and cones

Nine rises were not enough

A coarse sample of a band had never been wrong about whether anything changes inside it, and that record was the argument for trusting a negative from it. The third band cut whole makes it two of three, and the missed feature is one rise wide.
14 min read 20 figures State the claim, state the testCounted, not admiredOne parameter, one family

Worth reading first: Where a handover sits.

The design that cuts a band takes nine rises, evenly spaced in the logarithm of the rise, plus both ends and the handover itself. Nine was chosen for a quantity the design predicted to be constant, and a constant is tested at its extremes and at the crossing rather than by sampling the middle more finely.

The quantity turned out not to be constant. Since then the nine-rise design has been run beside three full sweeps, and the interesting number is not how many changes it finds but how often it is right about whether there are any.

What nine rises find on each band, against what the whole band holds. The coarse design cuts nine rises of a band, evenly spaced in the logarithm of the rise, and asks whether the family a cut keeps changes anywhere. On the golden 8/13 it finds five of nineteen changes and on the Lucas 7/11 it finds none of none, so it had never been wrong about whether anything changes. On the golden 5/8 there are two changes and it finds neither, both of them at single rises with the offset recovering on either side. Its record on that question is now 2 of 3.
Fig. 1 What nine rises find on each band cut whole, against what the whole band holds.

The record it had

On the golden 8/13 band the sample found five changes where the sweep found nineteen. Wrong about how many, right that something changes.

On the Lucas 7/11 band it found none where the sweep found none. Right about how many and right that nothing changes — and that was the more valuable of the two, because a negative from a sample is worth nothing unless the sample can be trusted to find a positive.

So after two bands the record read: right twice about whether, right once about how many. That reading is what let the round say a negative from nine rises means something, and the sentence was written down in those words.

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. 2 The coarse design’s record over the bands cut before this round, on both questions it can be asked.

What the third band did to it

The golden 5/8 band holds 112 rises. The nine-rise design visits ten of them at a step of fourteen. It finds no change.

The full sweep finds two: offset 5 keeps a family off its counted pair at rises 0.01625 and 0.01605, and keeps the ordinary one at the other twenty-six rises it wrecks at.

So the record is now two of three on the question the design exists to answer, and the band it is wrong on is the one it was asked about most recently.

All three bands cut at every rise, offset by offset. One row per wrecking offset on each band cut whole, one cell per rise, coarse on the left. A pale cell is a rise at which that offset's cut recovers and has no survivor; a dark cell is a cut that wrecks and keeps one of the band's own counted pair; a warm cell is a cut that keeps a family off the pair. The vertical rule on each row is that band's handover, where its two contact steps change places. The golden 8/13 band changes the family it keeps 19 times, the golden 5/8 twice and the Lucas 7/11 not at all.
Fig. 3 Every band cut whole, with the changes the coarse design would have missed on the third.

Why it missed them

Both changes sit at single rises with offset 5 recovering on either side. A feature one rise wide, sampled at a step of fourteen, is hit about one time in fourteen; two of them, at rises twenty apart in a band of 112, are missed together most of the time.

That is not a criticism of the sample. It is arithmetic, and the same arithmetic was already written down: the first band’s islands are one and two rises wide and the sample hit two of thirteen of them.

What is new is that on that band there were thirteen chances and on this one there were two. A design that undercounts thirteen features to five is a design that will report zero of two sooner or later, and this is sooner.

Offset 5 across the 5/8 band, rise by rise. The family this one offset keeps at each of the band's 112 rises, coarse on the left. It wrecks at 28 of them and keeps the 5-family and the 20-family at different rises. 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. 4 The one offset whose answer changes on the third band, and how narrow the two features are.

The two questions come apart

Does anything change? and how much? are different questions and the design answers them with different reliability, which is why they have been kept apart since the second band.

On how much it has always been poor: five of nineteen, none of none, none of two. The only band it gets exactly right is the one with nothing to count.

On whether it was perfect and is now not. And the failure mode is asymmetric in a way that matters: the design can only ever undercount, because a change it reports is a change between two rises it actually cut. So a positive from nine rises is sound and a negative is a bound.

What nine rises find on each band, against what the whole band holds. The coarse design cuts nine rises of a band, evenly spaced in the logarithm of the rise, and asks whether the family a cut keeps changes anywhere. On the golden 8/13 it finds five of nineteen changes and on the Lucas 7/11 it finds none of none, so it had never been wrong about whether anything changes. On the golden 5/8 there are two changes and it finds neither, both of them at single rises with the offset recovering on either side. Its record on that question is now 2 of 3.
Fig. 5 Both questions on all three bands, which are answered with different reliability.

What a bound is worth

Something, and it is worth stating rather than dismissing. Ten rises finding nothing on a band of 112 rules out any feature wide enough to have been stepped on — anything eight rises across or more would have been hit twice.

So a negative from nine rises is not nothing changes; it is nothing changes over a stretch wider than about seven rises. On the Lucas band the full sweep then confirmed the stronger statement, and on this band it did not.

The difference between the two readings is exactly the class of feature this thread keeps finding: islands one and two rises wide, with nothing continuous distinguishing them from their neighbours.

All one bands cut at every rise, offset by offset. One row per wrecking offset on each band cut whole, one cell per rise, coarse on the left. A pale cell is a rise at which that offset's cut recovers and has no survivor; a dark cell is a cut that wrecks and keeps one of the band's own counted pair; a warm cell is a cut that keeps a family off the pair. The vertical rule on each row is that band's handover, where its two contact steps change places. The golden 8/13 band changes the family it keeps 19 times, the golden 5/8 twice and the Lucas 7/11 not at all.
Fig. 6 The third band at full resolution, where the features are one rise wide and everything smooth is smooth.

The cost of the alternative

A full sweep of a band is between 1,120 and 1,890 cut stems, which is seventeen to thirty-eight minutes. The nine-rise design is about 140 stems and two minutes.

Three bands have been swept whole and three have not. Of the three that have not, two wreck nothing at all — nothing behind the front wrecks at the coarse end — so the real remainder is the Lucas 4/7 band at 86 rises and about half an hour.

That is affordable, and after it the ladder has no more bands. So the question of what a coarse sample is worth on a band is about to become a question about no bands, which is a reason to answer it now rather than later.

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 six bands the ladder holds, of which four can be cut and three have been.

Where the design is still right

At the ends. The coarse design reports what each end of a band keeps, and on all three bands that report matches the sweep exactly.

That is not trivial. A band’s two ends are the two lattices the whole handover thread compares, and everything said about the ordering of the two contact steps changing places rests on reading them correctly. Nine rises reads them correctly on every band tried.

So the design has a part that works and a part that does not, and they are separable: the ends are two readings and the middle is seven.

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. 8 The two ends of each band, which is the reading the coarse design has never got wrong.

And where it was never asked to work

The design was built to test a constant. If the surviving family had been constant across every band — which is what everything before the first full sweep suggested — then nine rises would have been the right number and sampling the middle finely would have been waste.

It is worth being fair about that. The design is not a bad sample of a varying quantity; it is a good sample of a constant, used on a quantity that turned out to vary. The mistake was in the model, and the design followed from it correctly.

What follows is that the design should not be reused unchanged on the fourth band, because the model it was built for is now known to be wrong.

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. 9 Every band the design was built to sample, and the model of a constant it was built for.

What a better design would look like

Not more rises evenly spaced. A step of seven instead of fourteen doubles the cost and still misses a feature one rise wide thirteen times in fourteen.

The feature that matters is narrow, so the only sampling that finds it reliably is one that does not skip. Between a full sweep and a nine-rise sample there is nothing useful: the quantity is an integer read off a run, it changes between adjacent rises with nothing continuous moving, and no interpolation is available.

That is a real conclusion about the object rather than a complaint about resources. A discrete answer that changes at single rises can only be found by cutting every rise, and the design should be either the full sweep or an honest bound.

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. 10 The rises a band holds, at which a discrete answer is either read or not read.

The same shape, elsewhere on this site

This is the third time a sampled instrument here has been swept and found to have been reporting its own step.

The reference search window decided answers when it was varied. The run length moved nineteen of twenty-four onsets when it was doubled. The reading window turned out to be measuring a drift rather than a level.

Four instrument settings have now been varied and all four decided something. That is a run long enough to be worth stating as a working assumption: a setting nobody has moved is a setting that is deciding an answer, until it has been moved.

Which cuts count as periodic, at each reading window. One row per wrecked cut and one column per window. A filled cell is a cut whose worst class spread is under the ten-degree line and is therefore called periodic. At 60 organs 28 of the 30 cuts are, at 120 organs 25, and at 180 organs 25. three rows change side, all of them losing their periodicity as the window widens, and they are marked.
Fig. 11 An instrument setting varied, which is what happened to the last three that were.

What the coarse design has decided elsewhere

Its step is also what the previous round’s location claims rested on. The changes on the first band were reported as sitting two to four sweep steps below the crossing, and one of its steps is fifteen rises — so that sentence located nineteen changes to within about sixty rises.

At full resolution they sit seven to fifty-seven rises below the handover and the three offsets cross at three different rises. The coarse statement was not wrong; it was a statement about the sample, phrased as a statement about the band.

That is the more insidious of the two failure modes, because it produces a number rather than a silence.

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. 12 The first band’s changes at full resolution, which the coarse design located to within its own step.

What is not in doubt

Every claim the sweeps make about what a band’s ends keep, because those are read directly.

Every claim about the wrecking set, because that is read at every rise of every band and does not depend on the sampling at all — which offsets wreck is a function of the rise on all three bands cut whole, and the coarse design would have found that too.

And every claim about a handover’s position, because a handover is geometry: it is computed from hop lengths rather than from cut stems, and it can be walked at whatever resolution is wanted for 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. 13 The wrecking set across a band, which is read at every rise and does not depend on the sampling.

The negative that is now weaker

No change of surviving family sits at a handover. That claim is scored over nineteen located changes on one band and none on another, and it stands.

But it stands on changes that were found, and the count of changes found is now known to be a lower bound on two bands rather than one. If the golden 8/13 band holds changes that even a full sweep at its own step missed — features narrower than two parts in a thousand in the rise — then the tally of nineteen is a floor as well.

Nothing suggests it does. The point is that the sweep’s own step is now the only thing standing between the tally and that possibility, and the sweep’s step has never been varied.

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. 14 One band read at two step sizes, which is the check the sweep’s own resolution has never had.

A setting this round did not move

The band sweep’s step is two parts in a thousand between consecutive rises. It was chosen because the ladder is geometric and a fixed step in the rise is a step that means different things at the two ends of it.

Nobody has cut a band at one part in a thousand. That would double the cost of every sweep already made and would answer one question: whether the features this thread calls one rise wide are one rise wide or narrower than the grid.

Given the last four settings varied, the expected answer is that it matters. That is a prediction with a direction attached — a finer grid finds more changes, never fewer — and it is the cheapest thing that would test the tally of nineteen.

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. 15 The grid a band is swept at, which is the one setting in this thread nobody has moved.

What to do with the fourth band

Sweep it whole. The nine-rise design would cost two minutes and produce a number nobody could use: a positive would be sound and a negative would be a bound that this round has just shown to be violated once in three.

The fourth band is the Lucas 4/7, and it is the test the surviving account of the three bands most needs. Running it at the coarse design and getting a negative would leave the account resting on a reading that has recently been wrong.

Half an hour, once, and the ladder’s bands are done.

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. 16 The Lucas branch, and the one band on it that has not been cut whole.

What the sample would have to be to work

Consider what it would take for nine rises to find a feature one rise wide with any reliability. A step of fourteen hits a given rise with probability about one in fourteen; to be more likely than not to hit at least one of two such features, the step has to come down to about three, which is 37 rises of 112 and a third of the cost of the full sweep.

At that point the argument for sampling has gone. A third of the cost for a coin-flip on the question, against the whole cost for the answer, is not a trade anybody would take once it is written down as numbers.

That is the general shape of sampling a discrete quantity with narrow features: there is no useful middle. Either the sample is dense enough that it is nearly a sweep, or it is a bound whose width is its own step.

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. 17 The rises a band holds against the rises a coarse design visits, which is where the middle would have to be.

What the sweeps have cost, and what they bought

Three bands, 4,622 cut stems and about ninety minutes of machine time in total. What they bought is four results that no sample produced.

That the alternation on the first band is speckle rather than a period; that its three changing offsets cross at three different rises; that a second band has no changes at all; and that a third has two, which is what makes the branch the only account left standing.

Not one of those is available from nine rises. The first two need the changes located, the third needs the negative at full strength, and the fourth needs a positive the sample missed.

All three bands cut at every rise, offset by offset. One row per wrecking offset on each band cut whole, one cell per rise, coarse on the left. A pale cell is a rise at which that offset's cut recovers and has no survivor; a dark cell is a cut that wrecks and keeps one of the band's own counted pair; a warm cell is a cut that keeps a family off the pair. The vertical rule on each row is that band's handover, where its two contact steps change places. The golden 8/13 band changes the family it keeps 19 times, the golden 5/8 twice and the Lucas 7/11 not at all.
Fig. 18 Everything the three full sweeps produced, none of which a nine-rise sample reports.

And what a sample is still the right tool for

The wrecking set, which the sweeps found moves on every band. A sample of nine rises would have found that too, because an offset that wrecks at 17 rises of 112 in seven stretches is not a narrow feature — it is a coarse texture, and any sampling sees it.

So the design is not wrong about everything. It is wrong about the one quantity that changes at single rises, and it is right about the one that changes over stretches.

The lesson generalises past this band: the resolution a design needs is set by the narrowest feature of the quantity it is reading, and nothing about the design says what that is until something is swept whole.

Which offsets wreck across the golden 8/13 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 22 of its 126 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. 19 The coarse texture a sample does see, on the band whose narrow features it missed.

What the record is worth as a record

Three bands is a small denominator, and the design’s record on them — two of three about whether, one of three about how many — is not a probability. It is three cases with names.

What makes it usable is that the failure is understood rather than merely counted. The design missed this band’s changes because they are one rise wide and its step is fourteen, and the same arithmetic says it will miss any feature of that width about thirteen times in fourteen. That is a prediction about the next band, and it does not depend on how many bands have been cut.

So the honest summary is not the design is right two thirds of the time. It is: the design finds features wider than its own step and misses narrower ones, which is what a sampling does, and this thread’s features are narrower than its step.

Where that leaves the uncut bands

The Lucas 4/7 is the only one left that wrecks, and it should be swept whole rather than sampled. The argument is now arithmetic rather than precautionary: a negative from nine rises on that band would be a bound of about seven rises, and the features this thread finds are one and two rises wide, so the bound would exclude nothing anybody wants excluded.

Half an hour buys the answer instead. That is the same trade the previous three bands took, and the only reason it was ever in question is that the coarse design’s record made a negative from it look like a result.

Stating the record

Three bands cut at every rise they hold. The nine-rise design run beside each of them.

Right about whether anything changes on two of three, right about how many on one of three, right about both ends on three of three. It cannot overcount, so a positive from it is sound and a negative is a bound of about seven rises.

That is what a coarse sample of a band is worth, measured rather than assumed, and the measurement took three full sweeps.

What nine rises find on each band, against what the whole band holds. The coarse design cuts nine rises of a band, evenly spaced in the logarithm of the rise, and asks whether the family a cut keeps changes anywhere. On the golden 8/13 it finds five of nineteen changes and on the Lucas 7/11 it finds none of none, so it had never been wrong about whether anything changes. On the golden 5/8 there are two changes and it finds neither, both of them at single rises with the offset recovering on either side. Its record on that question is now 2 of 3.
Fig. 20 The coarse design’s whole record, over every band that has been cut at every rise.

Where this goes next

All three bands cut at every rise, offset by offset. One row per wrecking offset on each band cut whole, one cell per rise, coarse on the left. A pale cell is a rise at which that offset's cut recovers and has no survivor; a dark cell is a cut that wrecks and keeps one of the band's own counted pair; a warm cell is a cut that keeps a family off the pair. The vertical rule on each row is that band's handover, where its two contact steps change places. The golden 8/13 band changes the family it keeps 19 times, the golden 5/8 twice and the Lucas 7/11 not at all. The third band, cut whole Two bands cut at every rise disagreed about whether the family a cut keeps ever changes, and three explanations were available for a difference between two things. The cheapest third band settles which of them survives, and it settles it against the account nobody was betting on. 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. 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. 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. A band with nothing inside it Five offsets wreck on the Lucas 7/11 band and every one of them keeps the same family at every rise it wrecks at. There are no islands, no transition region and no period to look for, which is what makes the picture from the other band a picture of that band.

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.

  • The hops cross once — both name claim testing, contact family, discretisation, handover, honest limits, instrument setting, negative result, resolution
  • The second band, cut whole — both name claim testing, handover, honest limits, negative result, replication, resolution, sampling
  • A list that was a rounding — both name artefact, claim testing, discretisation, honest limits, negative result, resolution
  • A period the grid invented — both name artefact, claim testing, discretisation, honest limits, negative result, resolution
  • A window nobody aligned — both name artefact, claim testing, honest limits, negative result, resolution, sampling
  • An offset that arrives — both name census design, claim testing, contact family, discretisation, honest limits, resolution

Named objects

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

ArtefactCensus designClaim testingContact familyDiscretisationHandoverHonest limitsInstrument settingNegative resultReplicationResolutionSampling