Phyllotaxis
the spirals are counted, not admired
Stems and cones

The branch is what is left

Four accounts of why one band's cuts change what they keep and another's do not were written down before a third band was cut. Three of them are now wrong on a band each, and the survivor is the one with no mechanism behind it.
14 min read 20 figures State the claim, state the testCounted, not admiredForm is not mechanism

Worth reading first: Where a handover sits.

Three bands on the ladder have now been cut at every rise they hold. Two of them produce cuts whose surviving family changes somewhere inside; one does not. The question is what separates them, and the useful part of the answer is what it eliminates.

Four accounts were written down, each naming something the first two bands differ in, and each of them fitted those two perfectly. That is what two cases do: they are consistent with every account that orders them the same way, and there is no shortage of such accounts.

Four accounts of which bands speckle, scored on the three cut whole. Each candidate explanation of why one band's cuts change the family they keep and another's do not, against what the three bands cut whole actually do. A tick is an account that puts that band on the side the sweep does. The branch the band sits on is right on all three; the size of the counted pair, the number of wrecking offsets and how much of its rung the band spans are each wrong on two. Three bands can eliminate and cannot confirm, and this eliminates three of the four.
Fig. 1 Four accounts of which bands change their surviving family, scored on the three cut whole.

The four

The branch. The golden 8/13 changes and the Lucas 7/11 does not, and one is golden and the other Lucas. A branch here is a seed angle and the additive sequence its counted pairs come from, so this account says the difference is in the stem’s own arithmetic.

The counted pair. The larger member is 13 on the changing band and 11 on the clean one, so a band changes when its pair is large enough. A larger pair means more families competing for the shortest hop, which is a reason as much as a correlation.

The number of wrecking offsets. Six against five. More offsets is more chances for one of them to change, so this account says the difference is one of sample size and not of kind.

The span. The changing band covers 72 per cent of its rung and the clean one 48, so a band changes when it is wide enough to reach a stretch of rise where something different happens.

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 Every band on the ladder, with the three cut whole and the three that are not.

Why a third band could separate them

Because the four accounts do not all order the six bands the same way, even though they order the first two identically. Choosing a band where they disagree is the whole of the design, and there was exactly one that disagreed cleanly.

The golden 5/8 sits on the branch with the changing band, so the branch account predicts it changes. Its pair is 5/8, its wrecking offsets are three, and it spans 23 per cent of its rung — smaller, fewer and narrower than either band already cut, so the other three accounts predict it does not.

That is a design in which every outcome is informative. Three of four accounts lose whatever happens, and which three is the finding.

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. 3 Every rung with a handover and the band grown around each, which is the supply the third band was chosen from.

What it did

It changes. Offset 5 keeps a family off the counted pair at two of the rises it wrecks at, and keeps the ordinary one everywhere else — and the family it keeps is a multiple rather than a divisor, which refutes a separate reading of its own.

So the branch account is right on all three bands and each of the other three is wrong on two. The pair account predicts no change here and no change on the Lucas band and is wrong on the first; the offsets account and the span account make the same two errors, because all three order the three bands identically and the answer does not.

That is worth saying carefully. The three failing accounts do not fail independently: they fail together, because on these three bands they are the same account written three ways.

Four accounts of which bands speckle, scored on the three cut whole. Each candidate explanation of why one band's cuts change the family they keep and another's do not, against what the three bands cut whole actually do. A tick is an account that puts that band on the side the sweep does. The branch the band sits on is right on all three; the size of the counted pair, the number of wrecking offsets and how much of its rung the band spans are each wrong on two. Three bands can eliminate and cannot confirm, and this eliminates three of the four.
Fig. 4 The account that survives all three bands, against the three that fail on the same two.

Which is why three bands is not three tests

A test that three accounts fail in the same way is one test. If the golden 5/8 had been wider, or carried more offsets, or a larger pair, the three would have come apart and the result would have been worth more.

No such band exists. On this ladder the golden bands are the wide ones and the Lucas bands the narrow ones, so the branch and the span are confounded across the whole design, and nothing inside it can separate them.

What broke the confounding here is that the golden 5/8 is narrow and golden, which is the one combination the ladder offers. It is the exception that made the round possible and it is the only one.

Two ways of predicting how wide a band is. A band ends where the settled divergence has moved 0.05° from its value at the handover, so the width should follow from how fast the divergence changes there. Reading that rate as the rung's average slope predicts widths that are wrong by factors of 0.20 to 5.92 — wrong in both directions, so no constant rescues it. Reading it as a curvature about a stationary point gives 0.41 to 1.08, with five of the six inside a third. The difference between the two is the difference between a curve and its average, and a band is exactly where the two are least alike.
Fig. 5 How wide each band is against how steep its rung’s divergence is, which is what makes width a measurement.

The account that survives explains nothing

A branch is a seed angle and a sequence. The golden branch’s stems settle near 137.5 degrees and its counted pairs are consecutive Fibonacci numbers; the Lucas branch’s settle near 99.5 and its pairs are consecutive Lucas numbers.

Neither of those has any visible connection to whether a cut’s surviving family changes across a stretch of rise. The survivor is read from two runs of azimuths and is the smallest lag whose hop the wrecked run held; the branch is a property of the intact stem it was cut from.

So the surviving account is a label. It says which bands do it and offers nothing about why, and a label that has beaten three explanations is still a label.

The golden 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 12 and 35 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; seven of them sit past the mark.
Fig. 6 The golden branch’s rungs, whose stems settle near one angle and whose pairs come from one sequence.

What a branch is confounded with

Everything a branch decides, which is a long list. The settled divergence, and therefore every hop length on every lattice. The counted pairs, and therefore which families compete. The spacing of the rungs, and therefore how much rise a rung covers. The depth of the front, and therefore how many offsets wreck at all.

Any of those could be the actor, and the design cannot tell them apart because they all change together when the branch does. The same confounding was found and named on a different quantity, where a rule that looked like it was about the ordering of two hops turned out to be about the offset.

The way out is a case where they come apart, and this ladder does not offer one.

The two contact steps change places inside the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not.
Fig. 7 One rung’s geometry, in which the branch decides the divergence, the pairs and the spacing at once.

The test that would move it

The Lucas 4/7 band. Eighty-six rises, about half an hour, and it is the second Lucas band — the only object that can make the branch account two of two on its own side rather than one of one.

If it changes, the branch account joins the other three and nothing on this ladder separates the bands. If it does not, the account is three golden bands and two Lucas ones, which is still a label but is a label with a real chance of having been wrong.

That is the whole of what is scheduled here, and it is worth stating that the negative outcome is the more informative one. An account that survives a fourth band has survived; an account that fails one has told the round something.

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

What the three failed accounts were worth

More than nothing, because each of them was a mechanism and the survivor is not.

The pair account had a reason: a larger counted pair means the two shortest hops are closer together in length, and a lattice where two candidates are nearly tied is a lattice where a small change in rise could switch which one a cut keeps. That is a prediction about why and it is now wrong on two bands.

The offsets account had a reason too, and a duller one: more offsets is more chances. It predicts a rate rather than a mechanism, and a band with three offsets changing twice while a band with five changes not at all is not what a rate predicts.

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. 9 How near the two contact steps come across each band, which is what the pair account was about.

And what the numbers say about the pair account in particular

It is the one worth burying carefully, because the quantity behind it is measured and does not do what the account needs.

The two shortest hops on the golden 8/13 band come within 0.6 per cent of each other at the handover; on the Lucas 7/11 within 0.03 per cent; on the golden 5/8 within 0.9 per cent. So the clean band is the one whose two candidates are most nearly tied, by more than an order of magnitude, and the account predicts the opposite.

That is a stronger refutation than the scoring. The account is not merely wrong about which bands change; it is wrong about the direction of the quantity it names.

Every rise of the 7/11 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 124 rises and 1612 cut stems. no of the five offsets change their answer somewhere inside, five never do, and the vertical rule is the handover — the rise where the two contact steps change places. Not one of the 0 changes is at it.
Fig. 10 The clean band at full resolution, whose two contact hops come nearest of the three and whose answer never changes.

A fifth account, which cannot be scored

That the changes are noise — that a surviving family read at a rise is a coin whose bias varies smoothly, and a band with nineteen changes and a band with none are two draws from distributions that differ only in degree.

It cannot be scored on three bands because it makes no ordering prediction, and it cannot be dismissed either. What speaks against it is the structure inside the first band: thirteen islands, three offsets crossing at three separate rises, and every change below the handover. Noise does not produce a boundary.

What speaks for it is this band, where two changes at two rises with nothing distinguishing them is what noise looks like. The same shape has appeared elsewhere in this thread and been called what it is: an alternation that fitted no period turned out to be speckle rather than structure, and a sample that hit two islands with a run between them produced a pattern out of nothing.

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. 11 The two changes on the third band, which are what a noise account would predict and what an ordering account would not.

The claim the whole thread rests on, still standing

That the rise where the two contact steps change places is not the rise where the survivor changes. Nineteen located changes on the first band and none on the second put nothing at a handover.

The third band cannot test it. Its one changing offset stops wrecking for thirty-four rises either side of the handover, so the return of its family is bracketed across a stretch that contains the handover and everything else, and the flag that fires is a statement about the wrecking set.

So the claim is unchanged: nineteen tests, all negative, one band unable to test it.

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 handovers sit inside their own rungs, which is the position a change would have to coincide with.

What replicates on all three

The wrecking set. Which offsets wreck is a function of the rise on every band cut whole, and on the third band not one offset wrecks everywhere — offset 3 wrecks at 17 of 112 rises in seven separate stretches and offset 5 at 28 in eleven.

That is the one result the design produces reliably and it is about a different quantity from the one this essay is about. A band changes what its cuts keep sometimes; a band changes which cuts wreck always.

Anything quantified over the offsets that wreck is therefore quantified over a set that moves, and no account of the changes has to explain that because every band does it.

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 moving across a band, which every band cut whole has shown.

What a fourth band costs and what it buys

Eighty-six rises at five offsets is about 430 cut stems and 86 controls, which is half an hour. The three bands cut so far cost 1,890, 1,612 and 1,120.

What it buys is the difference between an account resting on one band a side and an account resting on two a side. That is not a large difference in confidence and it is the largest one available, because the ladder holds six bands and two of them wreck nothing.

After that the design is exhausted. A fifth band would have to come from a different ladder, which means a different falloff exponent or a different geometry, and that is a different question rather than a finer answer to this one.

The families left standing, on both sides of every handover. One row per band, with every offset cut at rises spread across it and always at both ends and at the handover itself. The last column is every family left standing anywhere on that band. On three of the four bands that wreck at all it is a single family, unchanged across a rise at which the two contact steps swap places — so the step ordering is not what decides which family survives, and the result now rests on four counted pairs rather than on two. The shortest-hop reading scores 44 of 114 across the whole set, which is what a reading looks like when the quantity it is stated over is not in the mechanism.
Fig. 14 The whole supply of bands this question can be asked of, which is four rather than six.

How the accounts were scored

Each account is a function of the band that returns yes or no, and it is right on a band when its answer matches whether that band’s cuts ever change what they keep. That is the same scoring the survivor rules get and the same one the sign of the slot interaction gets: every candidate on every case, with the losers kept in the table.

Keeping the losers is the part that matters. An account reported alone is an account whose alternatives were never counted, and three accounts each wrong on two bands is a much more useful sentence than one account right on three.

The thresholds inside the three failing accounts were chosen to make them as strong as possible rather than as weak. The pair account is scored at a larger member of 11 or more, which is the cut that gets both original bands right; the span account at 45 per cent, the offsets account at five. Each is the most favourable line available, and each still fails on two.

Five accounts of the sign, on the 24 lattices that are measurements. Each bar is how many of the lattices an account puts on the right side of zero. The six rows where the second wall is free are left out, because their value is minus the first wall's cost by construction and any rule scores whatever it happens to say about them. Position inside the rung, the rise and the branch all fail. The larger counted number sorts 22 of the 24, and the misses are one rung's worth of rows rather than a scatter.
Fig. 15 The same scoring applied elsewhere on this site: every candidate rule on every case, with the losers kept.

What the bands have in common

More than the accounts suggest, and it is worth listing because a difference has to be found among the things that differ.

Every band is built the same way: outwards from a handover while the counted pair holds and the settled divergence stays within a twentieth of a degree. Every one is swept at a ratio of two parts in a thousand rather than at a fixed step. Every one is cut at every offset the front reaches, with a control at each rise sharing the history below the hole.

So the three bands are three instances of one instrument pointed at three lattices, and the difference between their answers is a difference between the lattices. That is what makes the elimination worth anything: none of the four accounts could be a property of the design.

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. 16 One band’s construction, which is identical on all three and is not among the things that differ.

The fifth thing a branch decides

The depth of the front, which is worth pulling out of the list because it is the one that nearly gets its own account.

A cut wrecks a stem only if the removed organ is close enough to the growing front to matter, and how far back that reaches is a measurement rather than a setting. The golden bands here reach three and six offsets and the Lucas band reaches five, and the front’s depth changes down a rung as well as between branches.

The offsets account is that quantity dressed as a count, so its failure is the front’s failure to predict the changes. What is left of it is a fact worth keeping: a band with three wrecking offsets can change its answer and a band with five need not, so the number of chances is not what decides.

The block is one of the two numbers, and not always the smaller. Every lattice a single organ was removed from, one row each: golden, rise 0.020 carrying 3/5; golden, rise 0.013 carrying 5/8; golden, rise 0.008 carrying 5/8; Lucas, rise 0.013 carrying 4/7. The two open ticks on each line are that lattice's own parastichy numbers; the filled dots are the blocks the stems that never recovered settled into, one per offset that failed. The claim this table was built to test is that the block is the smaller of the two, which held at the two rises it was first measured at. It does not hold here: 3/5 gives 5, 5/8 gives 5 and 8, 4/7 gives 4 and 7. What survives is weaker and still worth something — every filled dot but 0 sits on one of that row's own ticks, so the orbit carries a count of the lattice it was cut from, and which of the two it carries is decided by the offset rather than by the pattern.
Fig. 17 How far back a removal still wrecks a stem, which decides how many offsets a band has to change its answer at.

Why the survivor is the awkward quantity

Every other reading in this thread is continuous in the rise. The divergence a stem settles to moves smoothly, the hop lengths move smoothly, the interaction between two removals moves smoothly, and a claim about any of them can be checked by looking at whether it varies where it should.

The surviving family is an integer. It is the smallest lag whose hop the wrecked run held to within a tolerance, and it can change under an arbitrarily small change in the run without anything continuous changing at all. So the question what makes it change here and not there may have no answer of the kind the other questions have.

That is not a reason to stop asking. It is a reason to expect the answer, if it comes, to be about the discrete structure of the wrecked run rather than about any smooth property of the lattice it was cut from — and none of the four accounts is of that kind.

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. 18 The surviving family across the census, which is an integer read off a run rather than a smooth quantity.

What would count as a mechanism

Something that predicts a rise. Not which bands change but at which rises inside a band they change, because that is where the evidence is thickest: nineteen located changes on one band, three offsets crossing at three separate rises, thirteen islands.

No account here attempts that. The four scored are all about whether a band changes at all, which is one bit per band and three bits in total, and three bits cannot distinguish much. The nineteen changes carry far more and nothing has been fitted to them.

That is the shape of the next question rather than of this one, and it is a question about a single band rather than about the ladder — which is a reason to think it is answerable and not a reason to think the answer will generalise.

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 nineteen located changes on one band, which carry more information than the three bits the accounts were scored on.

Stating it as it stands

Three bands cut at every rise. Two change and one does not. Of four accounts written down first, three are wrong on two bands each and one is right on three.

The one that is right names the branch, and a branch is confounded with the divergence, the sequence, the pairs, the rung spacing and the depth of the front. So the finding is a correct prediction with no mechanism in it, standing on three cases, in a design that cannot separate its own survivor from five other things.

That is a smaller result than the branch decides it, and it is what the round can support.

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. 20 The three bands cut whole, which are the whole of the evidence behind the account that survives.

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. 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 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. 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.

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 second band, cut whole — both name claim testing, handover, honest limits, negative result, parastichy pair, replication
  • The side the census sat on — both name claim testing, handover, honest limits, negative result, parastichy pair, underdetermination
  • The wrecking set moves again — both name census design, claim testing, contact family, handover, honest limits, replication
  • A stem too fine to settle — both name claim testing, honest limits, negative result, parastichy pair, underdetermination
  • Every rise of a band — both name claim testing, handover, honest limits, negative result, parastichy pair
  • One offset, two answers — both name claim testing, honest limits, negative result, parastichy pair, underdetermination

Named objects

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

BranchCensus designClaim testingConfoundingContact familyDescription versus mechanismHandoverHonest limitsNegative resultParastichy pairReplicationUnderdetermination