Phyllotaxis
the spirals are counted, not admired
Stems and cones

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.
14 min read 19 figures State the claim, state the testCounted, not admiredOne parameter, one family

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

A band is the stretch of rise around a rung’s handover over which the counted pair holds and the settled divergence does not move. Cutting one at every rise it holds asks one question: does the family a wrecking cut leaves standing change anywhere inside it?

Two bands have been cut whole and they answered differently. The golden 8/13 band changes its answer nineteen times; the Lucas 7/11 band does not change it once. That is a difference between two things, and a difference between two things has as many explanations as anyone cares to write down.

The round that found the second answer said so plainly and stopped there, because two bands cannot choose between four accounts. Every account that names something the two bands differ in fits both of them perfectly, and there are more such things than there are bands.

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. 1 Every band cut at every rise it holds, one row per wrecking offset, with each band’s handover marked.

What a third band is for

Four accounts were written down before this one was cut, and each of them names something the two bands differ in.

The branch they sit on: one is golden and one is Lucas. The counted pair: 8/13 against 7/11, so the larger member differs. The number of wrecking offsets: six against five. And how much of its rung the band spans: 72 per cent against 48.

Two bands cannot separate four accounts, because every one of them puts the two bands on opposite sides. A third band can, and only if it is chosen to break the pattern.

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. 2 The four accounts of which bands change their answer, scored on the bands cut whole.

Why the golden 5/8

It is the only band on the ladder that separates the branch from everything else. It sits on the golden branch with the speckled band, so an account that names the branch predicts it changes; it carries a smaller pair, fewer offsets and a narrower span than either, so the other three accounts predict it does not.

That is a design with a prediction attached rather than a survey. Whichever way it comes out, three of the four accounts lose — and which three depends on the answer, so the design cannot fail to be informative in the way a confirmation can.

It is also the cheapest of the three remaining bands. At 112 rises and three wrecking offsets it needs 1,120 cut stems, against 1,890 for the band already done and 1,612 for the other. The two bands at the coarse end of the ladder are cheaper still and are worth nothing here, because nothing behind the front wrecks there and a band with no wrecked cut has no survivor to follow.

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 this band was chosen from.

The sweep

One hundred and twelve rises, every offset the front reaches at each of them, and a control at every rise sharing the history below the hole. Seventeen minutes.

The band is built the way every band here is built: outwards from the handover while the counted pair holds and the settled divergence stays within a twentieth of a degree of its value there. It reaches 0.01763 at the coarse end and 0.01413 at the fine, and the handover at 0.01558 sits 56 per cent of the way along it.

The rises are spaced by a ratio rather than by a step. Two parts in a thousand between one rise and the next, which is the same argument the ladder itself is swept with: a fixed step that is one per cent of the rise at one end of the ladder is half a per cent at the other, and a band swept that way is a band sampled twice as finely at one end as at the other.

The offsets tried at each rise are the ones the front reaches, which on this band is three — offsets 3, 4 and 5. Past the front every cut recovers and a recovered stem has no survivor to report, so there is nothing to gain by cutting further back.

Every rise of the 5/8 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 112 rises and 1120 cut stems. one of the three offsets change their answer somewhere inside, two never do, and the vertical rule is the handover — the rise where the two contact steps change places. Not one of the 2 changes is at it.
Fig. 4 The golden 5/8 band at full resolution, rise by rise, with what each cut leaves standing.

It changes

Twice. Offset 5 keeps the family 20 at two of the rises it wrecks at, and keeps 5 at every other.

So the branch account is the one left standing. Both golden bands change the family their cuts keep somewhere inside them and the Lucas band does not, and the three accounts that name the pair, the offsets or the span each put this band on the wrong side.

The arithmetic is worth doing out loud, because three of four eliminated is the kind of sentence that sounds stronger than it is. The pair account says a band changes when the larger of its counted numbers is 11 or more: it is right on the 8/13, wrong on the 7/11 and wrong here. The offsets account says a band changes when five or more of its offsets wreck: same three verdicts. The span account says a band changes when it covers 45 per cent of its rung or more: same again. All three fail on the same two bands, because all three order the bands the same way and the answer does not.

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. 5 The account that survives all three bands, which is the one nobody had a reason to prefer.

Two against nineteen

The support is thin and saying so is most of the work. Nineteen changes across three offsets is a transition region; two changes at one offset is two rises.

What replicates is that a change happens, not the shape it made on the first band. There are no islands here, no alternation to fit a period to, and nothing that would have prompted the question that band raised.

A reader who took the first band’s picture as what a golden band looks like would be wrong about this one in every particular except the sign. Thirteen islands and nineteen changes across three offsets is a texture; two changes at one offset is two rises, and two rises support no statement about texture at all.

The honest form of the finding is therefore conditional. Whether a golden band’s cuts ever change what they keep looks like a property of the branch, on three bands. How much they change looks like a property of the band, on the same three, and the two golden bands differ by a factor of ten in it.

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. 6 The one offset on this band whose answer changes, drawn across every rise the band holds.

What three bands can and cannot do

They can eliminate. Three accounts are now wrong on a band each, and an account wrong on a band cut at every rise it holds is wrong for good rather than pending a finer sweep.

They cannot confirm. One account being right on three bands is one account being right three times, and the branch is confounded with everything else a branch decides: its seed angle, its sequence, the divergence its stems settle to. Three golden bands and three Lucas ones would say more than a fourth of either.

The cheapest thing that would say more is the Lucas 4/7 band at 86 rises, because it is the second Lucas band. If it changes, the branch account goes the way of the other three.

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

The two ends were right

The coarse design cuts nine rises and reports what each end of the band keeps. On this band both ends keep 5, which is what nine rises said, and the full sweep confirms it at every rise outside the two.

That is worth recording because it is the half of the coarse design that has never been wrong. What a sample of a band gets right is the value at its ends; what it gets wrong is anything narrower than its own step.

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. 8 What nine rises find on each band, against what the whole band holds.

The wrecking set, for the third time

The one result that has replicated on every band cut whole: which offsets wreck is a function of the rise, not of the lattice.

Here it is more thorough than anywhere. Offset 3 wrecks at 17 of the 112 rises in seven separate stretches; offset 5 at 28 in eleven; and offset 4, which wrecks at 111 of 112, has a gap one rise wide in the middle of it.

So not one of this band’s three offsets wrecks everywhere. The golden 8/13 band has one that does and the Lucas 7/11 has three.

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. 9 The wrecking set moving across a band, which every band cut whole has shown.

Which makes a census a sample of a rise

The consequence is the same one the second band stated and it is sharper here. A census taken at one rise of this band cuts three offsets; a census taken at a rise fourteen steps away cuts one. Neither is wrong and they are censuses of different sizes.

Every claim in the ablation thread quantified over the offsets that wreck is therefore quantified over a set that depends on a number nobody chose deliberately. The claims survive because they are about what a cut keeps rather than about how many cut, and the distinction had not been forced until a band was cut whole.

It also puts a boundary on a result from the same thread. The offset mostly decides which family survives, scored over the census, and that score is over the offsets that happened to wreck at the ten rises the census holds. On a band where an offset wrecks at 17 rises of 112, the offset is a property of a rise as much as of a lattice — and a rule indexed by it is being asked to work on a quantity that is only sometimes defined.

Every stem that never repaired, and the lag it kept. The 19 offsets across six lattices at which a single removal leaves a stem that never returns to its divergence. For each one: which organ was removed, the period of the block of angles the stem settles into, the lag whose hop survived the cut unchanged, and how many whole turns the stem gains over one period of that lag. The block and the surviving lag are the same number in every row. The marked row is the one whose survivor is not a parastichy number of the lattice that was cut — a hop 6.8 times the length of a contact hop, which no census would report and which the rule held rigid all the same.
Fig. 10 The census the ablation thread is quantified over, whose membership depends on the rise it was taken at.

The band’s own geometry

Nothing about the lattice moves across the band, which is what makes the sweep a sweep of one thing. The counted pair is 5/8 at every rise. The settled divergence moves by 0.047 degrees end to end, which is a fifth of one step of the azimuth grid.

What does move is the ordering of the two contact steps, and that is what a band is built around: the 5-hop is shorter at the coarse end and the 8-hop at the fine end, crossing between 0.01564 and 0.01561.

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. 11 How near the two contact steps come across a band, which is what a handover is read from.

The span is a measurement and not a setting

This band spans 23 per cent of its rung against the golden 8/13’s 72 and the Lucas 7/11’s 48. That is not a choice: a band is grown until the divergence moves, so its extent measures how flat the divergence is at that rung’s handover.

A narrow band is a rung whose divergence is steep there. The prediction attached to that — that band width falls as the slope rises — is tested on all six bands and this one sits where it should.

So the span account being wrong here is not an accident of choosing a narrow band. It is the account being wrong. If the span had been chosen — if a band’s extent were a setting rather than a measurement — then a narrow band changing its answer would mean nothing, because the band could have been made wider and the extra rises might have held the changes. It cannot be. The band ends where the divergence starts moving, and past that the lattice being cut is a different lattice.

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. 12 How wide each band is against how steep its rung’s divergence is at the handover.

What the two changes are worth on their own

Almost nothing as evidence about transition regions, and something as evidence about survivors.

The family the cuts choose between here is 5 and 20, and 20 is not a member of the counted pair. That is the second band to keep a family off its own pair, and the second one does not look like the first: the golden 8/13 chose between 8 and 4, which is half of a counted number, and this one chooses a multiple of one.

The 17 rows of the exchange table, gathered by the lag they kept. One bar per surviving lag, its length the number of rows the table holds at that lag, with the hop that lag's stems keep written beside it. The hop is nearly constant inside a lag, so a correction fitted over 17 rows is fitted over four hops — which is the denominator that matters and is much smaller than the row count suggests. Adding a lag to the table is worth more than adding rows at a lag already in it.
Fig. 13 Every family a wrecking cut has been seen to keep, gathered by the lattice it was cut from.

Where the changes sit

Both are above the handover, at 0.01625 and 0.01605, on the coarse side of it. Every change the golden 8/13 band showed sits below its handover, seven to fifty-seven rises down from it.

That is one more thing this band does not replicate, and with two changes it is not evidence of anything. It is recorded because a later band with changes above its handover would make it two of three, and nobody would go back to look.

The two rises are twenty apart in the band’s own indexing and five apart in the sequence of rises where offset 5 wrecks at all, which is the only sequence in which they are adjacent. Between them the offset recovers five times, so whatever holds the family at 20 is not holding the cut.

Offset 5 on the 5/8 band, and the change that cannot be placed. Every rise of the band, coarse on the left, with offset 5's cut drawn at each: pale where it recovers, dark where it wrecks and keeps 5, warm where it wrecks and keeps 20. It keeps the off-pair family at two rises above the handover and then does not wreck again for 34 rises, so its return is bracketed across a stretch that contains the handover. The flag that says a change sits at a handover fires here for the first time, and it is a statement about where the offset stops wrecking rather than about the handover.
Fig. 14 Where this band’s changes sit against its handover, and how widely the second of them is bracketed.

The claim the thread rests on

That the rise where the two contact steps change places is not the rise where the survivor changes. Nineteen located changes on one band and none on another had never put a change at a handover.

Here the flag fires for the first time, and it fires on a change that is not located. The return from 20 to 5 is bracketed between two rises 108 steps of the grid apart, because offset 5 does not wreck at any rise between them — and the handover is inside that bracket because nearly everything is.

At the handover rise itself the only cut that wrecks is offset 4, which keeps 5 above and below. So this band cannot test the claim at its own handover, and saying so is the result.

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. 15 Where handovers sit inside their own rungs, which is the quantity a change is placed against.

What was cut and what it cost

Three bands are now cut whole: 126 rises, 124 and 112, for 1,890, 1,612 and 1,120 cut stems. Three of the ladder’s six bands remain, and the two at the coarse end wreck nothing at all, so the real remainder is one.

The Lucas 4/7 band is that one, and it is the test the branch account most needs. Eighty-six rises and about half an hour.

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, its rungs and the one band on it that has not been cut whole.

What the coarse design would have reported

Ten of this band’s rises, at a step of fourteen, and both changes missed. That is not a surprise once the changes are known — a feature one rise wide is invisible to a step of fourteen about thirteen times in fourteen — but it is the first time the coarse design has been wrong about whether rather than about how many.

Its record was the argument for reading a negative from it. On the golden 8/13 it found five changes of nineteen and on the Lucas 7/11 it found none of none, so it had been right twice about the only question anybody asked of it. It is now right twice in three, and what that costs is a separate matter because it is the design six uncut bands would otherwise be measured with.

What nine rises could see of 112. Above, offset 5'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 14. 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 absent on this offset, against a step of 14, so the sample can only land on one by accident.
Fig. 17 What a nine-rise sample sees of the one offset on this band whose answer changes.

The control at every rise

Each rise is cut with a control beside it: the same stem grown to the same length with no organ removed. That is what makes a displacement a displacement rather than a divergence, and it is why the sweep costs 1,232 stems rather than 1,120.

The control also carries the check that the band is one lattice. Its counted pair is read at every rise and is 5/8 at every rise; its settled divergence is read at every rise and moves by less than a fifth of one azimuth step. A band where either moved would be a band whose ends are two different objects, and the sweep would be comparing them rather than sweeping 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. 18 The golden 5/8 rung’s own geometry, from which the band around its handover is grown.

The shape of the answer

A round that cuts a band to test three accounts and eliminates three of them has done what it set out to do, and the account left standing is the one with the least behind it.

Nothing about a branch explains why a cut on it would change what it keeps. The branch is a seed angle and a sequence; the survivor is a contact family of a lattice; the connection, if there is one, is not in any file here. What the third band buys is the elimination of three explanations that could have been stated in one sentence each, and a fourth that cannot be.

All two 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. 19 The two golden bands cut whole, which are the two that change and the whole of the surviving account.

Where this goes next

The two widest bands on the ladder, each cut at every rise. One block per band, one row per offset that wrecks anywhere on it, one column per rise, coarse on the left. A filled cell is a cut that wrecks, and its tone is the family left standing; a pale cell is a cut that recovers. The golden 8/13 band above changes its answer at three of its six offsets, 19 times in all. The Lucas 7/11 band below changes it nowhere: every cut that wrecks on it keeps the 7 family, at every offset and every one of its 124 rises. The second band, cut whole One band was cut at every one of its rises and came back with a transition region — a stretch where three offsets change their answer, in short islands with uneven gaps. The obvious question is whether that is a picture of bands or a picture of that band. The other wide band answers it. Every rise of the 8/13 band, cut at every offset. One column per rise of the band, coarse on the left and fine on the right, and one row per offset that wrecks anywhere on it. A filled cell is the family the cut stem keeps; a pale cell is an offset that recovers at that rise and has no survivor to report. The band holds 126 rises and 1890 cut stems. three of the six offsets change their answer somewhere inside, three never do, and the vertical rule is the handover — the rise where the two contact steps change places. Not one of the 19 changes is at it. Every rise of a band A band is cut at nine rises because the quantity it was built to test is a constant, and a constant is checked at the ends and at the crossing. On the widest band that quantity turned out not to be constant, which makes nine the wrong number. This is all hundred and twenty-six. 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. 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.

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 offsets that never change — both name ablation, claim testing, handover, honest limits, negative result, parastichy pair, resolution, rung, sampling
  • One rise below the census — both name ablation, census design, claim testing, contact family, parastichy pair, replication, rung, sampling
  • The side the census sat on — both name ablation, claim testing, handover, honest limits, negative result, parastichy pair, rung, sampling
  • Two rises far apart — both name ablation, claim testing, contact family, handover, honest limits, negative result, resolution, rung
  • When nine rises are enough — both name ablation, claim testing, handover, honest limits, negative result, resolution, rung, sampling
  • A family that is a multiple — both name ablation, claim testing, contact family, honest limits, negative result, parastichy pair, resolution

Named objects

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

AblationCensus designClaim testingContact familyHandoverHonest limitsNegative resultParastichy pairReplicationResolutionRungSampling