Phyllotaxis
the spirals are counted, not admired
Stems and cones

The alternation is not a period

Nine sampled rises gave 8, 4, 8, 4 at one offset of one band, and a period was the obvious thing to look for. At full resolution it is thirteen islands one to three rises wide, with gaps of 1, 2, 3, 6, 7, 8, 9, 16, 31, 44 and 48 — and a fitted period buys exactly nothing.
12 min read 21 figures State the claim, state the testCounted, not admired

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

At offset 8 of the golden 8/13 band, the nine-rise design returned 8, 4, 8, 4 across four consecutive sampled rises. The obvious question was whether that alternation has a period.

Cutting every rise of the band answers it. It does not.

Offset 8 across the 8/13 band, rise by rise. The family this one offset keeps at each of the band's 126 rises, coarse on the left. It wrecks at 123 of them and keeps the 4-family and the 8-family at different rises. The ticks below mark ten islands — runs of 1 and 1 and 2 and 1 and 2 and 1 and 2 and 1 and 3 and 1 rises where the coarse family comes back inside the fine one. The handover is the taller rule and the change of answer is nowhere near it.
Fig. 1 Offset 8 at every rise of the band, coarse on the left. The alternation the sample saw is somewhere inside the speckled stretch on the right.

What is actually there

Below the rise where the offset first switches from the 8 family to the 4 family, there is a stretch of thirty-seven rises in which the 8 family comes back thirteen times. Every one of those returns is one to three rises wide.

The gaps between them are 1, 2, 3, 6, 7, 8, 9, 16, 31, 44 and 48 rises. No number divides that list and nothing in it repeats.

The three longest gaps are also the three at the ends of the stretch, which is what a margin does: dense in the middle, sparse where one family is clearly winning. A period would not care where in the stretch it sat.

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. 2 The whole band. The speckle is the broken stretch at the right-hand end of the bottom three rows.

Scored against no period at all

A list of gaps is a description. The test is a score, and the score has to be against something.

Every candidate period from two rises up to a third of the speckled stretch is fitted at its best phase, with the most generous rule available: in each residue class the period is allowed to predict whichever family is commonest there. That is the best a period could possibly do. It is scored against the baseline of naming the commonest family over the whole stretch and stopping.

A period fitted to the speckle, at every period it could have. Each mark is one candidate period, drawn at the share of rises it gets right when it is given its best phase and its best family in each residue class — the most generous reading of periodic there is. The flat rule is what saying nothing gets: name the commonest family and stop. The best period scores 76 per cent against 76 for no period at all, a gain of 0 points over 123 rises, so the alternation the coarse design reported is not a period being sampled badly.
Fig. 3 Every candidate period at its best phase, against the flat rule for no period at all.

The result is zero

The best period is 2, and it gets 75.6 per cent of the rises right. Saying the commonest family and stopping gets 75.6 per cent.

The gain is not small. It is nought, to the digit. A period fitted with a free phase and a free family per class recovers not one rise more than a constant does.

A period fitted to the speckle, at every period it could have. Each mark is one candidate period, drawn at the share of rises it gets right when it is given its best phase and its best family in each residue class — the most generous reading of periodic there is. The flat rule is what saying nothing gets: name the commonest family and stop. The best period scores 65 per cent against 58 for no period at all, a gain of 7 points over 110 rises, so the alternation the coarse design reported is not a period being sampled badly.
Fig. 4 The same scoring on a different offset of the same band, where there is less to fit and the answer is the same.

Why a period was worth looking for

Not idly. This thread has found periodic structure where nobody expected it before: the displacement above a hole is periodic at the lag the stem kept, which is a genuine period discovered by folding a sequence on a number measured elsewhere.

So a rhythm inside a band was a reasonable thing to hypothesise, and there was even a candidate for what it would be folded on — the band spans a factor of 1.28 in the rise and holds an integer number of something at every point. The hypothesis was specific enough to be worth an hour, and it is refused.

A period of 8, and the two classes that are not with the rest. The same wrecked stem, folded on the lag it kept: one row per residue class, each drawn at the mean displacement of its own organs against the level the rest of them share. The bar through each row is the spread inside that class, and the widest of them is 0.87° — so within a class the displacement is a constant. six classes sit at the common level. The two that do not sit at 134.3° and -134.2°, equal and opposite to within 0.0 per cent, and they are neighbouring residues. The stem's own divergence is 137.44°, so an exception is one organ's step.
Fig. 5 A genuine period found in this collection by folding a sequence on a measured number, which is the shape the band was hoped to have.

Which is a stronger negative than it looks

The scoring is rigged in the period’s favour in three ways. It is allowed its best phase. It is allowed to choose a family per residue class after seeing the data. And it is scored on the stretch that contains all the structure, rather than on the whole band where a constant would score above ninety per cent.

Under all three concessions it draws with a constant. A rule with more parameters that cannot beat one with none is not a rule that is nearly right. That is the same standard a rule scored against a coin had to meet here before, and it failed it in the same way: by being an account somebody found plausible rather than one the numbers had suggested.

The readings, and where in their rungs they fail. Each bar is one candidate account of which family a wrecked stem keeps, scored across every wrecked cut in the census. Under each bar are the positions inside their own rungs of the cuts it gets wrong, as percentages from the coarse end. The best of them is right 25 times of 30, and the positions of its failures are the point: two of them are the single lattice grown at the far fine end of its rung, which is also the only census row past three quarters of the way down. Nothing here rescues a reading. What it shows is that the table these readings were scored on varies a quantity nobody chose, over a range nobody stated.
Fig. 6 A different table’s candidate rules scored the same way, with the score for saying nothing drawn alongside.

What it is instead

A transition region. The offset’s answer changes from one family to the other over about thirty-seven rises rather than at a rise, and inside that stretch it is intermittent.

That is a perfectly ordinary thing for a quantity to do near a boundary, and it is not what the nine-rise reading suggested. An alternation implies a rhythm; speckle implies a margin.

And the three offsets that change have three different margins, at three different rises, so the band does not have one transition region with three rows in it. It has three, and they overlap.

What nine rises could see of 126. Above, offset 8's answer at every rise of the band. Below, the same row with only the rises a 9-cut design visits, which is one every 16. The design was built for a quantity expected to be constant and it reports the ends and the crossing correctly; what it cannot report is where inside the band the answer changes, or that it changes back. Every island here is 1 or 1 or 2 or 1 or 2 or 1 or 2 or 1 or 3 or 1 rises wide, against a step of 16, so the sample can only land on one by accident.
Fig. 7 The same offset at both resolutions. Two islands and a gap between them is what a step of sixteen rises turns into 8, 4, 8, 4.

How the sample produced the alternation

By landing on two islands. The nine-rise design visits one rise in sixteen, so across a thirty-seven-rise speckled stretch it visits two or three rises. Thirteen islands across thirty-seven rises means about a third of the rises in the stretch are islands.

Hitting two of them with a run of the other family between is therefore not unlikely at all, and it produces 8, 4, 8, 4 out of no periodicity whatever.

Offset 7 across the 8/13 band, rise by rise. The family this one offset keeps at each of the band's 126 rises, coarse on the left. It wrecks at 126 of them and keeps the 4-family and the 8-family at different rises. The ticks below mark one islands — runs of 2 rises where the coarse family comes back inside the fine one. The handover is the taller rule and the change of answer is nowhere near it.
Fig. 8 A different offset with two isolated rises of the coarse family deep inside its fine stretch, which is the same phenomenon at another place on the band.

The islands are not all the same family

Six of the thirteen are returns of the 8 family into a stretch of 4s, and seven are single rises of the 4 family inside stretches that are still mostly 8. So the speckle is symmetric in the sense that both families intrude on the other, which is what a margin looks like and not what an intermittent single state looks like.

That is a distinction worth keeping. A sequence of 4s with occasional 8s would suggest the 8 state is metastable and being fallen back into; a genuinely mixed region suggests neither state is preferred there and the rise is deciding by a margin too fine for the sweep to see.

Offset 8 across the 8/13 band, rise by rise. The family this one offset keeps at each of the band's 126 rises, coarse on the left. It wrecks at 123 of them and keeps the 4-family and the 8-family at different rises. The ticks below mark ten islands — runs of 1 and 1 and 2 and 1 and 2 and 1 and 2 and 1 and 3 and 1 rises where the coarse family comes back inside the fine one. The handover is the taller rule and the change of answer is nowhere near it.
Fig. 9 Offset 8 again. The islands of each family sit inside stretches of the other, in both directions.

The one structure in the speckle

It is not within an offset. At the rise 0.00541, offsets 7 and 8 both return to the 8 family on the same rise, having crossed to the 4 family forty-eight and twenty-nine rises earlier respectively.

That is one rise doing something rather than two offsets doing it independently. It is reported here and not explained, and it is the only thing in the speckle that looks like structure rather than margin.

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. 10 The band without the handover marked. The column where two rows return to the coarse family together is in the right-hand third.

Whether that is a coincidence

Two offsets sharing one island out of thirteen is not much to go on. Both offsets have a speckled stretch, both stretches overlap, and if islands were placed at random inside them a coincidence at one rise would happen fairly often.

What makes it worth a paragraph is that it is a rise-level event in a picture that is otherwise offset-level: every other feature here belongs to one row. A cheap check would be to cut the same band at more offsets and see whether the coincidence recurs, and the offsets past 9 all recover rather than wreck, so there are none to add — the front is where the census ends and it ends there for a reason.

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. 11 Which offsets wreck at a lattice and which recover. Past the front there is nothing left to cut.

What would count as a period

A gap list with a common divisor, or a score that beat the constant by enough to matter. Neither is here, and it is worth saying what “enough” would be: on 123 rises, a period recovering ten rises more than the constant would be about eight percentage points, which is well outside anything the fitting could manufacture.

Zero is not a marginal failure. It is the outcome that says the quantity has no periodic component at this resolution at all.

A period fitted to the speckle, at every period it could have. Each mark is one candidate period, drawn at the share of rises it gets right when it is given its best phase and its best family in each residue class — the most generous reading of periodic there is. The flat rule is what saying nothing gets: name the commonest family and stop. The best period scores 76 per cent against 76 for no period at all, a gain of 0 points over 123 rises, so the alternation the coarse design reported is not a period being sampled badly.
Fig. 12 The scoring again. Every mark sits on or below the rule, which is what no periodic component looks like.

The resolution qualification

At this sweep’s step. Two parts in a thousand between rises resolves an island of one rise, and a periodic structure at four parts in ten thousand would be invisible here exactly as these islands were invisible to nine rises.

That is not a hedge that can be removed by argument. It is removed by a finer sweep, which would be five hundred rises rather than a hundred and twenty-six, and it has not been run.

The two steps changing places inside the 5/8 band. Measured at every rise of a band on the golden branch, where a counter returns 5 and 8 spirals throughout. The settled divergence moves by 0.0469 degrees across the whole band, which is a fraction of the azimuth grid step and a fortieth of the slide across the rung it sits in. The ratio of the two contact steps does move: it falls to 1.0013 and the ordering changes hands at a rise of 0.0156, so above that rise the shorter step belongs to the 5 family and below it to the 8 family. Two of the three quantities that vary along a rung are therefore held here and the third is not, which is what makes the ends of this band a matched pair.
Fig. 13 A band on its own grid, where the sweep’s step is visible as the limit it is.

A margin means a quantity nobody is measuring

If two families are close to equally favoured across thirty-seven rises, something is nearly equal there, and this thread has a candidate: the two contact steps. The band is grown so that the counted pair holds and the divergence stays flat; what moves is the ratio between the two contact step lengths, and it passes through one at the handover.

But the handover is at the coarse end of the speckled stretch, twenty-nine rises above where offset 8 first switches. So the speckle is not sitting on the crossing, and whatever is nearly equal in it is not the two contact steps. That is a specific negative and it is the one this sweep can make cleanly.

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. 14 How the two contact steps compare across every band. The crossing is a point and the speckled stretch is not near it.

What is being counted

The family a cut stem keeps: the lag whose hop the stem holds unchanged from a control sharing its history. At every rise of this band that lag is either 8 or 4, and never anything else, at every offset that wrecks.

So the speckle is a two-valued signal and the period test is a test on a binary sequence. That is what makes the constant baseline available and the scoring straightforward, and it is worth noting that a three-valued signal would have needed a different null.

The two values are not arbitrary either: 8 is the larger of the band’s counted pair and 4 is half of it, and which of them a cut leaves standing is the census’s oldest question. What the band moves is the answer, and what this essay establishes is that it moves it without a rhythm.

One wrecked stem, lag by lag — golden, rise 0.013, organ 4 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 65 degrees. The lag-5 hop swings by 0.11 degrees and sits 0.09 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 5, which is the surviving lag and not a coincidence.
Fig. 15 The lag spectrum that identifies a surviving hop, which is the measurement behind each cell of the band picture.

What the previous round got right

That it flagged the alternation as a question rather than reporting a period. The sentence was that offset 8 changes answer “by alternating 8, 4, 8, 4 rather than switching once”, which is an accurate description of four sampled rises and stops there.

What it did not do was say that four sampled rises out of sixteen-rise steps cannot distinguish an alternation from two islands. That is the sentence this sweep supplies and it is a sentence about the design rather than about the band.

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. 16 The changes the nine-rise design reported across every band, which is the reading this sweep was built to check.

And what it means for the other bands

The five other bands were also cut at nine rises, and four of them reported no change of answer anywhere. That reading is now weaker than it looked: a band with a speckled stretch narrower than sixteen rises would report nothing at all under this sampling.

The narrow bands are narrow enough that a speckled stretch could not fit; the Lucas 7/11 is 124 rises wide and could easily hide one. Whether it does is two hours of runs and is the obvious extension.

Both bands, on two branches and two pairs. One row per band. Each runs from its coarse end on the left to its fine end on the right, with the rise at which the two contact steps change places marked, and the family that survives every wrecked cut written at the end. The 5/8 band on the golden branch keeps the 5 at all 24 of them and the 4/7 band on the Lucas branch keeps the 4 at all 31. Two branches, two counted pairs, one result: the quantity the band varies is not the quantity that decides the answer.
Fig. 17 Two bands drawn together. A speckled stretch in either would be invisible to the sampling that produced their published readings.

What the finding is worth

Modest and clean. One alternation reported as an open question turns out to be a transition region, the region is speckled, and no period accounts for the speckle.

The transferable part is the scoring rather than the answer. Fitting a period at its best phase and reporting the gain over a constant is four lines of arithmetic and it turns “does this repeat” from a matter of looking at a sequence into a number.

What nine rises could see of 126. Above, offset 6's answer at every rise of the band. Below, the same row with only the rises a 9-cut design visits, which is one every 16. The design was built for a quantity expected to be constant and it reports the ends and the crossing correctly; what it cannot report is where inside the band the answer changes, or that it changes back. Every island here is 2 or 1 rises wide, against a step of 16, so the sample can only land on one by accident.
Fig. 18 A third offset at both resolutions, where the sample sees a clean switch and the sweep sees a switch with an island after it.

What a finer sweep would cost

Five hundred rises at four parts in ten thousand, on this band alone, is about four times what this sweep cost — four and a half hours. What it would buy is the ability to say whether the islands have internal structure or whether they are single rises all the way down.

The prediction, if there is one, is that a finer sweep finds the islands are themselves speckled: a margin has no natural scale, so refining the sampling should reveal more of the same. If instead the islands resolve into clean stretches with sharp edges, then the transition has a structure and the two-parts-in-a-thousand sweep was under-resolving it in the same way nine rises under-resolved this one.

Both outcomes are informative and neither is in this round.

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 0.0537 degrees deep, with its floor at a rise of 0.0201. What it does not do is separate the ends, which still agree to 0.0195 degrees while carrying opposite step orderings. The matched pair the band is for survives the check that would have broken it.
Fig. 19 A band drawn on two grids, which is the comparison a finer sweep of this one would be making against itself.

What carries out of it

A method more than a result. Any claim that a sequence of measurements repeats can be scored the same way: fit every period at its best phase, let it choose its best value per class, and report the gain over the constant. It costs a few lines and it converts an impression into a number.

The impression here was reasonable, the number is zero, and the difference between those two is what the sweep was for.

A period fitted to the speckle, at every period it could have. Each mark is one candidate period, drawn at the share of rises it gets right when it is given its best phase and its best family in each residue class — the most generous reading of periodic there is. The flat rule is what saying nothing gets: name the commonest family and stop. The best period scores 76 per cent against 76 for no period at all, a gain of 0 points over 123 rises, so the alternation the coarse design reported is not a period being sampled badly.
Fig. 20 The scoring, which is the transferable part of this essay.

What the band was built to hold still

Worth restating, because the speckle is easy to read as the band failing. A band is grown outwards from a handover while two things hold: the counted pair, which is held exactly, and the settled divergence, which is held to within five hundredths of a degree. Everything else the rise controls is free to move across it.

So the band is doing its job at every one of the 126 rises. The pair is 8/13 throughout and the divergence moves by less than the width of the azimuth grid. What the speckle shows is that holding those two still is not enough to hold the survivor still, which is the finding the band design already made at nine rises and is here shown at a resolution that says what the failure looks like.

Two lines across the 5/8 rung, crossing once. The divergence the rule settles on, against the divergence at which the two contact steps would be exactly the same length. The second is arithmetic on the lattice and no stem is grown for it. Across this rung the balanced line moves 2.281 degrees and the rule's own line moves 1.262, so the shallower line crosses the steeper one, and it does so exactly once at a rise of 0.0154 — 16 per cent of the way down from the coarse end. That crossing is the handover: above it one family has the shorter step and below it the other does. So a rung has one handover, its position is fixed by the arithmetic rather than by any experiment, and a sweep of the rise carries a stem across it at a place nobody chose.
Fig. 21 The two quantities a band holds still, drawn across one. Neither of them moves, and the survivor does.

The one line

The 8, 4, 8, 4 that nine sampled rises found at one offset is, at every rise, a transition region thirty-seven rises long containing thirteen islands one to three rises wide, with gaps of 1, 2, 3, 6, 7, 8, 9, 16, 31, 44 and 48.

Every candidate period fitted at its best phase and allowed its best family per class scores 75.6 per cent, which is exactly what naming the commonest family and stopping scores.

Where this goes next

Every rise of the 8/13 band, cut at every offset. One column per rise of the band, coarse on the left and fine on the right, and one row per offset that wrecks anywhere on it. A filled cell is the family the cut stem keeps; a pale cell is an offset that recovers at that rise and has no survivor to report. The band holds 126 rises and 1890 cut stems. three of the six offsets change their answer somewhere inside, three never do, and the vertical rule is the handover — the rise where the two contact steps change places. Not one of the 19 changes is at it. Every rise of a band A band is cut at nine rises because the quantity it was built to test is a constant, and a constant is checked at the ends and at the crossing. On the widest band that quantity turned out not to be constant, which makes nine the wrong number. This is all hundred and twenty-six. The families left standing, on both sides of every handover. One row per band, with every offset cut at rises spread across it and always at both ends and at the handover itself. The last column is every family left standing anywhere on that band. On three of the four bands that wreck at all it is a single family, unchanged across a rise at which the two contact steps swap places — so the step ordering is not what decides which family survives, and the result now rests on four counted pairs rather than on two. The shortest-hop reading scores 44 of 114 across the whole set, which is what a reading looks like when the quantity it is stated over is not in the mechanism. The ordering on six bands A hundred and fourteen wrecked cuts across four bands, and at every offset of every one of them the family left standing is the same immediately above the handover and immediately below it. Where the answer does change — on the widest band, at three offsets — it changes somewhere else. Offset 7 across the 8/13 band, rise by rise. The family this one offset keeps at each of the band's 126 rises, coarse on the left. It wrecks at 126 of them and keeps the 4-family and the 8-family at different rises. The ticks below mark one islands — runs of 2 rises where the coarse family comes back inside the fine one. The handover is the taller rule and the change of answer is nowhere near it. Three offsets, three crossings The claim the band design rests on is that the survivor does not change where the two contact steps change places. It holds at full resolution: nineteen changes and not one at the handover. Where they are is three different rises, eight, nineteen and twenty-nine below it.

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.

  • A window nobody aligned — both name ablation, artefact, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, sampling
  • One offset, two answers — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, rigid hop, rise, rung
  • One way round, seventeen times — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, null model, resolution, rigid hop
  • Six lattices were not enough — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, rise, rung, sampling
  • The exception was already labelled — both name ablation, claim testing, control, handover, honest limits, lattice offset, measurement, negative result, rise, rung
  • The front deepens down a rung — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, rigid hop, rise, rung

Named objects

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

AblationArtefactClaim testingControlFittingHandoverHonest limitsLattice offsetMeasurementNegative resultNull modelResolutionRigid hopRiseRungSampling