Phyllotaxis
the spirals are counted, not admired
What a plant might be doing

An onset at the end of the run

One cut reported that its pattern began 299 organs into a 300-organ run. Given twice the room it reports no pattern at all. The reading was the run stopping, not the disturbance ending, and the definition guarantees one at the last organ of every run.
12 min read 21 figures State the claim, state the testForm is not mechanism

Worth reading first: Twice the run · The damage has a period · The organ that was taken away.

The onset is defined as the first lag from which every residue class stays within a few degrees of its own level for the rest of the run. It is the boundary between the disturbance healing and the arrangement it healed into, and it is what separates the two regimes a wrecked stem has.

Read that definition once more with the end of the run in mind. At the last organ, “for the rest of the run” is a condition on nothing. It is satisfied by every profile, always.

How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.
Fig. 1 The onset on every wrecked cut as the census reported it, over three hundred organs.

The row that took the offer

The golden 0.010 stem’s cut four organs back reported an onset of 299 organs into a 300-organ run — a lag of 303 from the removed organ, on a run that reaches 304.

Grown to six hundred organs the same cut reports no onset at all. Nothing about the cut changed; the run got longer and the condition stopped being free.

Where the pattern starts, measured at two run lengths. One row per wrecked cut. The small mark is the onset a 300-organ run reports and the ring is what a 600-organ run reports; a row with only one mark reports an onset at only one length. 19 of the 24 rows that report both move by more than twenty organs, and the largest move is from 135 to 435. The range the thread has been quoting, 7 to 303 organs, becomes 28 to 473.
Fig. 2 Every cut’s onset at both lengths. One row has a mark at the short length and nothing at the long one.

Why it is not a near miss

A row that settles at organ 290 of 300 and at organ 295 of 600 would be a slow settler measured twice. This is not that. Over six hundred organs the row’s classes never all stay on their levels for the rest of the run, at any starting point.

So the short run did not measure a late onset. It measured the absence of one, and reported it as a number because the definition has no way to say “never” until the run ends. The distinction between a late answer and no answer is the one thing the reading could not make, and it is the distinction the row needed.

One cut's profile over 600 organs, with both onsets marked. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control sharing its history. The two vertical rules are where the pattern is said to start: the left one is what a 300-organ run reports and the right one is what this 600-organ run reports. They are not both present. The onset is measured against the levels the classes hold at the end of the run, so a profile that drifts slowly is compared against different levels at each length and the reading moves with the length.
Fig. 3 The same cut’s profile over six hundred organs. Nothing in it settles, and the short run’s reading was taken at the point where it was cut off.

The shape of the trap

Any reading of the form the first index from which a condition holds for the rest of the sequence is trivially true at the last index. So such a reading always returns a value, and the value at the end of the sequence carries no information at all.

That is not a bug in this particular definition. It is a property of the form, and the form is a natural one: it is how anybody would write down “the point after which nothing changes”. It is also the form of every convergence test written in a hurry, and the reason numerical work carries a maximum-iterations count beside its tolerance.

How far every organ moved, 4 places back at a rise of 0.013. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control that shares its history. The collection has read two numbers out of profiles like this one — the largest displacement anywhere, and the first organ's — and never the profile. It is not a bump that decays. After about 3 organs it settles into a repeating pattern of five levels, one per residue class modulo 5, which is the lag whose hop this stem kept. three of those levels sit together and two do not.
Fig. 4 A profile whose onset is early. Reading from the left, the condition is genuine; reading at the far right it is not.

What the fix is

Refuse a reading at the end of the run. If the first index that satisfies the condition is the last one, or within a few of it, report no onset rather than that index.

That is a one-line change and it is now in the machinery, and it is worth saying that one line would have removed the 303 from every essay that quoted it. The value “303” was never a measurement.

How close to the end is close enough is a tolerance, and it is chosen the way the others in this thread are: the onsets over three hundred organs run 0, 1, 3, 14, 18 and up, so a guard at two organs from the end catches exactly the one row and touches nothing else. It sits in a gap rather than at a chosen point.

Where the pattern starts, measured at two run lengths. One row per wrecked cut. The small mark is the onset a 300-organ run reports and the ring is what a 600-organ run reports; a row with only one mark reports an onset at only one length. 19 of the 24 rows that report both move by more than twenty organs, and the largest move is from 135 to 435. The range the thread has been quoting, 7 to 303 organs, becomes 28 to 473.
Fig. 5 The onsets sorted by how far they moved between the two lengths. The row with only one mark is the one the refusal now catches.

How many essays quoted it

The range “7 to 303 organs above the hole” appears wherever this thread describes where a wrecked stem’s pattern begins, and the 303 is the top of it. Every use of that range was reporting an instrument’s edge as a measurement.

The corrected range over three hundred organs, with the refusal in place, ends at the next-highest onset. Over six hundred organs it is 28 to 473, and the top of that is not at a boundary.

How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.
Fig. 6 The onsets with the transients beneath them. The top of the range was the one value that could not be trusted.

What made it survive

It looked right. Three hundred and three is not a round number, it is not the run length, and it sits at the top of a range whose bottom is seven — a spread of a factor of forty, which is the kind of spread a real quantity has.

Had the reading come back as exactly 300, or as the run length in every row, somebody would have looked. A boundary artefact that lands one or two organs off the boundary is much harder to see than one that lands on it.

How constant the displacement is inside one residue class. One row per wrecked cut in the census, drawn at the widest spread found inside any one residue class when the profile is folded on the lag that stem kept. 25 of 30 rows sit between 0.12 and 6.09 degrees, which on a quantity whose between-class differences run past a hundred and fifty degrees is a constant. The five that do not sit from 10.3° up. There is nothing in between, so the line drawn at 10° could have been drawn anywhere in a wide interval.
Fig. 7 The census by stem. Nothing about the row that produced the 303 marks it out as unusual in any other column.

And it had a plausible story attached

A cut whose pattern takes three hundred organs to establish itself is a cut that barely establishes one, and this thread has a category for that: the five that never settle. A row at 303 reads as the sixth, the marginal case, the one just inside the boundary.

That story is right about the row and wrong about the number. The row is a non-settler; the 303 was not evidence of it and would have been reported whatever the row was doing.

How steady each class is, over 300 organs and over 600. Each mark is one wrecked cut, placed across at the widest spread found inside any one of its residue classes over the shorter run and up at the same reading over the longer one. A mark on the diagonal is a row the two lengths agree about. The rules are the 10 degrees that separates a profile called periodic from one that is not: three rows fall in different quadrants at the two lengths, two of them becoming periodic and one ceasing to be. The gap between the two groups narrows from 1.69 times to 1.27.
Fig. 8 How steady each row’s classes are at both lengths. The row in question is far from the settled group at both.

Which is the awkward part

A wrong number that supports a right conclusion is the hardest kind to find, because nothing downstream misbehaves. The essay that used the 303 was making a correct point — that the disturbance’s end is not a fixed distance above the hole — and the number it used to make it was an artefact.

This collection’s usual defence is that a claim should have a test it could fail, and that defence did not apply here. There was no claim about the 303; it was a descriptive maximum, and a number reported without a test attached to it is exactly the kind that survives being wrong.

Two readings of what a stem is, and one of them is wrong. The same runs read twice. On the left is the pair counted from the point positions by machinery that is never shown a divergence angle; on the right is the pair read from the divergence sequence, which is the column the depth thread has been quoting. Every exponent's stems are counted at 5/8 spirals from the points. Read from the angles, the deepest rule's stems come back as something else at every seed, and a pair with a 8 replaced in it is not a near miss but a family whose step is several times as long. The settled divergence moves by 0.118 degrees across all five exponents, so the rules are on one lattice and the disagreement is a defect in one of the two instruments.
Fig. 9 Two readings of one quantity, drawn together. Descriptive maxima are the readings least likely to carry a test.

A near relative, in a different thread

The same shape has appeared here before under a different name. The search for a reference organ asked which organ above a hole moved furthest, inside a window three times the larger counted number wide. It found a plateau rather than a peak, and a maximum over a plateau names an arbitrary organ.

That reading fails because its answer is under-determined; this one fails because its answer is over-determined. Both are the same kind of defect — a statistic whose value is produced by the shape of the window rather than by the data in it — and the collection has now met it from both directions.

Three references, scored on the same thirty cuts. The reading under test says the family that lost a member is the one left standing, and it needs a reference organ to say which family lost one. Taken from the growing tip it can be asked on 9 of the 30 wrecked cuts and is right on every one. Taken from the organ the cut disturbed most it can be asked on 8, a different set, and is right on 3 of them. Taken as which side of the tip the removed organ sat on — arithmetic on the divergence, needing no reference organ at all — it can be asked on all 30 and is right on 22, against 18 for naming the commoner family outright.
Fig. 10 The reference organ’s diagnostics, which is the other reading in this thread that a window rather than a stem was deciding.

What a window is for

A window exists because a run is finite and a claim about a pattern has to be made somewhere. Choosing one is unavoidable; the question is whether the choice is visible in the answer.

There are two tests for that and this thread has now applied both. Vary the window and see whether the answer moves — which is what running every cut twice as far does — and check whether the answer can be produced by the window with no data at all, which is what this essay is about. The second test is cheaper and it is the one nobody ran.

The order follows the window, so it was never the fractions'. The four fractions with a denominator of 34, ordered four ways. The left column puts them in order of how close the nearest other rational is — the crowding — with the most crowded at the top. The other three order them by the area of their dip, at windows of 50, 100, 200 scaled units. The residual claim this thread carried was that the most crowded fraction gives the widest dip, which would make all four columns the same order. They are not: the order changes between the first two windows and settles, from a window of 100 outwards, into 11/34 > 15/34 > 13/34 > 9/34 — which is not the crowding order either. A quantity that reverses when the measurement is stopped somewhere else is a property of the stopping.
Fig. 11 How a reading changes with the window it is taken in, which is the first of the two tests.

Where else the form occurs

Three other readings in this collection have it. The settling time is the first organ from which a run’s divergence stays within a tolerance of its final value for the rest of the run. The recovery lag is the first organ from which a cut stem’s divergences stay within a tolerance of the control’s. The block’s period is measured over a tail rather than by this form, so it is not one.

The first two are the same construction with different quantities in it, and both report a value at the last organ of any run whose condition never holds.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 12 The settling times over a fixed run length, which is a reading with exactly this shape and its own boundary.

Whether they have the same problem

The settling reading does not, and for a reason worth recording: it checks the tail’s spread before it looks for a first index, and refuses to report a time at all unless the run has genuinely settled. So the trivially-satisfied index is never reached on a run that has not settled.

The recovery reading has no such guard. It has not been audited here and it should be, and the audit is a grep rather than a sweep. Its answers feed the census of which offsets wreck, so if it has the same defect the consequences reach further than one number in one range.

Both edges of the front heal; the middle of it does not. The same removals, followed for 300 organs each. A cut one to three places back is undone within fifty organs and a cut ten to thirteen places back within sixty. A cut in between is never undone: the divergence sequence settles into an exactly repeating cycle of 5 angles and holds it for the rest of the run. The rule corrects a displacement and cannot correct a deletion.
Fig. 13 The recovery lag by offset, which is the other reading in this thread with the same construction and no guard in front of it.

The guard is the general fix

Ask the condition of the tail first. If the tail does not satisfy it, there is no first index to look for and the answer is “never” rather than “the last one”. Only if the tail satisfies it is a search backwards for the earliest index meaningful.

That ordering is what the settling reading does and what the onset reading did not, and the difference between the two is one conditional. Both were written in this thread, months apart, by the same reasoning.

The same table, grown 2.7 times as long. Every rise and every starting angle, grown to 1200 organs and then to 3200. The two middle columns are how many starting angles reached a lattice at each length, and they are the same column: of the 72 pairs of runs, 72 are identical organ for organ and 0 settle at the longer length after failing at the shorter one. Tripling the budget buys nothing anywhere. What the fine rises are short of is not run length: the share of starting angles that reach a lattice at all falls from 7 of 9 to 1, so the arrangements a stem could fall into have mostly stopped existing.
Fig. 14 The settling thread’s distinction between a run that ends and one that never does, which is what a guard in front of a first-index search preserves.

What the row actually is

The golden 0.010 stem’s cut four organs back is a wrecked stem whose profile never becomes periodic within six hundred organs. Its widest within-class spread is 7.9° at six hundred organs and 69.6° at three hundred — it crosses the periodicity line in the other direction — so it is a row that gets steadier without ever settling.

That is a more interesting thing than a late onset and it was hidden behind the artefact. A profile whose classes converge without ever holding still is not a transient and it is not a pattern.

The 3 cuts the two run lengths disagree about. Each block is one wrecked cut, with its widest within-class spread drawn at both run lengths and the 10 degrees that separates periodic from not marked by the rule. Two of these become periodic when the run is doubled, at spreads falling from about seventy degrees to about eight. One goes the other way, from six degrees to a hundred and seventy — and that one is the row an entirely separate reading of the same census independently reports as its worst fit.
Fig. 15 The rows the two lengths disagree about. This one becomes periodic by the spread reading and never settles by the onset reading.

Two readings that disagree about one row

That is worth pausing on. Over six hundred organs the spread reading calls this row periodic — its classes sit within eight degrees of their own means — and the onset reading says nothing ever settles.

Both are correct and they measure different things. The spread asks whether the classes are tight over the last hundred and twenty organs; the onset asks whether every organ from some point on is close to its class’s level. A profile that drifts slowly satisfies the first and fails the second forever.

A period of 5, 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 10.28° — so within a class the displacement is a constant. three classes sit at the common level. The two that do not sit at 143.0° and -147.6°, equal and opposite to within 3.2 per cent, and they are neighbouring residues. The stem's own divergence is 137.97°, so an exception is one organ's step.
Fig. 16 A row’s levels. Whether they are tight and whether every organ sits on one are two different questions with two different answers.

Which means the thread has two definitions of the same word

“Periodic” has meant both things at different points, and nothing marked the switch. The census’s headline count — twenty-five of thirty — is the spread reading. The onset’s count — twenty-five of thirty at three hundred organs, twenty-four at six hundred — is the other one, and the two agree at the shorter length by coincidence.

Separating them is the first thing a next round should do, and it is a naming problem rather than a measurement.

How constant the displacement is inside one residue class. One row per wrecked cut in the census, drawn at the widest spread found inside any one residue class when the profile is folded on the lag that stem kept. 25 of 30 rows sit between 0.12 and 6.09 degrees, which on a quantity whose between-class differences run past a hundred and fifty degrees is a constant. The five that do not sit from 10.3° up. There is nothing in between, so the line drawn at 10° could have been drawn anywhere in a wide interval.
Fig. 17 The census by spread, which is one of the two readings the word has been used for.

The cost of the artefact

One number in one range in several essays, corrected. No conclusion in the thread depends on it: the point it was used to support is supported by the rows either side of it, and the corrected range is wider rather than narrower.

That is a mild outcome and it is not the reason the artefact is worth an essay. The reason is the form, which is general, and the guard, which is one conditional and now appears in two places rather than one.

Where the pattern starts, measured at two run lengths. One row per wrecked cut. The small mark is the onset a 300-organ run reports and the ring is what a 600-organ run reports; a row with only one mark reports an onset at only one length. 19 of the 24 rows that report both move by more than twenty organs, and the largest move is from 135 to 435. The range the thread has been quoting, 7 to 303 organs, becomes 28 to 473.
Fig. 18 The whole comparison. Nineteen rows moved by more than twenty organs and one row’s reading disappeared entirely.

The count that did not change

It is worth being precise about what the correction costs the census, because the answer is nearly nothing. Twenty-five rows reported an onset over three hundred organs; with the guard in place, twenty-four do, and the twenty-fourth is the row this essay is about.

So the headline — most wrecked stems reach a pattern, a few do not — survives intact, and it survives at both run lengths. The artefact changed a maximum and a count by one, and it changed neither of them in a direction that mattered to any claim.

How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.
Fig. 19 The onsets with the guard applied. The distribution is the same distribution with one row removed from its top.

What a reader should take

That “the first point from which nothing changes again” is not a measurable quantity on a finite run without a guard in front of it, and that a reading which lands at the edge of an instrument should be checked by moving the edge before it is quoted.

Moving the edge cost two minutes here. Quoting the number cost five rounds — and the same arithmetic held for the window the exchange is read in, where the correction was likewise available from the day the reading was defined.

One cut's profile over 600 organs, with both onsets marked. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control sharing its history. The two vertical rules are where the pattern is said to start: the left one is what a 300-organ run reports and the right one is what this 600-organ run reports. They are 300 organs apart. The onset is measured against the levels the classes hold at the end of the run, so a profile that drifts slowly is compared against different levels at each length and the reading moves with the length.
Fig. 20 One profile at both lengths with both readings marked, which is what moving the edge looks like.

What it changes about the thread’s vocabulary

Two words are now doing double duty and both should be split. “Periodic” means either that a profile’s classes are tight over a window or that every organ from some point on sits on its class’s level, and those come apart on at least one row. “Onset” means either the first index satisfying the second condition or, with the guard, the same thing or nothing.

Neither split is a measurement. Both are naming, and naming is where this thread’s next hour is best spent — because a claim carried by a word that means two things is a claim nobody can check.

How steady each class is, over 300 organs and over 600. Each mark is one wrecked cut, placed across at the widest spread found inside any one of its residue classes over the shorter run and up at the same reading over the longer one. A mark on the diagonal is a row the two lengths agree about. The rules are the 10 degrees that separates a profile called periodic from one that is not: three rows fall in different quadrants at the two lengths, two of them becoming periodic and one ceasing to be. The gap between the two groups narrows from 1.69 times to 1.27.
Fig. 21 The spread reading at both lengths, which is one of the two things the word has meant.

The one line

A reading of the form the first index from which a condition holds for the rest of the run is satisfied trivially at the last index, so it returns a number on every run whether or not the condition ever holds. One row of thirty took the offer and reported 303 organs; over twice the run it reports nothing.

The guard is to ask the condition of the tail before searching for a first index — which the settling reading in this collection already does, and this one did not.

Where this goes next

Where the pattern starts, measured at two run lengths. One row per wrecked cut. The small mark is the onset a 300-organ run reports and the ring is what a 600-organ run reports; a row with only one mark reports an onset at only one length. 19 of the 24 rows that report both move by more than twenty organs, and the largest move is from 135 to 435. The range the thread has been quoting, 7 to 303 organs, becomes 28 to 473. Twice the run Five wrecked cuts never reached a pattern inside three hundred organs and one reached it at three hundred and three, which is a number asking to be checked. Run every cut in the census twice as far and three of the thirty change their answer. How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again. Two regimes above a hole Below the repeating pattern there is a transient, and the boundary between them is measurable: the first organ from which every class stays at its own level runs from 7 to 303 organs above the hole on twenty-five of thirty cuts, and five never reach it inside the run. The 3 cuts the two run lengths disagree about. Each block is one wrecked cut, with its widest within-class spread drawn at both run lengths and the 10 degrees that separates periodic from not marked by the rule. Two of these become periodic when the run is doubled, at spreads falling from about seventy degrees to about eight. One goes the other way, from six degrees to a hundred and seventy — and that one is the row an entirely separate reading of the same census independently reports as its worst fit. Three rows change sides Twenty-five of thirty wrecked cuts have a periodic displacement profile over three hundred organs and twenty-six do over six hundred. The count barely moves and the membership does: two rows join, one leaves, and the gap the threshold sits in narrows from 1.69 to 1.27.

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, census, claim testing, control, honest limits, measurement, negative result, resolution, selection effect, summary statistic, tolerance, transient
  • The plateau was a prediction — both name ablation, artefact, claim testing, control, honest limits, measurement, negative result, selection effect, summary statistic, transient
  • Which chains changed places — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, summary statistic, transient
  • A fifth of the hop — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, summary statistic
  • A list that was a rounding — both name artefact, census, claim testing, honest limits, measurement, negative result, resolution, selection effect, tolerance
  • The offsets that never change — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, selection effect

Named objects

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

AblationArtefactCensusClaim testingControlHonest limitsMeasurementNegative resultReproducibilityResolutionRim effectSelection effectSummary statisticToleranceTransient