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

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.
12 min read 21 figures State the claim, state the testCounted, not admired

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

A wrecked stem’s displacement profile is periodic at the lag the stem kept, and it takes a while to become so. Both readings are taken over a run of three hundred organs above the hole: the levels from the last hundred and twenty of them, the onset from the first lag at which every class settles onto its level and stays there.

Twenty-five of the thirty cuts reached an onset. The range was 7 to 303 organs — and 303 out of a possible 300 is a number that ought to be checked rather than quoted. A reading that lands at the edge of its own instrument is either a coincidence or the instrument’s edge showing through, and there is no way to tell which without moving the edge.

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. 1 Where the pattern starts on every wrecked cut, measured over three hundred organs and over six hundred.

The check

Grow every one of the thirty cuts, and its control, to six hundred organs instead of three hundred, and take the same two readings. Sixty runs, about two minutes, and the only change is the length.

The result is that all three answers move: how many cuts settle, where they settle, and which profiles are called periodic at all.

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. 2 One cut’s whole profile over six hundred organs, with the onset each run length reports marked on it.

The instrument first

Everything below is a difference between two lengths, so the first thing to prove is that the length is the only difference. The profile computed here at three hundred organs agrees with the one the thread has been reading, organ by organ, on every row compared: a largest difference of zero degrees.

That is asserted rather than assumed. A second implementation would have put the difference into the implementation, and this comparison would have measured two pieces of code.

The reason a second implementation was needed at all is that the reading the thread uses is written with the run length fixed. Opening it up meant writing the profile again with the length as an argument, and a rewritten computation is exactly where a silent divergence gets in — the same hazard a stem grown at the wrong starting angle demonstrated in this thread last time.

One rule at p = 1, cut three ways. loop cut at 3/√h: 8/13 at 137.62° with 0.58° of scatter. exponential cut-off, 3: 8/13 at 137.58° with 0.50° of scatter. no cut at all: no lattice, 44° of scatter. The first two agree to 0.03° — the prediction held — and the third is what the same rule does when nothing cuts it.
Fig. 3 The standing check that a continued run is the placement rule itself rather than a second version of it.

The number of cuts that settle goes down

Twenty-five of thirty reach an onset over three hundred organs. Over six hundred, twenty-four do.

That is the wrong direction for the obvious expectation. More room should let a slow profile finish settling, and one row did the opposite: it reported an onset over the short run and reports none at all over the long one.

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. 4 The onset on every row as the thread has been reporting it, over three hundred organs.

And no row gained one

Not one of the five that failed to settle over three hundred organs settles over six hundred. Two of them stop being ragged — their classes come much closer together — but neither reaches the point where every class stays on its level for the rest of the run.

So doubling the run did not rescue a single slow case. Whatever the five are doing, it is not settling slowly.

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. 5 How steady each class is at both lengths. The five that never settle sit far to the right at both.

Where the onsets move to

On the nineteen rows of twenty-four that report an onset at both lengths and move by more than twenty organs, they move a long way. The 0.016 stem’s cut four organs back reports 135 organs at three hundred and 435 at six hundred. The 0.010 stem’s three cuts report 18, 0 and 1 at three hundred and 128 at six hundred, all three.

The range the thread has been quoting — 7 to 303 organs — becomes 28 to 473.

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. 6 The same shifts in census order rather than sorted by size, so the rows that move together can be located.

Why an onset moves at all

Because it is measured against the levels the classes hold at the end of the run. The definition is the first lag from which every class stays within a few degrees of its own tail mean for the rest of the run, and the tail is wherever the run stops.

A profile that drifts slowly is compared against one set of levels at three hundred organs and a slightly different set at six hundred. Organs that were within a few degrees of the short run’s levels are not within a few degrees of the long run’s, and the onset moves up to find a place where they are.

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. 7 A profile’s first hundred and twenty organs. Whether an organ here counts as settled depends on levels measured three hundred organs later.

Which makes the onset a property of two things

The cut and the run length. That is not a fault in the measurement so much as a fact about what it is: “the first lag from which nothing changes again” is only defined relative to a horizon, and the horizon here is the end of a run somebody chose.

Every quantity of that shape has the same problem. The useful response is not to find a horizon-free definition — there may not be one — but to report the horizon beside the number, which the thread has not been doing.

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. 8 The onsets with their transients, the reading that was quoted for five rounds without the run length beside it.

The row at three hundred and three

One row reported an onset of 299 organs into a 300-organ run, at a lag of 303 from the hole. Over six hundred organs it reports no onset at all.

That is the cleanest result of the sixty runs. An onset at the last organ of a run is not a measurement of when a pattern started; it is a measurement of when the run stopped, because the condition “stays on its level for the rest of the run” is trivially satisfied by the last organ. It has an essay of its own.

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. 9 The shifts again. One row has a mark at the short length and none at the long one, which is the row this section is about.

The periodicity reading moves too

Whether a profile counts as periodic is decided by the widest spread inside any one class, against a line at ten degrees. Twenty-five rows are inside it at three hundred organs; twenty-six are at six hundred.

The count barely moves and the membership does. Three rows cross the line, two in one direction and one in the other, and that is the second essay.

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. 10 The three rows the two run lengths disagree about, drawn at their spreads at each length.

The gap narrows

The line at ten degrees was chosen because the two populations were separated by one: periodic rows measured 0.12° to 6.09° and the rest measured 10.3° and up, so the line sat in a gap of a factor of 1.69.

At six hundred organs the gap is a factor of 1.27. It is still a gap and it is narrower, which means the threshold is doing more work than it was. A tolerance chosen because it sat in empty space is worth re-checking whenever the space fills.

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. 11 The census by spread, which is where the gap the ten-degree line sits in can be read.

A definition that would not move

There is one, and it is worth writing down even though it has not been used. Define the onset against the levels measured over a fixed window — the organs between the two hundredth and the three hundred and twentieth, say — rather than against the tail of whatever run was grown. Then the levels are the same at both lengths by construction and the onset is a property of the cut alone.

What that definition loses is the thing the current one was for: it can no longer say that nothing changes again, only that nothing changed inside a chosen stretch. The two definitions answer different questions and the thread has been using one while quoting the other.

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 285 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. 12 A profile at both lengths. A fixed window would take its levels from the same organs whichever run it sits in.

The refusal that goes with the reading

A run shorter than the window has no window to read, and the machinery refuses it rather than returning a spread measured over five readings a class. At a surviving lag of eight, a hundred and twenty organs is fifteen samples a class; at forty organs it would be five, and five readings are steady whatever the profile does.

That refusal is the same one the periodicity reading already carried, applied to the run length rather than to the window. It is the check that stops the doubling comparison from being run in reverse against something too short to answer.

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. 13 The census by stem, with its surviving lags. The window has to hold several repeats of each of these for a class mean to be a mean.

What is not affected

The shape. The claim that a wrecked stem’s profile is levels rather than scatter survives the doubling on twenty-six rows of thirty, and the levels themselves are where they were: the exceptional pair, its size and its direction are unchanged on every row that carries one at both lengths.

So the thread’s central result is not run-length dependent. What is run-length dependent is every reading about when — the onset, the transient’s size, and the classification of the marginal rows.

That split is the useful thing to carry out of this. The exchange and its direction are readings about what, and they are stable. The readings that move are the ones that ask how long something took, and those have a horizon in their definition whether or not it is written down.

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. 14 The levels themselves, which are the same at both lengths on every row that has them at both.

Why six hundred and not more

Because doubling is the shortest length that can distinguish a reading about the cut from a reading about the run, and because the thread’s other two-length comparison — the settling test — uses the same trick for the same reason.

A longer run would say more and cost more. If the onsets move again between six hundred and twelve hundred, then no length is a horizon and the quantity has to be redefined rather than re-measured. That is the next test and it has not been run.

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. 15 The settling thread’s own two-length comparison, which is the design this one copies.

The cost, and why it was not paid earlier

Two minutes. Sixty runs at six hundred organs, on a machine that has spent hours on this census already.

The reason it was not paid is not cost. It is that “300 organs” had been a constant of the thread since it began, and a constant stops being visible: the readings were quoted with a range and without the length that produced the range, and nothing in the record flagged the 303 as the number it obviously is.

How far every organ moved, 7 places back at a rise of 0.005. 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 55 organs it settles into a repeating pattern of eight levels, one per residue class modulo 8, which is the lag whose hop this stem kept. two of those levels sit together and six do not.
Fig. 16 Another profile, on a fine stem. The window every reading is taken in is a choice that had not been varied.

Two minutes against five rounds

The comparison is sixty runs and takes about as long as reading this paragraph twice. The readings it corrects have been quoted in five separate essays across the thread’s whole history.

That ratio is the recurring shape of this collection’s leavings, and it is worth naming rather than apologising for. The expensive thing is never the run; it is noticing that a constant is a parameter. Three hundred organs had been in the code since the census was built, it appeared in no essay’s prose, and nothing in the record asked what would happen if it were four hundred.

What a cut moves, organ by organ. A stem counted at 5 and 8 spirals with the organ five places back from the tip removed, compared against a control that shares its history to the last digit. Each mark is one organ placed after the cut and how far its azimuth ended up from where the control put the same organ. The quantity folds at half a turn, so 152 degrees is near the largest displacement there is; and it does not decay with height, which is what a stem that never repairs means. There is therefore no organ that the cut disturbed most in any useful sense, and a reading that needs one has nowhere to stand.
Fig. 17 The profile as this thread first drew it, over sixty organs, at a time when the run length was not a quantity anybody was reading.

What a reader should carry

That the periodicity is robust and the timing is not. If an essay in this thread says a profile becomes a pattern after so many organs, the number is about a three-hundred-organ run, and doubling the run moves it by up to three hundred organs.

And that the direction of the surprise matters: fewer cuts settle over a longer run, not more. A reading that gets rarer when the instrument improves was probably being manufactured by the instrument — which is the same argument the reference organ failed on, one thread over, when a maximum turned out to be a plateau.

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 in one picture: small marks at three hundred organs, rings at six hundred, and lines joining what moved.

What it does to one row of another thread

The exchange thread scores four accounts of the exchanged pair’s size on seventeen rows and fits worst on the Lucas 0.013 stem’s cut four organs back. That row is one of the three this comparison moves, and it moves out of the periodic group.

Neither measurement knew about the other. One says a row fits a rule badly; the other says the same row’s reading was taken over a window still inside a transient. Agreement between instruments that were not looking for each other is worth more than either.

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. 19 The rows the two lengths disagree about, one of which an entirely separate scoring independently calls its worst fit.

The five that never settle

They are the golden 0.026 stem’s cut four organs back, the golden 0.010 stem’s cuts four and five organs back, and the Lucas 0.020 stem’s cuts four and five. Four of the five are cuts four or five organs back, which is the shallowest part of the census.

Whether that is a pattern or a coincidence is not decidable on five rows. What can be said is that they are not slow settlers: doubling the run does not settle them, and two of them become much steadier without ever reaching the condition. Something else is happening on those runs and this comparison does not say what.

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. 20 Recovery by offset at one of the lattices involved. The shallow offsets are where four of the five unsettled rows sit.

What the census would need

Twelve hundred organs on those five, which is another two minutes, and it would separate two accounts: a transient with a very long tail, and a profile that is genuinely not periodic at any length. The first would eventually settle and the second never will.

That is the obvious next run and it is not in this comparison, because the comparison was designed to check a number — the 303 — rather than to characterise a group. Designing to check one number and then finding a group is the ordinary way a sweep outgrows its own question.

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. 21 The settling thread’s version of the same distinction: a budget that eventually runs out against a wall that never does.

The one line

Thirty wrecked cuts, run twice as far: the shape survives, the timing does not. Nineteen of twenty-four onsets move by more than twenty organs and one moves by three hundred, the range 7 to 303 becomes 28 to 473, one row loses its onset entirely, and no row gains one.

The profile computed here at the old length reproduces the thread’s own to the last digit, so the only thing that changed is how far the runs were allowed to go.

Where this goes next

How far every organ moved, 6 places back at a rise of 0.01. 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 18 organs it settles into a repeating pattern of eight levels, one per residue class modulo 8, which is the lag whose hop this stem kept. six of those levels sit together and two do not. The damage has a period Every wrecked stem in the census has had two numbers read out of its displacement profile and the profile itself read out of none of them. Folded on the lag the stem kept, twenty-five of the thirty are constant inside each residue class to between 0.12° and 6.09°. 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. 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. 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.

What links here

Computed from the collection, not written here: the essays that point at this one.

Reads more easily once this is understood

Essays that name this one as worth reading first.

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, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, sampling, selection effect
  • The plateau was a prediction — both name ablation, artefact, claim testing, control, honest limits, measurement, negative result, rigid hop, selection effect, summary statistic, transient
  • The alternation is not a period — both name ablation, artefact, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, sampling
  • A list that was a rounding — both name artefact, census, claim testing, honest limits, measurement, negative result, resolution, selection effect, tolerance
  • Three offsets, three crossings — both name ablation, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, sampling
  • A period the grid invented — both name ablation, artefact, claim testing, honest limits, measurement, negative result, resolution, summary statistic

Named objects

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

AblationArtefactCensusClaim testingControlHonest limitsMeasurementNegative resultReproducibilityResolutionRigid hopSamplingSelection effectSummary statisticToleranceTransient