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

A window nobody aligned

Every reading this thread takes of a wrecked stem is taken inside a window, and there are three of them: a run of three hundred organs, a window of a hundred and twenty at its top, and a search window of fifteen to thirty-nine. None was aligned to anything, and one of them turned out to decide its own answers.
12 min read 20 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.

There are three windows in this thread and no essay has listed them together. A run of three hundred organs above the hole. A reading window of a hundred and twenty at the top of it, over which the levels are measured. And a search window of three times the larger counted number — fifteen to thirty-nine organs — inside which the most disturbed organ was once looked for.

Every number the census reports comes through at least one of them. This essay is the audit, and the reason for making it is that two of the three have now been shown to decide their own answers — one of them only after five rounds of quoting a number it had manufactured.

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. 1 The census read at two run lengths. Three rows fall in different quadrants, which is what a window deciding an answer looks like.

The search window went first

It was three times the larger counted number because that is the most generous reading of “near the hole” that is still a window. The organ that moved furthest was looked for inside it, and the search failed: the largest displacement turned out to be a plateau rather than a peak, so the maximum names an arbitrary organ.

Then it turned out to be worse than that. The disturbance has two regimes — a transient below the onset and a pattern above it — and the search window sits inside the transient on some rows and inside the pattern on others, because the onset runs from a few organs to a few hundred and the window is fifteen to thirty-nine.

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. 2 The reference organ’s diagnostics. The search window is the one this thread has already abandoned.

The run went second

Three hundred organs, and it had been a constant since the census was built. Running every cut twice as far moves nineteen of the twenty-four onsets by more than twenty organs, moves one by three hundred, and removes one entirely.

The reason is that the onset is measured against the levels the classes hold at the end of the run, so the levels move when the run does. That is a property of the definition and not of the stems.

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. 3 The onsets at both lengths. Nineteen rows of twenty-four move by more than twenty organs.

The reading window has not been moved

A hundred and twenty organs, chosen because it is the window the surviving hop is identified in, and for the same reason: at a surviving lag of eight it gives fifteen readings a class, which is enough for a class mean to be a mean.

Nobody has varied it. Sixty organs would give seven or eight readings a class and a hundred and eighty would give twenty-two, and both are available on runs already grown. It is the cheapest untested thing in this thread.

A narrower window makes every class look steadier, because a spread over fewer readings is smaller for the same underlying scatter. So the direction of the risk is known in advance: if the window matters, the periodicity count is too high and not too low, and the rows nearest the line are the ones it would move.

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. 4 The levels a hundred-and-twenty-organ window produces on one row. Nothing here says what sixty or a hundred and eighty would give.

Which readings depend on which

Worth setting out, because the thread’s essays do not distinguish them.

The first organ’s displacement depends on no window at all: it is one organ, named by the cut. The surviving lag, the block and the slip are read over the reading window. The onset and the transient’s size depend on the run and on the reading window together. The periodicity classification depends on both.

So of six quantities the thread reports, one is window-free, three depend on one window, and two depend on two. The exchanged pair’s size and direction belong to the middle group: they are read off the levels, so they inherit the reading window and not the run length, and they are unchanged on every row that carries a pair at both lengths.

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. 5 The census’s columns. Each of them comes through a different combination of the three windows.

Why the list was never made

Because the windows arrived one at a time, each for a good local reason, and each in a file that had no view of the others. The search window came from a question about where a disturbance is worst. The reading window was borrowed from the file that identifies the surviving hop, for its own good reason. The run length was a constant in the code that grows a cut.

Nothing wrong happened at any of those three moments. What is missing is the step where somebody asks what the three do together, and that step has no natural home: it belongs to none of the files and to all of them.

What a cut moves, organ by organ. A stem counted at 5 and 8 spirals with the organ seven 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 138 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. 6 A profile drawn over sixty organs, from the round when the run length was a constant nobody was reading.

The window-free one is the one that works

The displacement of the first organ placed after the removal separates cheap removals from expensive ones with a gap in the middle: 9° to 31° on one side and 63° to 168° on the other, with nothing between.

That is the cleanest result in the thread and it is the only reading that does not pass through a window. It is not obvious that those two facts are connected, and it would be strange if they were unconnected.

The honest version is weaker than it sounds. A reading of one organ has no window because there is nothing to average, and a quantity with nothing to average is a quantity with no noise suppression either. It is clean here because the effect it measures is enormous — tens of degrees against a grid of a quarter of one — and on a smaller effect the same reading would be the noisiest in the thread rather than the cleanest.

What a removal costs the next organ. One mark per wrecked cut in the census: how far the first organ placed after the removal ended up from where the control put it. The rows split by which organ was taken. Removing a direct chain-neighbour of the growing tip — an organ at a multiple of one of the two counted numbers — moves the next organ by between 8.9 and 30.7 degrees. Removing anything else inside the front moves it by between 62.8 and 167.6. Nothing lands between the two groups and the ratio across the gap is 2.05, so the line is a gap rather than a threshold. Taking away a neighbour is the cheap removal, which is the opposite of what the words suggest.
Fig. 7 The two populations of single removals with the gap between them, which is the thread’s cleanest separation and its only window-free reading.

What a window has to be aligned to

Something in the stem. A window of a hundred and twenty organs is a window of fifteen periods on a stem whose surviving lag is eight and of twenty-four on one whose lag is five, so the same window is a different instrument on different rows.

That is fixable and has not been fixed: measure over a fixed number of periods rather than of organs, and every row gets the same number of readings a class. Fifteen periods would be 60, 75, 105 and 120 organs at the four surviving lags in the census.

One wrecked stem, lag by lag — golden, rise 0.008, 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 64 degrees. The lag-5 hop swings by 0.00 degrees and sits 0.23 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. 8 The lag spectrum that names the period. A window measured in organs holds a different number of these on different rows.

Which might matter and might not

The prediction is specific: if the reading window is doing work, the rows with the shortest periods should look steadiest, because they have the most readings a class. The four surviving lags in the census are 4, 5, 7 and 8, and the spreads are 0.12° to 8.11°.

Sorting the spreads by lag is arithmetic on a table already computed, and it has not been done. If short-period rows are systematically steadier, the periodicity reading is partly a reading of how many samples each class got.

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. 9 The census by stem, with its surviving lags. The check this section asks for is a sort of this table by one column.

The one that is a length rather than a window

The runs start at organ four hundred. That is where the history ends and the continuation begins, and every cut in the census removes an organ from the last few of those four hundred.

Four hundred organs is long enough for the stem to have settled — the settling thread puts the slowest arrival at 290 organs at this falloff — so it is not an arbitrary number. It is the only one of the four that was chosen against a measurement.

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. 10 How long a stem takes to settle, which is the measurement the four-hundred-organ history was chosen against.

What a window costs when it is right

Nothing, and that is why they proliferate. A window that is comfortably wider than the structure it contains produces the right answer and leaves no trace in the result, so there is never any pressure to justify it.

The pressure arrives all at once, when a row turns up whose structure is wider than the window, and by then the window is in five essays as a constant. Every one of this thread’s three windows was harmless for several rounds before it was not, and the reference organ is the worked example: its search window was fine on every row until the transient was measured and turned out to run from seven organs to three hundred.

Which side of the tip the removed organ was on. Each row is one wrecked cut. The centre line is the azimuth the next organ would have taken; the two open marks are where the two contact families leave it, which are always on opposite sides because that is what makes them the two nearest neighbours of a lattice point. The filled mark is the organ that was removed. Reading the lost-member rule as a question about which side rather than about which chain makes it answerable on all 30 cuts instead of 9, reproduces the published reading on every one of those 9, and is right on 22 of 30 overall.
Fig. 11 A reading taken inside the search window, from a round when that window was still producing usable answers.

What an audit is worth

Two of the three windows in this thread have now been found to decide answers, and both were found by moving them rather than by reasoning about them. The third has not been moved.

That ratio is the argument for the audit. A window that has not been varied is not a window that has been shown to be harmless; it is a window nobody has looked at, and the two look identical from inside the results.

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. 12 The onsets and transients, both of which are quantities two windows are involved in producing.

Three windows and one grid

There is a fourth instrument and it belongs in the list even though it has caused no trouble. Every azimuth in every run is chosen from 1,536 candidates, so all of these readings are quantised at a quarter of a degree.

It has caused no trouble because the quantities are tens and hundreds of degrees and the grid is a quarter of one. Where it has come close is in the rows where a pair removal costs exactly what a single one costs, which agree to the last digit of the grid — and there the right response was to ask how large the effect would have to be to be visible, rather than to trust the agreement.

The measurement is limited by the protractor, not by the plant. The peak falls as the reading error grows, and it falls by an arithmetic factor with nothing fitted: a position error enters two consecutive divergences with opposite signs, adding variance at every lag while the pattern's signal sits at one. At a quarter of a degree the readout is right on all 5 runs; at half a degree on 2; at a degree on 1. Below the dashed floor the peak is the largest of thirty noisy numbers rather than a measurement.
Fig. 13 What the placement grid can and cannot resolve, which is the fourth instrument every reading here passes through.

The general form

A reading through a window is safe when the window is wide compared with whatever structure it is meant to contain, and dangerous when it is comparable. The search window failed because the structure — the transient — is sometimes ten times its width and sometimes a tenth.

The reading window’s structure is the period, and a hundred and twenty organs holds fifteen to thirty of them, so it is comfortably wide. The run’s structure is the transient again, and three hundred organs holds it on most rows and not on all.

How far every organ moved, 7 places back at a rise of 0.008. 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 41 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.
Fig. 14 A profile with its transient and its pattern. Whether a window is wide compared with the structure it contains is the whole question.

Which predicts where the next failure is

On the rows with the longest transients. The onset over six hundred organs reaches 473 organs, which is well past halfway, so the reading window on those rows sits close to the end of the transient rather than well inside the pattern.

That is checkable on runs already grown: compare the levels measured over the last hundred and twenty organs with those measured over the previous hundred and twenty. If they agree, the window is past the transient. If they do not, the row’s levels are still moving and its spread is a reading of the drift.

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. 15 A profile at both lengths, on the row whose onset moves furthest. Its reading window at either length sits near the end of its own transient.

What has been done about it

The run has been doubled and the results reported. The onset now refuses to report a value at the end of a run. And the levels are still measured over a hundred and twenty organs at the top of whatever run was grown, unchanged.

Two of three, which is the honest state. The third is a four-line change and a re-run, and it is written down here rather than left as an intention.

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. 16 The rows the doubling moved, which is what one of the two corrections produced.

The shortfall, stated

Vary the reading window at sixty, a hundred and twenty and a hundred and eighty organs on all thirty cuts at six hundred, and report the spread at each. Ninety readings on runs already computed, and it would say whether the periodicity classification is a property of the stems or partly of the window.

It is not in this round. Naming it is the least that can be done and the most that should be claimed, and it goes into the record beside the other checks this thread owes rather than into a sentence promising to get to it.

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. 17 A different quantity read at three window widths, which is the shape the check this thread still owes would take.

What a reader should carry away

That the shape of the damage — levels, one per chain, with an exchanged pair among them — survives every window this thread has moved. That the timing does not survive the run length. And that the reading window has not been moved and so is not known to be harmless.

The distinction between “checked and fine” and “not checked” is the whole content of an audit, and it is the distinction that is easiest to lose when results are quoted without their instruments.

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. 18 The comparison this audit came out of. Everything on the diagonal is a row two run lengths agree about, and the disagreements are the instrument showing through.

Two other threads with the same audit owing

The band sweep reads a survivor at every rise of a band, and the survivor is read from a run of its own with its own length. Nothing in the band work has varied that length, and a band’s speckle is exactly the kind of feature a marginal instrument produces.

The settling table reads a destination from the last stretch of a run and asks whether it has settled. That thread has done its own two-length comparison and is the only one in this collection that has — which is why its guard against reporting a settling time on a run that never settled was already in place when this thread needed one.

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. 19 The settling thread’s two-length comparison, the only instrument audit this collection had before this one.

What would make this permanent

A rule rather than an essay: every reading taken through a window reports the window beside it. That is a change to how the census prints rather than to what it computes, and it would have made the 7-to-303 range read as 7 to 303 organs, over runs of 300 — which is a sentence somebody would have questioned.

It costs nothing and nobody does it, because a constant that appears in every row looks like it does not need saying. That is exactly when it does.

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. 20 The onsets with both run lengths shown together, which is the shape every reading in this thread should be printed in.

The one line

Three windows: a search window that failed, a run length that moves nineteen of twenty-four onsets and changes three rows’ classification, and a reading window that nobody has varied.

The one reading in the thread that passes through no window at all — the displacement of the first organ placed after the removal — is also the cleanest result it has.

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. 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. 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.

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.

  • Three rows change sides — both name ablation, artefact, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic, tolerance, transient
  • The plateau was a prediction — both name ablation, artefact, claim testing, control, honest limits, measurement, negative result, rigid hop, selection effect, summary statistic, transient
  • Which chains changed places — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic, transient
  • A fifth of the hop — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic
  • Four accounts of one angle — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic
  • The alternation is not a period — both name ablation, artefact, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, sampling

Named objects

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

AblationArtefactCensusClaim testingControlHonest limitsMeasurementNegative resultResolutionRigid hopSamplingSelection effectSummary statisticToleranceTransient