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

The plateau was a prediction

Last round's search for a reference organ found that the largest displacement above a hole is a plateau rather than a peak, and reported it as a failure. A profile constant on each of k residue classes has exactly k levels, so its maximum is attained by a whole class — a ninth to a quarter of every window, forever.
13 min read 20 figures State the claim, state the testForm is not mechanism

Worth reading first: The damage has a period · The organ that was taken away · The survey this site cannot do.

The previous round wanted a reference organ. The census’s readings of which family survives a removal are stated in terms of the offset — how many places back the removed organ sat from the growing tip — and an offset is a coordinate chosen by the experimenter. A quantity read off the disturbance itself would be a coordinate the stem chose.

The obvious candidate was the organ that moved furthest. It does not exist, and this essay is about why the way it fails is now a consequence rather than a setback.

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. 1 The measurement that turns last round’s failure into a prediction: how constant the displacement is inside a residue class.

What was measured last round

For every wrecked cut, the largest displacement in a window three times the larger counted number wide, and the share of that window within a tenth of it.

The largest displacements ran 105° to 180° against a ceiling of 180°, and at sixteen of thirty rows a fifth of the window sat within a tenth of the maximum. Over the last sixty organs of each run — hundreds of organs above the hole — the displacement was still 70° to 179°.

Three readings, one conclusion: the disturbance has no far edge and no peak, so “the organ that moved furthest” names an arbitrary member of a wide flat.

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 reading that replaced the reference organ, which needs no most-disturbed organ at all.

What it was read as

A failed instrument. The round recorded that the repair for a reading answerable on nine rows of thirty did not work, and moved on to a different repair — which side of the tip the removed organ sat on — that is answerable on all thirty and scores 22.

That was the right move and the failure was left as an observation. A disturbance that does not decay is odd, it was reported as odd, and nothing followed from it.

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. 3 The sidedness reading, which is what the reference organ was replaced by.

What follows from it now

The displacement above a hole is constant inside each residue class modulo the lag the stem kept. A profile like that has exactly k values in it, not three hundred.

So its maximum is one of those k values, and the maximum is attained not by an organ but by every organ of one class. In a window of n organs that is about n/k of them, and k across this census is between 4 and 11.

A ninth to a quarter of any window sits exactly at the maximum, and the whole class sits within the spread of a class — between 0.12° and 6.09° — of it. A tenth of a maximum of 140° is 14°, comfortably more than that spread, so the share within a tenth of the maximum is the share of the window in that class.

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 One period’s k levels, of which the largest is attained by a whole class rather than by an organ.

The arithmetic against the measurement

A fifth of the window was what sixteen of thirty rows showed. One over k for k between 4 and 11 is between 0.09 and 0.25.

A fifth is 0.20, which is one over five, and five is the commonest surviving lag in the census. So the number reported last round as evidence that the maximum was a plateau is, to the precision it was quoted at, one over the surviving lag.

That is not a coincidence explained after the fact; it is what a periodic profile forces. And it is testable in the direction that matters: a row whose surviving lag is 8 should show an eighth of its window at the maximum, and a row whose lag is 11 an eleventh.

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. 5 The census with each row’s surviving lag, which is the denominator the plateau share should be one over.

Why “no far edge” follows too

The other half of last round’s finding was that the disturbance is still tens of degrees at the top of the run. A periodic profile explains that with nothing left over: the levels do not decay, so the displacement three hundred organs above the hole is the same as the displacement a hundred above it.

The smallest such tail in the census is 45° and most are far larger. A profile whose classes sit at 0° and ±134° has a tail of 134° whatever window it is read over, forever, because that is what the arrangement now is.

So “the damage does not decay” is not a statement about the damage spreading. It is a statement that the stem settled somewhere else and stayed there.

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.
Fig. 6 A profile read far above the hole, on which the levels are as flat at the top of the run as anywhere.

What a reference organ would have needed

A profile with a genuine peak: one organ, or a few, displaced much more than the rest, with the displacement falling away either side.

That shape is what a transient looks like, and there is one — below the pattern, where the stem is still moving. A reference organ read there would be reading a real feature.

The trouble is that last round’s window was three times the larger counted number, which at a 5/8 stem is twenty-four organs and at an 8/13 stem thirty-nine. The transient’s extent runs from 7 to 303 organs, so the window sometimes sits inside the transient, sometimes straddles the boundary, and sometimes is entirely in the pattern. A statistic taken over a window that is in a different regime on different rows is a statistic measuring different things on different rows.

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. 7 How far above the hole the pattern establishes itself, against the window the reference organ was looked for in.

Which is the sharper criticism

The reference organ did not fail because the disturbance is featureless. It failed because the window it was looked for in was not aligned to any feature.

That is a criticism of the instrument rather than of the object, and it is one this collection has made before in a different thread: a mean taken over a partial period of a repeating motif is biased by the angles it counts twice. Same shape of error — a window that does not respect the structure of what it is measuring — found the same way, by finding the structure first.

The block a wrecked stem settles into is the count it was cut from. A stem is cut in the middle of its front and followed for 300 organs. It does not come back to its lattice; what it does instead is repeat a fixed sequence of divergences exactly, to the resolution of the azimuth grid. Each row shows that sequence twice over, one bar per organ, drawn to the same scale. At the 5/8 rung the sequence is five angles long and advances -36.1° per block. Every block is the smaller number of the pair that was cut, so the second attractor carries the first one's count — and every precession is one part in twice the block, which makes each of these a two-jugate arrangement with 10 rows.
Fig. 8 The repeating motif whose period a window has to respect, in the divergence sequence rather than the displacements.

What could be measured instead

Three things, and all of them are now available.

The level of each class, which is k numbers rather than one, and which carries the whole of the difference between a wrecked stem and its control.

The transient’s extent, which is a genuine coordinate the stem chose: the first organ from which every class stays at its own level. It runs from 7 to 303 organs and five rows never reach it.

And the size of the exchange, which is one divergence step and is the same number on every row to within eight per cent.

None of those is a reference organ, and none of them needs one.

The exceptional pair, measured in divergences. One row per wrecked cut whose profile has exactly one pair of exceptional classes, drawn at the size of that pair divided by the stem's own settled divergence. Every row sits between 0.882 and 1.076, so the two chains that came apart moved by one organ's step rather than by two or by half of one. The residual is not scatter: rows are grouped by the lag the stem kept, and every lag sits wholly above the line or wholly below it. Why a surviving 5 or 7 overshoots and a surviving 4 or 8 falls short is not answered here.
Fig. 9 The size of the exchange, which is one of the quantities a wrecked stem does supply.

What the plateau share should be, row by row

The prediction has a specific form and the library asserts a weak version of it: that on every periodic row at least a thirteenth of the window sits at the maximum, since the largest surviving lag in the census is 11.

A stronger version — that the share is one over the surviving lag, row by row — would need the plateau share recomputed with a window aligned to the pattern rather than to the counted pair, which is a different measurement from last round’s and would not be comparable with the number it is meant to explain.

So the strong version is stated and not asserted, which is the honest way round.

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. 10 The rows the weak version is asserted over, being the twenty-five whose classes are constant.

What this does to last round’s conclusion

Nothing. The conclusion was that there is no reference organ and that a different reading was needed, and both remain true.

What changes is the status of the observation. “The maximum is a plateau” was an awkward fact about a measurement that did not work. It is now a consequence of the shape of the damage, and a consequence that could have been predicted from the shape before the search was run.

That is worth recording as an ordering: the shape was measurable last round from data that had already been computed, and it was not measured, and the search that failed was run instead. The nine thousand numbers were there.

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 153 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. 11 The profile as it was drawn last round, in a window too short to show that it repeats.

The general shape of the mistake

A summary statistic taken from a distribution whose shape has not been looked at.

The maximum of a set is a fine statistic when the set has a peak and a poor one when it has a plateau, and which it is cannot be read off the maximum. The site’s own second-statistic thread is fourteen essays about exactly this in a different subject, and the discipline it arrives at — look at the distribution before choosing a number to reduce it to — is the discipline that was not applied here.

It is not a subtle failure and it is not a rare one. It is what happens when a quantity is computed for a purpose and the computation is never plotted.

A count of m and n pins the divergence to 221°/mn. Each dot is one reported pair, and its height is the total width of the divergence angles that could have produced it at some rise. 2/3 leaves 38.8° open; 34/55 leaves 0.118°. The line is 221°/mn, taken from the three highest pairs and drawn back through the rest.
Fig. 12 A distribution shown rather than summarised, which is the discipline this thread failed to apply to its own profiles.

What the plateau does not explain

The five rows whose classes are not constant. Their spreads are 10.3° to 156.9°, so their profiles are not k levels and the argument above does not apply to them.

Two of the five are on the Lucas 0.020 stem and two on the golden 0.010 stem, and on those the maximum might well be a genuine peak. Nothing here has looked.

That is a small hole in the account and it is worth naming, because “the maximum is a plateau because the profile is periodic” is a claim about twenty-five rows and last round’s plateau statistic was computed over all thirty.

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 five rows the explanation does not cover, findable by their spreads.

The window that was chosen, and why it was reasonable

Three times the larger counted number. On a 5/8 stem that is twenty-four organs and on an 8/13 stem thirty-nine.

The reasoning behind it was sound at the time: a removal is felt across a front that runs to the larger counted number, so a window three times that is generous enough to contain the disturbance and short enough that the answer is about the cut rather than about the run length.

Both halves of that are about the front, which is a property of which offsets have a lasting effect. Nothing in it is about how far above the hole the effect settles down, because until this round nobody had measured that. The window was chosen against the one length scale the thread had.

On a rung the response is a run of offsets; near a transition it has a hole. One row per rise, from 0.032 at the top to 0.005 at the bottom, and one column per offset: the organ one place back at the left, 13 places back at the right. A cell is filled where removing that organ moves the next organ by more than 2.5°, and empty where it does not. On a rung the filled cells are a run from one to the larger parastichy number — 5, 8, 13 at the pairs shown on the left. Every rise shown here is in the middle of its rung, so every row is a run and nothing past it is felt — what happens between the rungs is a separate matter.
Fig. 14 The front’s depth by rise, which is the length scale the window was chosen against.

Why a plateau is not the same as no information

The maximum is uninformative and the levels are not.

A profile with k levels has k numbers in it, and the largest of them is one. Reporting only the largest throws away k − 1, and among those thrown away is the whole structure: the common level, the two exceptions, their sizes and their adjacency.

So the reference-organ search did not fail because the profile is featureless. It failed because it reduced a structured object to its extreme value, which is the one summary that cannot see structure. That is a stronger criticism than “the disturbance does not decay” and it points at what to do instead.

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 0.64° — so within a class the displacement is a constant. six classes sit at the common level. The two that do not sit at 135.1° and -135.0°, equal and opposite to within 0.1 per cent, and they are neighbouring residues. The stem's own divergence is 137.77°, so an exception is one organ's step.
Fig. 15 The levels a maximum reduces to one number, of which the largest is the one that was reported.

What the tails were saying

Over the last sixty organs of each run the displacement was 70° to 179°, and the smallest tail anywhere in the census is 45°.

Read as a decay rate that is a decay rate of zero. Read as a class level it is the size of the exceptional pair, which is one divergence step — 88° to 147° across the census, and 45° is below that only because a tail is the largest displacement over a window that may not contain an exceptional organ.

So the tail measurement, which was reported as showing the disturbance has no far edge, is a measurement of the exceptional level in disguise. It had the size of the exchange in it and nothing said so.

The exceptional pair, measured in divergences. One row per wrecked cut whose profile has exactly one pair of exceptional classes, drawn at the size of that pair divided by the stem's own settled divergence. Every row sits between 0.882 and 1.076, so the two chains that came apart moved by one organ's step rather than by two or by half of one. The residual is not scatter: rows are grouped by the lag the stem kept, and every lag sits wholly above the line or wholly below it. Why a surviving 5 or 7 overshoots and a surviving 4 or 8 falls short is not answered here.
Fig. 16 The size of the exchange on every row that has one, which is what a tail measurement was reading.

What was available and not used

Every number in this essay comes from runs the previous round had already grown. The profiles were computed, thirty of them, three hundred organs each; the surviving lag was measured for every one; and the fold is one line of arithmetic.

That is worth recording plainly rather than as a lesson. The round measured a maximum, measured a plateau share, measured a tail, and did not plot a profile — and every one of those three numbers is a consequence of a shape that was in the data.

The general form is: a summary computed for a purpose, three times, without the distribution being looked at once.

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. 17 A profile plotted rather than summarised, which is the step that was missing.

Two ways this could still be wrong

The first is that the plateau share and the periodicity are measured over different windows. The share was measured over three times the larger counted number and the periodicity over the top hundred and twenty organs, and on a row whose onset is 166 those two windows do not overlap at all.

So “the plateau is one over the surviving lag” is an argument about a window that was not the one measured. It holds where the search window sat inside the pattern and it does not where the window sat inside the transient — and which rows are which is now known.

The second is the five rows that are not periodic, whose maxima the argument says nothing about. Between them those two caveats cover a third of the census, and the claim is therefore that the plateau has an explanation on most rows rather than on all of them.

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. 18 The onsets, which decide whether a given row’s search window sat inside the pattern.

What replaced the reference organ

A reading that needs no organ at all. Which side of the tip the removed organ sat on is arithmetic on the settled divergence and the offset — the azimuth of the organ k places back, folded to a half turn either way — so it is answerable on every row rather than on the nine the tip’s chains reach.

It scores 22 of 30 against 18 for naming the commoner family, and it reproduces the published reading on all nine rows that reading covers.

Which is the shape a repair should have: not a better statistic taken from the same profile, but a quantity computed from something else entirely. The profile had been asked for a coordinate it does not supply, and the coordinate was available in the arrangement’s own geometry.

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. 19 The sidedness reading, which is answerable on every row and needs nothing from the profile.

The one line

A profile constant on each of k residue classes has k levels, so its largest value is attained by a whole class — a ninth to a quarter of any window, forever, with no decay above the hole. The plateau last round found and reported as a failed instrument is what that shape forces, and the shape was computable from data already on disk.

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 0.66° — so within a class the displacement is a constant. three classes sit at the common level. The two that do not sit at 145.9° and -148.4°, equal and opposite to within 1.7 per cent, and they are neighbouring residues. The stem's own divergence is 136.78°, so an exception is one organ's step.
Fig. 20 One more period, on which the maximum belongs to a class and not to an organ.

Where this goes next

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. The organ that moved furthest A reading that works on nine of thirty rows needs a reference organ, and the obvious repair is to measure one rather than to choose it. Measured, the disturbance turns out to have no far edge at all, so there is no organ that moved furthest in any sense the reading can use — and the generalisation that does work needs no reference organ. 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.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

  • One level and two exceptions — both name ablation, claim testing, control, description versus mechanism, measurement, negative result, prediction, rigid hop
  • A period the grid invented — both name ablation, artefact, claim testing, honest limits, measurement, negative result, summary statistic
  • Matching instead of correcting — both name artefact, claim testing, honest limits, measurement, negative result, selection effect, summary statistic
  • One offset, two answers — both name ablation, claim testing, control, honest limits, measurement, negative result, rigid hop
  • The clock a share cannot see — both name claim testing, control, honest limits, measurement, negative result, summary statistic, transient
  • The family that lost a member — both name ablation, claim testing, control, honest limits, measurement, negative result, rigid hop

Named objects

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

AblationArtefactClaim testingControlDescription versus mechanismExplanationHonest limitsMeasurementNegative resultPredictionRigid hopSelection effectSummary statisticTransient