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

The window nobody moved

Three instrument settings sit between the ablation census and every statement it makes. Two have been varied and both decided answers. The third is a hundred and twenty organs at the top of a run, it has never been moved, and moving it changes the verdict on three rows.
13 min read 25 figures State the claim, state the testCounted, not admired

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

Three numbers stand between a cut stem and every statement this thread makes about it. The run length — three hundred organs — decides how far the disturbance is followed. The offset decides which cuts there are to follow. The reading window — a hundred and twenty organs at the top of the run — decides what is read once they have been.

Two of the three have been varied. Doubling the run length moved nineteen of twenty-four onsets by more than twenty organs and changed the periodicity verdict on three rows. Sweeping the offset rather than sampling it turned out to decide which cuts wreck at all.

Which cuts count as periodic, at each reading window. One row per wrecked cut and one column per window. A filled cell is a cut whose worst class spread is under the ten-degree line and is therefore called periodic. At 60 organs 28 of the 30 cuts are, at 120 organs 25, and at 180 organs 25. three rows change side, all of them losing their periodicity as the window widens, and they are marked.
Fig. 1 Which cuts count as periodic at each of three reading windows, with the rows that change side marked.

The third has never been moved

A hundred and twenty organs, inherited from the file that reads rigid lags and kept for a stated reason: the quantity is a spread within a residue class, and a short window would make every class look constant.

That reason is correct and it is a prediction. Nobody had checked it, and checking it is cheap — the runs are already grown, so reading them at a different window costs a read rather than a stem.

How far every organ moved, 5 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 40 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. two of those levels sit together and three do not.
Fig. 2 A wrecked stem’s displacement profile, of which the window is the last stretch.

What the window does

A wrecked stem’s profile is folded onto the lag it kept, giving one residue class per lag. The spread is the worst class’s standard deviation over the window, and a row is called periodic when that spread is under ten degrees.

So the window is the sample the spread is taken over. It is the last N organs of the run, which makes it a tail rather than a slice — and that turns out to matter.

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 1.62° — so within a class the displacement is a constant. two classes sit at the common level. The three that do not sit at 67.7° and -149.5°, equal and opposite to within 75.3 per cent, and they are neighbouring residues. The stem's own divergence is 136.78°, so an exception is one organ's step.
Fig. 3 The classes a profile folds into, whose spread is the quantity the window is read over.

Three windows, ninety readings

Sixty organs, a hundred and twenty, and a hundred and eighty, at both run lengths, over the whole census. Thirty rows by three windows by two lengths is ninety readings, and they cost about four minutes because nothing has to be regrown.

A hundred and twenty is the middle of the three rather than an end of them, which is the same design the falloff exponent sweep uses for the same reason: a sweep starting at the working value can only ever say the answer moves one way.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 4 Every cut’s class spread at the three windows, on axes where proportionality is a line through the origin.

What the prediction said would happen

That a narrow window makes things look steadier. If a class is drifting or noisy, reading less of it gives a smaller spread and more rows fall under the ten-degree line.

That is what happens. At sixty organs twenty-eight of the thirty rows are periodic; at a hundred and twenty, twenty-five; at a hundred and eighty, twenty-five. The narrow window is the generous one.

Which cuts count as periodic, at each reading window. One row per wrecked cut and one column per window. A filled cell is a cut whose worst class spread is under the ten-degree line and is therefore called periodic. At 60 organs 28 of the 30 cuts are, at 120 organs 25, and at 180 organs 25. three rows change side, all of them losing their periodicity as the window widens, and they are marked.
Fig. 5 The same classification at the longer run length, where the count moves the same way.

What it did not say

The prediction was about degree and the finding is about kind. On the rows that keep one verdict at every window, the spread is very nearly proportional to the window: 0.8, 1.6 and 2.4 degrees on one row; 0.3, 0.7 and 1.0 on another; 3.0, 6.1 and 9.1 on a third.

A spread over a sample of a steady quantity is flat in the sample size. A spread that triples when the window triples is measuring a drift, and that is a different finding from the one the prediction contained.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 6 Every row’s spread against the window, including the ones whose verdict changes.

And three rows change side

g026 cut at offset 4, g010 at offset 4 and g010 at offset 5 are periodic at sixty organs and not at a hundred and twenty.

Their spreads do not scale. They go 5.1, 10.3 and 105.5 degrees; 3.9, 69.6 and 163.3; and 4.0, 69.5 and 162.9. That is not three times when the window triples; it is twenty and forty times, which is a different effect wearing the same units.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 7 The three rows the window moves, whose spreads grow far faster than in proportion to it.

Why they do that

Because the window is the tail of the run. A window of a hundred and eighty organs on a run of three hundred begins at organ 120, and these three rows have the census’s latest onsets — 229, 230 and 239 organs at the working window.

So the widest window starts a hundred organs below the point at which the profile settles into the arrangement it keeps. It is reading the disturbance healing and calling it the pattern.

Where each window begins, against where the pattern begins. The run is drawn left to right, one bar per cut. The pale stretch is the disturbance still healing, up to the organ from which the profile holds its levels for the rest of the run; the dark stretch is the pattern repeating. The marks are the organ each of the three windows starts at, since a window is the last N organs of a run. On these rows the widest window starts inside the pale stretch, so it is reading the healing and calling it the pattern — which is why widening the window takes them off the periodic list.
Fig. 8 Where each window begins against where each row’s pattern begins, on the rows that move.

Two effects, running opposite ways

That gives the reading its shape. Widening the window makes a genuinely periodic row look worse in proportion, because it is measuring more of a slow drift. Widening it past a row’s onset makes an unsettled row look far worse all at once, because it is measuring the transient.

Both are real and both are about the window rather than about the stem. They can be told apart by whether the spread scales: the drift multiplies by about three, the transient by twenty or forty.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 9 The same reading at the longer run length, where the drift is still proportional and the transient is further away.

So the classification is partly the window’s

The sentence the reading supports is narrow. Twenty-five of thirty rows are periodic at the working window; twenty-eight are at sixty organs; twenty-five are at a hundred and eighty. The verdict on three of thirty rows is decided by the window.

That is not a large fraction and it is not nothing. It is the same order as the three rows the run length moves, and it is a different three: one row overlaps.

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 rows a change of run length moves, which is the other instrument setting that decides verdicts.

What the ten degrees is doing

The threshold deserves its own look. A row is periodic when its worst class spread is under ten degrees, and ten was chosen because the measured spreads separate into two populations with a gap: 0.12 to 6.09 degrees on the periodic rows and 10.3 to 156.9 on the five that are not.

That gap is real at the working window. At sixty organs the same rows read 0.1 to 3.0 and 4.0 to 149.6, and at a hundred and eighty they read 0.2 to 9.1 and 12.2 to 163.3. The gap survives all three, which is the reassuring part.

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 spreads the classification is drawn on, and the gap the line sits in.

But it is a line on a quantity that scales

The threshold is a fixed number of degrees on a quantity proportional to the window. So at sixty organs it admits a drift rate three times steeper than at a hundred and eighty.

Nothing is wrong with that as a definition — it says this class moves less than ten degrees over the stretch that is read and it means it. It is not the window-free property the word periodic suggests, and the collection has used the word without the qualification.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 12 The spreads against the window, where a fixed threshold cuts a family of lines through the origin.

What would be window-free

A rate rather than a spread. Divide each class’s spread by the number of organs it was read over and the quantity stops depending on the window, at least on the drifting rows.

That is not proposed as a replacement here, because changing a definition mid-collection makes every earlier number incomparable. It is written down as what a window-free version would look like, so that anybody adopting it knows what they are giving up.

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. 13 One row’s profile at two run lengths, which is the other axis the same quantity depends on.

The two windows the collection already had

This is not the first time a window has decided something here. A residual that looked ordered turned out to be the window it was measured over, and a fragility attributed to a lattice turned out to belong to the reading.

Three instruments, three windows, three results that were partly about the reading rather than the object. That is a pattern in this collection and it is worth naming as one.

Two windows on a shoot at 700 nodes per rung. A stem grown at 700 nodes to the rung with a disturbance of 0.9, its rise falling from 0.4 to 0.004 over 3350 nodes. The upper window is the last 250 internodes — where a count would be made on a real plant — and the lower is the same length shifted down 125. The upper reads nothing; the lower reads nothing. The verdict is silent, and the window holds 0.36 of a rung.
Fig. 14 The same quantity read through two settings of one instrument, which is how each of these was caught.

Why a tail rather than a slice

One design choice worth defending. The window is the last N organs rather than a stretch in the middle, because the thing being measured is what the stem ends up doing.

That choice is what makes the interaction with the onset possible. A window taken in the middle would have a fixed relationship to nothing; a window taken at the end has a fixed relationship to the end of the run and a moving one to the onset.

Where each window begins, against where the pattern begins. The run is drawn left to right, one bar per cut. The pale stretch is the disturbance still healing, up to the organ from which the profile holds its levels for the rest of the run; the dark stretch is the pattern repeating. The marks are the organ each of the three windows starts at, since a window is the last N organs of a run. On these rows the widest window starts inside the pale stretch, so it is reading the healing and calling it the pattern — which is why widening the window takes them off the periodic list.
Fig. 15 Where each window begins against where each row’s pattern begins, over the whole census.

A window as long as the run is not a window

The instrument below this one takes the last N rows of a profile, and a slice longer than the array is the whole array. So a window of a thousand on a run of three hundred returns a reading over the whole profile, with an onset at its first organ by construction.

Nothing beneath refuses it. The refusal is in the file that moves the window, which is where it belongs, and it is checked rather than assumed — the same shape as an onset reported at the last organ of a run, which is a run length dressed as a measurement.

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. 16 Where each row’s pattern begins at two run lengths, and the row whose onset was the end of its run.

What the ninety readings cost

About four minutes, because profileAt regrows a stem and the readings at three windows share the same grown runs through the cache underneath.

That is the cheapest instrument variation this collection has run and it produced a finding. The comparison is with the run-length variation, which needed sixty fresh runs and two minutes, and with the offset sweep, which needed 1,890 cut stems and an hour.

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. 17 The runs the windows are read over, which are grown once and read three times.

Why nobody had done it

The honest answer is that the setting had a reason attached to it and a reason reads like a justification. A hundred and twenty organs was chosen so that a class would have fifteen samples at a lag of eight, which is a good argument, and a good argument for a setting is what stops anybody varying it.

The two settings that had been varied were varied because somebody doubted them. This one nobody doubted, which is exactly the condition under which a setting goes unchecked for several rounds of work.

Which cuts count as periodic, at each reading window. One row per wrecked cut and one column per window. A filled cell is a cut whose worst class spread is under the ten-degree line and is therefore called periodic. At 60 organs 28 of the 30 cuts are, at 120 organs 25, and at 180 organs 25. three rows change side, all of them losing their periodicity as the window widens, and they are marked.
Fig. 18 The classification at three windows, which is what four minutes of reading buys.

What is still not varied

The window’s placement is fixed at the end of the run and its shape is a flat count of organs. A window weighted towards the newest organs, or one placed relative to the onset rather than to the end, would both be defensible and neither has been tried.

Placing it relative to the onset is the interesting one, because it would remove the interaction that produces the three movers — and removing it is not obviously right, since whether a row has settled by the end of its run is a real property of the row.

The area never settles, so the number reported is the window. The equivalent width of the disorder dip — the area of the deficit divided by the dip's own depth — for four fractions with a denominator of 34, against the window it was integrated over. The window is in scaled units, u = δ·n²/q, so a head of any size is sampled at the same places on its own dip; 200 scaled units is about 0.0109° at these head sizes. A width would show as a flat stretch. There is none: every line climbs to 512 scaled units or so and then turns over, because past that the window has reached far enough towards the next rational that it is integrating the neighbour's dip instead of this one's background. The dip has a depth and a scale; it does not have an outer edge, and an integral of something with no outer edge is a statement about where it was stopped.
Fig. 19 A quantity read at several windows, which is the family of readings a different placement would change.

What the three movers have in common

They are not three arbitrary rows. All three keep a lag of 5, all three are on the golden branch, and all three have onsets past organ 220 at the working window — the three latest in the census.

That is a coherent group and it says the effect is not a lottery. A row whose pattern begins late has less pattern in the run than the window asks for, and widening the window reaches further into the part that has none.

Where each window begins, against where the pattern begins. The run is drawn left to right, one bar per cut. The pale stretch is the disturbance still healing, up to the organ from which the profile holds its levels for the rest of the run; the dark stretch is the pattern repeating. The marks are the organ each of the three windows starts at, since a window is the last N organs of a run. On these rows the widest window starts inside the pale stretch, so it is reading the healing and calling it the pattern — which is why widening the window takes them off the periodic list.
Fig. 20 The three rows whose widest window starts below their own onset, with each window’s start marked.

And what they do at the longer run

The obvious repair for a window that reaches too far back is a longer run, and it works. At six hundred organs the same three rows read 5.1, 10.3 and 15.4 degrees, and 3.9, 7.9 and 11.8, and 4.0, 8.1 and 12.2.

Those scale. So on a run twice as long the transient is far enough below every window that what is left is the drift, and the three rows join the population that behaves proportionally. The interaction is between the window and the run length together, not between the window and the stem.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 21 The same rows at the longer run length, where the three movers stop being a separate population.

Which is a small piece of good news

It means the classification is not unstable in some deep way. It is unstable exactly where a window reaches into a transient, and that is a condition anybody can check for by comparing the window’s start against the onset.

The check is arithmetic and costs nothing: a window of N organs on a run of M begins at organ M − N, and a row whose onset is above that is being read partly on its disturbance. Nothing in the thread had been making it.

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. 22 Where each row’s pattern begins at two run lengths, which is the number the check compares against.

The setting that is still doing the most work

Of the three numbers, the run length remains the one that moves the most. Doubling it moved nineteen of twenty-four onsets by more than twenty organs, took the quoted range of onsets from 7–303 to 28–473, and changed three rows’ periodicity in both directions.

The window moves three rows in one direction and scales a spread. The offset decides which rows exist. Ranking them by how much they change is not the same as ranking them by how much they matter, and the window’s finding — that the quantity underneath is drifting — is the one that changes how a reader should hear the word periodic.

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. 23 One row read at two run lengths, which is the setting that moves the most of the three.

What a reader should carry

That every verdict in this thread is periodic over the last hundred and twenty organs rather than periodic, and that the difference shows up on three of thirty rows.

And that the prediction attached to the setting was right — a narrow window does make everything look steadier — while being right for a reason nobody had stated, which is that the quantity it reads is drifting rather than noisy.

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality.
Fig. 24 Every row’s spread at the three windows, which is the whole of what four minutes of reading found.

What the picture at the top shows

Thirty rows, one per wrecked cut, and three columns, one per window. A filled cell is a cut whose worst class spread is under ten degrees.

Most rows are three filled cells or three pale ones. Three rows are filled at sixty organs and pale at a hundred and twenty and a hundred and eighty, and they are drawn in a second tone. The numbers on the right are each row’s spread at each window, and reading across them is where the proportionality shows.

Which cuts count as periodic, at each reading window. One row per wrecked cut and one column per window. A filled cell is a cut whose worst class spread is under the ten-degree line and is therefore called periodic. At 60 organs 28 of the 30 cuts are, at 120 organs 25, and at 180 organs 25. three rows change side, all of them losing their periodicity as the window widens, and they are marked.
Fig. 25 The classification once more, with the three rows the window decides marked.

The one line

The reading window was set at a hundred and twenty organs for a stated reason and never moved: at sixty organs twenty-eight of the census’s thirty cuts are periodic, at a hundred and twenty twenty-five, and at a hundred and eighty twenty-five.

Three rows change side, all of them losing periodicity as the window widens, and all three are rows whose widest window begins below their own onset.

Where this goes next

Each cut's class spread at three reading windows. The worst spread within a residue class, measured over the last 60, 120 and 180 organs of the same runs. A quantity that is steady with noise on it would give a flat line; a quantity drifting steadily along the run gives a line through the origin. Nearly every row gives the second, so the classes this thread calls steady are sliding slowly and the number that says how much depends on how much of the run is read. The dashed guide is exact proportionality. A spread that grows with its window A spread over a sample of a steady quantity does not depend on how big the sample is. These spreads triple when the window triples, on nearly every row of the census, which means the classes this thread calls steady are sliding — slowly, and invisibly at any single window. 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. Hold the neighbourhood and the denominator stops mattering. The equivalent width of the disorder dip — the area of the deficit divided by its own depth — for seven fractions whose nearest neighbours sit at the same distance and whose denominators run from 19 to 47. Each is measured at a head size chosen so that all of them share one scaled unit, which makes a window in scaled units the same window in degrees and the same fraction of the way to the neighbour for every member. At a window of 25 the seven widths are 38.8, 38.8, 39.0, 39.0, 38.7, 39.0, 39.0 — a spread of ×1.010 across a factor of 2.47 in denominator. The lines separate as the window widens, to ×1.147 at 200, and when they do they order by denominator rather than by crowding. So the residual this thread carried was the window: hold it and there is nothing left that belongs to the fraction. The residual was the window After the depth and the q over n squared scale are taken out of a disorder dip, something looked left over and looked ordered by how crowded the fraction's neighbourhood is. Measured on fractions whose neighbourhoods are identical by construction, seven widths across a factor of two and a half in denominator agree to one per cent. There is no residual; there was a comparison made at different effective windows.

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.

Named objects

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

AblationArtefactClaim testingDamage profileHonest limitsInstrument settingMeasurement errorOnsetPeriodicityReading windowResidue classResolution