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

Three rows a window moves

Three of the census's thirty cuts are periodic when a hundred and twenty organs are read and not when a hundred and eighty are. Their spreads do not grow in proportion to the window, they grow by twenty and forty times, and the reason is that a window is the tail of a run.
13 min read 25 figures State the claim, state the testCounted, not admired

Worth reading first: Two windows on one stem · The damage has a period.

Twenty-eight of the census’s thirty wrecked cuts are periodic when sixty organs are read. Twenty-five are when a hundred and twenty are, and twenty-five when a hundred and eighty are. The three that move are the subject here.

They are g026 cut at offset 4, g010 cut at offset 4, and g010 cut at offset 5. All three are periodic at the narrow window and not at the working one.

Three of thirty is a tenth of the census, which is small enough to be easy to dismiss and large enough that any count quoted from the classification should carry it.

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. 1 The three rows, with each window’s starting organ marked against where each row’s pattern begins.

Their spreads do not scale

On the rows that keep one verdict at every window, the spread is proportional to the window: 0.8, 1.6 and 2.4 degrees at sixty, a hundred and twenty and a hundred and eighty organs.

These three read 5.1, 10.3 and 105.5 degrees; 3.9, 69.6 and 163.3; and 4.0, 69.5 and 162.9. Those are ratios of twenty-one, forty-two and forty-one against a proportional prediction of three.

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. 2 The three rows’ spreads against the window, climbing far faster than proportionality allows.

So they are a different effect

A quantity that scales with the window is measuring a drift. A quantity that grows by forty times when the window triples is measuring something that is only present in part of the window.

That is the shape of a transient: a stretch at the start of the reading where the quantity is doing something else entirely, whose contribution to a spread grows enormously once it is included at all.

The distinction is available for nothing once three windows have been read, and it is not available at all from one. A single spread of 69.6 degrees says the row is not periodic and says nothing about why.

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. 3 Every row at every window, where two populations climb at two different rates.

Where the transient is

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 a window of sixty begins at organ 240.

So a wide window reaches further back into the run, and what it reaches back into is the part where the disturbance from the cut is still healing rather than the part where the stem has settled into the arrangement it keeps.

How far every organ moved, 4 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 299 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. 4 A wrecked stem’s profile, whose early stretch is the disturbance and whose late stretch is the pattern.

Which is exactly what these three rows have

The onset is the organ from which every class stays within ten degrees of its own tail mean for the rest of the run. These three rows have the census’s latest onsets: 229, 230 and 239 organs at the working window.

A window of a hundred and eighty begins at organ 120, which is over a hundred organs below every one of them. So the widest window on these rows is more than half disturbance.

Where the pattern starts, measured at two run lengths. One row per wrecked cut. The small mark is the onset a 300-organ run reports and the ring is what a 600-organ run reports; a row with only one mark reports an onset at only one length. 19 of the 24 rows that report both move by more than twenty organs, and the largest move is from 135 to 435. The range the thread has been quoting, 7 to 303 organs, becomes 28 to 473.
Fig. 5 Where each row’s pattern begins, with the three latest onsets in the census marked.

The check that would have caught it

It is arithmetic and costs nothing. A window of N organs on a run of M begins at organ M − N; a row whose onset is above that is being read partly on its transient.

Nothing in the thread had been making that comparison, because with one window and one run length the two numbers were both constants and the comparison never came up. It comes up the moment a second window exists.

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. 6 Every row’s onset against where each window begins, which is the comparison the check makes.

What happens on a longer run

The 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; 3.9, 7.9 and 11.8; and 4.0, 8.1 and 12.2 degrees.

Those scale. Ratios of 3.0, 3.0 and 3.1 against a prediction of three, which puts all three rows back into the drifting population.

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 same rows at the longer run length, where the three stop being a separate population.

So the effect is an interaction

Not between the window and the stem, but between the window and the run length together. A window of a hundred and eighty on a run of three hundred reaches into the transient; the same window on a run of six hundred does not.

That is a more comfortable finding than the alternative. It means nothing about these three stems is peculiar; what is peculiar is a reading whose start falls in the wrong place.

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. 8 One of the three rows at both run lengths, where the transient moves relative to the window.

What the three rows have in common besides that

All three keep a lag of 5, all three are on the golden branch, and all three sit at rises of 0.026 and 0.010.

That coherence is worth noting and worth not over-reading. Late onsets and a lag of five go together in this census for reasons the census does not explain, and three rows is three rows.

It is also not independent of which cuts wreck at all, since the census’s rows at a lag of five come from a particular stretch of the ladder rather than from across it.

How constant the displacement is inside one residue class. One row per wrecked cut in the census, drawn at the widest spread found inside any one residue class when the profile is folded on the lag that stem kept. 25 of 30 rows sit between 0.12 and 6.09 degrees, which on a quantity whose between-class differences run past a hundred and fifty degrees is a constant. The five that do not sit from 10.3° up. There is nothing in between, so the line drawn at 10° could have been drawn anywhere in a wide interval.
Fig. 9 The census’s spreads sorted by the lag each cut kept, where the movers sit.

The direction is always the same

Every one of the three loses its periodicity as the window widens. None gains it, which is not true of what a longer run does — there rows move in both directions.

That is what both effects predict, which is why they do not fight each other on any row: a wider window includes more drift, and a wider window may also include transient. Both push a spread up. Nothing in the census gets steadier when more of it is read.

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. 10 The classification at the three windows, where every change runs the same way.

Which makes the narrow window the generous one

At sixty organs twenty-eight of thirty rows are periodic. That is the largest count any window gives and it is the least informative, because a window of sixty organs on a run of three hundred is reading a fifth of it.

The prediction the setting was chosen against was exactly this — that a short window would make every class look constant. It is right, and it is right by three rows rather than by the whole table.

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. 11 The same classification at the longer run length, where the narrow window’s generosity persists.

What the working window is doing

A hundred and twenty organs on a run of three hundred begins at organ 180. Of the thirty rows, twenty-seven have onsets below that and three do not.

So the working setting is reading the pattern on twenty-seven rows and part of the disturbance on three. That is a defensible place to be and it is not what the thread had been claiming, which was that it was reading the pattern.

The same distinction — between what a setting does and what it was assumed to do — is what a residual turned out to be when its instrument was changed for one with no level in it.

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. 12 The three rows where the working window is not reading only the pattern.

Whether the three should be reclassified

They should not, and it is worth saying why rather than leaving it implied. A row whose pattern begins at organ 229 on a run of three hundred has less pattern in it than a row whose pattern begins at organ 7.

That is a real property of the row and calling it periodic on the strength of a window narrow enough to miss the disturbance would be hiding it. The right move is to say what the verdict is a verdict about, which is what these readings make possible.

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. 13 How many rows report an onset at each run length, and the row whose onset was the end of its run.

The same failure at the other end

There is a matching defect at the far end of the run and it has already been caught. An onset is the first index from which a condition holds for the rest of the run, and that reading is satisfied trivially at the last index — so it returns a number whether or not the condition ever holds.

The guard is to ask the condition of the tail before searching for a first index. One row of the census needed it. The window’s version of the same problem is at the beginning rather than the end, and its guard is to compare the window’s start against the onset.

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. 14 Where each row’s pattern begins at two run lengths, which is the reading both guards are about.

Three rows out of thirty

The size of the effect matters as much as its existence. Three of thirty is ten per cent, and the other twenty-seven rows keep their verdict at every window at both run lengths.

So the classification is mostly robust to the setting and specifically not robust on the rows whose onsets are latest. That is a better description than either the classification is stable or the classification depends on the window.

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. 15 The whole classification, of which three rows in thirty are decided by the window.

And three rows the run length moves

Doubling the run length also changed the periodicity verdict on three rows, in both directions: two joined as their spreads fell from about 70 degrees to about 8, and one left as its spread rose from 6.1 to 171.2.

One row appears in both sets. So of thirty rows, five have a verdict that one instrument setting or the other decides, and twenty-five are stable under both.

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 a change of run length moves, which overlap with the rows a change of window moves.

What five unstable rows out of thirty is

It is a sixth of the census, and it is the honest figure to attach to any statement of the form N of the census’s cuts are periodic. That number is twenty-five at the working settings and it is twenty-three to twenty-eight across the settings that have been tried.

Quoting the range rather than the point is what this reading makes possible, and it is cheaper than it sounds: the readings exist, so the range is a maximum and a minimum over numbers already computed.

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. 17 The spreads at two run lengths, from which the range of counts is read.

The row that is worst on both

g010 cut at offset 5 is the most sensitive row in the census. Its spread goes 4.0, 69.5, 162.9 across the windows and it moves at both run lengths.

It is worth naming because a reader tracing a claim through the thread will meet it several times, and because it is the row most likely to be reported differently by any two readings of the same table. Nothing about it is broken; it is simply the row whose pattern begins latest and whose transient is therefore the largest fraction of any window.

How far every organ moved, 5 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 a transient that does not end inside this run it settles into a repeating pattern of five levels, one per residue class modulo 5, which is the lag whose hop this stem kept. one of those levels sit together and four do not.
Fig. 18 The census’s most window-sensitive row, whose transient occupies most of a wide window.

What a transient is, on a cut stem

Worth naming, because the word is doing work. Remove an organ and the organs placed after it are placed against a neighbourhood that is missing one member. For a stretch the arrangement is unlike anything the control has and unlike what the cut stem will end up with.

Then it settles. The displacements fall into a pattern that repeats at the lag the stem kept, and from there on the profile is that pattern rather than the healing. The onset is the organ where the second begins.

So a transient is not noise and it is not error. It is a different regime of the same run, and a reading that spans both is averaging two things.

How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.
Fig. 19 Where a wrecked stem’s profile stops healing and starts repeating, which is the boundary a window can cross.

Why the onset is itself window-dependent

There is a circularity worth stating. The onset is found by comparing each organ’s displacement against its own class’s tail mean, and the tail mean is taken over the reading window.

So moving the window moves the onset as well as the spread. The onsets quoted here are at the working window, and at sixty organs they differ — which means the check comparing a window’s start against an onset is comparing two quantities that both depend on the setting being checked.

That does not make the check useless. It makes it a consistency reading rather than an independent one, and the honest form of it is that at the working window three rows have onsets above where the widest window starts.

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 Where each row’s pattern begins, a quantity that itself depends on the window it is read through.

What would break the circle

A definition of the onset that does not use the window. Comparing each organ’s displacement against the final value rather than against a windowed mean would do it, at the cost of using information from the end of the run to describe its beginning.

That is a trade rather than a fix, and the collection has made it in the other direction elsewhere: a settling time is measured against a tail mean for exactly the same reason, and the wandering runs are excluded first so that a mean of a wandering tail is never taken.

Two runs of the same rule from unrelated starting angles. Both settle at 137.5°, within 0.0° of the golden angle, from seeds 166° apart.
Fig. 21 A run read organ by organ: a transient, and then a divergence that holds flat — which is the shape both of these readings are about.

What the three rows cost the classification

Nothing, in the sense that the classification is unchanged at the settings everything else in the thread uses. Twenty-five of thirty rows are periodic and the same twenty-five as before.

What they cost is a sentence. Any statement of the form the damage falls into a pattern at the surviving lag on twenty-five of thirty cuts now needs at a hundred and twenty organs of a three-hundred-organ run, because three of the thirty are decided by that pair of numbers and not by the stems.

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. 22 The spreads the classification is drawn on, of which three rows sit differently at different windows.

What the same rows do to the exchange

Nothing at all, and that is worth checking rather than assuming. The exchange is defined on rows with a balanced pair of exceptional chains, and whether a row is periodic is a separate reading of the same profile.

Two of the three movers are unpaired and go to the set the exchange sets aside; the third carries a balanced pair and is in the exchange table at every window. So the window moves a periodicity verdict and moves no row in or out of the exchange.

The 13 cuts the exchange sets aside, chain by chain. One row per wrecked cut with no balanced pair, one cell per chain of the lag it kept, and the chains displaced away from the common level filled. The lag is printed at the left and the sum of the displacements at the right, folded into half a turn either way. seven of the 13 sum to within 12 degrees of nothing and six do not, and every row that closes kept a lag of 7 or 8 while every row that does not kept 4 or 5. The rows are stacked by that lag.
Fig. 23 The rows with no balanced pair, two of which are also rows the window moves.

What a reader should carry

That three of the census’s thirty cuts have a periodicity verdict decided by how many organs are read, and that they are the three whose pattern begins latest in the run.

And that the two effects a window has are separable by arithmetic: a spread that triples when the window triples is drift, and a spread that grows forty times is a transient reached into. Nothing needs a new measurement to tell them apart.

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, with the two populations climbing at two rates.

What the picture at the top shows

Three bars, one per row, each drawn as the run from left to right. The pale stretch is the disturbance still healing and the dark stretch is the pattern repeating, with the boundary at that row’s onset.

The vertical marks are where each of the three windows begins. On all three rows the widest window’s mark falls inside the pale stretch, and on two of them it falls a hundred organs inside it.

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. 25 The three rows once more, with the widest window starting inside the disturbance on each.

The one line

Three of thirty cuts are periodic at sixty organs and not at a hundred and twenty, and their spreads grow by twenty to forty times across the windows against a proportional three — because a window is the tail of a run, and on these three the widest window begins more than a hundred organs below where their pattern starts.

At six hundred organs all three scale like everything else, so the effect is an interaction between the window and the run length rather than anything about the stems.

Where this goes next

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. 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. 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. An onset at the end of the run One cut reported that its pattern began 299 organs into a 300-organ run. Given twice the room it reports no pattern at all. The reading was the run stopping, not the disturbance ending, and the definition guarantees one at the last organ of every run.

What links here

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

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.

ArtefactClaim testingDamage profileHonest limitsInstrument settingMeasurement errorOnsetPeriodicityReading windowResidue classRun lengthTransient