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

Three rows change sides

Twenty-five of thirty wrecked cuts have a periodic displacement profile over three hundred organs and twenty-six do over six hundred. The count barely moves and the membership does: two rows join, one leaves, and the gap the threshold sits in narrows from 1.69 to 1.27.
13 min read 20 figures State the claim, state the testCounted, not admired

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

A wrecked stem’s displacement profile is called periodic when the widest spread inside any one residue class is under ten degrees. Over three hundred organs, twenty-five of the thirty cuts in the census pass. Over six hundred, twenty-six do.

The count moved by one and the membership moved by three. Two rows that failed the test at the shorter length pass at the longer, and one that passed fails.

The reading itself is the thread’s central result: a wrecked stem’s damage is not three hundred scattered numbers but a few levels, one per chain, repeating. What is at stake here is not that shape but which rows have it.

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. 1 The three rows the two run lengths disagree about, drawn at their widest within-class spread at each length.

The two that join

The golden 0.010 stem’s cuts four and five organs back. Over three hundred organs their widest spreads are 69.6° and 69.5°, which is not marginal — it is seven times the line. Over six hundred they are 7.9° and 8.1°.

A factor of nine, on two rows, from nothing but a longer run. Whatever those profiles are doing over the first three hundred organs, they are not doing it over the next three hundred. Neither row is near the line at either length, which is the useful part: this is not a marginal case tipping over, it is a reading that was wrong by an order of magnitude and is now right.

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. 2 One of the two, over six hundred organs. The stretch a three-hundred-organ run holds is drawn paler than the rest.

Which is a transient with a long tail

The obvious reading, and it is the right one. The levels are measured over the last hundred and twenty organs of whatever run was grown. At three hundred organs that window sits between organs 180 and 300, which on these two rows is still inside the disturbance; at six hundred it sits between 480 and 600, which is past it.

So the shorter run was measuring a transient and calling it a failure of periodicity. The profile is periodic; the window was in the wrong place — which is the same diagnosis the onset reading got from the other direction, since both are computed against levels taken from the end of whatever run was grown.

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. 3 The onset and the transient’s size on every row. A window sitting inside a transient is what these two rows demonstrate.

Two rows or one stem

The two are cuts four and five organs back on the same stem, so they share a control and differ only in which organ was removed. Their spreads at both lengths agree to a tenth of a degree: 69.6° and 69.5°, then 7.9° and 8.1°.

That is one lattice behaving one way, observed twice, rather than two independent confirmations. Counting them as two rows is what the census does everywhere and it is worth flagging here, because two rows changing sides sounds like a pattern and one stem changing sides is an anecdote.

The next organ moves for the last 8, and for no others. One row per organ removed, counted back from the tip of a stem at a rise of 0.013 whose counted pair is 5 and 8. Removing any of the last 8 moves the next organ by 4.9° to 164.1°; removing an older one moves it by at most 0.70°, which is under the azimuth grid. The boundary is at 8, and 8 is the larger parastichy number — so the experiment counts the spirals without measuring an angle.
Fig. 4 The offsets cut at that lattice and what each costs. The two rows in question are neighbouring offsets on one stem.

And it is a selection effect in the census

The five rows that never reached a pattern within three hundred organs were named as a group and treated as a category — the cuts whose damage never settles. Two of the five are now rows whose damage settles late.

That leaves three, and three is a small enough number that the category may not be a category at all. It might be one row, twice, plus a genuine case. And one of the three is the row whose onset turned out to be its run ending, so the group has been carrying an artefact as well as a duplicate.

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 by spread. The group at the right-hand end is smaller than it was and its membership has changed.

The one that leaves

The Lucas 0.013 stem’s cut four organs back. Over three hundred organs its widest spread is 6.1°, comfortably inside the line; over six hundred it is 171.2°, which is past a half turn and as far outside as a number can be.

That is not a transient ending. It is a profile that looks steady over one window and is not steady at all over a longer one.

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. 6 The row that leaves, over six hundred organs. The stretch a shorter run would have measured is the paler one on the left.

What could make a profile look steady and not be

Two things, and they are distinguishable. A profile that drifts slowly has classes whose means move together; over a short window each class looks tight and over a long one it does not. A profile that jumps — settles onto one set of levels and later onto another — looks tight on either side of the jump and ragged across it.

The row that leaves is the second kind. Its spread over six hundred organs is 171.2°, which is not a drift accumulating; it is two regimes with a large step between them. Nothing in this thread had a category for a wrecked stem that settles twice.

How far every organ moved, 4 places back at a rise of 0.013. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control that shares its history. The collection has read two numbers out of profiles like this one — the largest displacement anywhere, and the first organ's — and never the profile. It is not a bump that decays. After about 3 organs it settles into a repeating pattern of five levels, one per residue class modulo 5, which is the lag whose hop this stem kept. three of those levels sit together and two do not.
Fig. 7 A profile’s first hundred and twenty organs. A row that settles and then settles again somewhere else would look exactly like this over its first window.

Which is the more awkward direction

A row that joins is a window in the wrong place, and the fix is a longer run. A row that leaves is a window that was in a place where a non-periodic profile happened to look periodic, and the fix is not obvious: a longer run found it, and a longer run still might find others.

The two directions have different implications for the census. One says the count of non-periodic rows is too high; the other says the count of periodic rows is too high. Both cannot be corrected by growing runs until nothing changes, because nobody knows where that is.

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. 8 Every row’s spread at both lengths. Marks on the diagonal are rows the two lengths agree about; the three that changed sides are drawn larger.

The row that leaves is the only one of its kind

Its surviving lag is 4, and it is the only wrecked cut in the whole census that keeps a four-hop. Every other row keeps a 5, a 7 or an 8.

That matters because a class of four residues is the shortest period in the census, so its classes have the most members and the least room to look accidentally steady. If any row were going to look periodic by luck, this is the one where luck should have had the least chance.

One wrecked stem, lag by lag — Lucas, rise 0.013, organ 5 backHow much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 119 organs. The lag-one hop is the divergence itself and it swings by 48 degrees. The lag-7 hop swings by 0.00 degrees and sits 0.53 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 7, which is the surviving lag and not a coincidence.0°30°60°12345678910111213141516lag, in organshow much that hop moves (°)lag 7: 0.00°Lucas, rise 0.013 · organ 5 back · block 7the surviving lag is 7
Fig. 9 A lag spectrum on the Lucas branch, which is where this row’s surviving four-hop is identified.

And a separate reading calls it out

Four accounts of the exchanged pair’s size are scored on the seventeen rows that carry a clean exchange, and the best of them fits worst on exactly this row — 4.0 per cent against 2.3 for the next worst and 1.0 for the mean.

Two measurements, made for unrelated reasons, neither aware of the other, agreeing about which row of the census is least trustworthy. That agreement is worth more than either measurement, and it is the kind this collection looks for deliberately.

What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17.
Fig. 10 The exchange’s residual against the surviving hop. The worst-fitting point is the row the two run lengths disagree about.

The gap narrows

The ten-degree line was chosen because the two populations were separated by one. At three hundred organs the periodic rows measured up to 6.09° and the rest started at 10.28°, a factor of 1.69. At six hundred the periodic rows reach 8.11° and the rest start at 10.28°, a factor of 1.27.

Still a gap and a much smaller one. A threshold that sat in empty space now sits in nearly-empty space, which means it has started to matter where it is.

How constant the displacement is inside one residue class. One row per wrecked cut in the census, drawn at the widest spread found inside any one residue class when the profile is folded on the lag that stem kept. 25 of 30 rows sit between 0.12 and 6.09 degrees, which on a quantity whose between-class differences run past a hundred and fifty degrees is a constant. The five that do not sit from 10.3° up. There is nothing in between, so the line drawn at 10° could have been drawn anywhere in a wide interval.
Fig. 11 The census by stem rather than by spread, so the rows now closest to the line can be located.

Which is a warning rather than a failure

Nothing here is decided by the tenth of a degree either side of ten. The row that leaves is at 171.2° and the rows that join are at 7.9° and 8.1°, so a line anywhere between about nine and ten gives the same table.

What has changed is the margin. At three hundred organs the classification was robust to a factor of 1.69 in the threshold; at six hundred it is robust to 1.27, and at twelve hundred it might not be robust at all. That is a thing to check rather than a thing to worry about now.

The margin is also the reason the threshold was ever quotable. A tolerance chosen because it sits in empty space is a tolerance nobody had to argue about, and the arguing starts as soon as the space fills. Watching a gap close is the cheapest early warning a thread with stated tolerances can have, and it is only available because the gap was reported alongside the line rather than in place of it.

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. 12 The same picture, with the ten-degree rules drawn on both axes so the narrowing gap can be read against them.

What the count hides

Twenty-five and twenty-six is a comforting pair of numbers and it is the least informative reading of the comparison. Three rows changed sides and the count moved by one because two changes cancelled one.

A census reported as a count is a census whose membership can turn over without anybody noticing, and this one turned over by ten per cent of its periodic rows while the headline moved by four.

The 1 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. 13 The row that leaves, on its own. Reporting only the count is what makes a row like this invisible.

What a second settling would mean

If a wrecked stem can settle onto one arrangement and later move to another, then the block a wrecked stem falls into is not a final state, and the census’s readings of what a cut leaves standing are readings of whatever the stem happened to be doing at organ three hundred.

That is a large claim to hang on one row and this essay does not hang it there. What it does is name the shape and say which measurement would settle it: the surviving lag, read separately over the first and second halves of a six-hundred-organ run. If the lag changes, the stem moved; if it does not, the profile drifted inside one arrangement.

Two readings on one row, on runs already grown. It is the cheapest open question in the thread.

Every stem that never repaired, and the lag it kept. The 19 offsets across six lattices at which a single removal leaves a stem that never returns to its divergence. For each one: which organ was removed, the period of the block of angles the stem settles into, the lag whose hop survived the cut unchanged, and how many whole turns the stem gains over one period of that lag. The block and the surviving lag are the same number in every row. The marked row is the one whose survivor is not a parastichy number of the lattice that was cut — a hop 6.8 times the length of a contact hop, which no census would report and which the rule held rigid all the same.
Fig. 14 What each wrecked cut keeps. If a stem can settle twice, this table is a snapshot rather than a result.

What the shape survives

The result the thread rests on — that a wrecked stem’s profile is a handful of levels rather than three hundred scattered numbers — is not in doubt at either length. Twenty-six of thirty rows have spreads under ten degrees against between-class differences past a hundred and fifty.

And the exchanged pair, its size and its direction, are unchanged on every row that carries one at both lengths. What moves is the classification of the marginal rows and every reading about timing.

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. 15 A row’s levels, which are the same at both lengths on every row that has them at both.

What a third length would settle

Twelve hundred organs on the thirty. Four minutes, and it answers two questions: are the two rows that joined still periodic, and does any further row change sides.

If nothing changes between six hundred and twelve hundred, the six-hundred classification is the one to quote and the three-hundred one was under-run. If more rows change, then the classification has no limit and the quantity needs a definition that does not depend on where a run was stopped.

That is the distinction the settling thread draws between a budget and a wall, and it is drawn there by exactly this comparison: grow to one length, grow to three times it, and see whether anything new arrives.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 16 The settling thread’s three-length comparison, which is the design this one would have to become.

Why it has not been run

The comparison in front of it was built to check one number — an onset reported at the last organ of its run — and finding a group of rows that change sides is something it did on the way. A sweep designed to check a number and then asked to characterise a group is always one length short.

That is not an excuse so much as a description of how this keeps happening. The cheapest useful thing is nearly always the next power of two.

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. 17 The comparison this one came out of, which was designed around a single suspicious reading.

What it means for the five

The five rows that never settle are now three, and one of the three is the row whose onset was an artefact. So the category “damage that never becomes a pattern” has one or two members that anybody has confidence in, and it has been carried as a group of five through several essays.

Rewriting that is the correction this comparison forces. The group is smaller, its membership is different, and the two rows that left it did so by becoming periodic rather than by being reclassified on a technicality.

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 again. The rows with no mark are the group this comparison shrinks.

The threshold’s other half

Ten degrees is one of two tolerances in this reading and the other has not moved. A class counts as an exception when it sits more than ten degrees from the level the rest share, and over the whole census the classes at the common level are within 5.55° of it while the exceptions are 12.4° and up.

That gap is a factor of 2.2 and the doubling does not narrow it, because it is a gap between levels rather than a gap in how steady a level is. So of the two tolerances in the periodicity reading, one is being squeezed by the run length and the other is not, and the essays quoting them have treated the pair as one number.

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. 19 The exceptional classes across the census, which is the reading the second tolerance belongs to.

Why a classification is worth this much attention

Because everything downstream is counted over it. The exchange thread works on the seventeen rows that carry a clean balanced pair, and whether a row carries one is decided on the same levels the periodicity reading uses. A row that changes sides here can change the denominator there.

In this case it does not: the row that leaves still carries a balanced pair at the shorter length, which is where the exchange was read. But it means the two threads share an instrument, and a change to the instrument moves both.

A period of 8, with the hole's own chain at the top. Each mark is one residue class of the displacement profile, placed round a ring at its own residue, with the chain the removed organ sat on at the top. The radius is how far that class sits from the level the rest of them share. six of the eight classes sit together at the middle ring; two do not, and on this row they are one pair, equal and opposite to within a twentieth. The forward one is chain 7 and the backward one is chain 0, one residue above it, which is the order every row of the census puts them in.
Fig. 20 The exchange, which is read off the same levels the periodicity reading is computed from.

The one line

Doubling every run moves three rows of thirty across the line between a periodic profile and a ragged one — two into the periodic group by a factor of nine, one out of it by a factor of twenty-eight — while the count moves by one because the changes partly cancel.

The row that leaves is the census’s only four-hop survivor and is independently the worst fit of a scoring made for an unrelated reason, and the gap the threshold sits in narrows from a factor of 1.69 to 1.27.

Where this goes next

Where the pattern starts, measured at two run lengths. One row per wrecked cut. The small mark is the onset a 300-organ run reports and the ring is what a 600-organ run reports; a row with only one mark reports an onset at only one length. 19 of the 24 rows that report both move by more than twenty organs, and the largest move is from 135 to 435. The range the thread has been quoting, 7 to 303 organs, becomes 28 to 473. Twice the run Five wrecked cuts never reached a pattern inside three hundred organs and one reached it at three hundred and three, which is a number asking to be checked. Run every cut in the census twice as far and three of the thirty change their answer. 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°. What the exchange misses one divergence step by, against the surviving hop. Each mark is one wrecked cut. Across: the angle from an organ to the one its surviving lag places above it, on the control. Down: how much the exchanged pair's size falls short of one divergence step. The relationship is a straight line through the origin with a slope of about 0.2, and its sign is right on every row of the census — the four surviving lags give hops on both sides of zero and the residual follows each of them. The honest limit is the four: the hop hardly moves inside a lag, so the line rests on 11 distinct positions and not on 17. A fifth of the hop The exchanged pair misses one divergence step by up to twelve per cent, and the miss is not scatter: every row keeping a lag of 5 or 7 overshoots and every row keeping a 4 or an 8 falls short. Subtract a fifth of the surviving hop's own angle and the worst row is four per cent.

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.

  • A window nobody aligned — both name ablation, artefact, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic, tolerance, transient
  • Which chains changed places — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic, transient
  • The plateau was a prediction — both name ablation, artefact, claim testing, control, honest limits, measurement, negative result, rigid hop, summary statistic, transient
  • Two regimes above a hole — both name ablation, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic, transient
  • The alternation is not a period — both name ablation, artefact, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop
  • The offsets that never change — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop

Named objects

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

AblationArtefactCensusClaim testingClassificationControlDiscriminationHonest limitsMeasurementNegative resultResolutionRigid hopSummary statisticToleranceTransient