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

Which chains changed places

A wrecked stem's displacement profile is a set of levels, one per chain, with two of them out of line — equal and opposite, on neighbouring chains. Nothing said which two. They are the hole's own chain and the one below it, on ten of the seventeen cuts that carry a pair.
13 min read 21 figures Counted, not admiredForm is not mechanism

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

Remove one organ from a settled stem, continue against a control that shares its history, and every organ placed afterwards sits some angle from where the same organ sits in the control. Folded on the lag the stem kept, those angles are not three hundred numbers but a handful of levels — one per chain, held to a few degrees, repeating for as long as the run goes on.

Most chains sit together and two do not: equal and opposite to within a twentieth, on neighbouring chains, on seventeen of the thirty wrecked cuts in the census. Two neighbouring chains displaced by equal and opposite amounts is what a pair of chains that have changed places looks like.

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. 1 One wrecked cut’s chains, drawn round its own period. Most sit on the middle ring and two do not.

What was left

The chains were labelled by residue, and a residue is a label. It said the two exceptional chains were adjacent and it said nothing about where they sit relative to the hole — whether the exchange happens next to the removed organ, or somewhere else, or in a different place each time.

That question is arithmetic on indices already computed. The profile is indexed by lag from the removed organ, so a chain’s label can be read as a lag rather than as a residue, and the chain the removed organ was on is the one whose lag is a multiple of the period.

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. 2 The same profile as a set of levels, which is the form the exceptional pair was first found in.

The convention, stated once

Chains are numbered from the removed organ: chain 0 is the chain it was on, chain 1 is the chain of the organ one place above it, and so on round the period. That is the control’s indexing, and it is the indexing the displacement is defined in — every row of the profile is a difference between the two runs at one control index.

There is a rival convention. The cut run has one fewer organ below the window, so numbering by its own ordinals shifts every label by one. It changes neither of the two results here, because both are differences: the orientation is a difference between two labels and the tally is over a fixed offset applied to all of them.

Take away the organ five places back, and the next one goes into the hole. The last 30 organs of a stem at a rise of 0.013, unrolled. The open circle is the organ removed — five places before the tip. The ring at the top is where the rule puts the next organ with every organ present; the filled mark beside it is where the rule puts it with that one missing. The two are 26.2° apart, against a local spacing of 41°, and the vacancy itself is 36.1° from the undisturbed answer. Nothing else differs between the two runs: same rise, same history, same rule.
Fig. 3 The removed organ and the two runs it separates, which is where the indexing this essay uses comes from.

The answer, and it is not one number

The backward-displaced chain is the hole’s own chain — chain 0 — on ten of the seventeen. On four more it is chain 2. On the remaining three it is chain 4, chain 5 and chain 6, once each.

Against a chain drawn at random from each row’s own period, ten at one value is against expectation of 2.8. So the exchange favours the hole’s own chain strongly and does not sit there always, and neither half of that is worth rounding away.

Which chain the backward exception sits on, over the census. Chains are numbered from the removed organ, so chain 0 is the chain the hole was on and chain 2 is two organs along it. The exchange is at the hole's own chain on 10 of the 17 rows that carry one, against 2.8 rows for a chain drawn at random from each row's own period. That is far more often than anywhere else and it is not every row, so the position is a tendency rather than a rule — and the file says so rather than rounding it up.
Fig. 4 The census tallied over the chains. Ten of seventeen at chain nought, against 2.8 for chance.

What a tendency is worth

A law would be better and this is not one. Ten of seventeen at a single value out of four to eight is a real concentration — it would happen by chance about once in a hundred thousand tables — and it leaves seven rows that need an account nobody has.

The temptation is to describe the ten and call the seven noise. They are not noise: their exchanges are as clean as the others, equal and opposite to within a twentieth, on adjacent chains. They are somewhere else.

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 How steady each class is over the census. The seven rows that put the exchange elsewhere are not the ragged ones.

It does not follow the offset

The obvious candidate is the offset that was cut — how far back the removed organ was. It fails. The 0.010 stem is cut six, seven and eight organs back and all three put the exchange at chain 0; the 0.008 stem is cut four, five and seven organs back and puts it at chains 2, 0 and 0.

So the exchange is not at a fixed place relative to the cut, and it is not at a place the cut’s own offset predicts. Two lattices, six cuts, and the two answers do not agree about which quantity is doing the work.

That is worth contrasting with what the offset does decide. It decides whether the stem wrecks at all, it decides how much the next organ moves, and on most lattices it decides which family survives. It is the strongest column in this whole thread and it is silent here.

The next organ moves for the last 8, and for no othersOne 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.organ removed, counted back from the tiphow far the next organ moves, in degrees1136.9°286.0°348.3°4164.1°526.2°692.3°7131.0°84.9°— the front ends here90.0°100.7°110.7°120.0°130.7°140.0°150.2°160.2°rise 0.013 · pair 5/8generated from a stated rule, not drawn to look right
Fig. 6 The offsets cut at one lattice and what each costs, which is the variable the exchange’s position does not follow.

Nor the surviving lag

The second candidate is the lag whose hop the stem kept, which is the period the profile is folded on. The chains at 0 come from cuts with surviving lags of 4, 5, 7 and 8; the chains elsewhere come from cuts with lags of 4, 5, 7 and 8.

Both lists are the same list. Whatever puts an exchange at chain 2 rather than chain 0 is not the period it happens inside.

One wrecked stem, lag by lag — golden, rise 0.013, organ 4 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 65 degrees. The lag-5 hop swings by 0.11 degrees and sits 0.09 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 5, which is the surviving lag and not a coincidence.
Fig. 7 The lag spectrum that identifies the surviving hop, measured independently of anything in this essay.

Nor the branch, nor the rise

Six of the ten chain-0 rows are golden and four are Lucas. Four of the seven others are golden and three are Lucas. The rises on both lists run from 0.005 to 0.026.

Four columns tried and four columns failing is the same shape the slot interaction produced on six rows before it was run on thirty, which is a reason to be careful: a quantity that sorts nothing on seventeen rows may be a quantity that has been sampled seventeen times.

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. 8 The census the seventeen rows are drawn from, with its lattices, offsets and surviving lags.

What would settle it

More rows. Seventeen is what thirty wrecked cuts yield once the rows whose exceptions are not a balanced pair are set aside, and thirty wrecked cuts is what twelve lattices yield. Twenty-four lattices would roughly double it.

That is a sweep this thread has already learned to want. The design that produced these thirty was built when the question was whether the damage had a shape at all, and it is now being asked a question about where inside that shape something sits.

Which lag survives, at every lattice and every offset. A row for each of twelve lattices and a column for each offset an organ is removed at, with the lag whose hop survived written in the cell. 30 cells never repair. The lag left standing is one of the two contact families at 29 of them, and at 17 it is the second shortest step on the lattice rather than the shortest, which is what refuses the reading that a rule holds its nearest neighbours. The ringed cells are the five the offset rule gets wrong. Two lattices carry no cells at all because no single removal wrecks them.
Fig. 9 Which offsets wreck a stem at each lattice, which is what sets how many rows a census of a given size yields.

The rows that carry no pair

Thirteen of the thirty do not have one exceptional pair. They have three exceptional chains, or four, or two that are not equal and opposite. Those rows are refused rather than reduced to their two largest exceptions.

Refusing them costs the census nearly half its rows and it is the right cost. A row with three exceptions is a different shape, and taking the largest two of three would manufacture a pair on every row that has one available — which is the failure mode a summary statistic computed past its own scope always has: it keeps returning a number after the thing it was a number for has gone.

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. 10 The exceptional chains on every row, with the rows that carry a balanced pair and the rows that do not drawn apart.

And they are not a separate population

The thirteen are not the ragged rows. Their profiles are as periodic as the seventeen — the widest spread inside a chain is under six degrees on most of them — and their exceptions are as far from the common level.

So the shape “one balanced pair” is not what damage does, with exceptions. It is one of at least two things damage does, and this thread has an account of the first and none of the second.

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 same census ordered by stem rather than by how steady it is, so the rows with and without a balanced pair can be located.

What the thirteen would need

An account of a profile with three exceptional chains is a different piece of work from an account of one with two, and this thread has not started it. The obvious first question is whether three exceptions are a pair plus something, or a three-cycle — three chains rotating rather than two swapping.

That is answerable from numbers already computed: a three-cycle displaces its members by amounts summing to zero, and a pair plus a stray does not. Nobody has added them up.

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. 12 The exceptional chains again, which is where the sums a three-cycle would have to satisfy are sitting uncomputed.

Which makes the ten a smaller claim

Ten of seventeen, out of thirty. Read against the whole census that is ten rows in three, and the sentence “the exchange is usually at the hole’s own chain” is a sentence about a subset chosen for having a clean exchange in the first place.

The subset is chosen by a property that has nothing to do with position — whether the two exceptions balance — so it is not obviously a biased sample. It is a selected one, and the difference between those two is exactly what cannot be checked without the rows that were dropped.

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. 13 The exceptional pairs by size rather than by lag, which is the property the seventeen were selected on.

What the position would mean

If the exchange were always at the hole’s own chain, the account would write itself: the chain that lost an organ is short by one, so the organs above the gap shift up a place and the chain next to it takes the strain. Ten rows out of seventeen is that account working most of the time.

The seven that put it two, four, five or six chains along have no such story. The organ that was removed is not on those chains and the exchange is not near it, so whatever propagates does so without leaving a mark on the chains in between.

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. 14 A whole profile, organ by organ, on one of the rows that puts its exchange away from the hole’s own chain.

The chains in between are undisturbed

That is the part worth pausing on. On a row whose exchange sits at chain 4, chains 1, 2 and 3 are at the common level to within a few degrees, and so are chains 5 and onwards.

So the disturbance does not spread from the hole and stop; it appears at a distance with nothing between. Every organ between the hole and the exchange is exactly where it would have been, and the two chains that moved are four chains away.

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 10.28° — so within a class the displacement is a constant. three classes sit at the common level. The two that do not sit at 143.0° and -147.6°, equal and opposite to within 3.2 per cent, and they are neighbouring residues. The stem's own divergence is 137.97°, so an exception is one organ's step.
Fig. 15 A profile whose exceptional pair is not next to the hole. The chains between sit on the common level.

Which is a mechanism-shaped fact

A displacement that appears at a distance with nothing in between is not what a local disturbance propagating outwards looks like. It is what a relabelling looks like: the arrangement above the hole is the control’s arrangement with two of its chains swapped, and where the swap happens is a property of the whole arrangement rather than of the hole.

That is a candidate account and it predicts something checkable: the position should depend on the arrangement’s own structure — its counted pair, its offsets — rather than on the cut. Four such columns have been tried here and none of them works, which is evidence against it.

The block is one of the two numbers, and not always the smaller. Every lattice a single organ was removed from, one row each: golden, rise 0.020 carrying 3/5; golden, rise 0.013 carrying 5/8; golden, rise 0.008 carrying 5/8; golden, rise 0.005 carrying 8/13; Lucas, rise 0.020 carrying 4/7; Lucas, rise 0.013 carrying 4/7. The two open ticks on each line are that lattice's own parastichy numbers; the filled dots are the blocks the stems that never recovered settled into, one per offset that failed. The claim this table was built to test is that the block is the smaller of the two, which held at the two rises it was first measured at. It does not hold here: 3/5 gives 5, 5/8 gives 5 and 8, 4/7 gives 4 and 7. What survives is weaker and still worth something — every filled dot but 1 sits on one of that row's own ticks, so the orbit carries a count of the lattice it was cut from, and which of the two it carries is decided by the offset rather than by the pattern.
Fig. 16 The orbit a wrecked stem falls into, which is the arrangement-level structure a relabelling account would have to depend on.

The instrument is not in doubt

Every number here comes out of the same two runs the rest of the thread uses, compared organ by organ at the same indices, over a window of a hundred and twenty organs at the top of a three-hundred-organ run. Nothing about the position reading is new machinery.

What is new is only the labelling, and the labelling is arithmetic that could have been done on the day the exceptional pair was found. It was not, which is the ordinary reason a leaving exists.

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. 17 Where the pattern starts on each row, which is the window every reading in this essay is taken above.

Except that the window is a choice

The levels are read over the last hundred and twenty organs of the run, and a longer run moves them. One of the seventeen rows here is a row that stops looking periodic when the run is doubled, which means its exchange is a reading taken inside a transient.

Removing it leaves sixteen rows and nine at chain 0, which changes nothing about the shape of the answer and is worth stating rather than leaving for a reader to work out.

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. 18 One profile at both run lengths, with the two readings of where its pattern starts marked.

What carries forward

An exchange has a definite location, it is at the hole’s own chain more often than anywhere else, and it is not always there. Four candidate accounts of the exception have been tried and none of them sorts the rows.

That is a partial regularity with an honest denominator, and it sits beside a result on the same seventeen rows that has no exceptions at all — the direction the exchange runs, which is the same on every one of them.

Which chain the backward exception sits on, over the census. Chains are numbered from the removed organ, so chain 0 is the chain the hole was on and chain 2 is two organs along it. The exchange is at the hole's own chain on 10 of the 17 rows that carry one, against 2.8 rows for a chain drawn at random from each row's own period. That is far more often than anywhere else and it is not every row, so the position is a tendency rather than a rule — and the file says so rather than rounding it up.
Fig. 19 The tally again, in chain order rather than by count, so the rows away from the hole can be read as a distribution.

A reading that is available and was not taken

The chains could have been labelled by azimuth rather than by index — ordered round the stem by where they sit, rather than by how many places above the hole their members are. The two orders are different: chains adjacent in index are one divergence step apart in azimuth, which is most of a turn.

Labelled that way, “adjacent chains” would become “chains a divergence apart”, and the exchange would look like a swap between two chains on opposite sides of the stem. The same measurement, described two ways, and the index labelling is chosen because the surviving hop is a lag and not a direction — every other reading in this thread is indexed the same way.

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 Another row’s chains round its period. The ring’s positions are indices, not azimuths, and the two orderings are not the same ordering.

What a reader should hold on to

That the exchange has a place, that the place is measured rather than assumed, and that it is at the hole’s own chain more often than anywhere else without being there always. Everything past that is a list of columns that do not sort the exceptions.

The list is short — the offset, the surviving lag, the branch, the rise — and it is the list anybody would have tried. What is missing is a column nobody has thought of, and this collection’s usual response to that is to widen the census rather than to think harder.

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. 21 The census as a spread, which is the table a wider sweep would be adding rows to.

The one line

The two chains that change places in a wrecked stem sit at the hole’s own chain on ten of the seventeen cuts that carry a clean exchange, at chain 2 on four more, and elsewhere on three — against 2.8 rows at any one chain by chance.

Which offset was cut, which lag survived, which branch and which rise all fail to sort the seven that are not at the hole, and the chains between the hole and a distant exchange are undisturbed.

Where this goes next

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°. 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. One level and two exceptions Inside a wrecked stem's period most residue classes sit at one level and a couple do not. On seventeen of the thirty cuts the exceptions are exactly two, equal and opposite to within five per cent — and on all seventeen they are neighbouring residues, which was not looked for. 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. One way round, seventeen times The two chains that change places in a wrecked stem are adjacent, which is symmetric and says nothing about direction. Label them by lag from the hole and the one displaced forwards is always the lower of the two — on every row of the census, without an exception.

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.

  • Two regimes above a hole — both name ablation, claim testing, control, description versus mechanism, honest limits, measurement, mechanism, negative result, resolution, rigid hop, summary statistic, transient
  • A window nobody aligned — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic, transient
  • Three rows change sides — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic, transient
  • A fifth of the hop — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, rigid hop, summary statistic
  • A step of one organ — both name ablation, claim testing, honest limits, lattice offset, measurement, mechanism, negative result, resolution, rigid hop, slip
  • An onset at the end of the run — both name ablation, census, claim testing, control, honest limits, measurement, negative result, resolution, summary statistic, transient

Named objects

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

AblationCensusClaim testingControlDescription versus mechanismDiscretisationHonest limitsLattice offsetMeasurementMechanismNegative resultResolutionRigid hopSlipSummary statisticTransient