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

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°.
13 min read 19 figures Counted, not admiredForm is not mechanism

Worth reading first: The organ that was taken away · Counting the spirals · A head is a set of points.

Remove one organ from a settled stem, continue the run against a control that shares its history, and every organ placed afterwards sits some angle away from where the same organ sits in the control. That is three hundred numbers per cut and there are thirty wrecked cuts in the census, so nine thousand numbers have been computed here across five rounds.

Two of them per cut have ever been read: the largest displacement anywhere, and the displacement of the first organ. This essay reads the rest.

How far every organ moved, 6 places back at a rise of 0.01. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control that shares its history. The collection has read two numbers out of profiles like this one — the largest displacement anywhere, and the first organ's — and never the profile. It is not a bump that decays. After about 18 organs it settles into a repeating pattern of eight levels, one per residue class modulo 8, which is the lag whose hop this stem kept. six of those levels sit together and two do not.
Fig. 1 One wrecked stem’s whole displacement profile, organ by organ above the hole, rather than the two numbers usually taken from it.

What the two numbers were for

The largest displacement was the search for a reference organ: something to measure a disturbance from, so that a removal could be described by where its effect is worst rather than by where the hole is. That search failed — the largest displacement turned out to be a plateau rather than a peak, and a maximum over a plateau names an arbitrary organ.

The first organ’s displacement was the search for a cost: how much a removal disturbs the rule that has to place the next organ. That one worked, and split the census cleanly: taking a chain-neighbour of the tip moves the next organ 8.9° to 30.7°, taking anything else 62.8° to 167.6°.

Neither reading looked at the shape.

What a cut moves, organ by organA stem counted at 5 and 8 spirals with the organ five places back from the tip removed, compared against a control that shares its history to the last digit. Each mark is one organ placed after the cut and how far its azimuth ended up from where the control put the same organ. The quantity folds at half a turn, so 152 degrees is near the largest displacement there is; and it does not decay with height, which is what a stem that never repairs means. There is therefore no organ that the cut disturbed most in any useful sense, and a reading that needs one has nowhere to stand.-200-1000100200204060organs above the removed oneazimuth moved (°)the removed organ was 5 places back5/8 at 0.013 · offset 5 · survivor 5generated from a stated rule, not drawn to look right
Fig. 2 The profile as it has been drawn before: a few organs above the hole, read for a maximum.

The reading that was missing

A profile is a sequence indexed by lag — how many places above the removed organ each one sits. A sequence indexed like that can be periodic, and the obvious period to try is the one the stem is known to have.

Because a wrecked stem does have one. Exactly one lag stays rigid: the angle from an organ to the one k places above it is unchanged from the control, organ by organ, while every other lag moves. That k is measured for every wrecked cut in the census and it is between 4 and 11.

So fold the profile on k. Take the last hundred and twenty organs of each run, split them into k residue classes, and ask how much the displacement varies inside a class.

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. 3 The lag spectrum that identifies the surviving hop, measured independently of anything in this essay.

The answer

On 25 of the 30 wrecked cuts, the widest spread inside any one residue class is between 0.12° and 6.09°.

The differences between classes on those same cuts run past a hundred and fifty degrees. So the profile is not a scatter of three hundred numbers; it is k levels, each one held to a few degrees, repeating for as long as the run goes on.

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. 4 Every wrecked cut, drawn at the widest spread found inside any one of its residue classes.

The five that are not

They are not near the line. Their spreads are 10.3°, 69.5°, 69.6°, 152.7° and 156.9°, against 6.09° for the widest of the twenty-five. Two populations with a gap between them, which means the threshold that separates them could have been drawn anywhere between about six and ten degrees.

That is the kind of separation this collection likes and it is worth naming the one soft edge: the nearest of the five is 10.3°, which is close enough to the 6.09° above it that a reader should know the gap is a factor of 1.7 and not a factor of ten. The other four are a factor of eleven clear.

The five are named rather than counted: one cut on the golden 0.026 stem, two on the golden 0.010 stem, and two on the Lucas 0.020 stem.

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 same census ordered by stem rather than by spread, so that the five can be located.

Two readings that name one number

The lag the profile is folded on is not fitted to the profile. It comes from a different measurement on a different quantity: the lag spectrum compares hops within the cut stem against hops within the control, while the profile compares the cut stem against the control organ by organ.

One is a difference taken inside a run; the other is a difference taken between runs. Neither is computed from the other, and they name the same k.

That is the two-routes check this site runs on everything it believes, and here it happened for free — the folding was tried on the surviving lag because it was the obvious thing to try, and the fact that it works is a second measurement of the same structure.

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. 6 One profile folded on the lag its stem kept, with the spread inside each class drawn as a bar.

What a period means here

A stem whose displacement is constant on each residue class modulo k is a stem whose arrangement differs from the control’s by something that repeats every k organs. Every organ in class 0 has moved by the same amount; every organ in class 1 by the same, different amount; and so on.

That is a much more specific object than “damaged”. It says the wrecked stem is still a regular arrangement — just a different one — and that the difference between the two is describable by k numbers rather than by three hundred.

It also says the damage does not heal and does not spread. A disturbance that was dying away would give classes whose means drift towards zero; a disturbance that was growing would give classes whose spread grows. Neither happens: the k levels are as flat at organ three hundred as at organ two hundred.

The block a wrecked stem settles into is the count it was cut from. A stem is cut in the middle of its front and followed for 300 organs. It does not come back to its lattice; what it does instead is repeat a fixed sequence of divergences exactly, to the resolution of the azimuth grid. Each row shows that sequence twice over, one bar per organ, drawn to the same scale. At the 5/8 rung the sequence is five angles long and advances -36.1° per block; At the 8/13 rung the sequence is eight angles long and advances 22.5° per block. Every block is the smaller number of the pair that was cut, so the second attractor carries the first one's count — and every precession is one part in twice the block, which makes each of these a two-jugate arrangement with 10 and 16 rows.
Fig. 7 The repeating motif a wrecked stem’s divergence sequence settles into, which is the same periodicity read in a different quantity.

Why 120 organs

The window has to hold several repeats of the lag or a class mean is a reading rather than a mean. At a surviving lag of 8 a window of 120 gives fifteen samples a class; at a lag of 11 it gives eleven.

It is the same window the lag spectrum uses, kept rather than re-chosen, so that the two measurements are made over the same stretch of the same runs. A different window would have made the agreement between them a comparison of two things measured at two places.

The check that the window is long enough is asserted rather than assumed: a call with a window under eight periods is refused. A class of four organs is steady whatever the profile does, and a measurement whose answer is guaranteed is not a measurement.

How far every organ moved, 7 places back at a rise of 0.008. 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 41 organs it settles into a repeating pattern of eight levels, one per residue class modulo 8, which is the lag whose hop this stem kept. six of those levels sit together and two do not.
Fig. 8 A second stem’s profile over the same window, on which the pattern is established well before the window begins.

Read against the reference-organ failure

The previous round’s search for a most-disturbed organ found a plateau: at sixteen of thirty rows, a fifth of the search window sits within a tenth of the largest displacement.

That is now a prediction rather than a disappointment. A profile with k levels attains its maximum on a whole residue class — every organ of one class, forever — so the share of any window within reach of the maximum is about one in k, which for k between 4 and 11 is a ninth to a quarter. A fifth is squarely inside that.

So the reference organ does not exist for a structural reason: the profile has no peak because it has no decay. Looking for the organ that moved furthest is looking for the first member of a class, and which member is first depends on where the window starts.

Three references, scored on the same thirty cuts. The reading under test says the family that lost a member is the one left standing, and it needs a reference organ to say which family lost one. Taken from the growing tip it can be asked on 9 of the 30 wrecked cuts and is right on every one. Taken from the organ the cut disturbed most it can be asked on 8, a different set, and is right on 3 of them. Taken as which side of the tip the removed organ sat on — arithmetic on the divergence, needing no reference organ at all — it can be asked on all 30 and is right on 22, against 18 for naming the commoner family outright.
Fig. 9 The reading that replaced the reference organ, which needed no most-disturbed organ at all.

What the levels are

Mostly one level, and a couple of exceptions. Between a quarter and six sevenths of the classes on a given cut sit together within 5.55° of each other, and the rest sit 12.4° to 160.4° away.

The exceptions are the subject of the next essay and they are where the mechanism is. What matters here is that they are few: a profile is not k arbitrary numbers, it is one number with two or three departures from it.

A period of 8, and the two classes that are not with the rest. The same wrecked stem, folded on the lag it kept: one row per residue class, each drawn at the mean displacement of its own organs against the level the rest of them share. The bar through each row is the spread inside that class, and the widest of them is 0.64° — so within a class the displacement is a constant. six classes sit at the common level. The two that do not sit at 135.1° and -135.0°, equal and opposite to within 0.1 per cent, and they are neighbouring residues. The stem's own divergence is 137.77°, so an exception is one organ's step.
Fig. 10 Another folded profile, on which most classes share a level and two do not.

What the common level is

A whole-stem offset. Every organ above the hole has moved by roughly the same amount, which is what happens when an arrangement slips: the pattern is intact and sitting somewhere else.

The size of that offset is not the same across the census — it runs from a fraction of a degree to more than a hundred and fifty — and it is not what this essay is about. It is the part of the damage that a single number describes, and the slip has already been measured: a wrecked stem’s surviving family slips by a whole number of turns per period.

The common level and the slip are the same fact seen from two sides.

A wreck is a whole number of extra turns. For each of the 19 stems that never repair, the slip of its settled divergence multiplied by the lag whose hop survived. Every value lands on a whole number of turns — the horizontal lines — with a largest departure of 2.97 degrees, against divergences that have moved between 0 and 103 degrees. 17 of the 19 close on exactly one turn. So a wrecked stem is the stem it was with one extra turn threaded through every period of the family that survived, which is a dislocation with a stated size rather than damage.
Fig. 11 The slip of every wrecked stem, in whole turns per period of the family it kept.

What this does not say

It does not say the profile is periodic everywhere. The window is the top hundred and twenty organs of a three-hundred-organ continuation, and below it the profile is doing something else — a transient, whose extent is its own measurement.

It does not say every wrecked stem is periodic; five of thirty are not, and nothing here explains them.

And it does not explain why the period should be the surviving lag rather than something else. That the two agree is a measurement. Why they agree is a statement about the placement rule, and this essay contains no such statement.

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. 12 How far above the hole the pattern takes to establish itself, which is the region this reading excludes.

What it changes about the census

Every number this thread has read off a displacement profile is now a number read off a level — or off the transient, or off the boundary between them.

The first organ’s displacement, which separates cheap removals from expensive ones, is in the transient. The largest displacement, which failed as a reference, is the extreme level. The survivor, the block and the slip are all properties of the pattern.

Sorting the readings that way is the useful thing this essay does. It means a future measurement on a profile has to say which regime it is in, and until now there was no reason to think there were two.

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. 13 The first organ’s displacement across every offset, which is a reading taken entirely inside the transient.

How the classes are numbered

From the end of the window, which is arbitrary and has to be said.

Splitting a hundred and twenty organs into k residue classes requires a starting point, and the one used is the first organ of the window rather than the removed organ. The two are three hundred organs apart and the number of organs between them is not a multiple of k, so class 0 here is not the removed organ’s own class.

Nothing in this essay’s claims depends on which class is which — a spread inside a class is a spread whatever the class is called — but two things later do. Whether the exceptional classes are adjacent is a statement about differences of class numbers, which survives renumbering. Anything that tried to relate a class to the cut would not, and nothing here does.

A period of 5, and the two classes that are not with the rest. The same wrecked stem, folded on the lag it kept: one row per residue class, each drawn at the mean displacement of its own organs against the level the rest of them share. The bar through each row is the spread inside that class, and the widest of them is 1.62° — so within a class the displacement is a constant. two classes sit at the common level. The three that do not sit at 67.7° and -149.5°, equal and opposite to within 75.3 per cent, and they are neighbouring residues. The stem's own divergence is 136.78°, so an exception is one organ's step.
Fig. 14 A folded profile with its classes numbered, which is a numbering from the top of the run rather than from the hole.

What the five exceptions have in common

Not much, and the honest answer is that they were not chosen and are not explained.

Two of the five are the two cuts on the Lucas 0.020 stem, which is the coarsest Lucas lattice in the census and the one whose wrecked stems land on a mirror rather than on an orbit. A mirror is a whole-arrangement reflection, and a reflection does not give a displacement that is constant on residue classes — it gives one that grows with the lag.

Two more are cuts four and five places back on the golden 0.010 stem, which are the only two of that stem’s five wrecked offsets that fail. The other three, at six, seven and eight, are among the cleanest rows in the census.

And the fifth is a cut four places back on the golden 0.026 stem, whose spread of 10.3° is the nearest of the five to the twenty-five that pass.

The wrecked stem is the lattice it was cut from, wound the other way. The divergence of each organ placed after two were removed from a settled stem at a rise of 0.032, over the 220 organs following the cut. The upper line is the divergence the undisturbed stem holds, 139.6875°; the lower is 220.3125°, which is 360° minus it and therefore the same lattice with the opposite handedness. The sequence is thrown by the cut, wanders for a few dozen organs, and settles on the second line to four decimal places — 220.3125° against 220.3125° — where it stays. A counter shown the positions afterwards returns 3 and 5, the pair the stem was cut from. Nothing about the pattern has been lost; its chirality has been reversed, which at this rung a single removal cannot do.
Fig. 15 A wrecked stem that has landed on the mirror of its own lattice, which is a shape a residue-class reading cannot describe.

What a real stem would have to show

An arrangement above a hole whose organs fall into chains, most of them where the undamaged pattern would put them and a couple of them one place along.

That is a photograph and a numbering, not a protractor. It is the same observation the specified ablation already calls for — the organs above the hole, identified and numbered — read for a different thing: not whether the pattern recovered but which chains ended up where.

It is also an observation that would be reported as “the pattern recovered” by almost any survey, since a stem with two chains exchanged still counts the same pair and still looks regular. That is the useful part of having the prediction: it says what to look at, and it is not what anybody would look at.

The two spiral families a counter finds between 0.68 and 0.92 of the radius. 34 spirals one way and 55 the other, found from the point positions alone — the counter is never told the divergence angle.
Fig. 16 An arrangement counted from its points, which is what a survey records and what two exchanged chains leave unchanged.

What the periodicity is not evidence for

That the wrecked stem is a lattice. A profile constant on residue classes says the difference between two arrangements repeats every k organs; it says nothing directly about whether the second arrangement is regular in its own right.

The second question has its own answer and it is measured elsewhere. A wrecked stem’s divergence sequence settles into a repeating motif, and a counter run over its points returns a pair — so it is a lattice by both of the tests this collection applies.

Keeping the two apart matters because the periodicity here is the easier measurement and the weaker statement. A difference that repeats could in principle be a difference between two irregular arrangements that happen to be irregular the same way.

The block a wrecked stem settles into is the count it was cut from. A stem is cut in the middle of its front and followed for 300 organs. It does not come back to its lattice; what it does instead is repeat a fixed sequence of divergences exactly, to the resolution of the azimuth grid. Each row shows that sequence twice over, one bar per organ, drawn to the same scale. At the 5/8 rung the sequence is five angles long and advances -31.4° per block. Every block is the smaller number of the pair that was cut, so the second attractor carries the first one's count — and every precession is one part in twice the block, which makes each of these a two-jugate arrangement with 11 rows.
Fig. 17 The repeating motif of a wrecked stem’s divergence sequence, which is the separate evidence that it is a lattice.

Why the fold was tried on that lag and not another

Because it is the only lag the stem is known to have.

A profile could in principle be periodic at any period, and finding one by searching is a different exercise with a different risk: a search over periods 2 to 24 on a three-hundred-organ sequence will find something, and what it finds will need a null model.

Folding on a lag measured independently avoids all of that. There is one candidate, it comes from a different quantity, and the test is a yes or no rather than a maximum over a range. The twenty-five rows that pass do so at a period nobody chose.

The exercise a search would have been is still available and would answer a different question — whether any other period also works — and it is not run.

One wrecked stem, lag by lag — golden, rise 0.005, organ 7 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 91 degrees. The lag-8 hop swings by 0.00 degrees and sits 0.23 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 8, which is the surviving lag and not a coincidence.
Fig. 18 The single lag a wrecked stem keeps, which is the only candidate period the fold was tried at.

The one line

Folded on the lag the stem kept, the displacement above a hole is constant within each residue class to between 0.12° and 6.09° on twenty-five of the census’s thirty wrecked cuts, against between-class differences past a hundred and fifty degrees. The lag is measured independently of the profile and the two agree. The damage is not a bump that decays; it is a repeating pattern of a few levels that never stops.

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. 19 The whole census on one axis, with the five rows that are not periodic separated from the twenty-five that are by a gap.

Where this goes next

What a cut moves, organ by organ. A stem counted at 5 and 8 spirals with the organ five places back from the tip removed, compared against a control that shares its history to the last digit. Each mark is one organ placed after the cut and how far its azimuth ended up from where the control put the same organ. The quantity folds at half a turn, so 152 degrees is near the largest displacement there is; and it does not decay with height, which is what a stem that never repairs means. There is therefore no organ that the cut disturbed most in any useful sense, and a reading that needs one has nowhere to stand. The organ that moved furthest A reading that works on nine of thirty rows needs a reference organ, and the obvious repair is to measure one rather than to choose it. Measured, the disturbance turns out to have no far edge at all, so there is no organ that moved furthest in any sense the reading can use — and the generalisation that does work needs no reference organ. One wrecked stem, lag by lag — golden, rise 0.005, 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 60 degrees. The lag-8 hop swings by 0.09 degrees and sits 0.19 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 8, which is the surviving lag and not a coincidence. The hop that survived A stem that never repairs after an organ is removed settles into an exactly repeating block of angles, and the period of that block is a spiral count of the lattice it was cut from. Nobody could say why. Read the wrecked stem by lags rather than by neighbours and the answer is one line: one family of the original lattice is still standing, organ by organ, and the block is its 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. 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.

What links here

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

Reads more easily once this is understood

Essays that name this one as worth reading first.

Shares its objects with

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

  • A step of one organ — both name ablation, claim testing, lattice offset, measurement, mechanism, resolution, rigid hop, slip
  • A removal that changes nothing — both name ablation, claim testing, control, discretisation, measurement, mechanism, resolution
  • The angle is not the actor — both name ablation, claim testing, control, description versus mechanism, lattice offset, mechanism, rigid hop
  • A period the grid invented — both name ablation, claim testing, discretisation, measurement, resolution, summary statistic
  • Both walls of the slot — both name ablation, claim testing, control, lattice offset, measurement, mechanism
  • One offset, two answers — both name ablation, claim testing, control, lattice offset, measurement, rigid hop

Named objects

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

AblationClaim testingControlDescription versus mechanismDiscretisationLattice offsetMeasurementMechanismResolutionRigid hopSlipSummary statisticTransient