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

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.
12 min read 19 figures Counted, not admiredForm is not mechanism

Worth reading first: The damage has a period · The organ that was taken away · A head is a set of points.

The displacement above a hole is periodic at the lag the stem kept, which turns three hundred numbers into k of them. This essay is about what those k numbers look like, and they are not k different numbers.

Most of them are the same number. The interesting part is which ones are not, and how many.

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. 1 One wrecked stem’s period, class by class, drawn against the level most of them share.

The common level

Take the k class means of a wrecked cut and take their median. Between a quarter and six sevenths of the classes sit within a few degrees of it, and across the whole census the largest departure among those is 5.55°.

The median is used rather than the mean for the obvious reason: a couple of classes sitting a hundred and forty degrees away would drag a mean into a value no class holds. The median is a value some class actually has.

What the common level is has already been measured from another direction. A wrecked stem’s arrangement has slipped: the family it kept slips by a whole number of turns per period, and a whole-stem slip displaces every organ by the same amount. The common level is that slip seen organ by organ rather than sequence by sequence.

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. 2 The slip of every wrecked stem in whole turns, which is the common level read as a property of the sequence.

The exceptions

Every cut has some. Across the census the classes that are not at the common level sit between 12.4° and 160.4° away from it, so there is nothing between 5.55° and 12.4° — the two groups are separated by a factor of more than two, and the line drawn at ten degrees could have been drawn anywhere between six and twelve.

How many exceptions a cut has varies. On 17 of the 30 wrecked cuts there are exactly two. On the rest there are three, four, five or six.

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. 3 The census by how constant each class is, which is the measurement that has to hold before classes can be counted.

Equal and opposite

On those seventeen rows the two exceptions are not two arbitrary departures. One is above the common level and one is below it, and their sizes agree to within five per cent of their mean.

That is a strong condition and it is stated as a bound rather than fitted. The claim being tested is that a pair of chains has been displaced in opposite directions by the same amount, which is what changing two chains over would do; a tolerance of five per cent is loose enough for the grid the azimuths are read on and tight enough that an unequal pair fails it.

Seventeen rows pass. The rows with three or more exceptions are not scored, because “one balanced pair” is not a statement about a set of five.

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. 4 The size of the balanced pair on every row that has one, measured against the stem’s own divergence.

And they are adjacent

This is the part that was not looked for.

On all seventeen rows the two exceptional classes are neighbouring residues — class c and class c + 1, modulo k. Not two classes some way apart, not a class and its opposite. Adjacent.

Nothing in the measurement encourages that. The classes are numbered by position in the window, which is arbitrary; the pair could have sat anywhere in the cycle; and seventeen rows agreeing on adjacency across four different values of k, on two branches and five rises, is not a coincidence anybody set up.

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. 5 A second stem’s period, on which the two exceptions again sit side by side.

What adjacent means

Two organs in adjacent residue classes are one place apart in the sequence: organ i and organ i + 1. So the structure is two chains of organs — the class of every organ congruent to c, and the class of every organ congruent to c + 1 — displaced from the control by equal amounts in opposite directions, with every other chain sitting where a whole-stem slip put it.

The obvious reading is that the two chains have changed places. If the organs of one chain now sit where the organs of the next one sat and vice versa, each is displaced by one step of the sequence and the two displacements are opposite by construction.

That reading makes a size prediction, and the size is measurable.

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. 6 The profile before folding, on which the two exceptional chains appear as two values recurring every k organs.

The rows with more than two

Thirteen of the thirty have three, four, five or six exceptions. They are not failures of the periodicity — most of them are among the twenty-five whose classes are constant to a few degrees — they are cuts whose difference from the control is more than one exchange.

Two of them are extreme: on the golden 0.005 stem, cutting seven or eight places back leaves six of eight classes away from the common level, at −45.0°, 132.7°, 90.5°, 90.9°, −135.5° and −133.8°. That is a stem whose arrangement differs from the control’s in most of its chains, and calling the remaining two the “common level” is a stretch the median makes possible and the reading should not lean on.

Those rows are reported and not scored, which is the right treatment for a row where the statistic is defined and the description is not.

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. 7 Every wrecked cut, so that the rows with many exceptions can be found rather than inferred.

Where the counts come from

A class counts as an exception if its mean sits ten degrees or more from the common level. That number is a gap rather than a threshold — measured, the non-exceptions reach 5.55° and the exceptions start at 12.4° — so a line anywhere in that interval gives the same table.

It is worth checking what a badly chosen line would do. At five degrees, a handful of classes at the top of the common group would be counted as exceptions and rows that have a balanced pair would report four. At twenty degrees, the smallest real exceptions would be absorbed and rows would report zero. Both would change the count of seventeen; neither changes the adjacency, because the classes involved are the same ones.

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. 8 A third stem, whose exceptions sit far enough from the common level that no line between six and twelve degrees moves them.

What the pair is not

It is not the surviving family. The surviving lag is k, the period itself, and the exceptional pair is a pair of classes within that period — so the two are different objects and the exceptions do not name a lag at all.

It is not the removed organ’s own class. The removal is at the bottom of the run and the classes are read at the top, three hundred organs later; there is no alignment between the two that this measurement preserves, because the classes are numbered from the end of the window.

And it is not the front. The front is the stretch of stem a removal is felt across, which is a property of the transient below the pattern rather than of the pattern.

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. 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 rows.
Fig. 9 The repeating motif of the divergence sequence, which is the same period seen in the angles rather than in the displacements.

What it would take to break the adjacency

One row. Seventeen rows agree and a single row with a balanced pair two residues apart would end the claim, which is what makes it worth asserting rather than observing.

The claim is asserted in the library over every row that has exactly one balanced pair, with the count of such rows also asserted to be at least twelve — so a future change that quietly reduced the population to two rows could not pass by having both of them agree.

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 balanced pairs sorted by size rather than by lag, which is the same seventeen rows in a different order.

Read against the whole-stem slip

The picture the two facts give together is specific. A wrecked stem is the control lattice with every organ displaced by one amount, except for two adjacent chains which are displaced by that amount plus and minus one step.

That is a much smaller object than “the arrangement was destroyed”. It has three numbers in it — the slip, the size of the exchange, and which two chains — and the first has been measured for two rounds, the second is measurable, and the third is not obviously predictable from anything.

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 0.85° — so within a class the displacement is a constant. six classes sit at the common level. The two that do not sit at 133.2° and -133.0°, equal and opposite to within 0.1 per cent, and they are neighbouring residues. The stem's own divergence is 137.84°, so an exception is one organ's step.
Fig. 11 A fine stem’s period, where the same shape appears with eight classes rather than five.

What a plant would show

Nothing measurable with a protractor, and something measurable with a ruler.

Two adjacent chains exchanged is a statement about which organ is where, not about angles. On a real stem with a hole in it, the prediction is that the organs above the hole fall into their usual chains except for two neighbouring ones that are swapped — which is a statement about the arrangement’s combinatorics and is exactly the kind of thing a photograph and a numbering can settle.

It is also the kind of thing that would be invisible to a count. A stem with two chains exchanged still has k chains and still returns its counted pair, so nothing in a survey that records counts would see it.

The two spiral families a counter finds between 0.43 and 0.67 of the radius. 21 spirals one way and 34 the other, found from the point positions alone — the counter is never told the divergence angle.
Fig. 12 What a counter reads off an arrangement, which is unchanged by two chains changing places.

The exceptions across the branches

Seventeen rows, four values of the surviving lag, and two branches, and the shape is the same on all of them.

On the golden branch the pairs sit at ±134° to ±147° and on the Lucas branch at ±88° to ±104°, which is the two branches’ divergences showing through — a golden stem settles near 137° and a Lucas one near 99°. That the exceptional pair scales with the branch’s own angle rather than being a fixed number is the next essay’s subject.

What does not vary is the adjacency or the balance. A Lucas 7/11 stem with a pair at +104.2° and −104.0° on classes four and five, and a golden 5/8 stem with a pair at +134.3° and −134.2° on classes five and six, are the same structure at two sizes.

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. 13 A coarser stem’s period, on which the same two-adjacent-exceptions shape appears over a different common level.

What is not measured about the pair

Which two chains. The classes are numbered from the top of the run and the numbering has no relation to the removed organ, so “classes five and six” is not a statement about where the exchange sits relative to the hole.

Recovering that is arithmetic on the run’s indices and it was not done. It would say whether the exchanged pair is at a fixed place relative to the cut, or drifts, or is set by the offset — and any of those would be a step towards an account rather than a description.

It is a leaving rather than an omission: the measurement is available in the same runs, it costs nothing, and it did not fit in the round.

How far every organ moved, 5 places back at a rise of 0.013. One mark per organ above the hole, at the angle it sits from where the same organ sits in a control that shares its history. The collection has read two numbers out of profiles like this one — the largest displacement anywhere, and the first organ's — and never the profile. It is not a bump that decays. After about 40 organs it settles into a repeating pattern of five levels, one per residue class modulo 5, which is the lag whose hop this stem kept. two of those levels sit together and three do not.
Fig. 14 A profile drawn from the hole upwards, on which the exchanged chains could be located relative to the cut.

Reading it with the surviving hop

A wrecked stem keeps exactly one lag whose hop is unchanged, and that lag is the period the profile is folded on. The exceptions are two classes inside that period.

Put together: the surviving family’s chains are all intact and all displaced alike, except that two neighbouring ones have swapped. A hop measured along the surviving family steps from a class to itself, k organs along, so it does not cross the exchange — which is exactly why the hop survives while the arrangement has been rearranged.

That is not a proof; it is a consistency, and it is the first time the two measurements have been read against each other. If it is right, the surviving lag is not a fact about which family is robust but a fact about which lag the exchange is invisible to.

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.
Fig. 15 The lag spectrum of a wrecked stem, on which one lag is unchanged and every other has moved.

What a class mean is a mean of

Between eleven and thirty organs, depending on the surviving lag, taken over the top hundred and twenty organs of a three-hundred-organ continuation.

At a lag of 4 a class has thirty members and at a lag of 11 it has eleven. That range matters for the spreads: a class of eleven has a noisier mean than a class of thirty, so the rows with long periods should show slightly larger spreads for no reason but sample size.

They do not, particularly. The spreads run 0.12° to 6.09° across the twenty-five periodic rows and the largest of them are not the long-period ones — the 6.09° is a lag-5 row and several lag-7 and lag-8 rows sit under a degree. So whatever varies within a class is not sampling noise, and the classes with fewest samples are not the noisiest.

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. 16 The spreads across the census, which do not order themselves by how many organs each class holds.

Where the common level comes from

A wrecked stem has slipped: its divergence sequence settles on a value that differs from the control’s, and the difference accumulates organ by organ into a whole-stem offset.

That offset is not small. Across the seventeen paired rows the common level runs from a fraction of a degree to more than a hundred and fifty, and the slip that produces it is a whole number of turns per period of the surviving family — 72° per organ on a stem keeping a 5, 45° on one keeping an 8.

So the two numbers this essay separates have very different provenances. The common level is the slip, which has been measured for two rounds from the divergence sequence. The exceptions are new and are not in the sequence at all: they are a rearrangement that a sequence of consecutive divergences averages over.

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. 17 The slips of every wrecked stem, which is the common level measured from a different quantity.

What the rows with several exceptions might be

Not measured, and there is an obvious guess.

If one exchange gives two exceptional classes, two exchanges elsewhere in the same period would give four, and three would give six. The rows with more than two exceptions have three, four, five or six of them — and four and six are what two and three exchanges would produce.

Testing it means sorting the exceptions into balanced pairs and checking that each pair is adjacent and one divergence step in size. With six exceptions there are fifteen ways to pair them up, so the test has to be stated carefully or it will find a pairing on any set.

The clean version: require that the exceptions partition into adjacent pairs with no class left over. On a row with five exceptions that is impossible, which is itself informative — three of the census’s rows have an odd number.

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 1.01° — so within a class the displacement is a constant. two classes sit at the common level. The six that do not sit at 132.7° and -45.0°, equal and opposite to within 98.7 per cent, and they are neighbouring residues. The stem's own divergence is 137.84°, so an exception is one organ's step.
Fig. 18 A period with several exceptions, on which the pairing question can be asked.

The one line

Inside a wrecked stem’s period most classes sit at one level, within 5.55° of each other across the whole census, and the exceptions sit 12.4° to 160.4° away. On seventeen of the thirty cuts there are exactly two exceptions, equal and opposite to within five per cent — and on all seventeen they are neighbouring residues, which is what two chains that have changed places would look like.

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. three classes sit at the common level. The two that do not sit at 141.1° and -142.3°, equal and opposite to within 0.8 per cent, and they are neighbouring residues. The stem's own divergence is 137.77°, so an exception is one organ's step.
Fig. 19 One more period, with its two adjacent exceptions marked against the level the rest share.

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°. 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. A step of one organ The balanced pair inside a wrecked stem's period measures 88.0° to 147.2° against divergences of 99.1° to 138.0° — one organ's step, to within twelve per cent on every row. The residual is not scatter: every stem keeping a 5 or a 7 overshoots and every stem keeping a 4 or an 8 falls short. 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.

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.

  • The rung that two organs wreck — both name ablation, claim testing, control, lattice offset, measurement, mechanism, nearest neighbour, negative result, prediction, rigid hop
  • Both walls of the slot — both name ablation, claim testing, control, lattice offset, measurement, mechanism, nearest neighbour, negative result, prediction
  • The organ that moved furthest — both name ablation, claim testing, control, lattice offset, measurement, nearest neighbour, negative result, parastichy, rigid hop
  • Removing a neighbour costs least — both name ablation, control, lattice offset, measurement, mechanism, nearest neighbour, parastichy, prediction
  • The angle is not the actor — both name ablation, claim testing, control, description versus mechanism, lattice offset, mechanism, negative result, rigid hop
  • The family that lost a member — both name ablation, claim testing, control, lattice offset, measurement, negative result, parastichy, rigid hop

Named objects

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

AblationClaim testingControlDescription versus mechanismLattice offsetMeasurementMechanismNearest neighbourNegative resultParastichyPredictionRigid hopSlipTolerance