Phyllotaxis
the spirals are counted, not admired
Stems and cones

The last of three quantities

The pair, the divergence and the step ordering move together when the rise is swept, and for four rounds no result could be attributed to any of them. Two designs later, two are ruled out as sufficient and the third has never been held still — because holding it is what a rung already does.
12 min read 20 figures State the claim, state the testForm is not mechanism

Worth reading first: The angle the ladder returns to · Where a handover sits · The organ that was taken away.

A rung is a range of rises over which a counter returns one pair. Sweep the rise inside it and three things move: the pair does not, by definition; the settled divergence slides; and which of the two contact steps is the shorter changes hands once. Sweep across a transition and the pair moves too.

For four rounds every result about which family survives a removal was scored on a census that varied all three at once. This essay is the bookkeeping: what two designs have now removed, what is left, and why what is left cannot be tested the same way.

Which rungs of the golden branch share a divergence. One row per pair of rungs. A pair whose divergence ranges overlap has a rise on each rung where the rule settles on the same angle; a pair whose ranges do not overlap has none, whatever the search. On this branch four of six pairs match, and three of those match to 0.0000° — the same value of a quantity read on a grid of 1,536 azimuths. The rises differ by factors of 1.48 to 5.09, so the design holds one angle while changing everything the rise controls.
Fig. 1 The second of the two designs: pairs of rises on different rungs that settle on one divergence.

The problem, stated as an experiment

Every experiment here is a removal. Take a stem, take out one organ, watch what the rule does next, and record which lag’s hop is still rigid three hundred organs later. Repeat at a range of rises and a range of offsets and the result is a census.

A census’s rows differ in the rise, and the rise is not one variable. It is three variables tied together by the rule, so a rule that predicts the survivor from the counted pair and a rule that predicts it from the ordering will agree on most rows for no reason at all. That is not a subtlety about statistics; it is a statement that the census cannot tell them apart.

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. 2 The census the readings were scored on, in which every row varies three quantities together.

How badly they agreed

The measure of the problem is a number the previous round produced. Positioned inside its own rung, every census row acquires a fraction nobody had computed, and eight of the ten lattices that wreck sit past their rung’s handover — twenty-five of thirty cuts.

So the census was not sampling the ordering. It was very nearly holding it. Two readings scored twelve and eleven of thirty and looked like rivals, and on most rows they were the same reading with two names.

That is what a design has to fix, and a design fixes it by holding something on purpose rather than by accident.

The column the census never carried. One row per lattice the ablation census was grown at. The bar shows where inside its own rung that rise sat, measured in the logarithm of the rise because the ladder is geometric, with zero the coarse transition and one the fine one. The mark on each bar is that rung's own handover, the rise where the two contact steps change places. Of the ten lattices that ever wreck, eight sit past their handover and one sit before it, with one sitting so close to one that the two steps differ by parts in a thousand. The rise was recorded in every table this collection has published; this fraction was in none of them.
Fig. 3 Where each census row sat inside its own rung, which is how a quantity turned out to be nearly held.

The first design

A band. Near the rise where the two contact steps change places, the settled divergence has a shallow floor, so a run of rises either side of that crossing share an angle to a twentieth of a degree while the counted pair holds and the ordering reverses.

Two quantities held, one moved. Fifty-five wrecked cuts on two branches, and the family left standing never changes — including at the rises where the survivor is the longer step, which is where the shortest-hop reading would have had to be right.

A divergence that does not move across the 5/8 band. Measured at every rise of a band on the golden branch, where a counter returns 5 and 8 spirals throughout. The settled divergence moves by 0.0469 degrees across the whole band, which is a fraction of the azimuth grid step and a fortieth of the slide across the rung it sits in. The ratio of the two contact steps does move: it falls to 1.0013 and the ordering changes hands at a rise of 0.0156, so above that rise the shorter step belongs to the 5 family and below it to the 8 family. Two of the three quantities that vary along a rung are therefore held here and the third is not, which is what makes the ends of this band a matched pair.
Fig. 4 A band: eighteen rises whose divergence moves by five hundredths of a degree, with the ordering changing hands in the middle.

The second design

A matched pair. Down the golden branch the divergence curve turns, so a value it takes on one rung it takes again on another. Five such pairs exist; four of them wreck at both rises.

One quantity held, one moved, and the third moved with it. At every one of the four the two stems keep different families, so the divergence is not sufficient to decide which one survives.

The same angle, two rungs, two answers. Each block is one matched pair: two rises whose stems settle on the same divergence and whose counters return different pairs. Under each is the family every wrecked cut leaves standing. On three of the four pairs both rises wreck at some offset, and on every one of those the two stems keep different families — so the divergence, which is held, is not what decides the survivor. The two stems keep exactly the counted numbers their two pairs share, including the pair that shares none and keeps none.
Fig. 5 The four usable matched pairs, with the families each stem keeps.

What each design cannot say

Neither is an account. A band says the ordering is not the actor and says nothing about what is; a matched pair says the divergence is not sufficient and says nothing about what is.

Both are negatives, and negatives of the weakest useful form: this quantity, on its own, does not determine the answer. Neither rules a quantity out of an account with several terms in it. What they rule out is the shape of explanation in which one of them is the whole story — and that is the shape most of the candidate readings here have taken.

Two lines across the 5/8 rung, crossing once. The divergence the rule settles on, against the divergence at which the two contact steps would be exactly the same length. The second is arithmetic on the lattice and no stem is grown for it. Across this rung the balanced line moves 2.281 degrees and the rule's own line moves 1.262, so the shallower line crosses the steeper one, and it does so exactly once at a rise of 0.0154 — 16 per cent of the way down from the coarse end. That crossing is the handover: above it one family has the shorter step and below it the other does. So a rung has one handover, its position is fixed by the arithmetic rather than by any experiment, and a sweep of the rise carries a stem across it at a place nobody chose.
Fig. 6 The two curves whose crossing makes a band, which is the arithmetic the first design rests on.

The third quantity has no design

The counted pair is what is left, and it is the one quantity here that has never been held still while the other two moved.

The reason is structural. Holding the pair is what a rung is, and a rung moves the divergence and the ordering together. There is no rise at which the pair is held and the divergence is varied independently of the ordering, because both are functions of the rise and the rise is one number.

So the pair is the last candidate standing and is also the least directly tested of the three. That is an uncomfortable position for a result to be in and it is better said than left implicit.

The divergence slides along the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not.
Fig. 7 The slide down one rung, on which the divergence and the ordering both move and the pair does not.

What a third design would need

A stem with the same counted pair, the same divergence, the same ordering, and something else different — or a way of changing the pair without changing the rise. Neither exists in the rule as written.

There is one candidate that does not need the rise at all. A jugate stem — organs arriving several at a time — is an ordinary lattice seen k times over, and its counted pair is k times an underlying one. That changes the pair by a factor while leaving the underlying arrangement in place, which is closer to a pair-only variation than anything the rise can do.

Whether a jugate cut is comparable to a single-jugate one is a separate question with its own measurement, and it is a real route rather than a hopeful one.

The round trip, for every jugacy. Each pattern is built from a divergence, the divergence is thrown away, the families are counted by tracing chains, the jugacy is read off the symmetry, and the divergence is recovered. The worst error over the four is 2.3e-13 degrees — and each answer is modulo the pattern's own period, 360/k, because adding that leaves the point set identical.
Fig. 8 A two-jugate lattice, whose counted pair is twice an underlying one at the same geometry.

Two designs and one census, compared

The census has thirty wrecked cuts and varies everything. The band has fifty-five and varies one thing. The matched pairs have four comparisons and vary one thing.

Weight of evidence and cleanliness of design run in opposite directions here, which is the ordinary situation. The census is where the readings were found and is the worst place to test them; the two designs are where they can be tested and carry far fewer rows. A result that survives both is a result that survived a large dirty sample and a small clean one, and neither on its own would be worth much.

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. 9 The census as a whole, which is the large dirty sample the two designs were built against.

The arithmetic of what is left

Three quantities, two ruled out as sufficient, one standing. That is not the same as one quantity being the mechanism, and the difference matters enough to write out.

If the survivor were a function of the counted pair alone, then every stem with one pair would keep one family whatever the offset. It does not: a 5/8 stem keeps the 5 at some offsets and the 8 at others. So the pair is not sufficient either, and the honest statement is that the survivor depends on the pair and the offset together, with the divergence and the ordering ruled out as sufficient on their own.

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. 10 The offset, which is the second variable no design here holds and which the survivor plainly depends on.

What the two designs agree about

They agree on something neither was built to test. A band’s cuts keep one family at each offset across every rise of the band; a matched pair’s two stems keep exactly the counted numbers their pairs share.

Read together, both say the answer is a property of the contact structure — which chains exist and which one the removed organ belonged to — rather than of any continuously varying quantity. A band varies two continuous quantities and changes nothing; a matched pair holds one continuous quantity and changes everything, by changing which chains there are.

Shared counted numbers against shared survivors. One row per matched pair, over both branches. The third column is the counted numbers the two rungs have in common and the fourth is the families both stems leave standing; on every row the two are the same set. The row whose rungs share no counted number is the one whose stems share no survivor, which is what makes this a claim about an intersection rather than a restatement that a survivor is usually a contact family. four rows, and the empty case is one of them.
Fig. 11 The intersection result, which is the clearest statement of the answer following the chains.

The cost of the two designs

Worth recording because it is the argument for doing this kind of thing again. The band cost a sweep of the rise at two parts in a thousand and a cut at every offset of every rise it found: several hundred grown stems. The matched pairs cost an overlap test that is free, a refinement of twenty-five stems a pair, and sixteen cuts at each of ten rises.

Against that, the census cost thirty cuts and settled nothing. The designs are ten times the compute and they are the only part of this thread that has closed anything.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 12 The cost of a single settled stem, which is the unit both designs are paid for in.

What a plant sees

A specimen has a counted pair and does not have a measured divergence. So of the three quantities, the one the field can report is the one still standing, and the two that have been ruled out are the two the field cannot see.

That is a convenient result and it should be said carefully, because convenience is where a reading gets over-read. The convenience is real and it is about the specification rather than the mechanism: the ablation this collection has specified records counts, and these two designs say the counts are the right thing to record.

The spiral counts four different divergence angles produce. Fibonacci counts come from one angle. The Lucas angle — which the same dynamical model reaches on a different branch — gives 47 and 76, and neither number is a Fibonacci number.
Fig. 13 What a counter returns, which is the measurement a specimen can supply and a protractor cannot.

Where the ordering result now stands

It was carried by two bands and it is now carried by more. Four of the ladder’s six handovers had never had a band built on them, and building them puts the same claim on four counted pairs rather than two.

That is worth separating from this essay’s arithmetic. The three-quantity bookkeeping is about which designs exist; the number of bands is about how much weight one of them carries. Both changed this round and they changed independently.

Where the two contact steps change places, on every rung. One row per rung of the two branches, drawn from its coarse end to its fine one on a logarithmic axis. The mark on each row is the rise at which the two contact steps change places — the rung's handover — and the shaded stretch is the band around it over which the settled divergence holds still. six of the eight rungs have a handover and every one of those sits between 6 and 40 per cent of the way down its rung, never past the middle. The two rungs without one are the coarsest on each branch, whose crossing is above the range this ladder reaches.
Fig. 14 Every handover on the ladder, which is how many bands the ordering result can rest on.

What is still open

The offset. It is the second variable in every one of these experiments, no design here holds it, and the survivor plainly depends on it. A reading that predicts the survivor from the offset scores twenty-five of thirty and fails at five, and the five are not a fine-end artefact.

And the mechanism. Nothing in these two designs says why the contact structure should decide what a removal leaves standing. The displacement profile of a wrecked stem is the nearest thing to an answer this collection has, and it is a description of the damage rather than of the rule that produced it.

How constant the displacement is inside one residue class. One row per wrecked cut in the census, drawn at the widest spread found inside any one residue class when the profile is folded on the lag that stem kept. 25 of 30 rows sit between 0.12 and 6.09 degrees, which on a quantity whose between-class differences run past a hundred and fifty degrees is a constant. The five that do not sit from 10.3° up. There is nothing in between, so the line drawn at 10° could have been drawn anywhere in a wide interval.
Fig. 15 The shape of the damage, which is where an account of the survivor would have to start.

What a fourth quantity would look like

Three is the number of things a rung moves, and it is worth asking whether it is the number of things there are.

A rung is defined by what a counter returns, so “the counted pair” is a label on a range of rises rather than a physical quantity. Inside that range the two contact steps have lengths, the neighbourhood has a size, the front has a depth, and each of those is a continuous function of the rise. Any of them could be the actor and none of them is separated by either design here.

So the three-quantity bookkeeping is bookkeeping over the quantities this thread has named, not over the quantities that exist. A band’s own widest member is the case where that bites: the survivor at a fixed offset changes somewhere inside it, with the pair and the divergence both held, so something the two designs do not name is varying.

On a rung the response is a run of offsets; near a transition it has a hole. One row per rise, from 0.032 at the top to 0.005 at the bottom, and one column per offset: the organ one place back at the left, 14 places back at the right. A cell is filled where removing that organ moves the next organ by more than 2.5°, and empty where it does not. On a rung the filled cells are a run from one to the larger parastichy number — 5, 8, 13 at the pairs shown on the left. Every rise shown here is in the middle of its rung, so every row is a run and nothing past it is felt — what happens between the rungs is a separate matter.
Fig. 16 The front’s depth by rise, which is one of the continuous quantities neither design holds.

The offset, which no design holds

Every experiment here has two coordinates and the designs hold one of them.

A cut is named by a rise and by an offset — how many places back from the growing tip the removed organ sat — and the survivor depends on both of them. The offset rule scores twenty-five of thirty on the census, which is far better than anything stated over the rise, and it fails at five rows that are not a fine-end artefact.

Neither a band nor a matched pair varies the offset in a controlled way. A band holds the offset and varies the rise; a matched pair varies both, because the two stems of a pair wreck at mostly different offsets. So the offset’s own account is scored on the census, with all the mixing that implies.

The newest member of the front is the weakest. For every cell of the design whose rung boundary is inside the range, how far the next organ moves when the organ exactly as many places back as the larger parastichy number is removed — the offset that arrived when the stem entered this rung — against how far below that boundary the stem sits. Each line is one lattice on one branch. The horizontal line is the threshold that decides whether an offset counts as felt, and the one cells below it are the one whose front reads one offset short. Nothing is a different kind of thing: the boundary is a step everywhere, and near the top of a rung its last stair is shallow.
Fig. 17 The same offset at four rises, which is the comparison the offset’s own account has to survive.

Why the counted pair is a strange thing to be left with

It is not a continuous quantity. A stem is 5/8 or it is 8/13; there is no rise at which it is halfway between, because the transition is where a counter’s answer changes.

That makes it an unusual candidate for a mechanism. The placement rule is a minimisation over distances and every quantity in it is continuous; the counted pair is a reading taken off the result, by machinery that is never shown the divergence. A discrete label being what decides a continuous process is either a sign that the label is standing in for something continuous, or a sign that the process has a genuinely combinatorial step in it.

The shape of the damage points at the second: two adjacent chains exchanged is a combinatorial event, not a small displacement, and the chains are what the counted pair counts.

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. 18 The damage as two chains changing places, which is a combinatorial description rather than a continuous one.

What this round did to the two designs

Both got wider and both got a limitation.

The band went from two to six, and the sixth one holds all three quantities — a control nobody had. The claim it carries narrowed from “constant across a band” to “constant across a handover”, because a wide band turned out not to be a narrow experiment.

The matched pair is new and its limitation is that it produces four comparisons. There is no fifth on this ladder: the coarse end wrecks at nothing and the fine end has no further rung to pair with.

Neither of those is a defect to be repaired. They are what the ladder holds, and a design tested on everything a ladder holds has no sample left over.

What each band holds, and what it moves. One row per band. The counted pair is held by construction and the settled divergence is held to a twentieth of a degree; the quantity a band exists to move is which of the two contact steps is the shorter. five of the six do move it — the ordering changes hands exactly once inside, and at both ends the two steps differ by enough for an ordering to mean anything. The remaining one changes hands three times and its two steps are never more than 0.0 per cent apart, so it holds all three quantities and is a control rather than an experiment.
Fig. 19 The six bands, of which five are usable and one is the control the design did not plan for.

The one line

The rise moves the counted pair, the settled divergence and the step ordering together. A band holds the first two and shows the third does not decide the survivor; a matched pair holds the second and shows it is not sufficient either. The counted pair is the one left, it is not sufficient on its own, and it is the one quantity here that no design can isolate — because holding it is what a rung already does.

The settled divergence down the golden branch. Every rise from 0.07 down to 0.00482, plotted against the divergence the rule settles on, with each rung drawn in its own stroke and the branch's limit angle marked. The curve does not slide: it turns three times in four rungs, climbing across one and falling across the next, so a value it takes on one rung it takes again on another. That is what makes a matched pair possible — two rises, different counted pairs, one angle — and it is the whole reason the design exists on this branch. The widest excursions from the limit angle, coarse rung first, are 3.195°, 3.352°, 0.961°, 0.422°.
Fig. 20 The curve both designs are cut out of, and the reason only one branch supplies the second of them.

Where this goes next

Shared counted numbers against shared survivors. One row per matched pair, over both branches. The third column is the counted numbers the two rungs have in common and the fourth is the families both stems leave standing; on every row the two are the same set. The row whose rungs share no counted number is the one whose stems share no survivor, which is what makes this a claim about an intersection rather than a restatement that a survivor is usually a contact family. four rows, and the empty case is one of them. Where the survivors meet At a matched pair the two stems keep exactly the counted numbers their two pairs have in common — the 5 where 3/5 meets 5/8, the 8 where 5/8 meets 8/13, the 7 where 4/7 meets 7/11, and nothing at all where 3/5 meets 8/13. Four rows, including the empty one. The ordering changes and the survivor does not. Every offset that wrecks, at every rise of the band, with the family left standing written in the cell. The counted pair is 5 and 8 at all 18 rises and the settled divergence is held to a twentieth of a degree, so the one quantity moving across the columns is which of the two contact steps is the shorter — and it changes hands at the marked rise. The cells do not: the 5 family survives at all 24 wrecked cuts, on both sides. Scored on this band, the reading that a wrecked stem keeps its shortest hop is right 14 times out of 24, for an answer that never changed. The ordering was not the actor Cut an organ out of every rise of a band where the counted pair and the settled divergence are held and the two contact steps change places, and the family left standing does not change. Fifty-five wrecked cuts on two branches, and the ordering reverses underneath every one of them. The 3/5 rung at a tenth of the ladder's step. Every rise of one rung, sampled ten times as finely as the ladder that found the locked band. All 23 are counted at 3 and 5 spirals and all 23 settle: the largest wander is 0.221 degrees, against the 0.5 degree threshold and against the 0.79 to 1.60 degrees the band on the coarse rung wobbles by. The divergence slides smoothly from 139.0625 to 136.7344 degrees with no rise stuck on a rational and none stuck on anything else. Whatever the band is, it is not something a coarser sampling was hiding here. The slide a counter holds constant Inside one rung the settled divergence moves by more than a degree, monotonically, with no flat stretch anywhere — measured at a thousandth on one rung and at half a ten-thousandth on another. A rung is a plateau in one reported number laid over a geometry that never stops moving.

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 shortest hop was a coin flip — both name ablation, claim testing, control, negative result, parastichy pair, rigid hop, rise, rung, underdetermination
  • Two rungs, one angle — both name control, divergence angle, identifiability, matched design, negative result, parastichy pair, rise, rung, underdetermination
  • One rung, two answers — both name ablation, control, negative result, parastichy pair, rigid hop, rise, rung, underdetermination
  • The family that lost a member — both name ablation, claim testing, control, negative result, parastichy pair, rigid hop, rung, underdetermination
  • The front deepens down a rung — both name ablation, claim testing, control, negative result, parastichy pair, rigid hop, rise, rung
  • The organ that moved furthest — both name ablation, claim testing, control, divergence angle, negative result, parastichy pair, rigid hop, underdetermination

Named objects

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

AblationClaim testingControlDescription versus mechanismDivergence angleIdentifiabilityMatched designMechanismNegative resultParastichy pairRigid hopRiseRungUnderdetermination