Phyllotaxis
the spirals are counted, not admired
Stems and cones

The angle is not the actor

Cut an organ out of two stems that settled on the same divergence and return different counted pairs, and the family left standing is different at every one of the four pairs where both stems wreck. The angle is held to a hundredth of a degree underneath.
12 min read 14 figures State the claim, state the testForm is not mechanism

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

Remove one organ from a settled stem and one of two things happens. The arrangement repairs, and every lag it had is still there; or it never repairs, and exactly one lag stays rigid — the angle from an organ to the one p places above it, unchanged from a control that shares the stem’s whole history, while every other lag moves by tens of degrees.

Which p is the question this thread has been chasing for five rounds. This essay closes one of the three candidate answers by holding it still.

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. 1 Each block is two rises settled on one divergence, with the families every wrecked cut leaves standing at each of them.

The three quantities, one more time

Sweeping the rise along a rung moves the counted pair, the settled divergence and the step ordering together. Every reading of the survivor ever scored here has been scored on a census that varied all three, so a rule that follows any of them scores the same as a rule that follows the others on most rows.

A band took the ordering out: hold the pair, hold the angle, reverse the ordering, and the surviving family does not change. That left two. A matched pair takes the angle out: hold the angle, change the pair, and see whether the answer moves.

If it does, the angle is not what decides the survivor and the pair is the only candidate left. If it does not — if two stems with different counted pairs keep the same family at the same divergence — the angle would be back in the running and the pair would be in trouble.

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. 2 The design that removed the ordering: one rung, one pair, one angle, and the two contact steps changing places inside it.

What is cut, and against what

At each rise of each matched pair, an organ is removed from a settled stem at every offset the front reaches, and the run is continued for three hundred organs against a control that shares its history to the last digit. A cut that repairs is recorded as repairing and has no survivor; a cut that never repairs has its whole lag spectrum measured, and the surviving lag is the shortest one whose hop is both steady and unmoved from the control’s.

That is the same instrument the census uses, unchanged, pointed at rises chosen for sharing an angle rather than for being convenient. Nothing about the measurement is new here. What is new is which stems it is pointed at.

Reusing the instrument without a single change is deliberate and it costs something. A measurement tuned for these particular stems might resolve them better; a measurement tuned for these particular stems is also a measurement whose tuning could be doing the work. The thresholds are the census’s — half a degree of spread before a hop is called rigid, three degrees of shift before it is called moved — and on these stems as on the others nothing sits near them. The rigid hops here measure hundredths of a degree of spread and the next steadiest lag in any of these wrecked stems measures tens.

The control matters as much as the cut. Both runs are continued from the same history by the same loop, so the two differ in exactly one organ and in nothing else — not in the seed, not in the grid, not in the heights, which are prescribed by the rise rather than by the pattern. A comparison between a cut stem and a freshly grown one would carry every difference two independent runs can accumulate, and none of them would be the removal.

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 of one wrecked stem, which is how a surviving family is identified rather than assumed.

The 3/5 and 5/8 pair

At 137.266°. The 3/5 stem is grown at a rise of 0.02251 and the 5/8 stem at 0.01074, a factor of 2.10 apart.

The 3/5 stem wrecks at one offset, four places back, and keeps the 5. The 5/8 stem wrecks at four offsets — three, four, six and seven places back — and keeps the 5 at the first two and the 8 at the other two.

So at one angle the two stems keep different sets: {5} against {5, 8}. The divergence is held to 0.0000°, the counted pair is not, and the answer follows the pair.

The 8 is the interesting half. It is a counted number of the 5/8 stem and it is not a lag the 3/5 stem has any particular relationship to — an 8-hop exists on a 3/5 stem, as every lag does, but it is neither of its two contact families and it is nowhere near the top of its length ranking. The 5/8 stem keeps it at two of its four wrecked offsets; the 3/5 stem keeps it at none.

Reading that the other way makes the point sharper. The two stems agree about the 5 — both keep it where they keep anything at the offsets they share — and they disagree about the 8, which one of them counts and the other does not. An account in which the divergence decides has no way to produce that asymmetry, because the divergence is the same number at both.

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. 4 The census’s own grid of offsets against surviving families, which the matched pairs are read against.

The 5/8 and 8/13 pair

At 137.844°, rises of 0.00739 and 0.00500 — the closest of the five, a factor of 1.48 apart, and the one where an objection is easiest to make.

The 5/8 stem wrecks at three offsets and keeps the 8 at two of them and the 5 at one. The 8/13 stem wrecks at five offsets and keeps the 8 at four of them and the 4 at one. So the two sets are {5, 8} and {4, 8}: they overlap in the 8 and differ in the 5 and the 4.

A reading that said “the angle decides it” would have to explain why the same angle produces a 5 at one rise and a 4 at the other. A reading that says the pair decides it has nothing to explain: 5 is a counted number of a 5/8 stem and 4 is not, and 4 is a counted number of nothing on that stem — which is a result the census already carries and is not this essay’s to settle.

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. 5 The matched pairs as arithmetic, with the two rises and the angle each holds.

The widest pair

At 137.859° and 137.844°, sixteen thousandths of a degree apart, with rises of 0.02547 and 0.00500 — a factor of 5.09. This is the pair where the two stems have least in common: different counted pairs, different fronts, different neighbourhood sizes, five times the spacing.

The 3/5 stem wrecks at one offset and keeps the 5. The 8/13 stem wrecks at five and keeps the 8 at four and the 4 at one. {5} against {4, 8}, sharing nothing.

It is the strongest of the four comparisons and the easiest to over-read. Two stems this different would be expected to differ in almost any measurement, so the fact that they differ here is not by itself surprising. What makes it informative is the pairing: the one quantity they share is the one being tested, and it is shared to a fifteenth of a grid step.

The width of the pair also buys a check that the narrow pairs cannot supply. If the surviving family were set by the divergence, the two ends of this pair should agree more than the two ends of the 1.48× pair, since the angle is what they have in common and everything else is further apart. It is the other way round: the 1.48× pair shares the 8 and the 5.09× pair shares nothing. The agreement tracks how alike the counted pairs are and not how alike the angles are, and the angles are equally alike at both.

None of that is a fit. There are four points and no parameter; what is being read off them is the direction of a difference, which four points can carry and a slope cannot.

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. 6 Where on the ladder the widest pair’s two rises sit, and the horizontal line that puts them at one angle.

The Lucas pair

At 99.273°, rises of 0.01046 and 0.00624. The 4/7 stem wrecks at four offsets and keeps the 7 at one of them and nothing at all at the other three. The 7/11 stem wrecks at six and keeps the 7 at three and the 11 at three.

The three cuts that keep nothing are worth pausing on. A wrecked stem with no rigid lag is not a lattice with a slip in it; it is a stem that is no longer any lattice, and multi-organ cuts produce them routinely while single-organ cuts almost never do. They are counted separately here rather than folded into a family, because folding a null into a set makes a claim about fewer rows than it says.

Counting them separately, the comparison still holds: {7} against {7, 11}, and the 11 is a counted number of the 7/11 stem and of nothing on the other.

Which rungs of the Lucas 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 one of six pairs match, and one 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 1.68 to 1.68, so the design holds one angle while changing everything the rise controls.
Fig. 7 The single Lucas match and the five pairs of rungs that cannot produce one.

The pair that says nothing

The 2/3 and 3/5 rungs match at 139.297°, and the comparison is empty. The 2/3 stem at 0.06151 wrecks at two offsets and keeps nothing at either; the 3/5 stem at 0.03065 wrecks at no offset at all.

That is the coarse end of this ladder doing what it has been measured to do. A 3/5 stem’s front is five organs, which is its own two edges with no middle, and a single removal has nothing to sever. So a fifth matched pair exists and produces no experiment.

It is in the table, and it is asserted to be there: a check requires that exactly one matched pair fail to wreck and that it be the one involving the coarsest rung. A design that quietly dropped its unusable cell would report four for four with nothing saying that a fifth was ever attempted.

Which rises are a lattice, from 0.04 to 0.13. How much the divergence wanders over the last sixty organs, at each rise up the coarse ladder. 13 of the 19 settle, scattering between 0.0000 and 0.3356 degrees. six do not: from 0.09 to 0.115 the divergence sticks on exactly 135.0000 degrees, which is three eighths of a turn, and wobbles about it by 0.79 to 1.60 degrees. A counter shown either kind returns the same pair, so the counts cannot tell them apart. The line is the threshold a rise has to pass before a cut is made on it, and it sits in the gap rather than among the measurements.
Fig. 8 The coarse end, where a single removal cannot wreck a stem, which is why the coarsest match produces no comparison.

Four for four

At every matched pair where both stems wreck, the two stems leave different families standing. Four pairs, three counted pairs of rungs on one branch and one on the other, angles held to 0.0000° at three of them and 0.0156° at the fourth.

The claim is a negative and it is worth stating in its weakest form, which is the form that is certainly true: the settled divergence is not sufficient to determine which family survives. Two stems at one divergence give different answers, so the answer is not a function of the divergence alone.

The weak form is the one the design supports, and it is stronger than it sounds. A quantity that is not sufficient cannot be the mechanism; it can at most be a term in one. And the divergence is the quantity every popular account of phyllotaxis is written in — the golden angle, its rational neighbours, how close a particular plant sits to 137.5° — so ruling it out as sufficient rules out the shape most explanations here take.

There is also a version of this that would have been much weaker and is worth naming so it is not confused with what was done. Growing two stems at two rises, noticing that their divergences happen to be similar, and comparing them, is not a matched design; it is a coincidence reported after the fact. The pairs here are found before anything is cut, by an overlap test and a refinement whose target is the midpoint of an interval — so which rises are compared is decided by the ladder and not by the answer.

That is all a matched pair can say. It cannot say the divergence is irrelevant — it might be one term in an account with other terms — and it cannot say the counted pair is the cause. What it can do is remove the divergence from the list of quantities that could, on their own, be the whole story.

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 one of the one 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. 9 The Lucas comparison drawn like the golden ones, with the cuts that keep nothing counted apart.

What the two designs together leave

Three quantities, two designs, and the arithmetic is simple. A band holds the pair and the angle and moves the ordering: the answer does not move, so the ordering is not it. A matched pair holds the angle and moves the pair: the answer moves, so the angle is not sufficient.

The counted pair is what is left, and it is left in an awkward position: it is the only one of the three that has never been held still while the others moved, because holding the pair is what a rung does and a rung moves the other two together. So the pair is the last candidate standing and is also the one quantity here that has not been tested in isolation.

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. 10 The two curves whose crossing makes a band, which is the other half of the pair of designs.

What a specimen would show

Nothing, and it is worth saying why the negative is useful anyway. Nobody measures a divergence on a plant; the protractor this would need is not an instrument anybody has, and the numbers in the literature are recovered from counts rather than measured directly.

So a result saying “the divergence is not the actor” removes from contention precisely the quantity that a field observer cannot see, and leaves standing the quantity a field observer can. That is the right way round for a collection whose survey specification is written in counts.

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. 11 What a counter returns across a range of angles, which is the only one of the two quantities a specimen supplies.

What could still overturn this

One thing, and it is not subtle. The comparison is between sets of surviving families, and a set is assembled over the offsets a stem wrecks at. Two stems that wreck at different offsets could produce different sets while agreeing perfectly at every offset they share.

They do not share any. A 3/5 stem wrecks at offset four and a 5/8 stem at three, four, six and seven — offset four is common, and the two keep the 5 there. Encouraging, and one row. The honest reading is that these four pairs establish the negative and do not establish anything positive about the offset, which is a separate reading with its own census.

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. 12 The offsets a cut wrecks a stem at, which is the axis the two stems of a matched pair do not share.

What the four pairs cost

Twenty-five grown stems per refinement to locate each pair, sixteen cut stems at each of ten rises, and a control beside every one of them. Against that, the whole of the search that decides which rungs can be paired at all is two comparisons of intervals.

Worth stating because the ratio is the argument for doing the arithmetic first. Of the twelve pairs of rungs across the two branches, seven cannot produce a match and were ruled out before a single stem was grown; five can, and only those were searched.

The design’s expense is therefore concentrated where it produces something, which is not the usual shape of a sweep here — a band pays for every rise it covers whether or not that rise wrecks.

Which rungs of the Lucas 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 one of six pairs match, and one 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 1.68 to 1.68, so the design holds one angle while changing everything the rise controls.
Fig. 13 The pairs of rungs that cannot match, which cost two comparisons each to rule out.

The one line

At four matched pairs the two stems settle on the same divergence, return different counted pairs, and leave different families standing. The divergence is held to a hundredth of a degree or better underneath every one of them, so it is not what decides the survivor — and with the ordering already removed, the counted pair is the only one of the three quantities left.

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. 14 The four usable pairs, with what each pair of rungs shares and what each pair of stems keeps.

Where this goes next

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. Two rungs, one angle Five pairs of rises settle on the same divergence while a counter returns different pairs at them, and four of the five agree 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. 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. 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.

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 organ that moved furthest — both name ablation, claim testing, control, divergence angle, falsifiability, lattice offset, negative result, parastichy pair, rigid hop, underdetermination
  • The family that lost a member — both name ablation, claim testing, control, falsifiability, lattice offset, negative result, parastichy pair, rigid hop, underdetermination
  • The organ that was nobody's neighbour — both name ablation, claim testing, control, falsifiability, lattice offset, negative result, parastichy pair, rigid hop, underdetermination
  • The shortest hop was a coin flip — both name ablation, claim testing, control, falsifiability, lattice offset, negative result, parastichy pair, rigid hop, underdetermination
  • Both walls of the slot — both name ablation, claim testing, control, lattice offset, matched design, mechanism, negative result, parastichy pair
  • One level and two exceptions — both name ablation, claim testing, control, description versus mechanism, lattice offset, mechanism, negative result, rigid hop

Named objects

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

AblationCounting blindClaim testingControlDescription versus mechanismDivergence angleFalsifiabilityLattice offsetMatched designMechanismNegative resultParastichy pairRigid hopUnderdetermination