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

Both walls of the slot

The growing tip sits between its two chain-neighbours. Removing either alone is a cheap removal on all six lattices — 2.3° to 41.7°. Removing both together throws the next organ past the expensive line on three of them, and the interaction runs from −25.8° to +132.9°.
13 min read 19 figures State the claim, state the testForm is not mechanism

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

Removing a chain-neighbour of the growing tip is the cheap removal: the next organ placed moves 8.9° to 30.7°, against 62.8° to 167.6° for removing anything else. Nine rows against twenty-one, with a factor of two in the empty space between them.

That is the opposite of what the placement rule’s own weighting suggests, and the sharp version of the question it raises is a two-by-two. Take away one wall of the slot the tip sits in; take away the other; take away both.

Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 5 places back and the organ 8 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 26.3° and 4.9°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 164.1°, against 31.2° for the two effects added, so the interaction is +132.9°. The slot is not two independent walls.
Fig. 1 The four cells at one lattice: nothing removed, the smaller wall, the larger wall, and both.

What the walls are

On a stem counted p and q, the organ p places back from the tip and the organ q places back are the tip’s two chain-neighbours: the nearest members of the two contact families, one on each side in azimuth.

Every other organ within reach is further away in the arrangement even where it is close up the stem. So the tip sits in a slot with two walls, and the two walls are named by the stem’s own counted numbers.

They are counted from the points of the very stem being cut rather than from a stem grown the same way somewhere else. That sounds like fussiness and is not: the whole design consists of naming two offsets, and an offset read off the wrong stem names an organ that is nobody’s neighbour while every other number in the row stays plausible.

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. 2 A counter reading the two families off an arrangement, which is where the two offsets come from.

The design

Four cells. Neither wall removed, which is the control and has a displacement of zero by construction. The p-wall removed. The q-wall removed. Both.

Run on six lattices spanning three counted pairs and both branches: golden stems at rises of 0.020, 0.013, 0.010 and 0.008, and Lucas stems at 0.013 and 0.008. That is twenty-four runs, of which eighteen are cuts and six are controls.

The measured quantity is how far the first organ placed after the cut sits from where the same organ sits in a control sharing the whole history. It is the same quantity the cheap-removal reading uses, unchanged.

Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 3 places back and the organ 5 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 41.7° and 20.6°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 47.8°, against 62.3° for the two effects added, so the interaction is -14.5°. The slot is not two independent walls.
Fig. 3 The same four cells at the coarsest lattice in the design, where both single removals heal.

Each wall alone is cheap

On all six lattices, both. The twelve single removals move the next organ by 2.3°, 4.9°, 8.9°, 11.3°, 12.0°, 18.3°, 20.6°, 25.3°, 25.8°, 26.3°, 29.5° and 41.7°.

Every one is inside the 45° that separates the two groups in the census, and ten of the twelve are inside 30°. That is the cheap-removal reading reproducing itself on six lattices it was not measured on, which is worth having as a check on the reading before anything is built on top of it.

The largest, 41.7°, is the 3-wall of the coarse 3/5 stem, and it is the one row that sits close to the line.

What removing both walls does that removing each does not. One row per lattice, drawn at the difference between how far the next organ moves when both walls of the slot are removed and how far the two removals move it separately added together. Zero would mean the walls act independently. five of six lattices are more than 10° from it, three above and two below — so the pair is reliably not the sum of its parts and is not reliably larger than it either. On four of them the pair throws the next organ past the 45° that separates a cheap removal from an expensive one, which neither wall alone comes near.
Fig. 4 The six lattices with what each single removal costs and what the two together cost.

Both together, on three of the six

At the golden 0.013 stem the two walls alone move the next organ 26.3° and 4.9°. Together they move it 164.1°.

At the golden 0.010 stem, 25.3° and 8.9° alone; 163.6° together.

At the Lucas 0.008 stem, 18.3° and 2.3° alone; 117.9° together.

Three rows on which two cheap removals make an expensive one. The prediction that the pair should move the next organ much further than either alone is right on those three by a wide margin, and it is right in the specific sense that matters: the fourth cell crosses the gap the census measured between the two kinds of removal.

Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 5 places back and the organ 8 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 25.3° and 8.9°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 163.6°, against 34.2° for the two effects added, so the interaction is +129.4°. The slot is not two independent walls.
Fig. 5 One of the three: two cheap removals, and a fourth cell six times either of them.

And not on the other three

At the golden 0.020 stem the two alone move it 41.7° and 20.6°, and together 47.8° — barely more than the larger of the two.

At the Lucas 0.013 stem, 29.5° and 11.3° alone; 34.9° together.

At the golden 0.008 stem, 25.8° and 12.0° alone; 12.0° together — which is exactly the larger removal’s own value, to the last digit. That row is worth an essay of its own.

So the prediction is right on half the lattices and wrong on half, and the way it is wrong is not “a bit less than expected” but “nothing happened”.

Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 4 places back and the organ 7 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 29.5° and 11.3°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 34.9°, against 40.8° for the two effects added, so the interaction is -5.9°. The slot is not two independent walls.
Fig. 6 One of the three where the pair is cheaper than the two removals added, and cheap in absolute terms too.

The interaction

With the intact stem as the origin, the two-by-two interaction is what the pair costs minus what the two singles cost added together. Across the six lattices it measures −25.8°, −14.5°, −5.9°, +97.3°, +129.4° and +132.9°.

Five of the six are more than ten degrees from zero, so the two walls are not independent almost anywhere. Three are strongly positive and two strongly negative.

An interaction that is large and of both signs is the least convenient result available. A consistently positive one would say two removals compound; a consistently negative one would say they partly cancel; this says the sign depends on the lattice, and nothing here says on what about the lattice.

What removing both walls does that removing each does not. One row per lattice, drawn at the difference between how far the next organ moves when both walls of the slot are removed and how far the two removals move it separately added together. Zero would mean the walls act independently. five of six lattices are more than 10° from it, three above and two below — so the pair is reliably not the sum of its parts and is not reliably larger than it either. On four of them the pair throws the next organ past the 45° that separates a cheap removal from an expensive one, which neither wall alone comes near.
Fig. 7 The interaction on each lattice, drawn either side of the zero a pair of independent walls would give.

What “much further than either alone” scores

Five of six. On every lattice but the golden 0.008 the pair moves the next organ further than the larger single removal does.

That is the weaker form of the prediction and it survives, while the stronger form — that the pair is more than the sum — scores three of six. The two forms differ because a sum of two cheap removals is itself sometimes past the expensive line even though neither part is, so exceeding the sum is a higher bar than exceeding either part.

Stating both scores is the point. A reading quoted as “five of six” and a reading quoted as “three of six” are the same six rows described with two different bars, and quoting only the flattering one is how a prediction gets credited.

What removing both walls does that removing each does not. One row per lattice, drawn at the difference between how far the next organ moves when both walls of the slot are removed and how far the two removals move it separately added together. Zero would mean the walls act independently. five of six lattices are more than 10° from it, three above and two below — so the pair is reliably not the sum of its parts and is not reliably larger than it either. On four of them the pair throws the next organ past the 45° that separates a cheap removal from an expensive one, which neither wall alone comes near.
Fig. 8 The same six rows, on which the two versions of the prediction score differently.

Why the ceiling matters

The displacement is an angle folded to a half turn, so it cannot exceed 180°. Two of the three positive rows sit at 164° and 163°, which is 91 per cent of the ceiling.

That has two consequences and both cut against over-reading the size of the effect. A quantity near a ceiling cannot grow much further, so “much larger than the sum” is bounded above by how much room is left. And an angle folded to a half turn treats 170° and 190° as the same displacement, so a pair that threw the organ most of the way round would be reported as one that threw it nearly half way.

The interaction is therefore a lower bound on those two rows. It is quoted as a difference rather than a factor for the same reason.

Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 7 places back and the organ 11 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 18.3° and 2.3°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 117.9°, against 20.6° for the two effects added, so the interaction is +97.3°. The slot is not two independent walls.
Fig. 9 The third positive row, at 118°, which is the one comfortably below the ceiling.

The cheap and expensive groups, redrawn

The census’s two groups were 8.9° to 30.7° for a neighbour and 62.8° to 167.6° for anything else, with nothing between 31° and 63°.

The six fourth cells here are 12.0°, 34.9°, 47.8°, 117.9°, 163.6° and 164.1°. Two are in the cheap group, two are in the expensive one, and two sit in the gap — 34.9° and 47.8°, in the empty space the census had nothing in.

So removing two neighbours produces displacements the census’s single removals never produced. That is a small result and it is the kind that is easy to miss: a gap in a distribution is evidence about the process only until a different process fills it.

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 census’s single removals across every offset, whose gap the two-organ cuts land in.

What the rule does with no wall

The placement rule puts each organ where a sum over its neighbours is least, with the weight falling as the cube of distance. With both walls gone the two nearest terms of that sum are missing, and what is left is a neighbourhood of organs further up the stem.

Two things could happen. The minimum could move a long way, because the terms that were pinning it are gone. Or it could hardly move, because the remaining terms were already deciding it and the two walls were nearly balanced against each other.

Both happen. Which one happens on a given lattice is what the sign of the interaction records, and the design does not say what selects it.

A stem gathers neighbours linearly; a growing disc barely gathers them at allOn a cylinder of circumference 1 with a rise of 0.02, the nodes within distance d number 2d/0.02 once d exceeds one turn — a fitted exponent of 1.020 and 100 per unit against the 100 the geometry fixes. In the disc model an element of age k sits at radius e^(0.4k), so each doubling of the distance adds the same two or three neighbours rather than twice as many.123-0.50000.50011.50distance from the node, log₁₀neighbours, log₁₀a stemslope 1a disca logarithmrise 0.02 · 6000 nodes · meristem growth 0.4slope 1.020 against slope 1
Fig. 11 The neighbourhood the rule sums over, from which two terms are removed.

What the pair leaves standing

The families a wrecked stem keeps are measured for every cell of the design, and the fourth cell is not like the others.

Four of the six pairs leave a family standing — the 5 at two golden lattices, the 7 at a Lucas one, and so on. Two of the six leave nothing at all: no lag whose hop is both steady and unmoved from the control’s, anywhere in the spectrum.

Single-organ cuts essentially never do that. It is a destination multi-organ cuts reach and one organ does not, and it is worth separating from the displacement result because a stem that keeps nothing is not a lattice with a slip in it.

Which hops survive one wall, the other, and both. One row per lattice. The last three columns are the lags whose hop the cut stem still holds, unchanged from a control that shares its history — the measurement that identifies what a wrecked stem has become. Removing a single wall always leaves something standing, which is what every single-organ cut in this collection does. Removing both leaves nothing at all on two of six lattices, including the coarse rung that no single removal can wreck. A stem that keeps no rigid hop is not a wrecked lattice with a slip in it; it is a stem that is no longer a lattice.
Fig. 12 The lags each cell of the design leaves standing, including the two pairs that leave none.

What was cut to make this possible

A limitation in the shared machinery. Cuts of several organs are made by the file that measures how many organs it takes, and it grew every stem from one fixed starting angle.

That put the whole Lucas branch out of reach of any multi-organ design: asking for a Lucas stem returned a golden one, silently, with the caller’s contact numbers read off a Lucas lattice and applied to it. The offsets then named organs that were nobody’s neighbour and the removal came back expensive — a plausible answer to a question that was never asked.

The first version of this table had Lucas rows reading 164° and 131° for single removals, which is how it was found. The starting angle is a parameter now, defaulting to what it was, so every stem previously grown is unchanged.

One rule, one rise, two branches that stay where they were put. The top 70 organs of two stems grown by the same placement rule at the same rise of 0.013, differing only in the stretch of ideal lattice each was started from. The left one was seeded at the golden angle and settles at 136.781° with the pair 5/8; the right one was seeded on the Lucas lattice and settles at 99.785° with 4/7. Neither drifts towards the other: 0.73° and 0.28° from where each was seeded, over four hundred organs. That is what makes an intervention on the right-hand stem a measurement about a different lattice rather than about a different rule — and 4 and 7 are not Fibonacci numbers, which is the property the experiment needs.
Fig. 13 Two stems at one rise on the two branches, which the machinery could previously only grow one of.

What the design does not vary

The offset. Every cut here is at one of the stem’s two counted numbers, so the design has two cells that are single removals and neither of them is a removal somewhere else.

That is what makes the singles comparable across six lattices: the 5-wall of a 5/8 stem and the 7-wall of a 7/11 stem are the same object named by different numbers. It also means the design says nothing about removals that are not walls, and the census’s expensive group — 62.8° to 167.6° — is made entirely of those.

So the four cells are a slice through a larger table rather than a table. The larger one, with every pair of offsets rather than the two walls, is the two-organ sweep the dose thread ran and it did not record displacements or spectra.

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. 14 Every offset’s single-removal displacement at one rise, of which the design uses two.

Why the walls are the right two offsets

Because they are the two the placement rule’s own nearest terms belong to.

The rule sums over neighbours with the weight falling as the cube of distance, so the contribution of an organ is set by how far it is from where the next one goes. The two chain-neighbours are the two nearest organs in the arrangement, which is what makes them the two largest terms — and the fact that removing either is the cheap removal is the standing puzzle, since the largest term going missing ought to move the answer most.

The two-by-two is the sharpest form of that puzzle. If two cheap removals make an expensive one, the cheapness of each is about the pair being intact rather than about either member.

On three of six lattices they do, which is the shape of an answer without being one.

A stem gathers neighbours linearly; a growing disc barely gathers them at all. On a cylinder of circumference 1 with a rise of 0.02, the nodes within distance d number 2d/0.02 once d exceeds one turn — a fitted exponent of 1.020 and 100 per unit against the 100 the geometry fixes. In the disc model an element of age k sits at radius e^(0.4k), so each doubling of the distance adds the same two or three neighbours rather than twice as many.
Fig. 15 How far each organ within reach sits from the next placement, on which the two walls are the two nearest.

What a bigger design would settle

Whether the sign of the interaction tracks anything. Six lattices give three positives and three negatives and no visible pattern; the rise does not order them, the counted pair does not, and the branch does not.

Twenty lattices would be eighty runs, of which twenty are controls shared with everything else at those rises, and would either produce a pattern or establish that there is not one. It is the cheapest unrun experiment this round leaves.

The candidate the design points at is position inside the rung, which is the column that turned out to be doing the work the last time this thread thought a quantity was varying freely.

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. 16 Where each census lattice sits inside its own rung, which is the column a bigger design would add.

What the fourth cell is not

A dose. The dose thread removed one, two, three, four and five organs from a stem and found that the share of arrangements which never repair climbs 0.25, 0.56, 0.71, 0.83, 0.98 — so removing more wrecks more, and it invents no new destination.

This design removes two organs and the two are named rather than swept. That makes it a different question: not how much damage, but which two organs, chosen for being the two the rule’s own nearest terms belong to.

The two questions meet in one place. A dose of two is one of sixty-four arrangements at each rung, and the two walls are one of them. Where the walls sit inside the dose sweep’s own distribution of outcomes is a comparison worth making and is not made here, because the dose sweep recorded fates and not displacements.

The mirror belongs to the lattice, not to the dose. How close the closest arrangement came to the mirror of the divergence it was cut from, against the share of the front that was removed. The marked point at 40 per cent is the coarse 3/5 rung with two organs taken, which reaches the mirror exactly. Every other point is a finer rung: five sizes of cut at 5/8 running from 13 to 63 per cent, and three organs at 8/13. Taking a larger share of a larger front than the coarse rung needs gets nowhere near, so the quantity that decides it is not the fraction of the neighbourhood removed.
Fig. 17 How the share of wrecked arrangements grows with the number of organs removed, which is the other two-organ question.

Where the design’s numbers can be checked

Against the census, which measured the same quantity on different stems.

The twelve single removals here are all at offsets equal to one of the stem’s counted numbers, and the census’s own reading of that case gives 8.9° to 30.7° over nine rows. This design gives 2.3° to 41.7° over twelve, which brackets it — wider at both ends, on stems the census does not contain, and with no row outside the 45° that separates the two groups.

That agreement is the check that the machinery is measuring the same thing. It is worth having because this design’s first version measured something else — Lucas offsets applied to golden stems — and the way it announced itself was single removals coming back at 164° and 131°, outside the cheap group by a wide margin.

What a removal costs the next organ. One mark per wrecked cut in the census: how far the first organ placed after the removal ended up from where the control put it. The rows split by which organ was taken. Removing a direct chain-neighbour of the growing tip — an organ at a multiple of one of the two counted numbers — moves the next organ by between 8.9 and 30.7 degrees. Removing anything else inside the front moves it by between 62.8 and 167.6. Nothing lands between the two groups and the ratio across the gap is 2.05, so the line is a gap rather than a threshold. Taking away a neighbour is the cheap removal, which is the opposite of what the words suggest.
Fig. 18 What a removal costs the next organ, split by whether the removed organ was a chain-neighbour of the tip.

The one line

The tip sits in a slot between its two chain-neighbours. Removing either alone costs 2.3° to 41.7° on all six lattices, which is the cheap group everywhere. Removing both costs 12.0° to 164.1°: past the expensive line on three lattices, in the gap between the two groups on two, and exactly the larger single removal’s own value on one. The interaction runs from −25.8° to +132.9° and is large on five of six, in both directions.

What removing both walls does that removing each does not. One row per lattice, drawn at the difference between how far the next organ moves when both walls of the slot are removed and how far the two removals move it separately added together. Zero would mean the walls act independently. five of six lattices are more than 10° from it, three above and two below — so the pair is reliably not the sum of its parts and is not reliably larger than it either. On four of them the pair throws the next organ past the 45° that separates a cheap removal from an expensive one, which neither wall alone comes near.
Fig. 19 The six interactions, which is the whole design in one column.

Where this goes next

What a removal costs the next organ. One mark per wrecked cut in the census: how far the first organ placed after the removal ended up from where the control put it. The rows split by which organ was taken. Removing a direct chain-neighbour of the growing tip — an organ at a multiple of one of the two counted numbers — moves the next organ by between 8.9 and 30.7 degrees. Removing anything else inside the front moves it by between 62.8 and 167.6. Nothing lands between the two groups and the ratio across the gap is 2.05, so the line is a gap rather than a threshold. Taking away a neighbour is the cheap removal, which is the opposite of what the words suggest. Removing a neighbour costs least Take away an organ that is a direct chain-neighbour of the growing tip and the next organ moves by under thirty-one degrees. Take away anything else inside the front and it moves by at least sixty-three. Thirty cuts, two groups, a factor of two between them and nothing in the gap. A second cut moves the next organ, and does not move the boundary. Every pair of organs that can be taken out of a settled stem at a rise of 0.032, where the pattern is 3/5. A row is the offset of the nearer organ removed, counted back from the tip; a column is how many further places back the second one sits. The shade is how far the next organ ends up from where it would have been, from under a degree in the palest cells to a half-turn in the darkest. The run of shaded rows ends at 5, which is the larger parastichy number, and every row below it is blank across the whole width — the largest displacement anywhere past the front is 2.34°. So the nearer of the two organs decides whether the removal is felt at all, and the second one, wherever it is put, cannot make the pattern notice an organ it was not going to notice. A cut of two organs One organ removed from a stem is felt out to the larger parastichy number and no further, and at the coarsest arrangement the stem always repairs itself — so the one rung where the interesting prediction could be checked had no experiment that could reach it. Two organs can. The second cut brings a parameter with it, and that parameter turns out to be a control. Removing one wall of the slot, then the other, then both. The growing tip sits in a slot between its two chain-neighbours — the organ 5 places back and the organ 8 places back. Each bar is how far the next organ placed moves when those are removed, against a control sharing the same history. Either alone is a cheap removal: 25.8° and 12.0°, both inside the 45° that separates taking a neighbour from taking anything else. Together they move it 12.0°, against 37.7° for the two effects added, so the interaction is -25.8°. The slot is not two independent walls. A removal that changes nothing On one of the six lattices, taking away both walls of the slot moves the next organ 11.953125° — which is exactly, to the last digit, what taking away the larger wall alone moves it. The smaller wall's removal contributes nothing at all when the larger one is already gone.

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.

  • One level and two exceptions — both name ablation, claim testing, control, lattice offset, measurement, mechanism, nearest neighbour, negative result, prediction
  • The ordering was not the actor — both name ablation, claim testing, control, lattice offset, matched design, nearest neighbour, negative result, parastichy pair, the placement rule
  • The organ that moved furthest — both name ablation, claim testing, control, lattice offset, measurement, nearest neighbour, negative result, parastichy pair, the placement rule
  • One offset, two answers — both name ablation, claim testing, control, lattice offset, measurement, negative result, parastichy pair, the placement rule
  • The angle is not the actor — both name ablation, claim testing, control, lattice offset, matched design, mechanism, negative result, parastichy pair
  • The family that lost a member — both name ablation, claim testing, control, lattice offset, measurement, negative result, parastichy pair, the placement rule

Named objects

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

AblationClaim testingControlThe range of the interactionLattice offsetMatched designMeasurementMechanismNearest neighbourNegative resultNeighbourhoodParastichy pairThe placement rulePrediction