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

When the second wall is free

On six of thirty lattices, removing both walls of the slot costs exactly what removing the larger one alone costs — 35.9° and 35.9°, 12.0° and 12.0°, agreeing to the last digit of the grid the azimuths sit on. The smaller wall is not a wall on those rows.
13 min read 21 figures State the claim, state the testForm is not mechanism

Worth reading first: Both walls of the slot · The organ that was taken away · Where a handover sits.

Take the smaller wall of the slot away and the next organ moves 104.3°. Take the larger wall away and it moves 35.9°. Take both away and it moves 35.9°.

Not approximately. The two agree to the last digit of the grid the azimuths are placed on, which is 1,536 samples of the circle and a quarter of a degree a step. Removing the smaller wall as well as the larger one changes nothing at all.

Where taking the second wall as well changes nothing. Each row is one lattice, with three marks: how far the next organ moves when the smaller wall alone is removed, when the larger alone is removed, and when both are. On these lattices the third mark sits on the second, to within two steps of the azimuth grid. The smaller wall is free — taking it away as well changes nothing — and on a row like that the interaction is minus the smaller wall's own cost by construction, which is arithmetic and not a measurement. Three of them are the whole of one rung and the others are the fine ends of two more.
Fig. 1 The six lattices where the third mark sits on the second. The two single removals are far apart and the pair costs what the larger costs.

What that does to the number being measured

The interaction is defined as the cost of removing both, less the costs of removing each added together. When the cost of removing both is the cost of removing the larger, the smaller’s cost cancels against nothing and what is left is minus the smaller’s cost.

So the interaction on these rows is not a measurement. It is a cell of the table copied and given a minus sign, and it is guaranteed negative and guaranteed large whenever the smaller wall is expensive to remove on its own.

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.
Fig. 2 One of the free lattices as a two-by-two. The fourth bar and the third are the same length, which is what makes the interaction arithmetic.

Six rows of thirty

Three of them are the whole of the Lucas 1/3 rung, at all three positions sampled on it. The other three are the fine ends of two longer rungs — the golden 5/8 at a rise of 0.008 and 0.00795, and the Lucas 3/4 at 0.0277 — whose coarse ends are not free at all.

The gap is clean. On the other twenty-four lattices the pair costs at least eight degrees more than the larger wall alone, and on most of them it costs more than a hundred. Nothing sits between half a degree and eight.

The slot interaction at 30 lattices, gathered by rung. One row per lattice, drawn at how much further the next organ moves when both walls of the slot are removed than the two single removals added together account for. Zero would mean the walls act independently. The pale rows are the ones where removing the second wall costs nothing at all, so their value is minus the first wall's own cost and is arithmetic rather than a measurement. Of the 24 rows that are measurements, 13 are strongly positive and 11 are not, and every rung falls on one side or the other with nothing straddling.
Fig. 3 All thirty lattices, with the free rows drawn pale. They are the three most negative entries in the table and none of them is a measurement of an interaction.

The obvious worry, checked

A number that agrees to the last digit of a grid is the shape of a discretisation artefact: two different quantities rounded onto the same sample. The azimuths here are chosen from 1,536 candidates, so a quarter of a degree of real difference would be invisible.

That worry is answered by the size of what is being compared. The smaller wall’s removal costs 104.3° on the Lucas 1/3 rung. If it were doing anything at all when the larger wall is already gone, it would have to be doing under a quarter of a degree of it — four hundred times smaller than what it does alone.

The measurement is limited by the protractor, not by the plant. The peak falls as the reading error grows, and it falls by an arithmetic factor with nothing fitted: a position error enters two consecutive divergences with opposite signs, adding variance at every lag while the pattern's signal sits at one. At a quarter of a degree the readout is right on all 5 runs; at half a degree on 2; at a degree on 1. Below the dashed floor the peak is the largest of thirty noisy numbers rather than a measurement.
Fig. 4 What the sampling grid can and cannot resolve, which is the instrument every displacement here is read on.

What “free” would mean

The obvious reading is that the smaller wall was never a wall. If the two organs named as walls are not both touching the slot, then one of them is a neighbour by arithmetic and not by contact, and removing it should cost nothing.

That reading is available and it is not quite right, because removing the smaller wall alone costs 104.3° — the most expensive single removal anywhere in the design. An organ that is not touching cannot cost a hundred degrees to take away, and which organ the rule actually feels has been separated from which organ is nearest for long enough here that the distinction is not available as an escape.

A stem gathers neighbours linearly; a growing disc barely gathers them at allOn a cylinder of circumference 1 with a rise of 0.013, the nodes within distance d number 2d/0.013 once d exceeds one turn — a fitted exponent of 1.009 and 154 per unit against the 154 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 0.013 · 9231 nodes · meristem growth 0.4slope 1.009 against slope 1
Fig. 5 The standing distinction here, drawn: which organ is nearest and which organ the rule actually feels are two different questions with two different answers.

So the order matters

Removing the smaller wall from an intact stem is expensive. Removing it from a stem that has already lost the larger wall is free. That is not a contradiction; it is what an interaction of exactly minus one hundred per cent looks like.

The two removals are not independent and they are not additive: the second one’s effect depends entirely on whether the first has happened. On these six rows the larger wall’s removal has already done everything the smaller one’s would have done.

Move the second organ far enough back and the experiment is the old one. The displacement of the next organ when two organs are removed — one three places back and one a further gap behind it — against that gap, at a rise of 0.013 where the pattern is 5/8. The dashed line is what removing the single organ three places back does on its own, computed by the earlier one-organ intervention and not by this one. Inside the front the two vacancies interact and the answer swings over 45°; from the gap that puts the second organ 2 places behind the front onwards it settles onto the single cut's -48.3°, within 0.7°. That limit is what makes the second parameter a control rather than a confound.
Fig. 6 A single removal against the gap it leaves, which is the quantity the second removal would have to find something left of.

Which is a mechanism, and it is drawn

The slot has two walls and the rule places the next organ at the minimum of a sum over its neighbours. Taking the larger wall away opens the slot on one side; the organ falls into the opening and comes to rest against something else. Taking the smaller wall away as well removes an organ it is no longer resting against.

That is a picture, not a measurement, and this collection is careful about the difference — a description is not a mechanism however well it fits. What is measured is that the second removal changes nothing; what is offered is a reason it might not.

Two answers 138° apart, and one organ holding the second one up. The repulsion the rule minimises, around the circumference of a stem at a rise of 0.008, at the height the next organ will sit at. It has two low points 138.3° apart: the slot the next organ takes, and the slot the organ after it will take. The runner-up is 13.6% higher. The organ 13 places back carries 14.6% of the energy at the winning slot and twelve places back carries 16.3% at the runner-up — and that is more than the gap, so taking that organ away makes the runner-up win and the next organ appears a whole divergence away. Neither guard is a contact of the organ being placed; twelve is one place inside the larger number of the pair this stem is climbing towards.
Fig. 7 The two slots a tip can sit in, which is the geometry the two walls belong to.

The prediction it makes

If the account is right, the free rows should be the rows where the larger wall’s removal moves the organ past the smaller wall. That is checkable: it predicts that the organ ends up on the far side of the slot, which is a displacement of about one divergence step rather than of a few degrees.

The three Lucas 1/3 rows move 35.9°, 37.3° and 38.9°, against a settled divergence on that rung of about 148°. That is not one step and the prediction fails as stated. The account survives only in the weaker form that the organ ends up somewhere the smaller wall does not reach.

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. 8 What a single removal costs, grouped: the two populations this thread has been separating since it began.

Position, and the one thing it decides

Position inside the rung was refused as an account of the interaction’s sign. It is not refused here. Both of the rungs that go free at one end go free at the fine end, at 84% and 85% of the way down, and their coarse ends are the most positive rows in the whole table.

So there is a transition inside a rung, and it is a transition in what the design is measuring rather than in the quantity it measures. That is the more awkward kind.

The interaction across each rung, coarse end to fine end. One line per rung, drawn against where in the rung each lattice sits — nought at the coarse end, one at the fine end, measured in the logarithm of the rise. The lines are flat. Inside a rung the interaction moves by 3.3 to 13.6 degrees, against a spread of 240 degrees across the ladder, and it falls from the coarse end to the fine one on 6 of the 7 rungs. Position inside a rung was the candidate this design was built to test and it is not what decides the sign.
Fig. 9 The golden branch’s rungs against position. One of them falls off its own line at the fine end, which is where it goes free.

Two rungs go free and two do not

The golden 8/13 rung and the Lucas 7/11 rung were both sampled at 85% and neither is free: their pairs cost 120.2° and 118.4° against larger walls of 2.6° and 4.9°. So the fine end of a rung is not where a slot loses a wall; the fine ends of two particular rungs are.

Which is the same shape as everything else in this table. A property that looks like a function of position turns out to be a function of the rung, with position deciding where on the rung it happens.

Five accounts of the sign, on the 24 lattices that are measurements. Each bar is how many of the lattices an account puts on the right side of zero. The six rows where the second wall is free are left out, because their value is minus the first wall's cost by construction and any rule scores whatever it happens to say about them. Position inside the rung, the rise and the branch all fail. The larger counted number sorts 22 of the 24, and the misses are one rung's worth of rows rather than a scatter.
Fig. 10 The five accounts of the sign, scored on the twenty-four lattices that are measurements. The free rows are left out of the scoring for the reason this essay is about.

What the Lucas 1/3 rung is

It is the shortest rung on either branch: it spans eight per cent in the rise against ninety-eight per cent for the Lucas 3/4 below it. Three positions on it are three rises within five thousandths of each other, and all three behave identically.

A rung that short is nearly a point, and a design that samples it three times has sampled one lattice three times. That is worth saying because three of the six free rows are that rung, so the six are really four lattices and one of them counted three times. It is the same accounting error a census that lists a lattice under two names makes, arrived at from the sampling side rather than from the labelling side.

The Lucas ladder, rung by rung. Each bar is one rung — a run of rises over which a counter returns one pair — drawn from its coarse end on the left to its fine end on the right, with the whole bar scaled to the same width so that positions inside different rungs can be compared. The mark on each bar is the rise at which the two contact steps change places, and it falls between 6 and 40 per cent of the way down on every rung that has one. The dots are the lattices this collection's ablation census was grown at, dropped onto the rungs they belong to. They are not spread across the bars; three of them sit past the mark.
Fig. 11 The Lucas branch’s rungs. The topmost is a sliver, and three of this table’s rows sit inside it.

Which is a warning about the design

The design put three lattices on every rung because three positions can show a trend. On a rung that spans eight per cent in the rise, three positions cannot show anything: they are one measurement with two confirmations.

Nothing in the design noticed. A rung is a rung to it, and the span was not a condition of sampling. It should have been, and a later sweep should weight rungs by their span rather than by their existence — which is the correction the band sweep already made when it moved from an absolute step in the rise to a proportional one.

Where a rung's handover sits, over six rungs. The rise at which the two contact steps change places, as a fraction of the way down each rung from its coarse end, over every rung on both branches that has one. All six fall in the coarse half and none in the middle: the positions measure 6, 12, 15, 33, 35, 40 per cent. That is a fact about the arithmetic of the lattice rather than about any experiment run on it, because the crossing is where the two step lengths are equal and both are functions of the rise and the divergence alone.
Fig. 12 How the ladder’s own features sit across a rung, which is the distribution a design sampling three positions is assuming something about.

The honest count

So the six free rows are four distinct lattices: one on a very short rung, counted three times, and three at the fine ends of two long ones. Stated that way the regime is thinner evidence than six rows of thirty sounds.

It is still a regime rather than a row. Four lattices agreeing to the grid, on three different rungs and both branches, is not a coincidence — and the twenty-four that are not in it miss by eight degrees or more.

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. 13 What each pair removal leaves standing. On the free rows the answer is what the larger removal alone leaves standing, which is the same fact from another direction.

The row that is free and wrecks

One of the four is stranger than the others. On the Lucas 3/4 rung’s fine end, either single removal heals — the stem recovers its divergence and its counted pair — and removing both wrecks it. The pair costs 23.0° against 23.2° for the larger alone, so the second wall is free by the measurement this essay is about, and the fate of the stem is different.

A displacement of the next organ and a wreck of the whole run are two different readings, and here they disagree. The cheap reading says nothing happened; the run says the arrangement never recovered.

Both edges of the front heal; the middle of it does not. The same removals, followed for 300 organs each. A cut one to three places back is undone within fifty organs and a cut ten to thirteen places back within sixty. A cut in between is never undone: the divergence sequence settles into an exactly repeating cycle of 5 angles and holds it for the rest of the run. The rule corrects a displacement and cannot correct a deletion.
Fig. 14 Recovery against displacement, offset by offset. The two quantities usually agree and this row is one of the places they do not.

Which is why the first organ is not the whole story

The first organ’s displacement was adopted as a cost because it separates cheap removals from expensive ones cleanly, and it does. What it does not do is predict what the run becomes, and that has been visible before: the displacement is a reading in the transient and the fate of the stem is a reading in the pattern.

Here they come apart on one row of thirty, which is exactly the rate at which a statistic that usually works stops working. The largest displacement came apart the same way, and for a related reason: both readings are taken inside a window nobody aligned to anything.

How far above the hole the damage becomes a pattern. One row per wrecked cut, drawn at the first organ from which every residue class stays at its own level for the rest of the run. On the 25 rows that reach it at all, it runs from 7 to 303 organs above the removed one; five rows never reach it inside the 300 organs each run is continued for. Below that point the stem is still moving, and the displacement of the first organ after the cut — the quantity that tells a cheap removal from an expensive one — is measured there. Above it, nothing changes again.
Fig. 15 The two regimes a wrecked stem has, and which readings belong in which. The displacement of the first organ is in the left one.

What this does to the older table

Two of the original six lattices are free rows. Their interactions were reported as −25.8° and, on the row that also wrecks, −49.2°, and both were read as negative interactions of the tip.

They are not. They are the smaller wall’s own cost, with a sign. So the earlier sentence — three lattices above zero and three below, with no ordering — was counting two arithmetic entries among its six, and the ordering it could not find was being hidden by them.

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. 16 The six lattices as they were first reported, in lattice order. Two of the three negatives are rows where the second wall is free.

What would settle it

A finer sweep of one rung: ten positions between 70% and 100% of the golden 5/8, asking where the pair stops costing more than the larger wall. If the transition is sharp, there is a rise at which the slot loses a wall, and that rise is a thing to name. If it is gradual, the free rows are the end of a slope and the word “free” is doing too much work.

Thirty runs, and it has not been run. The design that produced this table samples three positions a rung, which is the resolution that found the regime and cannot describe it. That is the standing shape of this thread’s shortfalls: a sweep at one resolution finds a feature and a finer one is needed to say what it is.

Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it.
Fig. 17 One rung at a fine grid, which is the shape the sweep this essay asks for would take.

What it costs to leave them in

A table of thirty with these six in it has a range of 240° and no rule that sorts it. The same table with them named and set apart has a gap of eighty-five degrees in the middle and a rule that sorts twenty-two of the remaining twenty-four.

That is the whole value of the distinction, and it is not a matter of cleaning data. The six rows are correct measurements of what they measure; they are just not measurements of the thing the column is called.

Five accounts of the sign, on the 24 lattices that are measurements. Each bar is how many of the lattices an account puts on the right side of zero. The six rows where the second wall is free are left out, because their value is minus the first wall's cost by construction and any rule scores whatever it happens to say about them. Position inside the rung, the rise and the branch all fail. The larger counted number sorts 11 of the 24, and the misses are one rung's worth of rows rather than a scatter.
Fig. 18 The accounts scored again, in a fixed order. Every one of them would score differently if the arithmetic rows were left in.

What a free wall does to the survivor

There is a third reading available on every row and it has not been used here: the lags whose hop the cut stem still holds. On the free rows the pair leaves standing what the larger removal alone leaves standing, which is the same statement as the displacement one and arrives by a different route.

On the Lucas 3/4 fine end the pair keeps the four-hop and its multiples where the singles keep everything, so the free row that wrecks is free by displacement and not free by what it destroys. Two routes, and they do not always name one number.

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. 19 The lags a cut stem keeps, across the census. A pair removal that keeps what its larger half keeps is the free case seen through a different instrument.

Why this was invisible on six lattices

The six-lattice design had two free rows in it and no way to notice. Free is a relation between two cells of one row — the third and the fourth — and the table was read down the interaction column, where a free row and a strongly negative interaction look identical.

Reading down a derived column and never across the row it came from is a specific habit, and it is the one the plateau caught in a different quantity: a statistic computed correctly, read as though it still meant what it meant when it was defined.

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. 20 A lattice that is not free, drawn as a two-by-two, for comparison with the one at the top of this essay.

What is worth carrying

Three things. A regime exists in which the slot behaves as though it has one wall, and it is found at the fine end of two rungs and across the whole of a third. Inside it the interaction is arithmetic and must be set aside before any claim is made about the rest.

And the reason it took thirty lattices to find is not that six were unlucky. It is that six rows read down one column cannot distinguish a measurement from a restatement, whatever the six happen to be.

The slot interaction at 24 lattices, gathered by rung. One row per lattice, drawn at how much further the next organ moves when both walls of the slot are removed than the two single removals added together account for. Zero would mean the walls act independently. Of the 24 rows that are measurements, 13 are strongly positive and 11 are not, and every rung falls on one side or the other with nothing straddling.
Fig. 21 The table with the free rows removed, which is the table every claim in this thread is now made over.

The one line

On four lattices — six rows, because the shortest rung is sampled three times — removing the second wall of the slot changes nothing to within the grid the azimuths sit on, and the interaction reported for them is the first wall’s own cost with a minus sign.

They are a regime rather than an artefact, and setting them aside is what lets the rest of the table say anything.

Where this goes next

The slot interaction at 30 lattices, gathered by rung. One row per lattice, drawn at how much further the next organ moves when both walls of the slot are removed than the two single removals added together account for. Zero would mean the walls act independently. The pale rows are the ones where removing the second wall costs nothing at all, so their value is minus the first wall's own cost and is arithmetic rather than a measurement. Of the 24 rows that are measurements, 13 are strongly positive and 11 are not, and every rung falls on one side or the other with nothing straddling. Six lattices were not enough The interaction between the two walls of a slot came back at −25.8° to +132.9° on six lattices, three above zero and three below, with no ordering by rise, by counted pair or by branch. A quantity that looks free on six rows is usually a quantity that has been sampled at six rows. 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. 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°. Five accounts of the sign, on the 24 lattices that are measurements. Each bar is how many of the lattices an account puts on the right side of zero. The six rows where the second wall is free are left out, because their value is minus the first wall's cost by construction and any rule scores whatever it happens to say about them. Position inside the rung, the rise and the branch all fail. The larger counted number sorts 22 of the 24, and the misses are one rung's worth of rows rather than a scatter. The rung decides the sign Twenty-four lattices where both walls of the slot are really there. Thirteen give a strongly positive interaction, at 85° to 135°; eleven give a negative or null one, at −25° to −0.5°. Nothing lies between. Every rung's lattices fall on the same side as each other.

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.

  • Every rise of a band — both name ablation, claim testing, control, honest limits, lattice offset, matched design, measurement, negative result, parastichy pair, rise, rung
  • Three offsets, three crossings — both name ablation, claim testing, control, honest limits, lattice offset, matched design, measurement, negative result, parastichy pair, rise, rung
  • One offset, two answers — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rise, rung
  • The alternation is not a period — both name ablation, artefact, claim testing, control, honest limits, lattice offset, measurement, negative result, rise, rung
  • The front deepens down a rung — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rise, rung
  • The offsets that never change — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rise, rung

Named objects

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

AblationArtefactClaim testingControlDiscretisationHonest limitsThe range of the interactionLattice offsetMatched designMeasurementNearest neighbourNegative resultParastichy pairRiseRungSummary statistic