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

The exception was already labelled

The larger counted number sorts twenty-two of twenty-four lattices by the sign of their slot interaction. Both misses are on the Lucas 3/4 rung — the one rung a different measurement had already singled out, for reasons with nothing to do with this one.
13 min read 20 figures State the claim, state the testTwo routes, one number

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

A rule that sorts twenty-two of twenty-four rows has two misses, and where the misses are matters more than how many there are. Two misses scattered across two rungs is a rule that is roughly right. Two misses on one rung is a rule that is right with an exception, and an exception is a thing that can be named.

Both misses here are on the Lucas 3/4 rung. Its larger counted number is 4, so the rule says its interaction should be small or negative, and its two measured lattices give +90.5° and +85.1°. They are not near the line: they sit with the golden 8/13 and the Lucas 7/11, at the positive end of a table whose other end they were predicted to be at.

The rung’s third lattice is not in the count at all, for a reason of its own, and that turns out to matter.

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. 1 The Lucas branch’s rungs against position inside them. One of these lines is on the wrong side of the middle for its counted numbers.

The rung was already marked

Not by this thread. The handover sweep grows a band around the rise at which a rung’s two contact steps change places, and it reported the Lucas 3/4 as unlike the other five it found.

Its two contact steps stay within four parts in a thousand of each other across the whole of its band. On every other rung the two steps separate cleanly on either side of the crossing; here they never separate at all, so the ordering between them changes hands three times inside one band, and the sweep calls the band a control rather than an experiment.

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. 2 How far apart the two contact steps get on each band. One band never opens a gap worth the name.

And it was marked twice

The same sweep predicts a band’s width from the curvature of the settled divergence at the handover, and on five of six rungs the prediction is right to within a third. On the Lucas 3/4 it is wrong by a factor of nearly six.

The reason is in the fit: the linear term in the divergence at the handover is 0.005 to 0.13 on the five rungs the prediction works for, and 2.19 on this one. The prediction assumes a stationary point and this rung does not have one.

Two ways of predicting how wide a band is. A band ends where the settled divergence has moved 0.05° from its value at the handover, so the width should follow from how fast the divergence changes there. Reading that rate as the rung's average slope predicts widths that are wrong by factors of 0.20 to 5.92 — wrong in both directions, so no constant rescues it. Reading it as a curvature about a stationary point gives 0.41 to 1.08, with five of the six inside a third. The difference between the two is the difference between a curve and its average, and a band is exactly where the two are least alike.
Fig. 3 The measured band widths against the widths predicted from curvature. Five points sit near the line and one does not.

So the same rung fails two unrelated tests

One is about how two step lengths behave across a rung and one is about what happens when two organs are removed from a stem. They share a rung and they share nothing else: no quantity, no design, no window.

The Lucas 3/4 is one rung of eight. If the two anomalies were independent, the chance of them landing together is one in eight. That is suggestive and it is not evidence, and stating it as one in eight is the honest form.

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. 4 Every rung with a handover in it, and where the handover sits. The one this essay is about carries the narrowest band on the ladder.

A third mark, from this thread

There is one more, and it belongs to this design rather than to the handover sweep. Of the eight rungs, only two go free at their fine end — only two have a lattice where removing both walls costs what removing the larger costs — and the Lucas 3/4 is one of them.

The other is the golden 5/8, which is the most-sampled rung in the whole census and the one nearly every earlier measurement in this thread was made on. So the two rungs that behave unusually at their fine ends are the one that was already flagged and the one that has been looked at hardest, which is exactly the pair a selection effect would produce.

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. 5 The golden branch’s rungs. The most-sampled of them is also the other one that goes free at its fine end.

What makes a rung strange

A rung is a stretch of rise over which a stem’s counted pair does not change. Inside it everything else does: the settled divergence slides, the two contact steps lengthen and shorten at different rates, and at some rise the shorter of the two becomes the longer.

On most rungs those two steps are clearly different lengths for most of the rung and equal only at a point. On the Lucas 3/4 they are nearly equal everywhere, which makes every quantity that depends on which of them is shorter into a quantity with no signal in it.

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. 6 The geometry inside a rung: how the divergence and the two contact steps move as the rise falls through it.

Which is a reason the interaction might be different

The slot’s two walls are the two contact numbers. If the two contact steps are the same length, the two walls are the same kind of thing and the slot is symmetric — and a symmetric slot has no smaller and larger wall, only two walls.

That is a candidate account and it makes a prediction: the two single removals should cost more nearly the same on this rung than on the others. They cost 45.7° and 8.7° at one position and 49.0° and 15.7° at the next, a ratio of five and three. The prediction fails.

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. 7 A two-by-two on the Lucas branch. The two single removals are the two walls, and how unlike each other they are is a property of the rung.

And a reason it might not be

The steps being close in length is not the same as the walls being close in effect. A step length is a distance across the surface; a removal’s cost is what the placement rule does when an organ is gone, and the rule sums over a neighbourhood with an exponent in it.

Two organs at nearly the same distance can be doing very different things if one of them is inside the reach and the other is at the edge of it. So the account is not refuted by the failed prediction; it is unsupported by it, which is a weaker and more accurate statement.

The neighbourhood of 8/13, and where its background was taken from. μ₂ across nine tenths of a degree either side of 8/13, on a head of 300 organs — 23 in each of its 13 rows. The deep notch at the centre is the dip. The shaded columns are the offsets this site now takes a background from: those at least 0.03° from every rational with a denominator of 60 or less, of which there are 70 here. The marked sample at 0.2° is the one the earlier work used, and it lands 0.262° from 37/60. It reads 0.596 against a floor of 0.059. The clear offsets give 0.501.
Fig. 8 The neighbourhood the placement rule sums over, which is where a step length turns into an effect.

The rung’s own fine end

The Lucas 3/4’s third lattice, at 85% of the way down it, is not in the table at all. It is one of the six rows where the second wall is free: the pair costs 23.0° and the larger wall alone costs 23.2°.

So this rung has two lattices with the largest interaction the rule fails to predict, and a third where the interaction is not a measurement. It is the only rung in the design that does both.

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. 9 The lattices where the pair costs what the larger wall costs. One of them is this rung’s fine end.

And that third lattice wrecks

Either single removal on it heals — the stem recovers its divergence and its counted pair. Removing both wrecks it. That is the strongest form the interaction takes, a rung no single removal can reach, and it happens on a row where the cheap reading says the second removal did nothing.

The cheap reading is the displacement of the next organ placed. The expensive reading is what the run becomes three hundred organs later. Here they disagree completely, and the essay that separated the transient from the pattern is the reason it is not a contradiction.

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 Displacement by offset at a rise on this rung, which is the cheap reading — and the one that says nothing happened.

What a control is for

The handover sweep calls this rung a control because a band whose ordering never resolves is a band where the design’s own variable does not vary. A control is worth having: it is where a claim that the ordering does not decide the survivor can be checked against a band that holds the ordering still.

A control in one design is not a control in another. Here the rung is not a control; it is the row the rule cannot sort, and the same property that makes it useful over there makes it awkward here. That is a general hazard with a shared vocabulary: the ordering was not the actor on the bands precisely because this rung held it still, and the holding-still is what breaks the sorting here.

The families left standing, on both sides of every handover. One row per band, with every offset cut at rises spread across it and always at both ends and at the handover itself. The last column is every family left standing anywhere on that band. On three of the four bands that wreck at all it is a single family, unchanged across a rise at which the two contact steps swap places — so the step ordering is not what decides which family survives, and the result now rests on four counted pairs rather than on two. The shortest-hop reading scores 44 of 114 across the whole set, which is what a reading looks like when the quantity it is stated over is not in the mechanism.
Fig. 11 The survivor family kept on every band, including the one whose ordering changes hands three times.

The selection worry

A reader should be suspicious of “the exception was already labelled”. The ladder has eight rungs and several of them have been called unusual for something at some point, and picking out the label that matches after the exception is known is how a coincidence becomes a story.

So the claim is stated narrowly. The Lucas 3/4 was singled out in writing, in a different thread, before this table existed, on two measurements neither of which involves an ablation. That is checkable and it is what “already labelled” is meant to mean.

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 Where the handovers sit inside their rungs. Nothing in this picture is about ablations, and it is where the rung was first singled out.

The other way to read it

There is a reading in which the rung is not an exception at all. If the interaction is really set by the width of the slot rather than by the counted number — the account the sign rule leaves open — then a rung whose two contact steps are the same length has a slot of a shape no other rung has, and it could sit anywhere on the scale without breaking anything.

On that reading the counted number was never the actor; it was a label on the several quantities that move together down a ladder, and the Lucas 3/4 is the rung where those quantities come apart. That would make it the most informative rung in the design rather than its awkward one.

Distinguishing the two readings takes a measurement nobody has made: the angular width of the slot after each removal, on stems already grown.

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. 13 An organ’s neighbours with their distances, which is where a slot’s width would be read from.

What is not claimed

That the rung’s strangeness explains the interaction. Nothing here connects a pair of contact steps that never separate to a pair of removals that cost more together than apart, and the one candidate connection — a symmetric slot — makes a prediction that fails.

What is claimed is smaller and more useful: the rule’s error is not distributed. It is concentrated on a rung that an unrelated measurement had already found to be unlike the others, so the rule should be read as holding on the ladder’s ordinary rungs and as untested on its strange one.

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. 14 The accounts of the sign scored on the twenty-four measured lattices. The best of them misses twice, on one rung.

The rung is short, and that matters

The Lucas 3/4 spans nearly the whole of one factor in the rise — from 0.064 to 0.0239 — so it is not short at all as a rung. Its band is short: sixteen rises, three per cent of the rung, against seventy-two per cent for the widest.

Those are different quantities and it is easy to slide between them. The rung is long and ordinary in extent; the stretch of it over which the two contact steps are in doubt is nearly the whole of it, which is why the band grown by the width rule comes out tiny.

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. 15 How each band ends: at the rung’s edge, at a change of pair, or where the divergence stops being flat. The shortest band ends for a different reason from the others.

A test the ladder cannot supply

The rule’s threshold sits between a larger counted number of 7 and one of 8, and there is no rung on either branch whose larger number is 9 or 10. So the threshold is fitted into a gap and cannot be probed from inside the ladder.

The Lucas 3/4 is the only rung that sits on the wrong side of it, and it does so by five counted numbers rather than by one. A rung with a larger number of 5 or 6 that came out positive would be a much sharper problem, and neither branch has one.

Every family but two is the sum of two others. Four heads and the contact families each actually has. An arc arrives at every family that is the sum of two smaller ones; the two with no arc arriving are the generators, and on every head they are the two smallest. whorled, 144°: 2, 3, 5 — golden, 137.508°: 8, 13, 21, 34, 55, 89 — Lucas, 99.502°: 11, 18, 29, 47, 76 — rational, 137.5°: 8, 13, 21, 34, 55, 89 — 137.0°: 8, 13, 21, 29, 50, 71, 92, 113. So once a pair is counted the rest is arithmetic, and a third counted family is a prediction rather than a second measurement.
Fig. 16 The two sequences the ladder is built from, and therefore the counted numbers a design confined to them can ever test against.

Where the numbers 9 and 10 do occur

On other additive sequences, which stems do settle on: the settling test reaches several of them, including one running 1, 4, 5, 9, 14 and another running 2, 7, 9, 16.

A stem grown to one of those and cut at its two contact numbers would put a larger counted number of 9 into this table. It is four runs and it has not been done, which makes it the sharpest untried test this thread has — sharper than a finer sweep of the rung, because it would put the rule’s threshold under a value rather than beside it. It is recorded as an outstanding check rather than as an intention.

Every destination in the settling table, counted. One row per destination, placed across at the divergence the stems settled on and labelled with the parastichy pair a counter returns at the top of the run. 117 settled runs land on 15 destinations when two values within half a degree are called one. five of them sit on the golden or the Lucas sequence and ten do not, and the ones that do not are reached at every falloff exponent including the shallowest. Nothing here refuses to count: the settling test does not admit arrangements this collection would decline to call lattices.
Fig. 17 The destinations that count on sequences the ladder does not carry, several of which have a 9 in them.

What this rung would need

Ten lattices on it rather than three, spread across its whole span, with the two single removals and the pair at each. If the interaction is positive down its whole length and collapses only at its very fine end, the rule has a genuine exception. If it crosses zero somewhere in the middle, the rung is not an exception at all — it is a rung with a transition in it, and the three sampled positions happened to straddle the transition.

Thirty runs. The design that produced this table samples three positions a rung because three is the smallest number that can show a trend, and three is not enough to find a crossing. That distinction — between a sweep that can show a trend and one that can locate a feature — is the same one a band cut at nine rises rather than every rise turns out to have got wrong in the other direction.

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. 18 One rung at a fine grid, which is the shape the sweep this essay asks for would take.

Why the exception is worth an essay

Because a rule that sorts a table is worth exactly as much as the account of its failures. Twenty-two of twenty-four with the misses named, on a rung that was independently strange, is a different claim from twenty-two of twenty-four with two rows shrugged at.

The first can be tested — grow the rung finely, or find a 9 elsewhere. The second cannot be tested at all, which is the whole objection to it.

The readings, and where in their rungs they fail. Each bar is one candidate account of which family a wrecked stem keeps, scored across every wrecked cut in the census. Under each bar are the positions inside their own rungs of the cuts it gets wrong, as percentages from the coarse end. The best of them is right 25 times of 30, and the positions of its failures are the point: two of them are the single lattice grown at the far fine end of its rung, which is also the only census row past three quarters of the way down. Nothing here rescues a reading. What it shows is that the table these readings were scored on varies a quantity nobody chose, over a range nobody stated.
Fig. 19 A different table’s accounts and their misses, scored the same way. Naming where a rule fails is what makes the score mean something.

What carries forward

The rule, with its exception stated. The Lucas 3/4 rung, now marked by three unrelated measurements rather than two. And two designs that have not been run: a fine sweep of one rung, and a stem grown on a sequence the ladder does not carry.

Neither is expensive. Both were available while this table was being built and neither was done, which is the ordinary reason a shortfall exists.

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. 20 The whole table, with the free rows pale and the exception’s two rows above the middle where the rule says they should not be.

The one line

The larger counted number sorts the slot interaction on twenty-two of twenty-four lattices, and both misses are the Lucas 3/4 rung — the one rung a sweep of contact steps had already called a control rather than an experiment, on measurements with no ablation in them.

One rung of eight, so the coincidence is one in eight, and the account that would connect the two anomalies makes a prediction that fails.

Where this goes next

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 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. A band that moves nothing One of the six bands holds the counted pair, holds the divergence, and does not move the ordering: its two contact steps stay within four parts in a thousand of each other across the whole of it, so the ordering changes hands three times and neither end has one worth the name. 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. 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.

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.

  • Six lattices were not enough — both name ablation, claim testing, control, honest limits, the range of the interaction, lattice offset, matched design, measurement, negative result, parastichy pair, rise, rung
  • The offsets that never change — both name ablation, claim testing, control, handover, honest limits, lattice offset, measurement, negative result, parastichy pair, rise, rung, selection effect
  • Three offsets, three crossings — both name ablation, claim testing, control, handover, honest limits, lattice offset, matched design, measurement, negative result, parastichy pair, rise, rung
  • The side the census sat on — both name ablation, claim testing, control, handover, honest limits, lattice offset, negative result, parastichy pair, rise, rung, selection effect
  • One offset, two answers — both name ablation, claim testing, control, honest limits, lattice offset, measurement, negative result, parastichy pair, rise, rung
  • Seven rises and two seeds — both name ablation, control, honest limits, ladder, lucas numbers, matched design, measurement, parastichy pair, rise, rung

Named objects

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

AblationClaim testingControlHandoverHonest limitsThe range of the interactionLadderLattice offsetLucas numbersMatched designMeasurementNegative resultParastichy pairRiseRungSelection effect