Phyllotaxis
the spirals are counted, not admired
Where the angle comes from

A steeper rule walls nowhere else

The account of the wall at the fine end was that the basin narrows because the neighbourhood deepens, which predicts a steeper falloff walling somewhere else. Grown at four exponents, the four columns settle 30, 31, 29 and 27 of 72 — a spread of 0.056 against an error of 0.058.
12 min read 19 figures State the claim, state the testOne parameter, one family

Worth reading first: How long a stem takes to settle · A pattern with a rate · The angle is an output.

Below a rise of about 0.004 a stem does not settle onto a lattice, and growing it three times as long changes nothing: the limit is a wall rather than a budget. What closes is the share of starting angles that reach a lattice at all — seven of nine at a rise of 0.030, three at 0.005, one at 0.004 and 0.003.

The account offered for that was that the basin narrows because the neighbourhood deepens. It makes a prediction, and the prediction is the reason this essay exists.

How many starting angles reach a lattice, by rise and by falloff. Every cell is 9 stems grown from 9 starting angles spanning 40° to 180°, at one rise and one exponent in the placement rule's falloff, for 1200 organs each. The number is how many of them settle onto a lattice. Reading down a column, the basin closes as the rise falls, which is the wall. Reading across a row, nothing much happens: the exponent-three column is the published settling table digit for digit, and the other three are the same column inside their own sampling error. The account that a steeper falloff should wall somewhere else does not survive the table it predicted.
Fig. 1 Eight rises by four falloff exponents, with how many of nine starting angles reach a lattice in each cell.

The parameter that had never been varied

The placement rule puts each organ where a sum over its neighbours is least, with each neighbour’s contribution falling as a power of the distance. That power is three in every measurement on this site and has been three since the collection’s first essays.

Nothing chose it as the physically right number; it was chosen once, as a working value, and everything since has been measured at it. So a result stated as a property of the rule is a result measured at one exponent, and whether it belongs to the rule or to the exponent is a question nothing here had asked.

For most results that question is idle. For the wall it is not, because the account of the wall is about the exponent: a steeper falloff makes the effective neighbourhood shallower in reach and sharper in weight, which is exactly what the account says closes the basin.

Four ways to count the rule's neighbourhood, and one ordering. How many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by three different definitions. Counting the organs that carry half the profile gives 33, 12, 3, 2, 1. Counting the organs that carry nine tenths gives 182, 120, 30, 8, 3. Counting the organs that carry the weighted mean lag gives 68, 45, 21, 14, 11. They disagree about the size of the neighbourhood by a factor of 10 — the widest moves by 61 times across the sweep and the narrowest by 6.1 — and every one of them falls as the exponent rises. So no choice among them changes which rule is the deeper, and no redefinition can turn a result about the depth around.
Fig. 2 How deep the placement rule looks, measured four ways, which is what the exponent sets.

The design

Four exponents — 2, 3, 4 and 5 — at each of the eight rises the settling table uses, from nine starting angles spanning 40° to 180°, twelve hundred organs a stem. Two hundred and eighty-eight runs.

Three is in the middle of the range rather than at an end of it, which is deliberate: a sweep that started at the working value could only ever say the wall moves one way. Below two the neighbourhood stops being local at these run lengths, and above five the sum is decided by one organ, which is a different rule rather than a steeper one.

The rises are the settling table’s own list, unchanged, so that the exponent-three column of this table and the table that found the wall are the same measurement rather than two designs being compared.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 3 The settling table as it was measured at one exponent, which is the column this sweep has to reproduce.

The control

It does reproduce it. At exponent three, 7 of 9 starting angles reach a lattice at a rise of 0.030, 7 at 0.013, 5 at 0.008, 3 at 0.005 and 1 at each of 0.0045, 0.004 and 0.003 — the published table digit for digit.

Its slowest settling is 290 organs, which is the same 290 the settling table reported.

That check costs nothing and is worth running because the alternative — four columns that disagree, with no way to tell whether one of them is the old measurement — is a table nobody can read.

How many starting angles reach a lattice, by rise and by falloff. Every cell is 9 stems grown from 9 starting angles spanning 40° to 180°, at one rise and one exponent in the placement rule's falloff, for 1200 organs each. The number is how many of them settle onto a lattice. Reading down a column, the basin closes as the rise falls, which is the wall. Reading across a row, nothing much happens: the exponent-three column is the published settling table digit for digit, and the other three are the same column inside their own sampling error. The account that a steeper falloff should wall somewhere else does not survive the table it predicted.
Fig. 4 The same grid, whose second column is the exponent every other measurement here is made at.

The prediction, and what happened

If a steeper falloff narrows the basin, the columns should differ: the exponent-5 column should close earlier than the exponent-2 column, or later, but not sit on top of it.

Pooled over the whole ladder the four columns settle 30, 31, 29 and 27 of 72 runs.

That is 0.417, 0.431, 0.403 and 0.375, a spread of 0.056. The binomial standard error on each column is 0.058. The four are the same column.

The share of starting angles that reach a lattice, by exponent. Each row pools one exponent's whole column — every rise, every starting angle — into one share, with a bar an error either side of it. Pooling is what makes the comparison possible: a single cell is 9 runs and carries an error of about 0.17, which is three times the whole spread between the four exponents. Pooled, the four sit at 30/72, 31/72, 29/72, 27/72 — a spread of 0.056 against an error of 0.058. Nothing in the falloff moves the basin. A five-angle version of the same sweep gave a clean monotone trend, which is what a null looks like when it is sampled too thinly.
Fig. 5 The four columns pooled, each with an error bar an error either side of it.

Why pooling is the only way to ask

A single cell is nine runs, and a share of nine has a standard error of about 0.17 near a half — three times the whole spread between the exponents. So a cell-by-cell comparison of four columns is four columns of noise.

Pooling costs the shape of each column and buys an error bar small enough to see the difference that is being asked about. It is the right trade here because the question is not where each column closes but whether the four close at different places, and a pooled share answers that with the smallest error the design supplies.

The shape is not thrown away — it is in the grid. What the pooled figure adds is a number the four columns can be compared as.

How many starting angles reach a lattice at all. The share of nine starting angles, spread from 40 to 180 degrees, whose stem settles onto a lattice at each rise. It falls from 7 of 9 at the coarse rises to 1 at the finest, and the runs that fail do not fail by running out of organs — grown three times as long they fail identically. So what closes at the fine end is the set of arrangements a stem can fall into rather than the time it has to find one, which is a different kind of limit and a more interesting one.
Fig. 6 The share of starting angles that settle, at one exponent, which is the quantity being pooled.

The five-seed version, which looked like a trend

Before the nine starting angles were run, a cheaper version of this sweep used five, at four rises rather than eight. It gave 3, 3, 2, 1 down the exponent-2 column; 3, 2, 2, 1 down the exponent-3; 3, 2, 1, 1 down the exponent-4; and 3, 2, 0, 0 down the exponent-5.

Read as a table that is a clean monotone trend: the steeper the falloff, the earlier the basin closes. It is the result the account predicted, arriving on the first attempt.

It is the same null with a bigger error bar on it. Five runs a cell gives an error of 0.22 on a share near a half, and the differences between those columns are one run.

How many starting angles reach a lattice, by rise and by falloff. Every cell is 9 stems grown from 9 starting angles spanning 40° to 180°, at one rise and one exponent in the placement rule's falloff, for 1200 organs each. The number is how many of them settle onto a lattice. Reading down a column, the basin closes as the rise falls, which is the wall. Reading across a row, nothing much happens: the exponent-three column is the published settling table digit for digit, and the other three are the same column inside their own sampling error. The account that a steeper falloff should wall somewhere else does not survive the table it predicted.
Fig. 7 The full grid at nine starting angles, in which the trend the five-angle version showed is not there.

Which is why nine is not negotiable

The nine starting angles are the settling table’s, and they span 40° to 180° with the golden angle among them as a control — because a run started at the destination settles at organ zero and measures nothing.

Nine is not many. It is enough to separate a share of 0.4 from a share of 0.9 and not enough to separate 0.40 from 0.43, and this sweep needed the second. Pooling eight rises is what makes it possible at all, and even then the error is the same size as the spread.

A version of this with twenty starting angles would halve the error and is affordable — 640 runs rather than 288 — and would either sharpen the null or overturn it. It is named in the leavings.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 8 The starting angles the table is grown from, one mark per angle per rise.

What a null of this shape is worth

It rules out an account, which is what this collection is for.

“The basin narrows because the neighbourhood deepens” was the only account offered for the wall, and it was offered as a plausible mechanism rather than as a measurement. Its one testable consequence is that the wall should move with the exponent, and the wall does not move with the exponent to within the error available.

So the wall is not about that parameter. What it is about is not answered here, and the honest position is that the fine end of the ladder is a place stems cannot reach for a reason nothing in this collection has yet identified.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.381, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 9 The ladder whose fine end the wall sits at, and which the exponent does not move.

What the columns do share

All four settle most of their starting angles at the coarse end — 0.67, 0.78, 0.56 and 0.44 at a rise of 0.030 — and nearly none at the fine end. The shape of the decline is the same and the position of it is the same.

That shared shape is what makes the null a null rather than an incomparability. If one column had settled everything and another nothing, the four would not be four measurements of one quantity and pooling them would be meaningless.

It is asserted: every exponent settles at least four of nine at the coarsest rise swept.

How many starting angles reach a lattice, by rise and by falloff. Every cell is 9 stems grown from 9 starting angles spanning 40° to 180°, at one rise and one exponent in the placement rule's falloff, for 1200 organs each. The number is how many of them settle onto a lattice. Reading down a column, the basin closes as the rise falls, which is the wall. Reading across a row, nothing much happens: the exponent-three column is the published settling table digit for digit, and the other three are the same column inside their own sampling error. The account that a steeper falloff should wall somewhere else does not survive the table it predicted.
Fig. 10 The grid again: down a column the share falls with the rise, and across a row it does not move.

What is not claimed

That the exponent does nothing. It does two things, both measurable and both in the next two essays: it changes how long settling takes, by a factor of nearly three, and it changes which divergences a stem can settle on.

Neither of those is visible in a share. A run that settles at organ 808 and a run that settles at organ 8 are the same entry in a table of shares, and a run that settles on 42.3° and a run that settles on 137.8° are also the same entry.

So the null is specific: the exponent does not move the share of starting angles that reach a lattice. Said any more broadly it would be false.

How many organs settling takes, by falloff exponent. One mark per stem that settles, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. At the exponent every other measurement here uses, the slowest is 290 organs — the number four rounds of this collection have carried as though it belonged to the rule. It belongs to the exponent: at 4 the slowest is 808 and at 5 it is 674. A table of shares cannot see this, because a stem that settles at organ 800 and one that settles at organ 8 are the same entry in it. And it is still not a shortage of time: of 72 pairs of runs grown to 1200 organs and then to 3200, 0 settle at the longer length after failing at the shorter.
Fig. 11 The quantity the exponent does move, which a table of shares cannot see.

What the exponent is in the rule

Each organ is placed at the azimuth where a sum over the organs already there is least, and each of those organs contributes an amount falling as the inverse cube of its distance from the candidate position.

Three is that cube. Raise it and the sum is dominated by the very nearest organ; lower it and organs further up the stem keep contributing, so the effective neighbourhood reaches further. How deep the rule looks has been measured four ways and the four agree on an ordering, which is what makes the exponent a meaningful knob rather than an arbitrary one.

So a sweep of the exponent is a sweep of neighbourhood depth, and neighbourhood depth is what the account of the wall is stated in.

Four ways to count the rule's neighbourhood, and one ordering. How many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by four different definitions. Counting the organs that carry half the profile gives 33, 12, 3, 2, 1. Counting the organs that carry nine tenths gives 182, 120, 30, 8, 3. Counting the organs that carry ninety-nine hundredths gives 284, 263, 149, 54, 9. Counting the organs that carry the weighted mean lag gives 68, 45, 21, 14, 11. They disagree about the size of the neighbourhood by a factor of 10 — the widest moves by 61 times across the sweep and the narrowest by 6.1 — and every one of them falls as the exponent rises. So no choice among them changes which rule is the deeper, and no redefinition can turn a result about the depth around.
Fig. 12 Four definitions of how deep the rule looks, which agree on an ordering across exponents.

Why 2 to 5 and not further

Below two the sum stops being dominated by anything local at these run lengths: organs a hundred places back contribute comparably to organs five places back, and the arrangement that results is not the object this collection studies.

Above five the sum is decided by the single nearest organ to within a per cent, so the rule becomes a matter of sitting as far as possible from one nearest neighbour, and the neighbourhood has one member. That is a different rule rather than a steeper version of this one.

Between those, four values with the working value in the middle. It is a narrow sweep and it is narrow on purpose: the question is whether the wall moves when the rule is perturbed, not what happens in a limit.

14 specimens separate 14.7% from 50%. The exact binomial power against sample size, for a one-sided test at 5 per cent. It is a staircase rather than a curve because the decision rule is a whole number of specimens: at 14 the cut sits at 5 and the power is 91.0 per cent. A normal approximation smooths that staircase away and reports a different answer.
Fig. 13 The weight an organ contributes against its distance, at the exponents the sweep covers.

What the columns’ own shapes say

Each column falls from most of nine at the coarse end to one or none at the fine end, and the fall is not smooth on any of them.

At exponent 3 the shares by rise are 0.78, 0.67, 0.78, 0.56, 0.33, 0.11, 0.11, 0.11 — which rises before it falls. At exponent 5 they are 0.44, 0.56, 0.78, 0.44, 0.22, 0.11, 0.44, 0.00, which goes up, down, and up again.

That is what a binomial over nine looks like on a declining trend, and it is the reason a “wall” located by the first crossing of a half is not a stable quantity: it lands at 0.0140, 0.0071, 0.0071 and 0.0087 on the four columns, and moving one run in one cell would move it by a factor of two.

How many starting angles reach a lattice, by rise and by falloff. Every cell is 9 stems grown from 9 starting angles spanning 40° to 180°, at one rise and one exponent in the placement rule's falloff, for 1200 organs each. The number is how many of them settle onto a lattice. Reading down a column, the basin closes as the rise falls, which is the wall. Reading across a row, nothing much happens: the exponent-three column is the published settling table digit for digit, and the other three are the same column inside their own sampling error. The account that a steeper falloff should wall somewhere else does not survive the table it predicted.
Fig. 14 The four columns, whose declines are not smooth enough for a crossing to be located reliably.

What would have counted as an effect

A pooled share differing by more than about 0.12 — two standard errors — between the extreme exponents.

At 0.417 and 0.375 the extremes differ by 0.042, which is under one. So the sweep had the power to detect a difference of about an eighth in the share and did not find one, which is a specific enough statement to be useful and a good deal weaker than “no difference”.

Anything smaller than an eighth is invisible here, and an account predicting a five-per-cent shift would be untouched by this measurement. Nothing offered one.

The share of starting angles that reach a lattice, by exponent. Each row pools one exponent's whole column — every rise, every starting angle — into one share, with a bar an error either side of it. Pooling is what makes the comparison possible: a single cell is 9 runs and carries an error of about 0.17, which is three times the whole spread between the four exponents. Pooled, the four sit at 30/72, 31/72, 29/72, 27/72 — a spread of 0.056 against an error of 0.058. Nothing in the falloff moves the basin. A five-angle version of the same sweep gave a clean monotone trend, which is what a null looks like when it is sampled too thinly.
Fig. 15 The four pooled shares with the error bars that set what this sweep could have found.

What else is measured at one exponent

Nearly everything here, and this sweep is the first time any of it has been asked whether it belongs to the rule or to the parameter.

The ladder of rungs, the transitions between them, the two branches, the settled divergences, the front’s depth, the survivor of a removal, the slip, the shape of the damage — all of them are exponent-three measurements. Some of them plainly do not depend on it: the ladder’s geometric spacing is arithmetic on the lattice rather than on the rule.

Others plainly might. The settling time nearly triples across this sweep, and it had been carried for four rounds as a property of the rule; there is no reason the front’s depth or the survivor should be different in kind.

That is not a call to re-measure everything. It is a note that “at exponent three” belongs in the statement of anything this collection measures about the placement rule, and until this round it was in none of them.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.381, 0.383, 0.381 of the previous rise — 1/φ² is 0.3820.
Fig. 16 The ladder, which is arithmetic on the lattice and does not depend on the rule’s exponent.

Why a null is reported at all

Because the prediction was made in public, in this collection, by the round that found the wall.

An account offered and then not tested is an account that quietly becomes an assumption. Testing it and finding nothing is worth the 288 runs precisely because the alternative is that “the basin narrows because the neighbourhood deepens” gets repeated in the next round’s framing as though it had been established.

The wall is real, it has been checked against run length twice, and it has no explanation. That sentence is shorter and more accurate than the one it replaces.

The same table, grown 2.7 times as long. Every rise and every starting angle, grown to 1200 organs and then to 3200. The two middle columns are how many starting angles reached a lattice at each length, and they are the same column: of the 72 pairs of runs, 72 are identical organ for organ and 0 settle at the longer length after failing at the shorter one. Tripling the budget buys nothing anywhere. What the fine rises are short of is not run length: the share of starting angles that reach a lattice at all falls from 7 of 9 to 1, so the arrangements a stem could fall into have mostly stopped existing.
Fig. 17 The check that established the wall, whose result the exponent sweep does not disturb.

What the wall still is

A place a stem cannot reach, established twice and explained not at all.

The first establishment was against run length: grown to 1,200 organs and again to 3,200, all seventy-two pairs of runs are identical, so the fine end is not a place stems take longer to arrive at. The second is the same comparison at every exponent, which gives the same answer on another 72 pairs.

What closes is the share of starting angles that reach a lattice at all, and the survivors down there settle on 106.1° and 151.3° — angles no branch of this ladder reaches. So a stem below the wall either fails to settle or settles somewhere that is not on the ladder, and neither of those is explained by anything measured here.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 18 The settling table at the length the wall was established against, on which the fine rises settle almost nothing.

The one line

Grown at exponents 2, 3, 4 and 5, the settling table’s four columns settle 30, 31, 29 and 27 of 72 runs — a spread of 0.056 against a standard error of 0.058 on each. The exponent-three column reproduces the published table digit for digit. The account that a steeper falloff should wall somewhere else predicted a difference that is not there, and a five-angle version of the same sweep showed one.

The share of starting angles that reach a lattice, by exponent. Each row pools one exponent's whole column — every rise, every starting angle — into one share, with a bar an error either side of it. Pooling is what makes the comparison possible: a single cell is 9 runs and carries an error of about 0.17, which is three times the whole spread between the four exponents. Pooled, the four sit at 30/72, 31/72, 29/72, 27/72 — a spread of 0.056 against an error of 0.058. Nothing in the falloff moves the basin. A five-angle version of the same sweep gave a clean monotone trend, which is what a null looks like when it is sampled too thinly.
Fig. 19 The four pooled shares with their error bars, which is the whole of the null.

Where this goes next

The same table, grown 2.7 times as long. Every rise and every starting angle, grown to 1200 organs and then to 3200. The two middle columns are how many starting angles reached a lattice at each length, and they are the same column: of the 72 pairs of runs, 72 are identical organ for organ and 0 settle at the longer length after failing at the shorter one. Tripling the budget buys nothing anywhere. What the fine rises are short of is not run length: the share of starting angles that reach a lattice at all falls from 7 of 9 to 1, so the arrangements a stem could fall into have mostly stopped existing. A wall and not a budget Below a rise of about 0.005 this collection's stems stop settling onto a lattice, and the limit has been written up four times without anybody asking which kind of limit it is. Grown three times as long, the table is identical row for row: not one stem that failed to settle succeeds. The floor is a wall. How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest. How long a stem takes to settle Every result here is grown on a stem that has settled onto a lattice, and settling has always been tested for and never timed. Timed, it takes between nothing and two hundred and ninety organs — against the four hundred every ablation run grows before it cuts anything, and the nine hundred the noise runs carry. How many organs settling takes, by falloff exponent. One mark per stem that settles, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. At the exponent every other measurement here uses, the slowest is 290 organs — the number four rounds of this collection have carried as though it belonged to the rule. It belongs to the exponent: at 4 the slowest is 808 and at 5 it is 674. A table of shares cannot see this, because a stem that settles at organ 800 and one that settles at organ 8 are the same entry in it. And it is still not a shortage of time: of 72 pairs of runs grown to 1200 organs and then to 3200, 0 settle at the longer length after failing at the shorter. The clock a share cannot see Settling takes nothing to 290 organs, and four rounds of this collection have carried that as a property of the rule. At a steeper falloff the slowest is 808 — and not one of the 72 pairs of runs settles at 3,200 organs after failing at 1,200, so the fine end is still a wall.

What links here

Computed from the collection, not written here: the essays that point at this one.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

  • The share was not the thing — both name attractor, basin, control, falsifiability, measurement, negative result, null model, the placement rule
  • Six of six is not a measurement — both name claim testing, control, exponent, measurement, negative result, neighbourhood depth, null model
  • The drift goes the other way — both name falsifiability, measurement, negative result, neighbourhood depth, null model, the placement rule, prediction
  • A removal that changes nothing — both name claim testing, control, measurement, negative result, the placement rule, sample size
  • A wreck has a short list — both name attractor, basin, falsifiability, measurement, negative result, the placement rule
  • Both walls of the slot — both name claim testing, control, measurement, negative result, the placement rule, prediction

Named objects

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

AttractorBasinClaim testingControlExponentFalsifiabilityMeasurementNegative resultNeighbourhood depthNull modelThe placement rulePredictionSample sizeSettling time