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

A list that was a rounding

Three destinations only a steep falloff reaches, read at two grid steps. Thirteen, read at a tenth of a degree. And four arrangements, read by what the counter returns rather than by the angle — a different four, with one the angle reading hides.
12 min read 20 figures State the claim, state the testCounted, not admired

Worth reading first: How long a stem takes to settle · Counting the spirals · The angle is an output.

Three destinations are reached at falloff exponents four and five and at neither of the shallower ones: 42.3°, 47.9° and 148.1°. That list has been quoted through two rounds.

It depends entirely on how the destinations are grouped, and there are three defensible groupings. They give three, thirteen and four.

None of the three is a mistake and none is obviously the right one. What is a mistake is quoting a list without saying which grouping produced it, and that is what the thread has been doing — the number three has appeared in two rounds of essays with no tolerance attached to it, and the tolerance is doing all the work.

The destinations a steep rule reaches, read two ways. Above: the list read to a tenth of a degree, which is what the settling table prints — 13 entries. Below: the same list read at half a degree, which is two steps of the grid the azimuths are placed on — 3. The 10 entries the finer reading adds are destinations a shallow exponent also reaches, a tenth or two of a degree away, and a tenth of a degree is a fifth of one step of that grid. The pale marks are every destination in the table, for scale.
Fig. 1 The same list read at a tenth of a degree and at two steps of the grid the azimuths are placed on.

Thirteen, at a tenth of a degree

That is the resolution the settling table prints. Read that way, the destinations only a steep exponent reaches are 42.3°, 47.9°, 78.6°, 78.9°, 79.2°, 99.0°, 99.6°, 99.7°, 136.6°, 138.8°, 139.0°, 148.1° and 151.5°.

Ten of those thirteen sit within a fraction of a degree of a destination a shallow exponent also reaches: 99.0° against 99.1°, 139.0° against 139.1°, 136.6° against 136.8°. They are the same arrangement measured on runs grown at slightly different rises.

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. 2 Every destination in the table with its pair. Several of the entries are within a fraction of a degree of each other and count the same way.

Why they differ at all

Because a settled divergence is not a constant of an arrangement. It slides across a rung: a stem on the golden ladder settles at 136.8° at one rise and 139.1° at another, and both count as 3/5.

So two runs reaching “the same arrangement” at different rises report different angles, and a grouping at a tenth of a degree separates them. The list of thirteen is mostly a list of rises.

The slide is not incidental to this subject; it is most of what a rung is. A stem holds one counted pair across a stretch of rise and its divergence moves by two or three degrees inside that stretch, so any grouping of divergences finer than the slide will split a single rung into several entries.

Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.382, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 3 The ladder: one angle across a range of rises, and the settled divergence sliding as the rise falls through each rung.

Three, at two grid steps

Half a degree is two steps of the grid the azimuths are chosen from, which is 1,536 samples of the circle and a quarter of a degree a step. It is the smallest gap at which two settled values can be told apart at all.

Grouped that way the fifteen destinations of the table are fifteen, and the steep-only ones are three. It is a stated tolerance rather than a rounding, and it is the one the counting essays use.

What the 3 steep-only destinations count as. Each block is a divergence that only the two steeper falloff exponents settle a stem on, with the pair a counter returns and the sequence that pair belongs to. All three count. None is a rung of either ladder. Two of them are the only members of their sequence anywhere in the table; the third, at 148.1 degrees, is the coarsest rung of a sequence that supplies another destination at every exponent — so it is not a new kind of arrangement but a coarser one of a kind the table already reached.
Fig. 4 The three destinations that survive the coarser grouping, with the pair each counts and the sequence that pair belongs to.

Which of the two angle groupings is right

Neither, and that is the point of the essay rather than a dodge. Half a degree is defensible because it is the instrument’s own floor; a tenth is defensible because the settled value is a mean over sixty divergences and a mean can resolve finer than one step of the grid it is built from.

Both are answers to “how far apart must two settled values be before they are two values”, and that question has no answer independent of what the values are for. For “has the rule found somewhere new”, two angles a tenth of a degree apart on the same arrangement are not somewhere new. For “does the settled divergence slide across a rung”, they are the whole measurement.

A round trip on four heads of 900 primordia: the divergence angle recovered from each. The counter is shown the points and nothing else. The worst recovery across the four is 0.012°.
Fig. 5 The angle recovered from a settled run, which is a measurement whose useful resolution depends entirely on what is being asked of it.

Which is still indexed by an angle

And that is the deeper problem. A destination is a divergence; an arrangement is a counted pair; and the two do not correspond. Three of the fifteen destinations return more than one pair, and several pairs occur at more than one destination.

So a list of destinations only a steep rule reaches is not a list of arrangements only a steep rule reaches, however carefully the destinations are grouped.

The 20 arrangements the settling table reaches, and which falloffs reach them. One row per counted pair, with a mark under each exponent that reaches it. Indexed this way the table has 20 arrangements rather than 15 destinations, because the same pair occurs at several divergences as the rise moves along its own rung and the same divergence occurs with several pairs. four arrangements are reached only by the two steep falloffs and two only by the two shallow ones, which is the same question asked in both directions.
Fig. 6 Every arrangement the table reaches, with a mark under each falloff exponent that reaches it.

Four, indexed by arrangement

Group the 117 settled runs by the pair the counter returns and the table has twenty arrangements. Four of them are reached at exponents four and five and at neither shallow one: 8/9, 8/15, 9/16 and 2/5.

Three of those are the three destinations. The fourth, 9/16, does not appear on the angle list at all.

The 10 sequences the settling table's destinations belong to. Every pair the counter returns is two consecutive terms of a sequence in which each term is the sum of the two before it, which is the ladder's own rule. two of these are the sequences this collection is built on. The rest are not, and they are not rare: the 1, 4, 5, 9, 14 sequence and the 2, 5, 7, 12, 19 sequence each supply several destinations, at every falloff exponent the table is grown at. The right-hand column is the divergences that land on each.
Fig. 7 The sequences behind the table. The arrangement the angle list hides is on the 2, 7, 9, 16 sequence.

Why the angle reading hides it

9/16 is reached at 157.7°, and 157.7° is also reached at exponent two — with 7/9. Two runs, one angle to within a tenth of a degree, two different arrangements: one found by a shallow rule and one only by a steep one.

So the angle says “a shallow exponent gets here too” and the count says “not to this”. Both are true and they are answers to different questions, and only one of them is the question the thread was asking.

The pair of runs is worth looking at directly. At exponent two a stem at one rise settles on 157.6° and counts 7/9; at exponents four and five stems at other rises settle on 157.7° and count 9/16. Seven and nine and sixteen are consecutive terms of one sequence, so this is not two arrangements that happen to share an angle — it is two adjacent rungs of one ladder, whose divergences differ by a tenth of a degree because the ladder’s angle barely moves between them.

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. 8 The destinations off the ladder, including the one where two exponents arrive at the same angle and count differently.

Which is a small and clean correction

The list of steep-only findings is four rather than three, and the fourth is on the same sequence as one of the others — 2, 7, 9, 16 supplies both 7/9 and 9/16, so the steep rule reaches a finer rung of a sequence the shallow rule already reaches a coarser rung of.

That is a shape worth naming, because it is the same shape 148.1° turned out to have: a coarser rung of a sequence the table already had. Two of the four steep-only arrangements are new rungs on old sequences and two are new sequences.

The 20 arrangements the settling table reaches, and which falloffs reach them. One row per counted pair, with a mark under each exponent that reaches it. Indexed this way the table has 20 arrangements rather than 15 destinations, because the same pair occurs at several divergences as the rise moves along its own rung and the same divergence occurs with several pairs. four arrangements are reached only by the two steep falloffs and two only by the two shallow ones, which is the same question asked in both directions.
Fig. 9 The arrangements in a fixed order rather than by how many runs reach them, so the rungs of one sequence can be read together.

And the question was never asked in the other direction

If a steeper falloff reaches arrangements a shallow one does not, the mirror question is whether a shallow falloff reaches arrangements a steep one does not. Nobody asked it, and the answer is not zero: 9/14 and 4/9 are reached at exponents two or three and never at four or five.

Two each way. The asymmetry the thread has been describing — that a steeper rule finds new destinations — is, at the level of arrangements, no asymmetry at all.

And the two shallow-only arrangements are as well-evidenced as the weaker two of the steep-only ones: 9/14 at two runs and 4/9 at one, against 9/16 at two and 2/5 at one. So the four-against-two split is really one well-evidenced steep-only arrangement and five findings of one or two runs each, three of which point one way and two the other.

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 four columns of the settling table. The question of what each column reaches that the others do not has two directions and had been asked in one.

How that happened

Because the sweep was run to test whether a steeper falloff moves the wall, and the destinations were a by-product noticed while reading the table. A by-product gets described in whatever direction it was noticed in.

The mirror question needed no extra runs at all — it is a filter over a table already computed — and it took a round to be asked because nobody was looking that way.

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. 11 The reading the sweep was built to make. The destinations were noticed on the way past it.

Both directions are weakly evidenced

Four steep-only arrangements and two shallow-only ones, and the run counts are 4, 2, 2, 1, 2 and 1. Every one of them is a handful of runs from nine starting angles a cell.

So the honest statement is that the exponent moves which arrangements appear in a sparse sample, in both directions, by about the amount a sparse sample would move anyway. Whether the effect is real needs twenty starting angles rather than nine, which is 640 runs against 288.

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. 12 The share of starting angles that settle at all, the reading whose binomial error already forced this thread to stop reading shares.

The same lesson as the shares

The settling shares were four columns of one column: a spread of 0.056 against a binomial error of 0.058, reported as a difference between exponents. The response was to stop reading the share and read the clock and the map instead.

This is the same lesson one level down. A destination is a measured angle with a resolution; an arrangement is a count with a run behind it; and a list of either is a list at a stated tolerance from a stated sample. Neither is a fact about the rule until the tolerance and the sample are quoted with it.

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. 13 The clock, which is the reading that survived the same problem when it was found in the shares.

What a run count adds

Every list in this essay is more honest with a run count beside it, and none of them had one. Three destinations reads as three findings; three destinations at four, two and one runs reads as one finding and two hints.

That is a change to how the table prints rather than to what it computes, and it is the same correction the ablation thread made about run lengths — report the instrument beside the number, because a constant that appears in every row stops being visible.

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 table cell by cell, which is where a run count for each destination would come from.

What the three groupings agree about

More than they disagree about. All three say the steep exponents reach 42.3° and 47.9° and the shallow ones do not, and all three say those two are countable and off both ladders.

The disagreements are entirely at the margins: whether near-duplicate angles are one destination or several, and whether an arrangement hidden behind a shared angle counts. That is the usual shape of a resolution problem and it is worth saying that the core is not in dispute.

What the 3 steep-only destinations count as. Each block is a divergence that only the two steeper falloff exponents settle a stem on, with the pair a counter returns and the sequence that pair belongs to. All three count. None is a rung of either ladder. Two of them are the only members of their sequence anywhere in the table; the third, at 148.1 degrees, is the coarsest rung of a sequence that supplies another destination at every exponent — so it is not a new kind of arrangement but a coarser one of a kind the table already reached.
Fig. 15 The two destinations every grouping agrees about, with the third that two of the three do.

What to quote

The arrangement list, with its run counts. Four arrangements reached only at the steep exponents, at 4, 2, 2 and 1 runs; two reached only at the shallow ones, at 2 and 1. Twenty arrangements in all, from 117 settled runs.

That is longer than “three destinations only a steep rule reaches” and it is the sentence that does not change when somebody re-groups the table — which is the test a quotable sentence has to pass, and the one a summary statistic that outlives its scope always fails.

The 20 arrangements the settling table reaches, and which falloffs reach them. One row per counted pair, with a mark under each exponent that reaches it. Indexed this way the table has 20 arrangements rather than 15 destinations, because the same pair occurs at several divergences as the rise moves along its own rung and the same divergence occurs with several pairs. four arrangements are reached only by the two steep falloffs and two only by the two shallow ones, which is the same question asked in both directions.
Fig. 16 The arrangement list itself, which is what the essays in this thread should be quoting.

What it does to the wall

Nothing, which is worth saying because the wall is the thread’s substantial result and the destinations are the ornament on it. Below a certain rise no starting angle reaches any lattice at all, at any of the four exponents, and that is a statement about runs that fail rather than about which arrangement the successes land on.

Every grouping in this essay leaves it untouched. A regrouping that changed a negative result about failures would be a much more serious matter than one that changes a list of successes, and this is not one.

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 wall, which no grouping of the destinations touches.

The general form

A list of things one condition produces and another does not is a list of set differences, and a set difference is exquisitely sensitive to how the members are identified. Group them finely and the difference grows; group them coarsely and it shrinks; change what the members are and it changes shape.

There is no neutral grouping. What there is instead is the obligation to say which one is in use, and to check that the finding survives the others — which here it mostly does, and where it does not is stated.

It is worth noting how little this cost. All three groupings run over the same 117 counted runs, which were computed once; re-grouping them is a second and produces a different answer. A finding that changes when the grouping does, at that price, has no excuse for being quoted at one grouping only.

The destinations a steep rule reaches, read two ways. Above: the list read to a tenth of a degree, which is what the settling table prints — 13 entries. Below: the same list read at half a degree, which is two steps of the grid the azimuths are placed on — 3. The 10 entries the finer reading adds are destinations a shallow exponent also reaches, a tenth or two of a degree away, and a tenth of a degree is a fifth of one step of that grid. The pale marks are every destination in the table, for scale.
Fig. 18 The same list at two resolutions, which is the cheapest form of that check.

What is not claimed

That a tenth of a degree is a wrong resolution to print. It is the right resolution to print a measured angle at; it is the wrong one to group by, and printing and grouping had not been separated.

Nor that twenty arrangements is the table’s true number. A denser sweep of starting angles would very likely find more, and a different counting window might merge or split some of these.

The spiral counts four different divergence angles produce. Fibonacci counts come from one angle. The Lucas angle — which the same dynamical model reaches on a different branch — gives 47 and 76, and neither number is a Fibonacci number.
Fig. 19 Counts against angle, where the relationship between a divergence and a pair is one-to-many in both directions.

The arrangement grouping has its own tolerance

It is worth being even-handed. Grouping by counted pair looks tolerance-free — a pair is two whole numbers — and it is not: the counter reads a window of two hundred organs and a count depends on where it is taken. Two runs on the same arrangement can return different pairs if the window catches different parts of the pattern.

There is direct evidence of that in the table. At 79.2° the counter returns 4/5, 4/9 and 5/9 across nineteen runs, and 4/9 skips a term of the sequence the other two sit on — which is what a count taken at the wrong place produces. So the arrangement grouping has a soft edge too, in a different place from the angle grouping’s.

Neither grouping is clean. What can be said is that they fail differently, and a finding surviving both is a finding that has been checked.

The spiral counts, band by band, in one head. The same flower gives 13/21, 21/34, 34/55, 55/89 at different radii. A caption saying "34 and 55 spirals" is a statement about one annulus.
Fig. 20 How a count moves with where in the pattern it is taken, which is the arrangement grouping’s own soft edge.

The one line

The list of destinations only a steep falloff reaches is three at half a degree, thirteen at a tenth of a degree, and — indexed by the arrangement a counter returns rather than by the angle — four, one of which the angle grouping hides behind a divergence a shallow exponent also reaches.

And the mirror question, never asked, has an answer: two arrangements are reached only by the shallow exponents, so at the level of arrangements the exponent’s effect is symmetric.

Where this goes next

What the 3 steep-only destinations count as. Each block is a divergence that only the two steeper falloff exponents settle a stem on, with the pair a counter returns and the sequence that pair belongs to. All three count. None is a rung of either ladder. Two of them are the only members of their sequence anywhere in the table; the third, at 148.1 degrees, is the coarsest rung of a sequence that supplies another destination at every exponent — so it is not a new kind of arrangement but a coarser one of a kind the table already reached. What a steep rule counts as Exponents four and five settle stems on 42.3°, 47.9° and 148.1°, and the two shallower ones reach none of them. All three count: 8/9, 8/15 and 2/5. None is a rung of either ladder, and one of them is the coarsest rung of a sequence the table already had. 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. A counter on the settling table The settling table has reported an angle for every run that reaches a lattice, and nobody has ever counted one. A hundred and seventeen settled runs, regrown and counted: every one of them has a parastichy pair, and sixty-five of them land on sequences the ladder does not 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.

  • Destinations only a steep rule reaches — both name attractor, basin, claim testing, discretisation, divergence angle, exponent, honest limits, measurement, negative result, resolution, selection effect
  • A period the grid invented — both name artefact, attractor, claim testing, discretisation, divergence angle, honest limits, measurement, negative result, parastichy pair, resolution
  • An onset at the end of the run — both name artefact, census, claim testing, honest limits, measurement, negative result, resolution, selection effect, tolerance
  • Ten sequences, two of them the ladder's — both name attractor, basin, census, claim testing, divergence angle, honest limits, measurement, parastichy pair, selection effect
  • Twice the run — both name artefact, census, claim testing, honest limits, measurement, negative result, resolution, selection effect, tolerance
  • A band that moves nothing — both name discretisation, divergence angle, measurement, negative result, parastichy pair, resolution, selection effect, tolerance

Named objects

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

ArtefactAttractorBasinCensusClaim testingDiscretisationDivergence angleExponentHonest limitsMeasurementMeasurement errorNegative resultParastichy pairResolutionSelection effectTolerance