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

Destinations only a steep rule reaches

A settled stem's divergence is the sharpest instrument in the sweep and the cheapest: it is already in every run, it does not average, and it is read on one grid at every exponent. Exponents 4 and 5 reach 42.3°, 47.9° and 148.1°, and the two shallower ones reach none of them.
13 min read 21 figures Counted, not admiredThe angle is an output

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

Two hundred and eighty-eight stems, four falloff exponents, and the share that reach a lattice does not move. How long they take does, by a factor of nearly three.

There is a third reading in the same runs and it is the sharpest of the three. Every settled stem has a divergence, and where a stem ends up is a much finer question than whether it got there.

Every divergence a stem settles on, by falloff exponent. One row per exponent, one mark per distinct settled divergence anywhere in that column, read to a tenth of a degree. This is the sharpest instrument in the sweep and the cheapest: it is already in every run, it does not average, and it is read on the same azimuth grid at every exponent. The steeper falloffs reach 42.3°, 47.9°, 148.1° — values that appear nowhere in the two shallower columns — while the traffic the other way is empty. So the exponent changes the map of where a stem can end up while leaving alone the share of starting angles that end up anywhere, which is a result a measurement of "did it settle" was never going to find.
Fig. 1 Every distinct divergence a stem settles on, by falloff exponent, read to a tenth of a degree.

Why a destination is a good instrument

Three reasons, and all three are about it costing nothing.

It is already computed. A settling time is read off a run’s divergence sequence, so the settled value is a by-product of the measurement that was already being made.

It does not average. A share of nine runs has an error of 0.17; a settled divergence is one number per run and two runs either agree or they do not.

And it is read on the same grid at every exponent — 1,536 candidate azimuths, 0.234° a step — so a comparison across exponents is a comparison of quantities measured the same way.

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. 2 What a counter returns across a range of divergences, which is the other reading a settled stem supplies.

What the columns hold

At exponent 2 the twenty distinct destinations run from 79.6° to 157.6°. At 3, seventeen from 65.2° to 151.3°. At 4, eighteen from 42.3° to 157.7°. At 5, twenty from 42.3° to 157.7°.

Most of them are shared. Every column has values near 79°, near 99° to 102°, near 137°, and near 150° — which are the two branches of the ladder and two other places stems fall into.

six limit divergences, all of them 137.5078 over a whole number. The golden angle is the k = 1 member of a family. Real bijugate plants — teasel, Cephalaria — are reported near 68.75°, which is exactly half of it, and the pairs they are counted at are the Fibonacci pairs doubled.
Fig. 3 The limit angles the two branches converge on, which account for two of the four clusters every column holds.

The values only the steep columns reach

42.3°, 47.9° and 148.1°.

None of them appears within a degree of anything in the exponent-2 or exponent-3 columns. 42.3° and 47.9° appear at both 4 and 5; 148.1° appears at 5 only.

They are not near-misses of anything. The nearest thing the shallow columns hold to 42.3° is 65.2°, twenty-three degrees away; to 47.9° the same; and to 148.1° the nearest is 150.4°, which is two and a quarter degrees — nine grid steps, and the one of the three worth being careful about.

Every divergence a stem settles on, by falloff exponent. One row per exponent, one mark per distinct settled divergence anywhere in that column, read to a tenth of a degree. This is the sharpest instrument in the sweep and the cheapest: it is already in every run, it does not average, and it is read on the same azimuth grid at every exponent. The steeper falloffs reach 42.3°, 47.9°, 148.1° — values that appear nowhere in the two shallower columns — while the traffic the other way is empty. So the exponent changes the map of where a stem can end up while leaving alone the share of starting angles that end up anywhere, which is a result a measurement of "did it settle" was never going to find.
Fig. 4 The same rows with the two low destinations included, which are the two the shallow columns come nowhere near.

And the traffic the other way is empty

Every destination the two shallow columns reach, a steep column reaches within a degree. Nothing is lost by steepening the falloff; three things are gained.

That asymmetry is the part that makes this a result rather than a difference of samples. Two columns drawn from one set of destinations would each miss some by chance and the misses would go both ways. Here they go one way.

It is worth saying how strong that is with 72 runs a column: not very. Nine destinations appear in exactly one column somewhere in the table, and three of them are the three above. The claim is that the pattern of which column has the extras is one-sided, and nine is not many.

Every divergence a stem settles on, by falloff exponent. One row per exponent, one mark per distinct settled divergence anywhere in that column, read to a tenth of a degree. This is the sharpest instrument in the sweep and the cheapest: it is already in every run, it does not average, and it is read on the same azimuth grid at every exponent. The steeper falloffs reach 42.3°, 47.9°, 148.1° — values that appear nowhere in the two shallower columns — while the traffic the other way is empty. So the exponent changes the map of where a stem can end up while leaving alone the share of starting angles that end up anywhere, which is a result a measurement of "did it settle" was never going to find.
Fig. 5 Each column with the destinations no other column reaches drawn in a second stroke.

Where the new destinations come from

The fine rises, and specifically the corner of the table where the basin is nearly closed. 42.3° appears at exponents 4 and 5 at rises of 0.0045 and 0.004; 47.9° at rises of 0.005 and 0.0045; 148.1° at exponent 5 at a rise of 0.030.

Two of the three are therefore at the fine end, which is where the slow settling also sits. The third is at the coarsest rise in the sweep, which is not.

So it is not simply that a nearly-closed basin admits odd arrivals. One of the three is at the far end of the table from that.

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. 6 The grid, on which the cells the new destinations come from can be located.

What 42.3° is not

It is not on either branch. The golden branch converges on 137.5078° and the Lucas on 99.5016°, and the ladder’s rungs sit between 136.6° and 140.7° on one and 99.1° and 104.8° on the other.

It is not a mirror of anything on them: 360° − 42.3° is 317.7°, which folds to 42.3° again, and neither branch reaches it.

And it is not a rational angle of small denominator. 42.3° is not 360/8.5 or 360/9 (40°) or 3×360/25 (43.2°) to the resolution the grid gives. The nearest simple rational is a ninth of a turn, 1.8° away — eight grid steps.

So it is a place a stem settles that this collection has no name for and no ladder containing.

How nearly each angle is a simple fraction of a turn. A dip means a rational approximation and a lattice with visible rows. The golden angle's floor is 0.008; the rational angle reaches exactly zero.
Fig. 7 How well an angle is approximated by rationals of small denominator, against which a settled divergence can be placed.

What 148.1° and 150.4° are

The 150° cluster is not new. It is in every column — 150.4°, 150.6°, 150.8°, 150.9°, 151.0°, 151.1°, 151.3°, 151.5° — and it is where the fine end’s few survivors settle: 151.3° is the divergence the single surviving starting angle reaches at a rise of 0.003.

148.1° sits two and a quarter degrees below that cluster and appears once, at exponent 5 and a rise of 0.030. Whether it is a member of the cluster or a separate destination is not answerable from one run.

That is the honest reading and it takes one of the three claimed values down to a maybe. The two low destinations do not depend on it.

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 settling table at one exponent, where the 150° cluster’s members are the fine end’s survivors.

What it means for the wall

The wall is where the basin closes, and the share that reaches a lattice at all does not move with the exponent. What moves is the map: which arrangements exist to be reached.

That is a different quantity from the one the account of the wall was about. The account said the basin narrows; the measurement says the basin is the same size and contains different things.

A narrower version of that is testable and is not tested here: whether the number of distinct destinations differs by exponent. It is 20, 17, 18 and 20, which is flat, and a count of distinct values over 72 runs is a statistic with an awkward distribution that this essay is not going to lean on.

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. 9 The share that reaches a lattice, which is flat, drawn from the same runs as the destinations that are not.

What a specimen would show

A count, not an angle. A stem settled on 42.3° would return some counted pair, and that pair is what a plant supplies — the divergence is the harder measurement and the count is the easy one.

So the honest transfer of this result to anything observable is: a placement rule with a steeper falloff produces arrangements whose counted pairs are not on either ladder. Whether it does is not measured here; the counter was not run on the runs that settled on 42.3°.

That is one line of code and it is a leaving rather than an omission, because a counter needs a stem grown long enough to count and the settling runs were grown for a different purpose.

The two spiral families a counter finds between 0.43 and 0.67 of the radius. 21 spirals one way and 34 the other, found from the point positions alone — the counter is never told the divergence angle.
Fig. 10 A counter reading an arrangement, which is the measurement that would place a new destination on or off the ladder.

What the three readings say together

The exponent does not change how many starting angles find a lattice. It nearly triples how long they take. And it adds three destinations and removes none.

Read as one sentence: the exponent changes the map without changing how much of it is reachable. That is a specific enough statement to be wrong, and it is the kind of statement a table of shares could never have produced — which is the methodological point the three essays share.

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 second of the three readings, which the first could not see either.

The four clusters every column holds

Around 79°, around 99° to 102°, around 137°, and around 150°.

The second and third are the two branches of the ladder: a Lucas stem settles near 99.5° and a golden one near 137.5°, and the rungs spread each of those over a few degrees. Those two account for most of the entries in every column.

The 150° cluster is the fine end’s own destination — the divergence the last surviving starting angles reach below the wall — and it appears at every exponent. The 79° cluster is the fourth and this collection has no account of it either; it appears at every exponent too, so it is not new here.

So the map has four regions at every exponent and the steep ones add points below all four.

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. 12 What a counter returns across the range the four clusters sit in.

How a destination is read

As a mean over the last sixty divergences of a run, folded so that a stem ending on 220.9° is recorded at 139.1° — the same arrangement traversed the other way.

That fold matters for a comparison across exponents, because without it a column could appear to reach a destination another does not simply by having wound the other way. It is the same fold the settling table uses, and it is applied before any mean is taken rather than after, since a mean over both representations is a number belonging to neither.

The values are then rounded to a tenth of a degree, which is under half the azimuth grid’s own step, so two runs reported at the same destination did land in the same neighbourhood rather than being rounded together.

How nearly each angle is a simple fraction of a turn. A dip means a rational approximation and a lattice with visible rows. The golden angle's floor is 0.008; the rational angle reaches exactly zero.
Fig. 13 An angle placed against the rationals of small denominator, which is one way to ask what a destination is.

The count of destinations

Twenty, seventeen, eighteen and twenty across the four columns, from 72 runs each of which 30, 31, 29 and 27 settle.

So the number of distinct destinations is roughly two thirds of the number of settled runs at every exponent, and it is flat. That is a statistic with an awkward distribution — it depends on how many runs settled, which itself varies — and it is not leaned on.

What it does rule out is a picture in which the steeper rules simply have more places to go. They have the same number of places and three of them are different places, which is a change in the map’s contents rather than in its size.

Every divergence a stem settles on, by falloff exponent. One row per exponent, one mark per distinct settled divergence anywhere in that column, read to a tenth of a degree. This is the sharpest instrument in the sweep and the cheapest: it is already in every run, it does not average, and it is read on the same azimuth grid at every exponent. The steeper falloffs reach 42.3°, 47.9°, 148.1° — values that appear nowhere in the two shallower columns — while the traffic the other way is empty. So the exponent changes the map of where a stem can end up while leaving alone the share of starting angles that end up anywhere, which is a result a measurement of "did it settle" was never going to find.
Fig. 14 The four columns with their counts, which are flat while their contents are not.

Why this is the reading to have taken first

It costs nothing and it is the sharpest.

Every run in the sweep already had its settled divergence computed, because that is how a settling time is defined. Reading the list of distinct values takes no additional computation at all, and it separates the four exponents where a share of 72 runs cannot.

The order the three readings were taken in was share, then time, then destination — which is the order of decreasing cost and increasing information, run backwards. That is worth noticing rather than moralising about: the share is the quantity the question was asked in, and asking a question in a quantity is how a measurement gets chosen.

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 reading the question was asked in, which is the least informative of the three and the one that was run first.

What a destination is not

A rung. The ladder’s rungs are ranges of rises at which a counter returns one pair, and a destination is a divergence a stem settles on — so the two are different objects and a new destination does not add a rung.

Whether it adds one is answerable and is the obvious next measurement. Grow stems at exponent 5 across a range of rises, count each one’s pair from its points, and see whether 42.3° sits on a run of rises returning one pair. If it does there is a ladder down there nobody has walked; if it does not, 42.3° is an isolated arrangement.

The ladder sweep already does exactly that at exponent three, in 288 runs a branch. At exponent five it is the same 288 runs and it was not part of this round’s slate.

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. 16 The ladder as it is measured at exponent three, which is the sweep that would say whether a new destination has one.

Why the shallow columns’ emptiness is the load-bearing half

Three values appearing only in the steep columns could be three values the shallow columns happened not to sample. Three values appearing only in the steep columns while nothing appears only in the shallow ones is harder to produce that way.

If both sets were drawn from one pool, each column would miss some by chance and the misses would be roughly symmetric. Four columns, 20, 17, 18 and 20 distinct values, and a one-sided difference of three is not decisive at these numbers — but it is the shape a real difference has and a sampling artefact usually does not.

The honest weight is: suggestive, one-sided, three values, and testable in one sweep.

Every divergence a stem settles on, by falloff exponent. One row per exponent, one mark per distinct settled divergence anywhere in that column, read to a tenth of a degree. This is the sharpest instrument in the sweep and the cheapest: it is already in every run, it does not average, and it is read on the same azimuth grid at every exponent. The steeper falloffs reach 42.3°, 47.9°, 148.1° — values that appear nowhere in the two shallower columns — while the traffic the other way is empty. So the exponent changes the map of where a stem can end up while leaving alone the share of starting angles that end up anywhere, which is a result a measurement of "did it settle" was never going to find.
Fig. 17 The four columns, whose one-sided difference is the claim’s own evidence.

What the destinations say about the two branches

Nothing changes about them. Every column reaches the golden neighbourhood between 136.6° and 139.3° and the Lucas one between 99.0° and 101.8°, and the values inside those bands are the ladder’s rungs at whichever rises settled.

That is the control the comparison needs. If the steeper exponents had lost the branches, or shifted them, the columns would not be four measurements of one thing and the extra destinations would be a different phenomenon.

They do not. The branches are where they are at every exponent, to within the spread a handful of rises produces, and the three extra values sit outside both of them by twenty degrees or more.

four limit divergences, all of them 137.5078 over a whole number. The golden angle is the k = 1 member of a family. Real bijugate plants — teasel, Cephalaria — are reported near 68.75°, which is exactly half of it, and the pairs they are counted at are the Fibonacci pairs doubled.
Fig. 18 The limit angles the two branches converge on, which every column reaches.

Reading the three together

The share does not move; the clock nearly triples; the map gains three points and loses none.

The three are read from one set of 288 runs and they cost, in order, nothing extra, nothing extra, and nothing extra — every one of them is already computed by the time a settling time exists. So the difference between finding an effect and not finding one, in this sweep, was entirely a matter of which of three free readings was taken.

That is the methodological content and it generalises past this table. A sweep produces runs; a run has many readings; and the question a sweep was designed around picks one of them, usually the coarsest.

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. 19 The second free reading, which the first could not see.

What a stem settling on 42.3° would look like

An arrangement with organs a ninth of a turn or so apart in azimuth, which is a much coarser rotation than either branch’s.

Whether that is a lattice with a countable pair, a whorled arrangement with organs effectively arriving together, or something with no regularity a counter can read, is not answered — the counter was not run on those runs, because the settling sweep was built to time arrivals rather than to classify them.

It is one line of code and two of the three things it could return would be interesting. A countable pair at 42.3° would be a rung off the ladder; no countable pair would say the settling test admits arrangements the rest of this collection would not call lattices.

The two spiral families a counter finds between 0.68 and 0.92 of the radius. 34 spirals one way and 55 the other, found from the point positions alone — the counter is never told the divergence angle.
Fig. 20 A counter reading an arrangement, which is the classification the new destinations have not had.

The one line

Exponents 4 and 5 settle stems on 42.3°, 47.9° and 148.1°, none of which appears within a degree of anything the exponent-2 or exponent-3 columns reach — while every destination those two reach, a steep column reaches. The share that settles does not move and the map does. The weakest of the three values is 148.1°, which sits nine grid steps from an existing cluster and appears once.

Every divergence a stem settles on, by falloff exponent. One row per exponent, one mark per distinct settled divergence anywhere in that column, read to a tenth of a degree. This is the sharpest instrument in the sweep and the cheapest: it is already in every run, it does not average, and it is read on the same azimuth grid at every exponent. The steeper falloffs reach 42.3°, 47.9°, 148.1° — values that appear nowhere in the two shallower columns — while the traffic the other way is empty. So the exponent changes the map of where a stem can end up while leaving alone the share of starting angles that end up anywhere, which is a result a measurement of "did it settle" was never going to find.
Fig. 21 The four columns of destinations, with the one-sided traffic between them.

Where this goes next

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. 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. 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. The rule, 26 steps in, at a growth of 0.40. The next primordium goes where the repulsion is least — the marked minimum at 216°. Nothing in the rule refers to any particular angle. The angle is an output 137.5° is not a constant of nature. It is where a rule settles — a rule that places each new element as far as it can from the ones already there, contains no reference to the golden ratio, and reaches the same answer from starting angles a hundred and sixty degrees apart.

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.

  • A period the grid invented — both name attractor, claim testing, discretisation, divergence angle, honest limits, measurement, negative result, the placement rule, resolution, summary statistic
  • The pair read from the angles — both name claim testing, divergence angle, exponent, honest limits, measurement, resolution, summary statistic
  • The slide a counter holds constant — both name claim testing, divergence angle, honest limits, measurement, negative result, the placement rule, summary statistic
  • The stem that changed hands — both name attractor, basin, divergence angle, honest limits, initial condition, measurement, the placement rule
  • Two-ranked, by two different routes — both name attractor, discretisation, divergence angle, honest limits, initial condition, measurement, the placement rule
  • A band that moves nothing — both name discretisation, divergence angle, measurement, negative result, resolution, selection effect

Named objects

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

AttractorBasinClaim testingDiscretisationDivergence angleExponentHonest limitsInitial conditionMeasurementNegative resultThe placement ruleResolutionSelection effectSummary statistic