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

A basin has a width

A destination reached from one starting angle is a presence. A destination reached from seven consecutive starting angles spanning forty-five degrees is a basin with an extent, and nine angles could not have measured one — they were too far apart to have two of them land in the same place.
12 min read 23 figures State the claim, state the testCounted, not admiredOne parameter, one family

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

A basin is the set of starting angles from which a stem ends up at one destination. The word has been used in this collection since the settling table was built and nothing had measured one, because measuring an extent needs two points inside it.

At nine starting angles spanning 40 to 180 degrees the spacing is fifteen to twenty degrees. At twenty it is 6.25 to 10, and that is close enough for neighbouring angles to land in the same place.

The eleven extra angles were grown for two reasons and this is the second of them. The first was the error bar; this one needed the spacing rather than the count.

Where a stem started at each of twenty angles ends up, at a falloff exponent of 3. One column per starting angle and one row per rise, coarse at the top. A filled cell is a run that reached a lattice, its tone the destination it reached; a pale cell is a run that never settles. Runs of one tone across neighbouring columns are basins, and the widest of them spans 7 consecutive angles. The angles are 6.25 to 10 degrees apart, so a basin narrower than that cannot be seen here and a single filled cell says nothing about how wide its basin is.
Fig. 1 Where a stem started at each of twenty angles ends up, with runs of one tone across neighbouring columns.

What is measured

Across the table’s thirty-two cells — four falloff exponents by eight rises — there are a hundred and fifty-one runs of consecutive starting angles that reach one destination.

Thirty-five of them hold more than one angle. The widest holds seven, and the widths run from 6.24 to 45 degrees.

A hundred and sixteen hold one angle each, and what those are is the harder half of this reading.

Placements that went to a different minimum, per thousand. The rule's own counterfactual, run beside it: where would this node have gone with the noise taken away and everything else left alone? Placement noise displaces the node after the argmin, so the answer is always "here" — 0.2°, 0.4°, 0.8° all give zero. A jostle and a field perturbation are upstream of the choice and change one or two placements in a thousand while the lattice is still intact.
Fig. 2 The widths of the runs of starting angle that reach one destination, at the spacing the table samples.

What a width means here

Less than it sounds and more than nothing. A run of seven consecutive angles from 120 to 165 degrees all reaching 139.1 degrees says that a stretch of at least forty-five degrees of starting angle ends up in one place.

It does not say the basin is forty-five degrees wide. The true edges lie somewhere in the gaps either side — between 110 and 120 on one side and between 165 and 172.5 on the other — so the measured width is a lower bound and the true one is between forty-five and about sixty-two.

That is the same shape as every bracket in this collection: a transition located between two consecutive rises is located to the step, and quoting the midpoint as though it were the answer is the error to avoid.

Where a stem started at each of twenty angles ends up, at a falloff exponent of 3. One column per starting angle and one row per rise, coarse at the top. A filled cell is a run that reached a lattice, its tone the destination it reached; a pale cell is a run that never settles. Runs of one tone across neighbouring columns are basins, and the widest of them spans 7 consecutive angles. The angles are 6.25 to 10 degrees apart, so a basin narrower than that cannot be seen here and a single filled cell says nothing about how wide its basin is.
Fig. 3 The table with the widest run in it, whose true edges are somewhere inside the gaps at its ends.

And a single angle says nothing

That is the more important half. A destination reached from one starting angle and no neighbour has a measured width of nothing, and its true width could be anything from a tenth of a degree to nearly twenty.

So the hundred and sixteen single-angle runs are not a hundred and sixteen narrow basins. They are a hundred and sixteen destinations whose basins the sampling could not resolve, which is a different statement and the one the table supports.

Placements that went to a different minimum, per thousand. The rule's own counterfactual, run beside it: where would this node have gone with the noise taken away and everything else left alone? Placement noise displaces the node after the argmin, so the answer is always "here" — 0.4°, 0.8° all give zero. A jostle and a field perturbation are upstream of the choice and change one or two placements in a thousand while the lattice is still intact.
Fig. 4 The destinations reached from one starting angle alone, whose widths this sampling cannot report.

Why that distinction is worth insisting on

Because a count of narrow basins would be a result and it is available for the taking. Nothing in the numbers stops anybody writing most basins are narrow, and the sentence would be consistent with every cell of the table.

It would also be consistent with every basin being nine degrees wide and the sampling missing them. A sampling at ten degrees cannot distinguish a nine-degree basin from a tenth-of-a-degree one, and both look like a single filled cell.

How far each starting angle sits from somewhere a stem could settle. Each of the twenty starting angles against its distance to the nearest destination the table ever reaches. Several of the original nine sit within two degrees of one — 100 is beside 101.6, 80 beside 79.2, 150 beside 151.0, and the golden angle is a destination — while the angles placed halfway between them mostly are not. A run that starts next to where it would settle has almost nowhere to go, which is the account of why the original nine settle half as often again as the angles between them.
Fig. 5 The spacing the twenty angles are sampled at, which is the floor on any width this table reports.

The floor, stated

Six and a quarter degrees between the closest pair, ten between the widest. So a basin narrower than about six degrees is invisible and a basin between six and ten is visible only if it happens to contain two sampled angles.

Every width in this reading is quoted against that floor, and the widths worth believing are the ones several times it: the runs of five, six and seven angles.

Placements that went to a different minimum, per thousand. The rule's own counterfactual, run beside it: where would this node have gone with the noise taken away and everything else left alone? Placement noise displaces the node after the argmin, so the answer is always "here" — 0.2°, 0.4°, 0.8° all give zero. A jostle and a field perturbation are upstream of the choice and change one or two placements in a thousand while the lattice is still intact.
Fig. 6 The measured widths against the spacing, where only the largest are several times the floor.

Where the wide basins are

At the coarse rises. The widest run — seven angles, forty-five degrees — is at a rise of 0.030 at the working exponent, and the next widest are at 0.020 and 0.013.

At the fine rises there are no wide basins because there is almost nothing to have a basin: at a rise of 0.004 one of twenty angles settles, and one angle is one filled cell with no neighbours. That is the wall, and it is a property of the rise rather than of the sampling.

Where a stem started at each of twenty angles ends up, at a falloff exponent of 2. One column per starting angle and one row per rise, coarse at the top. A filled cell is a run that reached a lattice, its tone the destination it reached; a pale cell is a run that never settles. Runs of one tone across neighbouring columns are basins, and the widest of them spans 7 consecutive angles. The angles are 6.25 to 10 degrees apart, so a basin narrower than that cannot be seen here and a single filled cell says nothing about how wide its basin is.
Fig. 7 The table at the shallowest exponent, where the coarse rises hold the wide runs and the fine ones hold none.

Which is the wall seen sideways

The wall is the rise below which most starting angles stop reaching a lattice, read as a share. The basins say the same thing as a geometry: as the rise falls, the stretches of starting angle that lead somewhere shrink and then disappear.

Those are two readings of one table and they agree, which is worth having because the first needs a threshold and the second does not.

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. 8 The share of starting angles that settle against the rise, which is the same fact read as a number.

What a basin is a basin of

Worth being precise, because the word carries a lot from elsewhere. A basin here is a set of starting angles that reach one settled divergence, at a fixed rise and a fixed falloff exponent.

It is not a basin in a state space with a fixed rule: the rise and the exponent are held, so what varies is one number in the initial condition. A stem started at 120 degrees and a stem started at 165 are two runs of the same rule from two initial conditions, and they arrive at the same place.

Which is what makes the divergence an output of this rule rather than a parameter of it. A basin width is the strongest form of that statement available: a whole stretch of inputs producing one output.

Two runs of the same rule from unrelated starting angles. Both settle at 137.0°, within 0.5° of the golden angle, from seeds 166° apart.
Fig. 9 A run arriving at a divergence, which is what a whole stretch of starting angles does together.

What the basins are not separated by

A gap, in most cases. Between a run that reaches 139.1 degrees and a run that reaches 101.4 there is often a single angle that settles somewhere else, or one that does not settle at all.

So the picture is not a clean partition of the starting angles into a few wide basins. It is a few wide runs with interruptions, and whether an interruption is a narrow basin between two wide ones or a stretch that fails to settle is exactly what the spacing cannot say.

Where a stem started at each of twenty angles ends up, at a falloff exponent of 5. One column per starting angle and one row per rise, coarse at the top. A filled cell is a run that reached a lattice, its tone the destination it reached; a pale cell is a run that never settles. Runs of one tone across neighbouring columns are basins, and the widest of them spans 7 consecutive angles. The angles are 6.25 to 10 degrees apart, so a basin narrower than that cannot be seen here and a single filled cell says nothing about how wide its basin is.
Fig. 10 The table at the steepest exponent, where the runs are interrupted more often.

The unsettled angles are the interesting interruptions

At the working exponent and a rise of 0.013, the twenty angles give: nothing from 10 to 70, then 79.6 twice, then 99.8, then 136.8, then 79.6, then 136.8 twice, then 150.9, then 156.9, then 79.6, then nothing, then 79.6.

That row alternates. Three separate single angles reach 79.6 degrees with other things between them, which either means one basin cut into pieces by the sampling or three arrivals at one destination from three separated stretches.

Where a stem started at each of twenty angles ends up, at a falloff exponent of 3. One column per starting angle and one row per rise, coarse at the top. A filled cell is a run that reached a lattice, its tone the destination it reached; a pale cell is a run that never settles. Runs of one tone across neighbouring columns are basins, and the widest of them spans 7 consecutive angles. The angles are 6.25 to 10 degrees apart, so a basin narrower than that cannot be seen here and a single filled cell says nothing about how wide its basin is.
Fig. 11 The table at the working exponent, whose middle rows alternate between destinations.

Which of those it is matters

If a basin is one connected stretch, then a destination reached from separated angles means the sampling is stepping over other basins in between. If basins are genuinely disconnected, the picture is more interesting and the word basin is doing less work.

Neither can be told from twenty angles ten degrees apart. Forty angles would halve the spacing and would say whether the gaps fill in, which is the single most informative thing a further refinement could do.

The same alternation appears in a different thread and was resolved the same way: islands of one answer inside runs of another turned out to be speckle rather than a period once every step was swept rather than sampled.

Placements that went to a different minimum, per thousand. The rule's own counterfactual, run beside it: where would this node have gone with the noise taken away and everything else left alone? Placement noise displaces the node after the argmin, so the answer is always "here" — 0.2°, 0.4°, 0.8° all give zero. A jostle and a field perturbation are upstream of the choice and change one or two placements in a thousand while the lattice is still intact.
Fig. 12 The single-angle destinations, whose neighbours a finer sampling would fill in or not.

What forty angles would cost

Another 640 runs, about twenty-five minutes, and a spacing of three to five degrees. That would resolve basins down to about three degrees and would turn most of the single-angle destinations into runs or leave them alone.

It would also grow the destination list again, since every refinement so far has found off-ladder destinations, and it would halve the error on every cell a second time. Of the three that is the basin question that most needs it.

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. 13 The destinations the table reaches, a list that has grown at every refinement of the sampling.

What the widest basin is

Seven consecutive angles — 120, 128.75, the golden angle, 143.75, 150, 157.5 and 165 — all reaching 139.1 degrees, at a rise of 0.030 and a falloff exponent of two.

That destination is on the golden ladder, and the golden angle itself is one of the seven angles in it. So the widest measured basin in the table is the one around the angle the whole subject is about, which is a satisfying result and is one cell of one table.

It is also the reason the golden angle is a poor starting angle to sample with: a run started at a destination has nowhere to go, and its being inside the widest basin makes that worse rather than better.

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 destinations that sit on a ladder sequence, of which the widest basin’s is one.

And whether that is a pattern

It is not, on this evidence. The other wide runs reach 101.4 degrees and 136.9 and 151.0, of which only one is on a ladder sequence.

So the widest basin being a ladder destination is one cell rather than a rule, and the honest reading is that the on-ladder destinations do not obviously have wider basins than the off-ladder ones. Testing that properly would need the finer sampling.

Everywhere a cut of one to five organs can send a 5/8 stem. Every settled divergence reached by any arrangement of up to five organs removed from a stem at the 5/8 rung, on one axis. There are six of them and no more. three are slips of the lattice the stem was cut from: each keeps the lag-5 family intact and sits a whole number of turns of it from the next, which is the ladder marked below the axis with rungs 72.0 degrees apart. The other three keep no lag at all and sit near a fraction of a turn, marked above: 175.0 degrees near 1 of 2 turns, 190.0 degrees near 1 of 2 turns, 235.0 degrees near 2 of 3 turns. A stem that is cut either slides along the ladder it was on or leaves it for a lattice with files in it.
Fig. 15 Every destination with how many runs reach it, which is the count a width would have to be compared against.

A count of runs is not a width either

That is a third thing to keep apart. A destination reached by forty-one runs across the whole table is reached often; a destination whose basin is forty-five degrees wide in one cell is reached from a wide stretch in that cell.

The two are different and they do not track each other: the most-visited destination in the table is reached from single angles in most cells and from a run of three in one.

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. 16 The destinations with their visit counts, which is a different quantity from a basin width.

What this adds to the settling picture

One number, and a way of asking for more. The picture before was a list of destinations with counts; it is now a list of destinations with counts and, in thirty-five cases, an extent of starting angle attached.

Thirty-five of a hundred and fifty-one is not a lot. It is the first time any of them has had one, and it came free with a sampling change made for a different reason.

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. 17 The basins each exponent produces, which is a reading nine angles could not have supported.

What the counter says about the destinations

Each of the eighteen destinations has an arrangement attached to it, because every settled run has been regrown and counted. So a basin is a stretch of starting angle reaching one counted pair as well as one angle.

That is worth noting because three of the destinations return more than one pair across the runs that reach them. At 137.8 degrees some runs count 5/8 and others 8/13, which is not a failure of the counter: the pair a patch shows depends on the rise as well as on the divergence, and the runs at one destination are grown at different rises.

Within one cell of the table the rise is fixed, so a basin measured inside a cell does have one pair. The ambiguity is between cells rather than inside them.

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. 18 The arrangements the table reaches, of which three sit at destinations that return more than one.

Whether a basin’s edge is sharp

Unknown, and worth naming as the next question after the spacing. A basin could end abruptly — one angle reaching 139.1 and the next reaching 101.4 — or there could be a stretch between them where runs fail to settle at all.

The table has both patterns in it. At some transitions the two destinations are adjacent columns; at others there is a pale column between them. Which of those is generic would need the finer sampling, and it is the same question as whether the interruptions are basins.

Where a stem started at each of twenty angles ends up, at a falloff exponent of 2. One column per starting angle and one row per rise, coarse at the top. A filled cell is a run that reached a lattice, its tone the destination it reached; a pale cell is a run that never settles. Runs of one tone across neighbouring columns are basins, and the widest of them spans 7 consecutive angles. The angles are 6.25 to 10 degrees apart, so a basin narrower than that cannot be seen here and a single filled cell says nothing about how wide its basin is.
Fig. 19 The table at the shallowest exponent, where some destination changes are abrupt and some have a gap.

What a width would be worth if it were tight

The reason to want basin widths is that they are the thing a claim about robustness needs. A stem reaches the golden lattice is a statement about a rule; a stem reaches the golden lattice from any starting angle in a forty-five degree stretch is a statement about how much that depends on where it began.

The second is the one a reader coming to phyllotaxis from outside is usually after, because the question behind the subject is why a pattern that could be anything is so often one thing. A wide basin is part of an answer to that and a list of destinations is not.

What is available so far is one wide basin at one coarse rise, which is a long way from the answer and is the first measurement in its direction.

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. 20 The basins at each exponent, which is the reading a robustness claim would be built from.

And what the fine rises say about it

That whatever robustness there is belongs to the coarse end. Below the wall almost nothing settles from any starting angle, so there are no basins to be wide.

That is not a small caveat. The rises where a real seed head or a real stem sits are a question this collection has not settled, and if they are at the fine end then the wide basins are a property of a regime nothing lives in.

The settling share at every rise, sampled at twenty angles. How often a stem reaches a lattice, against the rise, at each of the four falloff exponents. The horizontal rule is the half share the wall is read at. At nine starting angles the four exponents cross it at rises spanning a factor of 1.96 and order themselves 2, 5, 3, 4; at twenty they span a factor of 1.18 and order themselves 5, 4, 2, 3. The ordering reverses and the spread collapses, so the conclusion that the exponents do not separate is confirmed and the numbers that had suggested otherwise were the coarser sampling's own error.
Fig. 21 The share of starting angles that settle, across the rises, which is where the basins can exist.

What a reader should carry

That thirty-five stretches of starting angle now have a measured extent, that the widest spans forty-five degrees across seven consecutive angles, and that the spacing puts a floor of about six degrees under all of them.

And that a destination reached from one angle alone is a destination whose basin was not measured, rather than a narrow basin. The difference is the whole of what a sampling can and cannot say.

Placements that went to a different minimum, per thousand. The rule's own counterfactual, run beside it: where would this node have gone with the noise taken away and everything else left alone? Placement noise displaces the node after the argmin, so the answer is always "here" — 0.4°, 0.8° all give zero. A jostle and a field perturbation are upstream of the choice and change one or two placements in a thousand while the lattice is still intact.
Fig. 22 The measured widths, every one of which is a lower bound set by the spacing either side.

What the picture at the top shows

Twenty columns, one per starting angle, and eight rows, one per rise. Each filled cell’s tone names where that run ended up; a pale cell is a run that never settles.

A basin is a run of one tone across neighbouring columns. The top rows have several of them, some five and six and seven columns long; the bottom rows have almost no filled cells at all, which is the wall.

Where a stem started at each of twenty angles ends up, at a falloff exponent of 3. One column per starting angle and one row per rise, coarse at the top. A filled cell is a run that reached a lattice, its tone the destination it reached; a pale cell is a run that never settles. Runs of one tone across neighbouring columns are basins, and the widest of them spans 7 consecutive angles. The angles are 6.25 to 10 degrees apart, so a basin narrower than that cannot be seen here and a single filled cell says nothing about how wide its basin is.
Fig. 23 The table once more, in which every run of one tone is a stretch of starting angle with an extent.

The one line

Thirty-five of the settling table’s hundred and fifty-one runs of consecutive starting angles hold more than one angle, the widest holds seven and spans forty-five degrees, and the measured widths run from 6.24 degrees upwards.

Every one of those is a lower bound, because the angles are 6.25 to 10 degrees apart — so a destination reached from a single angle has a basin this sampling did not measure rather than a narrow one.

Where this goes next

Where a stem started at each of twenty angles ends up, at a falloff exponent of 3. One column per starting angle and one row per rise, coarse at the top. A filled cell is a run that reached a lattice, its tone the destination it reached; a pale cell is a run that never settles. Runs of one tone across neighbouring columns are basins, and the widest of them spans 7 consecutive angles. The angles are 6.25 to 10 degrees apart, so a basin narrower than that cannot be seen here and a single filled cell says nothing about how wide its basin is. Twenty angles instead of nine Every claim in this collection about where a stem ends up rests on nine starting angles a cell, and the file that uses them says so — it computes a binomial error of 0.17 and declines to read a spread against it. Eleven more angles halve that error and change what several of the numbers were. How often each group of starting angles reaches a lattice. The share of runs that settle, for the nine angles the table was grown from, for eight angles placed halfway between them, and for three angles below the nine's lowest. The nine settle far more often than either. So it is the refinement rather than the extension that drags the share down, and the settling share this collection reports is biased upwards by the choice of angles rather than by their range. Round numbers are not a sample The nine starting angles the settling table was grown from reach a lattice four times in ten. Eight angles placed exactly halfway between them reach one a quarter of the time. The difference is not noise and it is not the range — several of the nine sit next door to somewhere a stem could settle. The settled divergence down the golden branch. Every rise from 0.07 down to 0.00482, plotted against the divergence the rule settles on, with each rung drawn in its own stroke and the branch's limit angle marked. The curve does not slide: it turns three times in four rungs, climbing across one and falling across the next, so a value it takes on one rung it takes again on another. That is what makes a matched pair possible — two rises, different counted pairs, one angle — and it is the whole reason the design exists on this branch. The widest excursions from the limit angle, coarse rung first, are 3.195°, 3.352°, 0.961°, 0.422°. The angle the ladder returns to Down the golden branch the settled divergence climbs across one rung and falls across the next, turning three times in four rungs. That is why the same angle is reached at two different rises — and why the Lucas branch, which turns once, almost never offers the same thing.

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.

Named objects

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

AttractorBasinClaim testingDivergenceHonest limitsIdentifiabilityMeasurement errorResolutionSample sizeSamplingSettlingStarting angle