A basin that doubled

A stem grown from a starting angle either settles somewhere or does not. Consecutive starting angles that settle to the same place form a basin — a stretch of angle from which the rule goes to one arrangement.

At twenty starting angles the widest basin held seven consecutive angles and spanned 45 degrees. That was reported with a caveat attached: a basin measured on a grid is a lower bound in both directions, and a stretch of seven adjacent readings is exactly what an accident of sampling can produce.

The test #

Halve the spacing and look again. If the basin is real, the number of angles inside it roughly doubles and its width stays where it was. If it was an accident of the grid — seven neighbours that happened to agree — the new angles between them will disagree and both numbers will move.

That is the whole design, and it is available for nothing: the settling table has been grown from forty angles for a different reason, and the basins are arithmetic on the same runs.

The spacing goes from 6.2–10 degrees to 3.1–5.

The answer #

Fourteen angles and 47.5 degrees. The count doubled exactly and the width moved by two and a half degrees, which is less than one spacing.

So the widest basin is real. Every angle added inside it agrees with the ones around it, and the two and a half degrees the width gained are the new angles at its edges falling inside where the old ones had left a gap.

That is the cleanest positive this round produces, and it is a confirmation of the useful kind: a quantity that could have failed a specific test and did not.

What a width still is #

A lower bound. The basin’s true edges lie somewhere in the gaps either side of its outermost angles, so the width is at least 47.5 degrees and at most 47.5 plus two spacings — about 57.

Halving the spacing halves that uncertainty, which is the second thing the doubling bought. The width is now known to within about ten degrees where it was known to within about twenty.

Nothing here locates an edge. Doing that would mean sampling densely near one, which is a targeted run of a few dozen stems and has not been done.

The other half of the table #

Goes the other way, and it is the more surprising result. Of the 233 runs of consecutive angles in the forty-angle table, 170 hold a single angle. At twenty angles it was 116 of 151.

So the number of destinations reached from one starting angle with no neighbour agreeing went up by fifty-four when the sampling doubled.

If most destinations had basins wider than the spacing, the singletons would fill in and their count would fall. It rose, which means most of what this table reaches, it reaches from stretches of angle narrower than three degrees.

Which is not a contradiction #

A table can hold one very wide basin and a great many very narrow ones, and this one does. The widest holds fourteen angles and the median run holds one.

The distribution matters more than either number. Sixty-three of the 233 runs hold more than one angle, and their widths run from 3.1 to 47.5 degrees — so there is a spread rather than two populations.

What the doubling shows is that the top of that spread is real and the bottom is still unresolved. A run of one angle is a basin this sampling cannot measure, not a narrow basin.

What a singleton means #

That one starting angle settles somewhere and the angles three degrees either side of it do not settle there. They might not settle at all, or might settle elsewhere.

Both happen. About seventy per cent of runs do not settle, so most of a singleton’s neighbours are runs that never stopped changing — and a basin bounded by non-settling angles is a different object from one bounded by another basin.

The reading does not distinguish them, which is a real limitation. A run of consecutive non-settling angles is recorded as a gap and the basins either side are counted separately, whatever is going on between them.

The widest basin’s destination #

150.96 degrees, which counts a 5/7, 7/12 or 12/19 pair and sits on the 2,5 additive sequence. It is off both ladders.

That is worth noticing. The rule’s largest basin — the widest stretch of starting angle that goes to one place — leads not to the golden angle or the Lucas angle but to an ordinary off-ladder lattice.

It is also the destination most of the wrecked runs in the slot table finish at: fourteen of the twenty-nine that land on a destination land on this one. Two tables that share no run agree on which place the rule most easily ends up in.

Where the golden angle sits #

It is a destination and it is reached by nine runs of the 384 that settle, against 74 for 101.6 degrees and 54 for 151.0.

So the arrangement the entire subject is about is not the one the rule most readily falls into from an arbitrary start. That was already visible at nine angles and it is worth restating here because a wide basin makes it concrete: there is a 47-degree stretch of starting angle leading to an off-ladder lattice, and nothing comparable leading to 137.5.

The golden angle’s own basin at forty angles is narrow. It is reached from angles near itself, which is part of why the original nine were a biased sample — one of them is the golden angle.

What the doubling did not buy #

Any narrowing of the singletons. Halving the spacing from about eight degrees to about four resolved nothing at the bottom of the distribution, which means the basins there are narrower than four degrees and possibly much narrower.

Getting to one degree would be another two doublings — 160 angles, 5,120 runs, about three hours — and there is no reason to think it would terminate. A rule whose destinations are reached from stretches of a fraction of a degree is a rule whose basin structure is finer than any sampling this collection will do.

So the honest position is: one basin measured, sixty-two more bounded below, and a hundred and seventy unmeasured.

A targeted test instead #

Cheaper and better aimed. Take the widest basin’s two edges — around 116 and 164 degrees at one cell of the table — and sample densely across each, say every quarter of a degree over ten degrees.

That is eighty runs and it would locate both edges to a quarter of a degree, turning a bound of 47.5 to 57 into a measurement.

It would also say what an edge looks like: whether the destination changes abruptly to another destination, or fades into a stretch of non-settling angles. Nothing here can tell those apart and they are different claims about the rule.

What a basin is not #

A guarantee. A run inside the basin settles to that destination at that cell — one falloff exponent and one rise — and the basins are computed per cell.

So the widest basin is a property of one exponent at one rise, and the table holds thirty-two cells. Nothing here claims a stretch of starting angle that reaches one place across the whole table.

That is why the count of runs is per cell and the total of 233 is a sum over thirty-two cells. Reading it as 233 basins of the rule would be reading a table as an object.

What is confirmed and what is not #

Confirmed: the widest basin, which holds twice as many angles at half the spacing and the same width. Every width the previous round quoted stands, because each is a lower bound and a lower bound cannot be refuted by finer sampling.

Not confirmed: anything about the narrow end. The count of single-angle runs rose from 116 to 170, so halving the spacing found more unresolved basins rather than resolving the ones it had.

Newly bounded: the widths’ own uncertainty, which was two spacings and is now one — about ten degrees rather than twenty.

Why this is the round’s cleanest result #

Because the test was stated in advance with both outcomes named, the test was cheap, and the outcome was one of the two.

Most of what this round produced is a boundary put round an earlier claim — a list that shrank, a wall that was never located, a correction that fails at a large hop. Those are useful and none of them is a confirmation.

This one is. A quantity that could have failed a specific test did not fail it, and the failure mode was described before the test was run.

How a basin is read #

The starting angles are sorted, and consecutive ones are put in the same run when their settled values are within half a degree of each other — the same tolerance that calls two settled values one destination.

A run of angles that do not settle is also a run, and it is discarded. So a basin is a maximal stretch of adjacent angles all settling to one place, with non-settling angles acting as boundaries rather than as members.

That is the definition the previous round used, unchanged, so the two tables are comparable. Changing it would have made the comparison an argument about two readings.

Why the count doubling is the test #

Because it is the one prediction the two hypotheses disagree about sharply.

If a stretch of angle really goes to one destination, then putting an angle between every pair inside it adds an angle that also goes there — so seven becomes thirteen or fourteen, and the width is unchanged because the outermost angles are the same.

If seven adjacent readings agreed by accident, the new angles between them have no reason to agree. Some would land elsewhere, the run would break into pieces, and both the count and the width would fall.

Fourteen from seven is the first prediction exactly. Nothing about it is a matter of interpretation.

What the twenty-angle round could not say #

It reported the widest basin at seven angles and 45 degrees and said plainly that a width is a lower bound and that a basin narrower than the spacing is invisible. What it could not say is whether the seven were a basin at all.

That is the difference one doubling makes and it is worth naming: the previous round measured a quantity and this one tested whether the quantity exists. Both are necessary and only the second can fail.

A reader of that round who had taken 45 degrees as a measurement would not have been misled — the number stands — but would have been believing something that had not been checked.

What the basins say about the destinations #

Read the other way round, the basin table is a weighting of the destination list. A destination with a wide basin is one the rule reaches easily; one reached from a single angle is one it reaches with difficulty.

The most-reached destinations are 101.6 degrees with 74 settled runs, 151.0 with 54 and 139.1 with 50. The least-reached include the one destination this round added to the list, at 147.3 degrees, with one.

So the destination list and the basin table are two readings of the same 384 settled runs, and the second is what says which members of the first are worth anything. A list of nineteen values with no weights would put a destination reached 74 times and one reached once on the same footing.

A quantity that converges and one that does not #

Two readings of this table have now been doubled twice, and they behave differently.

The settling share converges: 40.6 per cent, 32.3, 30.0, with steps of 8.3 and 2.3. The count of runs of consecutive angles does not: 151 at twenty and 233 at forty, which is roughly proportional to the number of angles because most runs hold one.

That is the signature of a quantity measuring the sampling rather than the object. A count of basins will go on growing as long as most basins are unresolved, and it will start converging only when the spacing is finer than the typical basin.

Which is a useful thing to know before running eighty angles: the share will move by about a point and the basin count will go to about 450.

Whether a basin is the right word #

It carries a claim the reading does not make. In the ordinary sense a basin is the set of initial conditions leading to an attractor, and it is a set rather than an interval — it can be disconnected, it can have a complicated boundary, and it need not be a stretch of angle at all.

What is measured here is a run of consecutive sampled angles agreeing, which is an interval by construction. A destination whose true basin is two separate stretches would show as two runs, and nothing here would say they belong together.

That is not a reason to change the word, since the object is what the word usually means restricted to one dimension and one resolution. It is a reason to keep lower bound attached to every width, which the reading does.

What a lower bound is worth #

A great deal here, and it is worth being precise about why. Every basin width in this table is a lower bound in both directions: the run of angles that agree is inside the true basin, and the true edges lie somewhere in the gaps beyond the outermost agreeing angles.

A lower bound cannot be refuted by finer sampling. Halving the spacing can only find that a neighbour agrees — which widens the bound — or that it does not, which leaves the bound where it was and narrows the uncertainty above it. So every width the previous round quoted stands, and will go on standing however finely anybody samples.

What finer sampling can refute is the claim that a run of agreeing angles is a basin at all, and that is the claim this round tested. It survived on the widest, and the test was available only because the previous round had quoted a number that could be wrong.

The half of the table that does not converge #

A hundred and seventy runs of a single angle, out of 233. Each is a destination reached from one starting angle whose neighbours three degrees away do not reach it.

Some of those are basins narrower than three degrees. Some are wider basins whose neighbours happen not to settle at all — about seventy per cent of runs do not settle, so a basin bounded by non-settling angles is common, and the reading cannot distinguish the two cases.

That is a real limit and it is not one more sampling will fix. Resolving it means asking a different question of the angles that do not settle: not where do they go but how near a destination do they pass, which is a reading nothing in this thread takes.

What is claimed #

That the widest basin in the settling table holds seven consecutive starting angles spanning 45 degrees at twenty angles, and fourteen spanning 47.5 at forty — so halving the spacing doubled the count and left the width where it was.

That the basin’s true width is therefore at least 47.5 degrees and at most about 57, an uncertainty of one spacing rather than two.

And that the number of destinations reached from a single angle with no neighbour rose from 116 of 151 to 170 of 233, so most of what the table reaches it reaches from stretches narrower than three degrees, which remain unmeasured.