A counter on the settling table

The settling table grows a stem from each of nine starting angles at each of eight rises and four falloff exponents, and asks where each run ends up. Its answer is an angle: 137.8°, 99.0°, 42.3°. A hundred and seventeen of the two hundred and eighty-eight runs settle onto one.

An angle is not an arrangement. This collection’s whole position is that a pattern is counted rather than admired, and for five rounds the settling thread has reported angles and counted nothing. That is not a small oversight in a thread; it is the site’s own premise not being applied to one of its tables.

The design #

Regrow every settled run from the three numbers that define it — the rise, the falloff exponent, the starting angle — and count the parastichies over its top two hundred organs. That is countedAbove, which is the counter every other measurement on this site uses, applied to a table it had never been pointed at.

Two seconds a run. A hundred and seventeen runs. It has been available since the table was first grown, and the whole exercise costs four minutes against the several hours the table itself took.

Every one of them counts #

All 117 return a pair. Not one refuses, and the refusals available are real: the counter asserts that its patch has two families crossing, and it declines a patch that does not.

That answers the first half of the question this design was set up for. If a steep rule settled a stem on 42.3° and no counter could count it, the settling test would be admitting arrangements this collection would refuse to call lattices. It is not.

Fifteen destinations, not forty-nine #

Two settled values are the same destination when they are within half a degree of each other, which is two steps of the grid the azimuths are placed on. Read that way the 117 runs land on fifteen destinations, and the grouping is not delicate: the gaps inside a group are tenths of a degree and the gaps between groups are whole ones, so anything from a quarter of a degree to a degree gives the same fifteen.

Read to a tenth of a degree, which is what the table prints, they land on forty-nine. The difference between those two numbers is an essay of its own, and half a degree is the resolution used here throughout.

Five on the ladder, ten off it #

Of the fifteen, five return pairs on one of the two sequences this collection’s ladder is built from — the golden 1, 2, 3, 5, 8, 13 and the Lucas 1, 3, 4, 7, 11. The other ten do not.

That is the result. The settling rule has been reaching arrangements off both ladders at every exponent it has been grown at, including the exponent every other measurement on this site uses, and nothing asked what they were because the table reported an angle.

And they are the popular ones #

Counting runs rather than destinations makes it sharper. The two most-visited destinations in the whole table are 151.0°, reached by twenty-three runs, and 79.2°, reached by nineteen. Neither is on the ladder.

Sixty-five of the 117 settled runs land off it. A majority of everything the settling test has ever produced is an arrangement this collection has no rung for, and the two that account for a third of the table on their own are the two that were never looked at.

Sixty-five is not sixty-five stems #

The runs are not independent of each other in the way rows of a census usually are. Nine starting angles at one rise and one exponent are nine runs that share everything but where they began, so a destination reached by nine of them is one basin found nine times rather than nine findings.

Counted as basins the picture is the same shape and smaller: 151.0° is reached at several rises and exponents, 79.2° likewise, and both are reached from more starting angles than any golden destination. But the honest denominator is cells of the table rather than runs, and no essay in this thread has been quoting it.

Which is not a failure of the rule #

It is a failure of the reading. The placement rule was never told about Fibonacci numbers: it puts each organ where the sum over its neighbours is least, and what comes out is an output rather than an input. That it comes out on the golden angle from some starting angles and on 151.0° from others is a fact about basins, not about the rule preferring anything.

What the collection has been doing is reading the table through a vocabulary with two sequences in it. Sixty-five runs did not fit the vocabulary and were reported as angles.

An angle is a perfectly honest thing to report and it is not nothing: 151.0° is a measurement, and every essay quoting it was quoting something true. What an angle cannot do is be recognised. A reader who knows 137.5° recognises it instantly and would not recognise 151.0° as a member of anything, which is how sixty-five runs became a set of numbers rather than a finding.

The off-ladder pairs are not a rabble #

Every pair the counter returns is two consecutive terms of a sequence in which each term is the sum of the two before it — the ladder’s own rule, on different starting terms. 4/5 and 5/9 and 9/14 belong to 1, 4, 5, 9, 14. 5/7 and 7/12 and 12/19 belong to 2, 5, 7, 12, 19.

Ten such sequences account for all fifteen destinations, and two of them are the ladder’s.

A destination does not name a pair #

Three of the fifteen return more than one. At 137.8° some runs count 5/8 and others 8/13; at 151.0° the counter gives 5/7, 7/12 and 12/19; at 79.2° it gives 4/5, 4/9 and 5/9 across nineteen runs.

That is not a failure of the counter. The pair a patch shows depends on the rise as well as on the divergence — a finer stem at the same angle shows a finer pair — and the runs at one destination are grown at different rises. It is worth saying because the settling table is indexed by angle, and an angle names a lattice only with a rise beside it.

Which makes the fifteen an undercount of the arrangements #

Fifteen destinations, but more than fifteen arrangements: the pairs seen number seventeen across the table, and the same angle produces different ones at different rises.

A destination is a property of the rule — a value the divergence converges on — and a pair is a property of a patch. The settling table has been reporting the first and this collection cares about the second, which is exactly why the counting was worth doing and why its result is not a simple relabelling.

What the counter is #

The same one. countedAbove reads the top two hundred organs of a run, builds the neighbour graph from the positions, and returns the two families that cross. It does not know the divergence and it is not given one.

Using the site’s own counter rather than a special one is the point. A count produced by a different instrument would make every comparison with the ladder a comparison of instruments, and this collection has been caught by that before — three functions sharing a name and doing different things is the standing example.

The window it reads through #

Two hundred organs at the top of a twelve-hundred-organ run. That is the counter’s own default and the window every other count on this site is taken over, so a pair reported here and a pair reported by the ladder sweep are the same measurement.

It has not been varied here, which is a gap of the same shape as the reading windows in the ablation thread. Counting the same runs over a hundred organs and over four hundred is cheap and would say whether any of the multi-pair destinations are a window effect rather than a rise effect — and the counting radius is known to move a count on a disc, so the question is not idle.

What this does to the wall #

Nothing. The wall is the finding that below a certain rise no starting angle reaches any lattice at all, and it is about the runs that fail to settle. Counting the ones that succeed does not touch it.

What it does change is the description of what lies above the wall. It is not a basin structure over the ladder with some strays; it is a basin structure over ten sequences, of which the ladder’s two are neither the largest nor the smallest.

What it does to the destinations essay #

That essay reported three destinations only the steep exponents reach and asked what they would count as. The answer is 8/9, 8/15 and 2/5, and it is not the interesting half.

The interesting half is that the shallow exponents have been reaching off-ladder destinations all along, in larger numbers, and nobody had asked. A question about the new column turned out to be a question about the old ones.

Three readings of one table #

The settling table now carries three. The share of starting angles that reach a lattice, which is the reading it was built for and the noisiest — a binomial over nine runs, with a standard error of about 0.17 a cell. The time each run takes, which separates the exponents cleanly. And now the arrangement each run reaches, which is the only one of the three that says what came out.

Three readings of one grown table is the pattern this collection keeps arriving at, and the arrangement is the one the site’s own premise should have demanded first.

What a wider sweep would add #

Nine starting angles is a coarse sample of a continuum. The ten sequences are the ones nine angles at eight rises reach, and there is no reason to think a hundred angles would find the same ten.

The prediction, if there is one, is that a denser sweep finds more sequences with smaller basins — that the number of destinations grows with the sampling rather than saturating. Twenty angles is 640 runs against 288, and it would say whether ten is a count or a floor.

The cheap thing that was not done #

Two seconds a run, on a table that had already cost hours to grow. The reason it was not done is that the table’s own reading — the share of starting angles that settle — is a share, and a share does not need the arrangement named.

Once the thread started asking about destinations rather than shares, the counting became obviously necessary, and it stayed undone for one more round because the question was phrased about three specific angles rather than about the table.

What a reader of the ladder essays should adjust #

The ladder is a description of what a stem grown at the golden angle does as the rise falls, and it is unaffected: its rungs are still its rungs and the counted pairs on them are still what a counter returns.

What changes is the sentence that puts the ladder at the centre of the rule’s behaviour. The rule does not preferentially produce the golden ladder from arbitrary starting angles; it produces one of several sequences depending on where the run began, and the golden one is the one the collection has been growing stems onto by choosing their seed.

What the ladder’s own share looks like #

Fifty-two runs land on the ladder’s two sequences: twenty-seven on the golden and twenty-five on the Lucas. So the golden ladder — the sequence this whole subject is named for — accounts for twenty-three per cent of the settled runs in the table.

That number needs its caveat immediately. The starting angles are nine values spread across the useful half-circle, chosen without reference to any sequence, so the share is a share of an arbitrary sample rather than of anything a plant does. What it is not is evidence that the rule prefers the golden angle, which is a claim the table has sometimes been read as supporting.

What is not claimed #

That these arrangements occur in plants. Nothing here is a specimen, and the collection’s position on that is unchanged: the rule produces what it produces, and whether any of it is on a real stem is a survey nobody has done.

Nor that ten sequences is the complete list. It is the list this table reaches from nine starting angles at eight rises and four exponents, which is a sample of a continuum of starting angles, and a denser sample would very likely find more.

The one line #

Every one of the 117 settled runs in the settling table has a countable parastichy pair, so the settling test does not admit arrangements this collection would refuse to call lattices.

And sixty-five of them — a majority, including the table’s two most-visited destinations — land on sequences the ladder does not carry, at every falloff exponent including the one every other measurement here uses.