Ten sequences, two of them the ladder's

Counting every settled run in the settling table returns seventeen distinct parastichy pairs across fifteen destinations. Only five of the pairs are ones this collection has a name for.

The other twelve are not arbitrary. Every pair the counter returns — all seventeen — 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 started from different terms.

Which is almost no constraint #

Worth getting out of the way first. Any two whole numbers are consecutive terms of exactly one such sequence: walk backwards, subtracting the smaller from the larger, until the result stops being positive and smaller than what follows it. So “on an additive sequence” is true of every pair the counter could possibly return.

What carries information is which sequence, and how many destinations each one supplies. A table whose seventeen pairs sat on seventeen different sequences would be a table with no structure in it at all.

Ten sequences, not seventeen #

They sort into ten. Four of the ten supply more than one destination and six supply one each.

The two largest are 1, 4, 5, 9, 14 — which supplies 77.7°, 79.2° and 80.5°, with the pairs 4/5, 4/9, 5/9 and 9/14 — and 2, 5, 7, 12, 19, which supplies 148.1° and 151.0° with the pairs 2/5, 5/7, 7/12 and 12/19. Both appear at every one of the four falloff exponents the table is grown at.

The ladder’s two are third and fourth #

The golden 1, 2, 3, 5, 8, 13 supplies three destinations — 136.8°, 137.8° and 139.1° — with the pairs 3/5, 5/8 and 8/13. The Lucas 1, 3, 4, 7, 11 supplies two, at 99.5° and 101.6°, with 3/4, 4/7 and 7/11.

So of the four sequences the table reaches more than once, the collection’s two are not the top pair. Measured by runs rather than destinations the picture is the same: 151.0° alone is reached by twenty-three runs, which is more than either ladder sequence’s total.

The golden sequence’s three destinations take twenty-seven runs between them and the Lucas sequence’s two take twenty-five, so the collection’s two account for fifty-two of the hundred and seventeen. That is a large minority and it is a minority, and the sentence the thread has been carrying — that the rule settles onto the ladder — was written when nobody had counted the other half.

The six with one destination each #

1, 5, 6, 11, 17 at 65.2° with 6/11. 3, 7, 10, 17, 27 at 106.1° with 10/17. 2, 7, 9, 16, 25 at 157.7° with 7/9 and 9/16. 4, 9, 13, 22 at 79.2° with 4/9. 1, 8, 9, 17, 26 at 42.3° with 8/9. 1, 7, 8, 15, 23 at 47.9° with 8/15.

The last two are the ones only a steep falloff reaches. The other four are reached at three or four exponents each, including the shallowest.

One of the six does not survive the rest of this essay. 4, 9, 13, 22 is supported by a single run at 79.2°, and that run’s two counted families wind the same way.

Two sequences at one destination #

79.2° is reached by nineteen runs and the counter returns 4/5, 4/9 and 5/9 among them. 4/5 and 5/9 sit on 1, 4, 5, 9, 14; 4/9 sits on 4, 9, 13, 22, because 4 and 9 are not adjacent in the first sequence.

Something has skipped a term, and it is worth taking seriously rather than tidying away. The account first offered here was that the count had been taken over a slightly different patch, on the reasoning that a pair skipping a term of its own sequence is what a count taken at the wrong radius produces on a disc, and that the same could be true up a stem.

That account is wrong. The skip has nothing to do with where the count was taken. The counter works out whether its two chosen families wind in opposite directions — a parastichy pair is two families that cross — and the reading that takes the count drops the answer; one of the nineteen runs here returns two families of one handedness, and 4/9 is what such a reading looks like written as a pair. The measurement that settles it puts the same skip on two further destinations nobody had questioned.

Which means the ten is an upper bound #

If 4, 9, 13 is a reading artefact rather than a sequence, the table has nine. It is one, and the table has nine.

The test named for it was to count the same runs over a hundred organs and over four hundred and see whether 4/9 survives. That test has since been run, at seventeen widths from twenty organs to eight hundred, and 4/9 survives every one of them — as does every other pair in the table, which does not move with the counting window at all. So a fairly named test would have returned the wrong verdict, because the quantity it varied is not the quantity that produced the pair.

The reverse is also possible. Two of the nine that remain are represented by a single destination reached by a handful of runs, and a denser sweep of starting angles would very likely add more. So nine is a number this sample produced and not a property of the rule, and any use of it should carry the sample with it — the same discipline a count taken at one radius needs.

Each sequence has a limit angle, and the destinations sit at it #

The sequences are usually described by their integers, and describing them by their angles turns a list into an explanation — because each one has a limit divergence, computable from its first two terms, and the destinations on it cluster there.

The arithmetic is the ladder’s own. An additive sequence’s consecutive terms are the denominators of the convergents of a number whose continued fraction ends in all ones, and that number is what a divergence converges on as the ladder is climbed. Working it out needs only the seeds.

Take 1, 4, 5, 9, 14. Its first partial quotient is four and the rest are ones, so the limit is one over four-plus-one-over-φ, which is 0.21654 of a turn — 77.96°. The three destinations on that sequence are 77.7°, 79.2° and 80.5°, sitting just above it in the way a ladder’s rungs sit above their limit.

Take 2, 5, 7, 12, 19. Its quotients are two, two, then ones, giving 0.4198 of a turn — 151.1°. Its two destinations are 148.1° and 151.0°.

And the two the collection names come out where they should: the golden sequence’s limit is 137.51° against destinations at 136.8°, 137.8° and 139.1°, and the Lucas sequence’s is 99.50° against 99.5° and 101.6°.

Which makes the count a count of attractors #

Read that way the essay’s finding is sharper than a census of integers. The rule has at least nine attracting divergences, and this collection has been writing about two of them.

Each sequence is a limit angle with a ladder attached, and each destination is a rung of that ladder — so a sequence supplying three destinations is one whose ladder has been climbed three rungs in this sweep, and a sequence supplying one is one whose ladder has been reached once. That immediately explains the shape of the table. The four sequences with several destinations are not more common than the six with one; they are the ones a range of rises walked further along.

It also disposes of the worry about 4/9 without needing a rerun. A pair skipping a term of its own sequence would read as a rung of a different ladder, and 4, 9, 13, 22 is a ladder whose quotients are four, two and then ones — a limit of 0.22821 of a turn, or 82.2°. The two readings are therefore separated by which limit the destination sits near, and 79.2° is 1.2° from 77.96° and 3.0° from 82.2°. So 4/9 at that destination is the first sequence read a term apart, and the table has nine sequences rather than ten.

That is a stronger conclusion than the essay set out to reach, it follows from arithmetic already in hand, and it is right: reading the discarded crossing flag on every settled run takes ten sequences to nine, at the same sampling, by a route with nothing in common with this one.

What the limit angles do not supply is the mechanism, and the wording above claimed more than they can carry. They compare two candidate ladders after a reader has noticed an odd pair, and a comparison of that kind endorses whichever limit sits nearer whether or not the pair is real. It cannot say why an odd pair was returned, and it found neither of the two further same-handed readings at other destinations, because nothing about them looked odd enough to compare. The conclusion here rests on that measurement; the arithmetic is why anyone went looking. Which is still the argument for computing a limit angle for every sequence rather than listing its integers: an integer sequence is a description and a limit angle is a prediction about where a stem will settle.

What the sequences have in common #

Each has a limit ratio — the value its consecutive terms converge to — and a limit angle, which is a full turn divided by that ratio squared, folded into half a turn. Those limit angles are what the destinations approximate.

The golden sequence’s limit angle is 137.5° and its destinations are 136.8°, 137.8° and 139.1°. The 2, 5, 7, 12, 19 sequence’s is near 151°, and its destinations are 148.1° and 151.0°. So a sequence is a family of nearby destinations rather than a single one, which is exactly what the golden ladder already is: the ladder is one angle across a range of rises, giving a different pair at each rung, and the three golden destinations here are three readings of that.

And what makes them different #

The limit ratios are different irrationals. The golden sequence converges on the golden ratio, the Lucas one converges on the same ratio from different terms, and the others converge on their own values.

That is the part a reader coming from the standard account should notice. The usual story privileges the golden ratio because of a property of its continued fraction — it is the hardest number to approximate — and several of these sequences converge on ratios with no such property, and the rule reaches them anyway.

Which is not a refutation of that account #

It is a measurement it has to accommodate. The standard account says an arrangement whose divergence is poorly approximated by simple fractions packs better, and it predicts that such arrangements should be preferred. It does not say others are unreachable.

What the table shows is that a placement rule minimising a sum over neighbours reaches nine of them from nine starting angles, and lands on the golden one from about a quarter of its settled runs. Whether the golden one is preferred in any sense that matters is a question this table can be asked and has not been.

There is also a measurement on this site that already complicates the standard story from a different direction: the most irrational angle is not the most disordered one by the site’s own disorder statistic. Two independent readings, neither of which refutes the account and both of which say it is doing less work than it is usually given.

The question the table can be asked #

Whether the basins differ in size. Each destination’s basin is the set of starting angles that reach it, and the table samples nine of them per cell. Counting how many of the nine reach each destination is a measurement of basin width at that resolution.

At nine angles the answer is noisy — a binomial with a standard error of about 0.17 — and it is the same instrument the settling shares already broke on. Twenty angles halves the error at 640 runs against 288, and it is the obvious next sweep.

What this changes about the site’s premise #

Nothing about the premise and something about a habit. The premise is that spirals are counted rather than admired, and counting is exactly what produced this: the sequences were invisible while the table reported angles.

The habit is quoting the golden ladder as though it were what the rule does. It is what the rule does from the starting angles this collection has been seeding stems with, which is a different sentence and the one the table supports.

The sequences are not equally spread across the exponents #

Four of them appear at all four falloff exponents: the golden, the Lucas, 1, 4, 5, 9, 14 and 2, 5, 7, 12, 19. Three appear at three of them. Two appear at exponents four and five only, and one appears at exponents two, four and five and not at three, which is a gap that is almost certainly the sample rather than the rule.

That last one is worth naming because it is a small warning about reading a column as absent. 157.7° is reached at exponents two, four and five, and its absence from the exponent-three column is one cell of nine runs not finding a destination that neighbouring columns find. Nothing follows from it, and if it had been the only column it was absent from somebody could have made something of it.

Where these sequences have names #

Outside this collection they do. Sequences of this kind, and the arrangements they describe, have been catalogued in the botanical literature for a long time under several naming conventions, and 2, 5, 7, 12 and 1, 4, 5, 9 are among the ones that recur.

This essay does not use those names, for the reason the collection does not use borrowed names anywhere: a name imported without its definition is a claim that two things are the same, and checking that is a piece of work nobody here has done. The same caution applies to the pairs: 5/7 in a description of a plant and 5/7 out of this counter are two readings that would have to be shown to be one, and that is a survey rather than a footnote.

What a sequence is, as an object #

Worth being precise, because two things are being called one. A sequence here is a list of whole numbers with a rule. A destination is a measured divergence. A pair is what a counter returns from a patch.

The chain from one to the next is not automatic. A pair determines a sequence uniquely, by the walk backwards. A destination does not determine a pair — three of the fifteen return more than one — and a sequence certainly does not determine a destination, since four of them supply several. So the grouping in this essay is a map from measurements to a structure, and it is many-to-one in both directions in places.

What the collection’s own ladder is, restated #

A rung of the golden ladder is a range of rises over which a stem grown at the golden angle shows one counted pair. The ladder is one angle and a sequence of pairs down it.

A sequence here is the same thing seen from the other end: a family of angles, each carrying its own ladder of rungs. So there are nine ladders in the table, of which the collection has built machinery for two, and every measurement in this collection about rungs, bands, handovers and cuts has been made on one of the two.

That is not a criticism of those measurements. It is a statement about their scope, and it is the sort of statement that is easy to make once and hard to remember to attach to results.

The honest limits #

Nine starting angles, eight rises, four exponents, and a counter reading two hundred organs. Nine sequences is what that sample reaches; a denser sample reaches more, and the counting window merges and splits nothing at all — every settled run in the table returns one pair at every width from twenty-five organs to eleven hundred, which was not known when the sentence that used to sit here was written.

And the sample of starting angles is not a sample of anything in particular. They span the useful half-circle at even intervals, which is a reasonable thing to do and is not a model of how a stem chooses where to begin. A sweep that sampled starting angles the way a meristem might would give a different distribution over these nine and this table says nothing about what it would be.

What a basin over nine sequences would look like #

Worth sketching, because the picture the collection has been carrying is a basin structure over one ladder with strays. The measured picture is different in a specific way: several sequences with several destinations each, and a run’s starting angle deciding which sequence it lands on before it decides which rung.

That predicts something checkable and unchecked. Starting angles near a sequence’s own limit should land on that sequence, and starting angles between two limits should be the ones that fail to settle at all. Nine angles cannot show it and the table records which angle produced which destination, so a denser sweep would settle it in one picture.

What carries forward #

Nine sequences, four of them reached more than once, two of them the collection’s own and neither of those the largest. A pair that skips a term, which turned out to be a counting artefact of a kind that had no name here and which cost the census exactly one sequence. And a basin-width sweep at twenty starting angles, which would turn a list of destinations into a distribution over them.

The one line #

Every parastichy pair the settling table produces sits on an additive sequence, which constrains nothing; they sort into ten such sequences, and nine once the one supported by a pair whose families never meet is set aside, which constrains a good deal.

The golden and the Lucas are two of the nine, they supply five of the fifteen destinations and fifty-two of the hundred and seventeen settled runs, and the single most-visited destination in the table is on neither of them.