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

What a steep rule counts as

Exponents four and five settle stems on 42.3°, 47.9° and 148.1°, and the two shallower ones reach none of them. All three count: 8/9, 8/15 and 2/5. None is a rung of either ladder, and one of them is the coarsest rung of a sequence the table already had.
12 min read 19 figures Counted, not admiredThe angle is an output

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

Growing the settling table at four falloff exponents found that the exponent does not move the share of starting angles that reach a lattice — 30, 31, 29 and 27 of 72, a spread inside the binomial error — and does move two other things: how long a run takes to settle, and where it can end up.

Three destinations appear at exponents four and five and at neither of the shallower ones: 42.3°, 47.9° and 148.1°. The question left was what a counter would say about them, and it has two answers rather than one.

What the 3 steep-only destinations count as. Each block is a divergence that only the two steeper falloff exponents settle a stem on, with the pair a counter returns and the sequence that pair belongs to. All three count. None is a rung of either ladder. Two of them are the only members of their sequence anywhere in the table; the third, at 148.1 degrees, is the coarsest rung of a sequence that supplies another destination at every exponent — so it is not a new kind of arrangement but a coarser one of a kind the table already reached.
Fig. 1 The three destinations only a steep falloff reaches, with the pair a counter returns at each and the sequence that pair belongs to.

They count

All three. 42.3° counts 8/9, 47.9° counts 8/15, and 148.1° counts 2/5, over the same two hundred organs and with the same counter every other measurement on this site uses.

So the first half of the question is answered and answered negatively: a steeper falloff does not produce arrangements this collection would refuse to call lattices. It produces lattices with unfamiliar numbers on them.

That was the more interesting of the two outcomes the earlier essay set up, and it is the one that did not happen. A settled stem that no counter could count would have said the settling test admits things the collection’s own vocabulary cannot describe, which would have been a problem for the test rather than a curiosity about a rule.

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. 2 What the counter does: two families of spirals read off the positions, with nothing told to it about the angle.

None is a rung

8/9 is not two consecutive Fibonacci numbers and it is not two consecutive Lucas numbers. Neither is 8/15, and neither is 2/5.

Every one of them is, however, two consecutive terms of some additive sequence — a sequence in which each term is the sum of the two before it, which is the ladder’s own rule on different starting terms. 8 and 9 sit inside 1, 8, 9, 17, 26. 8 and 15 sit inside 1, 7, 8, 15, 23. 2 and 5 sit inside 2, 5, 7, 12, 19.

That is not a coincidence and it is not much of a constraint either. Any two whole numbers are consecutive terms of exactly one such sequence, walked backwards until it stops. So “on an additive sequence” is true of every possible pair, and what carries information is which sequence — whether it is one the table reaches elsewhere or one it reaches once.

Every family but two is the sum of two others. Four heads and the contact families each actually has. An arc arrives at every family that is the sum of two smaller ones; the two with no arc arriving are the generators, and on every head they are the two smallest. whorled, 144°: 2, 3, 5 — golden, 137.508°: 8, 13, 21, 34, 55, 89 — Lucas, 99.502°: 11, 18, 29, 47, 76 — rational, 137.5°: 8, 13, 21, 34, 55, 89 — 137.0°: 8, 13, 21, 29, 50, 71, 92, 113. So once a pair is counted the rest is arithmetic, and a third counted family is a prediction rather than a second measurement.
Fig. 3 The two sequences the ladder is built from, and the rule they share with every other sequence in this essay.

Which splits the three in half

Two of them are new and one is not.

The 1, 8, 9 sequence appears nowhere else in the table: 42.3° is its only representative at any exponent. The same is true of 1, 7, 8, 15 and 47.9°. But 2, 5, 7, 12, 19 is one of the table’s busiest sequences — it supplies 151.0°, which twenty-three runs reach, at every one of the four exponents including the shallowest.

So 148.1° is not a new kind of destination. It is a coarser rung of a sequence the table already had, found by a steeper rule for the same reason a coarser rung of the golden ladder is found at a coarser rise. And its run was grown at 0.03, which is the second coarsest rise the table uses — so the coarseness of the rung and the coarseness of the rise agree, which is what the ladder does everywhere and is a reason to expect this rather than a reason to be surprised by it.

The 10 sequences the settling table's destinations belong to. Every pair the counter returns 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. two of these are the sequences this collection is built on. The rest are not, and they are not rare: the 1, 4, 5, 9, 14 sequence and the 2, 5, 7, 12, 19 sequence each supply several destinations, at every falloff exponent the table is grown at. The right-hand column is the divergences that land on each.
Fig. 4 The ten sequences behind the table, with the destinations each supplies. One of the three steep-only destinations belongs to a well-populated row of this.

How many runs each of them is

This is where the result needs its denominator, and the denominator is small.

42.3° is four runs: two rises and two exponents. 47.9° is two runs — the same rise and the same starting angle at exponents four and five, so one basin found twice. 148.1° is one run.

Three destinations, seven runs, and one of them a single run out of two hundred and eighty-eight. The table’s other destinations are reached by up to twenty-three runs each, so these three sit at the thin end of a distribution rather than being typical members of it.

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. 5 The settling table cell by cell. The steep-only destinations occupy a handful of cells in the right-hand columns.

Which makes 148.1° the weakest of the three

One run: rise 0.03, starting angle 150°, exponent five. It reports a settled divergence of 148.1° and it reports settling at organ nine of twelve hundred, which is by a wide margin the fastest settling anywhere in the table.

A run that settles in nine organs at a coarse rise is a run that locked onto something almost immediately, and it is exactly the shape a reader should look at twice. It is not obviously wrong — a coarse rise has few neighbours and little to negotiate — and it is one run.

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. 6 How long each run takes to settle. The one that produces this destination is at the extreme left of the distribution.

And 47.9° the second weakest

Two runs at one rise from one starting angle, differing only in the exponent. The exponent is the variable the sweep is about, so finding the same destination at four and five is a confirmation of the rule’s continuity rather than a second observation of the destination.

Counted as basins it is one. Counted as cells of the table it is two. Counted as runs it is two, and the essay reporting “three destinations” was counting the third way.

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. 7 How a basin moves when the exponent does, which is what a destination found at two adjacent exponents is showing.

So the honest statement is about one destination

42.3° is the one that survives all three ways of counting. Four runs, two different rises, two different starting angles, two exponents. It counts 8/9, on a sequence nothing else in the table reaches, and it is the only steep-only destination this collection would defend if the sweep were re-run with a different set of starting angles — which is the standard an untested claim has to meet before it is quoted.

That is the result: a steeper falloff reaches at least one arrangement the shallower ones do not, and the arrangement is a countable lattice on a sequence beginning 1, 8, 9, 17.

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. 8 The off-ladder destinations, of which 42.3° is the one only a steep rule reaches and that more than one cell of the table finds.

What 8/9 is

Two consecutive integers, which is what a divergence near 360/8.5 gives. The organs fall into eight chains one way and nine the other, and the counter has no difficulty with it: the two families cross cleanly and the neighbour graph is unambiguous.

A pair of consecutive integers is the coarsest possible relationship between two counted numbers and it is entirely ordinary as a lattice. It is unfamiliar here only because the collection’s ladder does not contain a rung with 8 and 9 adjacent.

There is one thing about it worth flagging for a later round. A pair of consecutive integers means the two families cross at nearly the same spacing, which is the condition under which the two contact steps stop being separated — and a lattice whose two shortest steps are within a per cent of each other is one where every claim about which is shorter does no work. Whether 8/9 is such a lattice has not been checked.

What a divergence picked at random gives, at a rise of 0.008. Fibonacci pairs take 14.6% of the circle at this rise, and the share falls as the rise does. The claim that Fibonacci counts are what nature "prefers" needs the preference to come from somewhere, and it is not from the geometry being generous.
Fig. 9 A counted pair read from a stem’s own points. Nothing in the reading privileges one pair of numbers over another.

Why a steeper rule finds it

A guess, and it is offered as one. A steeper falloff means the sum over neighbours is dominated by fewer of them, so the placement is decided more locally. A more local rule can settle into an arrangement that a rule seeing further would be pulled out of.

That predicts that the steep-only destinations should be the ones with the shallowest basins — reachable from few starting angles and easily left. Four runs, two runs and one run is consistent with it and nowhere near enough to test it.

It also predicts something that can be checked on runs already grown: a shallow basin should be a slow one, because a run that barely finds a destination should take longer to settle into it. The settling times for the seven runs are 273, 254, 238, 206, 147, 140 and 9 organs, against a table whose median is nearer a hundred. Six of the seven are slow and the seventh is the fastest run anywhere, which is not a clean answer in either direction.

Where the lattice ends, for two falloff shapes at p = 1. Both shapes are read in the same unit — the distance at which the weight has halved — and they still disagree, by 50%: the exponential holds a lattice out to about 3.75 spacings and the gaussian only to about 2.25. So the range is not what decides whether there is a pattern.
Fig. 10 How far a placement rule reaches at different exponents, which is the quantity this guess is about.

The test it suggests

Denser starting angles at exponent five, on the two or three rises where 42.3° occurs. If its basin is narrow, twenty angles find it once or twice; if it is not, they find it five or six times.

That is sixty runs at one exponent, and it would turn a destination found four times into a basin with a measured width. This thread’s usual complaint about itself applies: the sweep was designed to compare columns and is being asked about one cell.

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. 11 The share of starting angles reaching a lattice, which is the reading a denser sweep would refine at the level of individual destinations.

What this does to the earlier essay

It answers its question and moves the interest elsewhere. The earlier essay asked what three steep-only destinations would count as, and the honest answer is that one of them is well-evidenced, one is a single basin seen twice, and one is a single run — and that all three count.

Meanwhile the shallow exponents have been reaching off-ladder destinations all along, in far larger numbers. A question about the new column turned out to be a question about the old ones.

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. 12 The destinations by exponent. The steep-only ones are a few marks at the edges of a table whose middle nobody had counted.

And to the phrase “only a steep rule”

It needs a resolution attached to it. Read to a tenth of a degree the list of steep-only destinations has thirteen entries rather than three, and ten of those are destinations a shallow exponent reaches a tenth or two of a degree away.

That is an essay of its own, and the reason it matters here is that the three-entry list quoted throughout this one is itself the product of a stated tolerance. Change the tolerance and the list changes.

The destinations a steep rule reaches, read two ways. Above: the list read to a tenth of a degree, which is what the settling table prints — 13 entries. Below: the same list read at half a degree, which is two steps of the grid the azimuths are placed on — 3. The 10 entries the finer reading adds are destinations a shallow exponent also reaches, a tenth or two of a degree away, and a tenth of a degree is a fifth of one step of that grid. The pale marks are every destination in the table, for scale.
Fig. 13 The same list read at two resolutions. The three entries this essay is about are the ones that survive both readings.

Two of the three are the same shape as an old mistake

A destination reached by one run and a destination reached by twenty-three are reported by the same table in the same column, and nothing in the table’s own output distinguishes them. That is the shape the settling shares already had: a spread of 0.056 read against a binomial error of 0.058, which is four columns of one column reported as four.

The response there was to stop reading the share and read the clock and the map instead. The response here is smaller and the same in kind: report how many runs reach a destination beside the destination, which the table can do and does not.

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. 14 The settling table read for its shape rather than its counts, which is the reading that survived the binomial error.

What a single run can and cannot support

It can support an existence claim. One run settling on 148.1° and counting 2/5 is enough to say the rule reaches that arrangement at exponent five, because a single positive instance is all an existence claim needs and the run is deterministic and reproducible from three numbers.

It cannot support anything comparative. That 148.1° appears at exponent five and not at exponent three is a statement about one cell of a table with two hundred and eighty-eight cells in it, and the sampling is nine angles a cell. The existence is solid and the exclusivity is not — the same asymmetry the settling wall rests on, where a single settled run at a fine rise would break the wall and no number of unsettled ones can establish it.

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 The destinations a rule of this kind reaches across a dense range of starting angles, which is the sweep a claim about exclusivity would need.

What is not claimed

That 8/9 or 8/15 occurs on a plant. Nothing in this thread is a specimen and the collection’s position is unchanged: the rule produces what it produces, and whether any of it grows is a survey that has not been done.

Nor that these are the only arrangements a steeper rule reaches. Nine starting angles at eight rises is a coarse sample, and a destination found by one run is a destination that a slightly different sample would have missed entirely — which cuts both ways, and the sample that missed it would have reported two steep-only destinations with equal confidence.

Every open question here needs under 28 specimens. The sample size at which each comparison reaches 80 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives.
Fig. 16 What a survey would need to say anything about which arrangements occur, which is not what this table is.

What the counting cost

Four minutes for the whole table, of which the three destinations here are seven runs and about fifteen seconds. The table itself took hours.

The reason it was not done in the round that produced the destinations is that the question was phrased about three angles, and answering a question about three angles by counting a hundred and seventeen runs is a step nobody was pushed to take. It is the step that turned out to matter.

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. 17 The settling clock without its markers, which is the reading the earlier round used to separate the exponents.

Where the exponent’s real effect is

Not here. The exponent’s clearest effect on this table is on the clock: the slowest settling is 160, 290, 808 and 674 organs across the four columns, so “settling takes at most 290 organs” — carried for four rounds as a property of the rule — is a property of the exponent.

Beside that, three destinations reached by seven runs is a small effect measured weakly. The essay that found both reported the clock as the sharper of the two and it was right, and this essay is the confirmation from the other side: counting the three destinations does not make them stronger evidence, it makes their denominator visible.

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. 18 The settling clock across the four exponents, which is where the exponent’s effect on this table is unambiguous.

What carries forward

One well-evidenced steep-only destination at 42.3°, counting 8/9, on a sequence nothing else in the table reaches. Two weaker ones. And a test — sixty runs at exponent five with twenty starting angles — that would turn a destination found four times into a basin with a width.

And a habit worth keeping: report the number of runs beside the destination. It costs a column and it would have made three of this essay’s fifteen sections unnecessary.

What the 3 steep-only destinations count as. Each block is a divergence that only the two steeper falloff exponents settle a stem on, with the pair a counter returns and the sequence that pair belongs to. All three count. None is a rung of either ladder. Two of them are the only members of their sequence anywhere in the table; the third, at 148.1 degrees, is the coarsest rung of a sequence that supplies another destination at every exponent — so it is not a new kind of arrangement but a coarser one of a kind the table already reached.
Fig. 19 The three destinations again, which is the list this essay both confirms and shrinks.

The one line

The three destinations only a steep falloff reaches all count: 8/9, 8/15 and 2/5, none of them a rung of either ladder and all three two consecutive terms of an additive sequence.

One of them is a coarser rung of a sequence the table already had at every exponent, one is a single basin found at two exponents, and one is a single run that settled in nine organs — so of three destinations, one is well-evidenced.

Where this goes next

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. 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. 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. A counter on the settling table The settling table has reported an angle for every run that reaches a lattice, and nobody has ever counted one. A hundred and seventeen settled runs, regrown and counted: every one of them has a parastichy pair, and sixty-five of them land on sequences the ladder does not carry. The 10 sequences the settling table's destinations belong to. Every pair the counter returns 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. two of these are the sequences this collection is built on. The rest are not, and they are not rare: the 1, 4, 5, 9, 14 sequence and the 2, 5, 7, 12, 19 sequence each supply several destinations, at every falloff exponent the table is grown at. The right-hand column is the divergences that land on each. Ten sequences, two of them the ladder's Every parastichy pair the settling table produces is two consecutive terms of a sequence in which each term is the sum of the two before it. Ten such sequences account for all fifteen destinations, and the two the collection is built on are neither the largest nor the smallest.

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, divergence angle, honest limits, measurement, negative result, parastichy pair, the placement rule
  • The stem that changed hands — both name attractor, basin, divergence angle, honest limits, initial condition, measurement, parastichy pair, the placement rule
  • Two accounts of one number — both name attractor, claim testing, honest limits, ladder, measurement, negative result, parastichy pair, the placement rule
  • A wall and not a budget — both name attractor, basin, honest limits, initial condition, ladder, measurement, negative result
  • A wreck has a short list — both name attractor, basin, honest limits, measurement, negative result, parastichy pair, the placement rule
  • The band was not the sampling — both name claim testing, divergence angle, honest limits, measurement, negative result, parastichy pair, the placement rule

Named objects

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

AttractorBasinClaim testingCounting radiusDivergence angleExponentFibonacciHonest limitsInitial conditionLadderMeasurementNegative resultParastichy pairThe placement ruleSample sizeSelection effect