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

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.
12 min read 25 figures State the claim, state the testCounted, not admired

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

A stem is grown from a starting angle, and where it ends up is the question the settling thread exists to answer. The table it answers from is grown from nine of them a cell: 40, 60, 80, 100, 120, the golden angle, 150, 165 and 180 degrees.

Nine is a small number to carry that much, and the file that uses them says so. It computes a binomial standard error of about 0.17 on each cell and declines to read a spread of 0.056 against it.

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, at the falloff exponent every other measurement uses.

Eleven more, of two kinds

The eleven added here are not one group and the distinction is load-bearing.

Eight midpoints — 50, 70, 90, 110, 128.75, 143.75, 157.5 and 172.5 — halve the spacing inside the range the nine already cover. That is a refinement, and it is what a basin width can be read off.

Three below forty — 10, 20 and 30 — extend the range downwards. The nine start at forty and nothing says why. That is an extension, and it answers a different question.

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. 2 The twenty starting angles, with each one’s distance to the nearest destination the table reaches.

Why the nine are kept

Because a table that re-sampled from scratch would have made every comparison an argument about two designs. The twenty contain the nine, so every number the old table reported is answered from the same cache rather than measured again.

That is the discipline the slot design followed when it went from six lattices to twenty-four, and it is what makes a difference between the two tables a difference in the sampling rather than in the code.

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. 3 The settling table the nine angles produce, which the twenty contain unchanged.

The cost

Four falloff exponents by eight rises by twenty starting angles is 640 runs against 288, each grown to twelve hundred organs. The three hundred and fifty-two new ones took about twelve minutes.

Two hundred and seven of the 640 settle, against 117 of the 288. So the count of settled runs nearly doubles and the share falls, which is the first thing the extra angles say.

How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest.
Fig. 4 How often runs settle, which is the share the extra angles move.

The share falls at every rise

At the working exponent the nine give 78, 67, 78, 56, 33, 11, 11 and 11 per cent across the eight rises. The twenty give 70, 60, 60, 30, 20, 10, 5 and 5.

Every one of those is lower. A fall at one rise would be sampling; a fall at all eight is a bias, and finding out which of the two groups of new angles caused it is why they were counted apart.

The settling share at every rise, sampled two ways. 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. 5 The settling share against the rise at both samplings, with the half level the wall is read at.

And it is the refinement, not the extension

The nine settle at 40.6 per cent over the whole table. The eight midpoints settle at 25.8 per cent. The three below forty settle at 25.0 per cent.

So the angles outside the original range behave like the angles between the original ones, and both settle at about two thirds the rate of the nine themselves. Extending the range did nothing unusual; refining it did, and that is the finding this sampling produced.

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.
Fig. 6 How often each group of starting angles reaches a lattice, with the original nine drawn apart.

What the extension was for

It is worth saying what the three below forty were expected to do, since they did nothing special. The nine span 40 to 180 degrees and a divergence can be anything from nothing to a half turn once folded.

So the range had a floor with no reason attached to it. Ten, twenty and thirty are the obvious continuation of the nine’s own spacing, and the answer is that a stem started at ten degrees settles about as often as one started at fifty.

five limit divergences, all of them 137.5078 over a whole number. The golden angle is the k = 1 member of a family. Real bijugate plants — teasel, Cephalaria — are reported near 68.75°, which is exactly half of it, and the pairs they are counted at are the Fibonacci pairs doubled.
Fig. 7 The range a divergence can take, of which the original nine covered the upper four fifths.

What the refinement was for

Two things. The first is the error bar: a share over twenty runs has a binomial standard error of about 0.11 where a share over nine has 0.17, so every cell of the table is read more precisely.

The second is basins. With the angles twice as close together, a destination reached from several consecutive starting angles has a width that can be quoted, and some of them are wide.

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. 8 The same table at the steepest exponent, where runs of one tone across neighbouring angles are basins.

The error, halved

Over the whole table the binomial standard error on a pooled share falls from 5.8 per cent to 3.7. On an individual cell it falls from about 17 to about 11.

That is what doubling a sample buys and it is worth stating that it is not more. Twenty runs a cell is still a small number, and the readings this table supports are orderings and presences rather than precise shares.

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. 9 The share of starting angles that reach a lattice, with the error a sample of this size carries.

Which is why the destinations matter more than the shares

The settling file’s own reading makes this point: the share of starting angles that reach a lattice is a binomial over nine runs, and the four exponent columns are the same column inside its error.

The times and the destinations are not shares. A destination is a measured angle and either the table reaches it or it does not, so a list of them is not a statistic with an error bar on it in the same way.

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. 10 The destinations the table reaches, which are presences rather than shares.

And the destination list grows

At half a degree the table reaches eighteen distinct destinations against fifteen. The three new ones are 55.2 degrees counting 6/7, 132.5 degrees, and 162.1 degrees counting 9/11.

Every one of the three is off both ladders, and the count of destinations that sit on the golden or Lucas sequence is five at nine angles and five at twenty. So the on-ladder targets were already enumerated and the off-ladder ones were not.

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. 11 Every destination the table reaches, with the ones on a ladder sequence marked.

The sequences, which were a lower bound

The destinations sort into additive sequences — each pair two consecutive terms of a rule where every term is the sum of the two before it — and the count was ten.

At twenty angles it is thirteen, and the arrangements go from twenty to twenty-four. Ten was what nine angles found, not what the rule reaches.

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. 12 The additive sequences the destinations sort into, of which nine angles found ten.

And the steep-only list

Three destinations were reported as reachable only by a steep falloff — 42.3, 47.9 and 148.1 degrees. At twenty angles the list runs to five, with 55.2 and 162.1 joining it.

Both additions are countable and both are two consecutive terms of an additive sequence, so what that list is like survives. How long it is was a property of the sampling, which is the second time a list in this thread has turned out to be.

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. 13 The destinations only a steep falloff reaches, a list whose length depends on the sampling.

The first time was a rounding

Worth recalling, because the two are different failures. The steep-only list runs to thirteen entries when the destinations are read to a tenth of a degree rather than half a degree, and ten of those thirteen are destinations a shallow exponent reaches within a tenth or two of one.

That was a resolution problem. This is a sampling problem, and a list can suffer from both at once: it is thirteen at a tenth of a degree with nine angles, five at half a degree with twenty, and seventeen at a tenth of a degree with twenty.

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. 14 The same list read at two resolutions, which is the other way its length moves.

So a list has two settings attached to it

Any count of destinations in this collection depends on the resolution it is grouped at and on the number of starting angles it is grown from. Both have now been varied and both change it.

The honest form is to quote the count with both settings: eighteen destinations at half a degree from twenty starting angles. That is longer than fifteen destinations and it is the number that means something.

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. 15 The same destinations grouped at a tenth of a degree, where the count is different again.

What did not change

The five on-ladder destinations. Doubling the sampling found no new destination on either the golden or the Lucas sequence, which is the one count in this table that appears to be complete.

That is worth as much as the three that were found. A list that grows when the sampling grows is a lower bound; a list that does not is a candidate for being the whole of something.

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. 16 The destinations that sit on a ladder sequence, whose count is the same at both samplings.

And the wall

The wall — the rise below which fewer than half the starting angles reach a lattice — is where it was. What changed is the spread between the four exponents’ walls, which falls from a factor of 1.97 to 1.18, and their ordering, which reverses completely.

Neither ordering means anything, and that is the point. The conclusion that the exponents do not separate is confirmed and the arithmetic that had looked less settled than the conclusion was noise.

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. 17 The four exponents’ settling shares at twenty angles, where their crossings sit close together.

What twenty angles cannot do

They cannot resolve a basin narrower than the spacing, which is 6.25 to 10 degrees over most of the range. A destination reached from one angle alone says nothing about how wide its basin is.

And they cannot turn a share over twenty runs into a precise number. The table is still a table of presences and orderings, read a little more finely.

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. 18 What a basin width can and cannot be read as at a stated spacing.

What forty would buy

Halving the spacing again would be another 640 runs and about twenty-five minutes, and it would resolve basins down to three degrees and take the error on a cell to about 8 per cent.

Whether that is worth it depends on whether the questions left are about widths. The destination list would probably grow again — every refinement so far has found off-ladder destinations — and the wall would not move.

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. 19 The twenty angles and their distances to destinations, at the spacing a further refinement would halve.

Where the nine came from

They are not arbitrary and they are not a design either. The list is 40, 60, 80, 100, 120, the golden angle, 150, 165 and 180 — five round tens, then a step of seventeen and a half to the golden angle, then fifteen, then fifteen, then fifteen.

That is what a list looks like when it is written to span a range and to include the value somebody wants to see. The golden angle is in it because a stem started at the golden angle is the case the whole subject is about, and the round numbers are in it because round numbers are what a person types.

Neither reason is bad. Both are reasons for a list to be unrepresentative of the circle, and the midpoints are what shows it.

four limit divergences, all of them 137.5078 over a whole number. The golden angle is the k = 1 member of a family. Real bijugate plants — teasel, Cephalaria — are reported near 68.75°, which is exactly half of it, and the pairs they are counted at are the Fibonacci pairs doubled.
Fig. 20 The range a divergence can take, and where the original nine sit inside it.

What a random sample would have been

The alternative design is twenty angles drawn at random from the range, which would have no structure to be unrepresentative in. It was not used and it is worth saying why.

A random list cannot contain the old one, so every number would have been re-measured and the comparison would have been between two designs. And a random list makes a basin width harder to read, since the spacing is irregular by construction.

The midpoint design keeps the old table inside the new one and gives an even spacing. What it gives up is the ability to say anything about the circle as a whole, which this table has never been claiming.

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. 21 The twenty angles against their distances to destinations, where the even spacing is visible.

What the table is for, said again

It is not an estimate of how often a stem settles. It is a comparison across rises and across falloff exponents, made by holding the starting angles fixed and moving one thing at a time.

For that purpose a biased list of angles is harmless as long as it is the same list in every cell, which it is. The bias cancels in a comparison and does not cancel in a share, which is why the shares are the numbers that moved and the orderings are the numbers that did 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. 22 The table as a comparison across rises and exponents, in which the same angles appear in every cell.

And where a share was being read anyway

One place, and it is the wall. The wall is defined as the rise where the share of starting angles that reach a lattice falls through a half, so it reads a share and reads it against a level.

That is the one number in the thread the bias could have moved, and it turns out not to have moved it — the four exponents’ walls sit closer together at twenty angles than at nine, not further apart. The level is arbitrary enough that a uniform bias shifts every column the same way.

The settling share at every rise, sampled at nine 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. 23 The four exponents’ shares at nine angles, where the wall was originally read.

What a reader should carry

That every settling number in this collection is from twenty starting angles rather than from the angles, and that the count was nine until now.

And that the eleven added split into a refinement and an extension for a reason: the share fell at every rise, and separating the two groups is what showed the fall came from the refinement rather than from the wider range.

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.
Fig. 24 The three groups of starting angles and how often each settles, which is what the split was for.

What the picture at the top shows

Twenty columns, one per starting angle, and eight rows, one per rise, coarse at the top. A filled cell is a run that reached a lattice and its tone names the destination; a pale cell is a run that never settles.

The short marks above the top row are the nine angles the table was grown from. Runs of one tone across neighbouring columns are basins, and the pale region growing towards the bottom of the picture is the wall.

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. 25 The same table at the shallowest exponent, where the pale region begins lower down.

The one line

The settling table is now grown from twenty starting angles rather than nine — eight midpoints and three below the old floor — which is 640 runs against 288 and halves the error on every cell from about 17 per cent to about 11.

The destination list goes from fifteen to eighteen with every addition off both ladders, the additive sequences from ten to thirteen, and the settling share falls at every rise for a reason that turns out to belong to the original nine.

Where this goes next

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 settling share at every rise, sampled two ways. 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. A wall that stopped moving Four falloff exponents were reported not to move the rise below which stems stop reaching a lattice. Their measured walls spanned a factor of two and ordered themselves 2, 5, 3, 4. At twenty starting angles they span a fifth of one and order themselves 5, 4, 2, 3 — so the conclusion was right and its arithmetic was noise. How many organs a stem needs before it is on a lattice. One row per rise, one mark per starting angle, placed at the organ from which every later divergence stays within a degree and a half of the run's own final value. Where a stem settles at all it does so between 0 and 290 organs in, against the 400 every ablation run here grows before it cuts anything. Not one row needs the length it is given. What changes down the table is the count on the right: how many of the nine starting angles reach a lattice at all, which falls from 7 at the coarse rises to 1 at the finest. How long a stem takes to settle Every result here is grown on a stem that has settled onto a lattice, and settling has always been tested for and never timed. Timed, it takes between nothing and two hundred and ninety organs — against the four hundred every ablation run grows before it cuts anything, and the nine hundred the noise runs carry.

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.

BasinClaim testingDivergenceFalloff exponentHonest limitsMeasurement errorNull modelResolutionSample sizeSamplingSettlingStarting angle