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

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.
13 min read 24 figures State the claim, state the testCounted, not admiredForm is not mechanism

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

The settling table was grown from nine starting angles: 40, 60, 80, 100, 120, the golden angle, 150, 165 and 180 degrees. Every share this collection reports about how often a stem reaches a lattice is a share over those nine.

Eight more were added, at the midpoints between them. The nine settle at 40.6 per cent and the eight midpoints at 25.8 per cent, over the same rises and the same falloff exponents.

That was not the finding the extra angles were grown for. They were grown to halve an error bar and to give basins a width; the bias came out of counting the two groups separately, which was done only so that a refinement could be told from an extension.

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. 1 How often each group of starting angles reaches a lattice, over the whole table.

That is not noise

Two hundred and eighty-eight runs against two hundred and fifty-six, which puts a binomial standard error of about three per cent on each share. The gap is fifteen points.

It is also not a fluke of one cell. The nine settle more often at every rise and at every one of the four falloff exponents, which is thirty-two comparisons all running the same way.

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. 2 The settling share against the rise at both samplings, where every curve moves the same way.

And it is not the range

Three more angles were added below the nine’s lowest — 10, 20 and 30 degrees — because the list started at forty for no stated reason. Those settle at 25.0 per cent.

So the three angles outside the original range behave like the eight inside it. If the fall were about extending the sampling into unfamiliar territory, the three below forty would look different from the eight midpoints, and they do not.

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. 3 The three groups again, of which the two additions agree with each other and not with the nine.

What the nine have that the midpoints do not

Several of them sit next door to a destination. The table reaches a lattice at 101.6 degrees and one of the nine is at 100; it reaches one at 79.2 and one of the nine is at 80; it reaches one at 151.0 and one of the nine is at 150.

And the golden angle is itself a destination, to within a third of a degree: runs settle at 137.8 and the ninth starting angle is 137.508.

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. 4 Each starting angle’s distance to the nearest destination the table reaches, with the nine marked.

A run that starts next to where it would settle

That is the account and it is not complicated. Settling is a stem’s arrangement stopping changing, and a stem started at an angle a lattice already sits at has almost nothing to change.

The measured settling times bear it out: the runs that settle fastest are the ones started nearest a destination, and several of the nine settle at organ zero — meaning the run was at its destination from the first organ after the seed.

Where a stem settles at all it does so within about 290 organs, so a settling time of zero is not a fast run; it is a run that had nowhere to go.

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. 5 How long a run takes to settle, which is shortest for runs that start where they will finish.

Which makes the nine a biased list

Not deliberately, and not by much of a mechanism. Round numbers in degrees are what a person types, and the golden angle is in the list because the whole subject is about it.

Both are good reasons for those angles to be in a list and neither is a reason for them to be representative. What makes them unrepresentative is that the destinations this rule produces are themselves near round numbers — 79.2, 101.6, 137.8, 151.0 — for reasons that have nothing to do with anybody’s typing.

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. two 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. 6 The destinations the table reaches, several of which sit within two degrees of a round number.

So the coincidence has to be checked

It is worth pausing on, because round numbers are near destinations could be a coincidence of four out of nine. The measurement is how many of each group sit within two degrees of a destination.

Of the nine, four do. Of the eight midpoints, one does. That is the account in one comparison, and it is thin — nine and eight are small groups — which is why the settling times are quoted beside it.

The destinations themselves are measured rather than assumed: eighteen of them at half a degree, of which five sit on a ladder sequence and thirteen do not.

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. 7 The distances again, with the two-degree line drawn on.

What it does to every share

Every settling share in this collection is about fifteen points too high, and the correction is uniform rather than selective: the shares fall at every rise and at every exponent by roughly the same amount.

That is the good case. A uniform bias cancels in any comparison — which is what the table is for — and survives only in a share quoted on its own.

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. 8 The table as a comparison across rises and exponents, in which a uniform bias cancels.

And to the one number that is a share

The wall is a share read against a level: the rise at which fewer than half the starting angles still reach a lattice. That is the one place in the thread where a share is doing work rather than being compared.

It survives, and more comfortably than before. The four exponents’ walls sit closer together at twenty angles than at nine, which is what the file’s own conclusion said they should — and the wall is a wall rather than a budget at both samplings, since tripling the run length changes no row of the table.

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. 9 The four exponents’ shares at twenty angles, with the half level the wall is read at.

Why a uniform bias does not move a level crossing much

Because the shares fall by about the same amount everywhere, and the rise at which a falling curve crosses a half moves by however far the curve shifts divided by its slope.

The curves here are steep near the wall — the share falls from 60 to 30 per cent over one step of the rise at the working exponent — so a fifteen-point shift moves the crossing by well under one rise of the eight.

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. 10 The share against the rise, whose slope near the crossing is what a bias has to be divided by.

What would not have survived

A claim of the form a stem reaches a lattice about four times in ten. That is a share quoted on its own and it is now a share over a list of angles chosen partly because they are round.

Nothing in the collection makes that claim, which is luck rather than foresight. The file that grew the table read its own error bar and confined itself to comparisons for that reason, which turns out to have protected it from a bias it had not noticed.

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. 11 The settling share, which is the number a claim like that would have been made from.

The general shape of it

A list of parameter values chosen for readability will sit near whatever the system’s own special values are, if the system’s special values are near readable numbers. That is not obvious in advance and it is not rare.

The check is cheap: place a second list between the first one’s members and compare. It costs one more run per cell and it is the only way the bias shows, since every cell of the original table is internally consistent.

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. 12 The comparison that shows it, which needed eight extra angles and no new machinery.

What a fair list would look like

Twenty angles drawn at random from the range would have no structure to be unrepresentative in, and it is not what was used. A random list cannot contain the old one, so every number would have been re-measured and the comparison would have become an argument about two designs.

The midpoint design keeps the old table inside the new one, which is what makes the nine settle more often than the eight between them a sentence about one table rather than two.

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. 13 The table with both lists in it, where the original nine and the midpoints alternate.

What the midpoints are not

They are not a control in the strict sense. Eight angles halfway between nine others are themselves a structured list, and their structure is as far from the round numbers as possible, which could bias them the other way.

That is a real worry and the three angles below forty are what answers it. They are not midpoints of anything and they settle at the same rate as the midpoints, so the midpoints are not unusually far from destinations either.

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. 14 Every angle’s distance to a destination, where the midpoints and the three below forty overlap.

The rate the two additions agree on

25.8 per cent and 25.0 per cent, from two groups constructed differently, is the best estimate this table has of what an unbiased list would give.

It is still not an estimate of the circle. Eleven angles is eleven angles, and both groups avoid the round numbers by construction, so the honest reading is that the true rate is somewhere at or below the nine’s forty per cent and near the twenty-five the others give.

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. 15 The share with the error a sample of this size carries, which bounds how precise any of this can be.

What this says about the angle itself

There is a temptation to read the golden angle is a destination as a result and it is not one — it is what the whole settling thread found, and the angle is an output of the rule rather than an input to it.

What is new here is narrower: because the golden angle is a destination, a table that includes it among its starting angles is including one starting angle that cannot fail to settle. That is a sampling remark rather than a claim about the angle.

The same distinction has to be kept when reading what a counter says about the settled runs: a destination is where runs go, and a starting angle at a destination is a run that was already there.

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 on a ladder sequence, one of which is the angle the table starts a run from.

And about the ones that never settle

The other end of the table is worth a sentence. At the finest rises almost nothing settles at any starting angle — one of twenty at a rise of 0.004 — so the bias has nothing to work on there.

That is the wall, it is a property of the rise rather than of the sampling, and it is the part of the table the extra angles leave exactly as it was.

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. 17 Both samplings across the rises, converging at the fine end where nothing settles.

Why this is a selection effect and not an error

Nothing was measured wrongly. Every one of the 288 original runs is a correct run, every share is a correct share of the angles it is a share of, and no number has to be withdrawn.

What is wrong is a reading: this share is about the rule when it is about the rule and a list. That is a selection effect, and selection effects are the class of problem that survives every check on the measurement itself, because the measurement is fine.

The collection has met one before. An ordering in a residual belonged to the method that measured it rather than to the heads it was measured on, and it too passed every check that asked whether the numbers were right.

Both vary; only one of them varies enough to find. Each organ's step exponents, divided by its own mean so the two are comparable. The ogive's run over 15 per cent of their mean across 5 rings. The convex head's run over 1.15 per cent across 5 — inside the band a 10 per cent error on each ring position leaves, so no ruler separates it from a flat disc.
Fig. 18 What a measurement can and cannot report about itself, which is the class this belongs to.

What made it visible

One design decision, and it was made for a different reason. The eleven new angles were split into eight midpoints and three below the old floor because a refinement and an extension answer different questions, and mixing them would have made it impossible to say which had moved a share.

Had all eleven been counted together, the answer would have been the share falls by eight points when eleven angles are added, which reads as an ordinary sampling correction and prompts nothing.

Splitting them turned one number into three, and the three do not agree with each other in the way an ordinary correction would.

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. 19 The three groups, whose disagreement is what a single combined number would have hidden.

What else is grown from a chosen list

Worth asking, since the same shape could be elsewhere. The rises the table is grown at — 0.030, 0.020, 0.013, 0.008, 0.005, 0.0045, 0.004, 0.003 — are also round numbers, and the ladder’s rungs sit at particular rises.

That list is not obviously biased in the same way, because a rise is not a thing a run settles onto: the rungs are stretches rather than points, and a rise near a rung boundary is not a rise where anything special happens to a run. But nobody has checked it by putting eight rises between these eight, which would cost the same twelve minutes.

Which family survives, along the 5/8 rung. The family a wrecked stem keeps, at every offset that wrecks and every rise on one rung. A counter shown any of these stems returns 5 and 8 spirals, at all twelve rises, so nothing in the pair distinguishes the columns. The cells say otherwise: at an offset of 4 the stem keeps the 5-family at the coarse end of the rung and the other one at the fine end. The offsets that wreck at all grow from 1 to 5 as the rise falls, because the front deepens, and the extra offsets are the ones that keep the larger number. So the rule stated over the offset alone is a rule with the rise left out of it.
Fig. 20 Where the table’s rises sit against the ladder’s rungs, which is the other chosen list in it.

The three that agree, and what they are worth

Two independent constructions gave 25.8 and 25.0 per cent. That agreement is the strongest thing in this reading and it is worth saying why rather than leaving it as a coincidence.

The eight midpoints were built to be as far from the nine as possible; the three below forty were built to extend a range and have no relation to the nine’s spacing at all. Two lists with different construction rules and no members in common landing within a point of each other is what a shared property looks like.

The property they share is that neither is near a destination. One of the eight is within two degrees of one and none of the three is, against four of the nine.

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 distances the three groups sit at, where the two additions overlap and the nine do not.

What it would take to be sure

Forty angles, evenly spaced, with the nine somewhere inside them. That is another 640 runs and about twenty-five minutes and it would give the share over a list with no structure worth speaking of.

It is not obviously worth doing, because the share is not a number this collection uses. What it would settle is whether 25 per cent is the rate or whether the two additions are also unrepresentative in some way nobody has thought of, and the answer to that changes nothing that is currently claimed.

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. 22 The table at the steepest exponent, where a further refinement would be read the same way.

What a reader should carry

That the nine starting angles the settling table was grown from settle half as often again as angles placed between them, and that four of the nine sit within two degrees of somewhere a stem could settle.

And that this cost nothing to find. Eight angles, three hundred and fifty-two runs, twelve minutes — and the check is the same one that has caught every instrument problem in this collection: put a second value beside the first and compare.

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. 23 The three groups and their shares, which is the whole finding in three numbers.

What the picture at the top shows

Three bars. The top one is the nine starting angles the table was grown from and reaches 40.6 per cent. The second is the eight midpoints between them, at 25.8. The third is the three angles below the nine’s lowest, at 25.0.

The counts beside each bar are how many runs of how many settled. The nine and the midpoints cover the same rises at the same exponents, so nothing but the angles differs between the top two bars.

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. 24 Why the top bar is higher: several of the nine start beside a destination and the others do not.

The one line

The nine starting angles the settling table was grown from reach a lattice 40.6 per cent of the time; eight angles placed halfway between them reach one 25.8 per cent of the time, and three below the old floor 25.0 per cent.

Four of the nine sit within two degrees of a destination and one of the eight does — so the settling share this collection reports is biased upwards by the choice of angles, uniformly enough that every comparison built on it survives.

Where this goes next

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. 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. 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. A basin has a width A destination reached from one starting angle is a presence. A destination reached from seven consecutive starting angles spanning forty-five degrees is a basin with an extent, and nine angles could not have measured one — they were too far apart to have two of them land in the same place. The same angle, two rungs, two answers. Each block is one matched pair: two rises whose stems settle on the same divergence and whose counters return different pairs. Under each is the family every wrecked cut leaves standing. On three of the four pairs both rises wreck at some offset, and on every one of those the two stems keep different families — so the divergence, which is held, is not what decides the survivor. The two stems keep exactly the counted numbers their two pairs share, including the pair that shares none and keeps none. The angle is not the actor Cut an organ out of two stems that settled on the same divergence and return different counted pairs, and the family left standing is different at every one of the four pairs where both stems wreck. The angle is held to a hundredth of a degree underneath.

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.

ArtefactBasinClaim testingDivergenceHonest limitsMeasurement errorNull modelSample sizeSamplingSelection effectSettlingStarting angle