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

Forty angles, and a limit

Nine starting angles turned out to be a biased sample of the circle, and doubling to twenty said by how much. Doubling again says the estimate is converging — to a smaller correction than one doubling extrapolated to.
14 min read 21 figures State the claim, state the testCounted, not admiredOne parameter, one family

Worth reading first: How long a stem takes to settle.

The settling table grows stems from a set of starting angles at four falloff exponents and eight rises, and records how often the divergence stops changing. Every claim this collection makes about where a stem ends up rests on it.

It was grown from nine angles for several rounds. Adding eleven more showed the nine were not a fair sample of the circle: four of them sit within two degrees of a destination the table reaches, so they settle easily, and every settling share was biased upwards.

How often a stem settles, at three samplings of the starting angle. The share of runs that reach a lattice, over the whole table of four falloff exponents by eight rises. The upper three bars are the pooled share at nine, twenty and forty starting angles; the lower three are the share for each group of angles on its own. The pooled figure falls 40.6 to 32.3 to 30.0 per cent, by 8.3 points and then 2.3, so it is converging. The eleven angles added at twenty and the twenty added at forty settle at 25.6 and 27.7 per cent, which differ by less than their own error.
Fig. 1 The settling share at nine, twenty and forty starting angles, with each group of added angles beside it.

What one doubling cannot say

Whether an estimate is converging. Nine angles gave 40.6 per cent; twenty gave 32.3. The difference is 8.3 points and there is no way to tell from two numbers whether the next doubling takes another 8 points off or one.

The previous round’s own summary said about fifteen points high, which is an extrapolation from one step and was labelled as one. This round is the test of it.

The design is the same as before: the nineteen midpoints of the twenty, plus one angle below the lowest, so the forty contain the twenty and the old table is a sub-table rather than a rival. Spacing halves to 3.1 to 5 degrees.

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. 2 Where each starting angle’s run ends up, at the sampling the previous round established.

The cost

Four exponents by eight rises by forty angles is 1,280 runs, of which 640 are new. Twenty-two minutes.

Each run is a stem grown to twelve hundred organs with a placement rule that puts each organ at the minimum of a sum over its neighbours across 1,536 candidate azimuths, so a run is not cheap and the table is most of what this thread costs.

Everything downstream — destinations, arrangements, basins, the wall — is arithmetic on the same 1,280 runs.

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. 3 One run of the table, grown until its divergence stops changing or until it does not.

The answer

The pooled settling share goes 40.6 per cent at nine angles, 32.3 at twenty, 30.0 at forty. It falls by 8.3 points and then by 2.3.

So it is converging, and to a correction of about ten and a half points against the whole table rather than the fifteen one doubling suggested.

The twenty angles added this round settle at 27.7 per cent on their own. Against that the nine are 13.0 points high, which is the sharper of the two ways to state it.

How often a stem settles, at three samplings of the starting angle. The share of runs that reach a lattice, over the whole table of four falloff exponents by eight rises. The upper three bars are the pooled share at nine, twenty and forty starting angles; the lower three are the share for each group of angles on its own. The pooled figure falls 40.6 to 32.3 to 30.0 per cent, by 8.3 points and then 2.3, so it is converging. The eleven angles added at twenty and the twenty added at forty settle at 25.6 and 27.7 per cent, which differ by less than their own error.
Fig. 4 The pooled share at each sampling, falling by eight points and then by two.

The two refinements agree

The eleven angles added at twenty settle at 25.6 per cent. The twenty added at forty settle at 27.7. The difference is 2.1 points against a standard error of 3.3, so it is nothing.

That is the check that matters and it goes the right way. A story in which the nine are high because they are round numbers sitting beside destinations predicts every later angle to be an ordinary angle, and predicts the two refinements to agree.

A story in which the share simply falls as the sampling gets finer — because finer sampling finds more of the circle that does not settle — predicts the second refinement to settle lower than the first. It does not.

How often a stem settles, at three samplings of the starting angle. The share of runs that reach a lattice, over the whole table of four falloff exponents by eight rises. The upper three bars are the pooled share at nine, twenty and forty starting angles; the lower three are the share for each group of angles on its own. The pooled figure falls 40.6 to 32.3 to 30.0 per cent, by 8.3 points and then 2.3, so it is converging. The eleven angles added at twenty and the twenty added at forty settle at 25.6 and 27.7 per cent, which differ by less than their own error.
Fig. 5 The two refinements’ own shares, which agree to within their error.

Why that distinction matters

Because the first story bounds the correction and the second does not. If the nine are special, the true share is whatever the ordinary angles give — about 27 or 28 per cent — and further sampling will not move it.

If the share falls with resolution, there is no true value to converge to and every number in the thread is a function of how finely somebody sampled.

The two refinements agreeing, and the pooled share’s steps falling from 8.3 to 2.3, both point at the first. That is two independent readings of the same table agreeing, which is worth more than either.

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 Each group of starting angles’ own settling share, which is where the two stories differ.

What is special about the nine

They are 40, 60, 80, 100, 120, the golden angle, 150, 165 and 180 degrees. Seven of the nine are multiples of ten or fifteen and one is a constant of the subject.

Four of them sit within two degrees of a destination the table reaches: 100 beside 101.6, 80 beside 79.2, 150 beside 151.0, and the golden angle is the destination at 137.8 to within a third of a degree.

A run that starts next to where it would settle settles quickly and easily. So the nine are not a random sample of the circle; they are a list of round numbers, and round numbers are where the interesting values are because the interesting values were named by people who round.

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. 7 The destinations the table reaches, four of which sit within two degrees of one of the original nine angles.

The one angle below the range

Five degrees, which continues the spacing at the bottom by halving it. It settles at 31.3 per cent over its thirty-two runs, which is above the pooled figure and well inside the error on thirty-two runs.

The previous round added three angles below forty — 10, 20 and 30 — and they settled at 25.0 per cent, which is the refinement’s rate rather than anything special. Adding a fourth confirms that: extending the range downwards finds nothing.

That is a small negative and it is worth one sentence. The nine started at forty degrees and nothing said why; the answer is that it did not matter.

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. 8 The starting angles including those below the original range, which behave like every other angle.

What the error does

Halves as it should. The standard error on a pooled column goes 5.8 per cent at nine angles, 3.8 at twenty, 2.6 at forty, which is the square-root behaviour a binomial sample has.

So the sampling is doing what a sampling does, and the numbers it produces are getting more precise at the rate arithmetic predicts.

That matters because the next essay is about a quantity where it does not help: the wall, which does not settle down as the error falls. Establishing that the error behaves normally is what makes that finding a statement about the wall rather than about the sampling.

How often a stem settles, at three samplings of the starting angle. The share of runs that reach a lattice, over the whole table of four falloff exponents by eight rises. The upper three bars are the pooled share at nine, twenty and forty starting angles; the lower three are the share for each group of angles on its own. The pooled figure falls 40.6 to 32.3 to 30.0 per cent, by 8.3 points and then 2.3, so it is converging. The eleven angles added at twenty and the twenty added at forty settle at 25.6 and 27.7 per cent, which differ by less than their own error.
Fig. 9 Each share with one standard error drawn, which halves as the sampling doubles.

What the corrected share is

About 28 per cent, if the ordinary angles are the honest estimate, or 30 if the pooled figure over the whole forty is taken.

The difference between those two is the question of whether to keep the nine in. They are in, because dropping them would make the forty not contain the twenty and would turn every comparison into an argument about two designs.

So the number to quote is 30.0 per cent with the note that the nine inflate it by about two points, and the number to use for anything about ordinary angles is 27.7.

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 settling share read against the rise, which is the shape every claim in this thread uses.

What the share is a share of

Runs, not angles. Each cell of the table is one exponent, one rise and one angle, and the share is over all cells.

That matters because the share depends strongly on the rise: at 0.03 it is around a half to three quarters and at 0.003 it is under a tenth. So the pooled figure is an average over a range that was chosen for coverage.

Any claim about how often a stem settles is therefore a claim about this table’s own range of rises. What the pooled figure is good for is comparisons — between exponents, between samplings, between groups of angles — and those are what it is used for.

The same table, grown 2.7 times as long. Every rise and every starting angle, grown to 1200 organs and then to 3200. The two middle columns are how many starting angles reached a lattice at each length, and they are the same column: of the 72 pairs of runs, 72 are identical organ for organ and 0 settle at the longer length after failing at the shorter one. Tripling the budget buys nothing anywhere. What the fine rises are short of is not run length: the share of starting angles that reach a lattice at all falls from 7 of 9 to 1, so the arrangements a stem could fall into have mostly stopped existing.
Fig. 11 The share against the rise, which falls by a factor of five across the range the table covers.

What settles and what does not

A run has settled when every later divergence stays within a tolerance of the mean over its last stretch. About seventy per cent do not, which sounds like a failure and is a measurement.

Below a rise of about 0.011 most starting angles never settle at all, and that is a wall rather than a budget — tripling the run length does not move it. A run that does not settle in twelve hundred organs does not settle in three thousand two hundred either.

So the seventy per cent are runs that have somewhere to be and never get there, not runs that were stopped too early.

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. 12 How long a settling run takes, against which the runs that do not settle are not merely slow.

What the refinement does not change

The comparisons. Every claim this thread makes that compares two things — the four exponents against each other, one rise against another, a steep falloff against a shallow one — is between two columns that are equally biased.

The bias is uniform because it is carried by the nine angles and those nine are in every column. So a share that is fifteen points high in one column and fifteen high in another supports the same comparison as two unbiased shares would.

That is why the previous round’s conclusions survived the first doubling and why this one changes no comparison either.

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. 13 The four exponents’ columns, every one of which carries the same nine angles and the same bias.

The refusal

A sampling this file compares against must contain the one before it. A list of forty angles evenly spread over the same range would have more angles in it and would not be a refinement of the twenty, so every difference between the two would be an argument about two designs rather than a measurement.

That is checked rather than assumed: such a list is constructed, shown not to contain the twenty, and the comparison is refused.

It is the same rule the slot table followed when it grew from six lattices to twenty-four, and the extended census when it added a lattice rather than replacing a list.

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. 14 The forty angles with the twenty inside them, which is what makes the comparison an addition.

How the midpoints were chosen

By halving. The twenty sorted are 10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 110, 120, 128.75, the golden angle, 143.75, 150, 157.5, 165, 172.5 and 180, and the nineteen midpoints of that list are the nineteen angles added.

That produces a spacing of 3.12 to 5 degrees, uneven because the twenty were themselves uneven — the golden angle sits at 137.5077 and the midpoints either side of it inherit its awkwardness.

The alternative would have been forty evenly spaced angles, which is tidier and is exactly the comparison the refusal above forbids. An even list would not contain the twenty and could not say what changed.

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. 15 The forty angles as sampled, uneven because they are the twenty and their own midpoints.

What a settling share is used for

Three things, and they need different amounts of precision.

The wall — the rise at which half the angles stop reaching a lattice — needs the share to be precise near a half, which is where it is least precise. Destination counting needs only that a run settled at all, so the share is a denominator rather than a measurement. Comparisons between exponents need the bias to be uniform, which it is.

So the correction this essay reports matters most to the first of the three and least to the second, and it turns out to matter to the first in a way the error bar does not capture.

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. 16 The table the share is read from, whose three uses need different amounts of precision.

What the nine were chosen for

Coverage, when nothing was known. Nine angles from 40 to 180 degrees at roughly twenty-degree spacing is a reasonable first sampling of a circle whose interesting region was not yet known, and the round that chose them said nothing about why forty was the floor.

There is no criticism in that. A first sampling is chosen before the answer and every later one is chosen after, which is why the later ones are refinements and the first is a guess.

The mistake, if there is one, is having quoted shares from it for several rounds without varying it. That is the fourth instrument setting on this site to have been varied and the fourth to have decided something.

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. 17 The original nine against every angle added since, which is the whole of what varying the setting bought.

A setting that has not been varied

The run length. Every run in this table is twelve hundred organs, and a run that has not settled by then is recorded as not settling.

That one has been checked, on a smaller table: tripling it to three thousand two hundred changes not one row at exponent three, and the check was repeated at the other exponents where it could bite. So the share is a wall rather than a budget.

Which leaves the tolerance — how close later divergences must stay to count as settled — as the setting nobody has moved. It is stated as a gap rather than a threshold, and nothing has tested whether it is one.

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. 18 The check that the run length is not the constraint, run at three thousand two hundred organs.

What the destinations do

The list of places a settled run ends up goes from fifteen at nine angles to eighteen at twenty to nineteen at forty. So the second doubling adds one where the first added three, which is the same converging shape the share shows.

The count of destinations sitting on the golden or Lucas sequence is five at every sampling, over a fourfold refinement. So the rule’s on-ladder targets were enumerated by the first nine runs and everything later sampling finds is off both ladders.

That is a separate reading with its own surprise, and it is mentioned here because it is the second quantity in this table that converges rather than drifting.

Destinations found at nine, twenty and forty starting angles. How many distinct divergences the settling table reaches at each sampling, split into those sitting on the golden or Lucas sequence and those off both. The on-ladder count is 5 at every sampling, over a fourfold refinement of the starting angle — so the rule's on-ladder targets were enumerated by the first nine runs. The off-ladder count goes 10, 13, 14, and the one the last doubling found is off both ladders.
Fig. 19 The destination list at three samplings, growing by three and then by one.

And what the basins do

A basin here is a run of consecutive starting angles that all reach one destination, and its width is a lower bound in both directions because the true edges lie in the gaps either side.

Halving the spacing doubles the number of angles in the widest basin — seven to fourteen — and leaves its width where it was, at 45 degrees against 47.5. That is what a real basin does and what a run of angles produced by the sampling does not.

So the same doubling that shrinks the share’s correction confirms the widest basin outright, which is the one clean positive this refinement buys.

The widest basin, at twenty starting angles and at forty. The starting angles of one cell of the table, drawn as ticks, with the run of consecutive angles that all reach one destination filled. At twenty angles it holds 7 angles and spans 43.75 degrees; at forty it holds 14 and spans 47.5. Halving the spacing doubled the count and left the width where it was, which is what a real basin does and what a run of angles produced by the sampling does not. The other half of the same table goes the other way: 170 of 233 runs are a single angle, against 116 of 151.
Fig. 20 The widest basin at both samplings: twice the angles, the same width.

What a doubling costs and what it buys

Twenty-two minutes and 640 new runs, against the same again for the previous doubling. So four samplings of the starting angle have cost about three quarters of an hour in total, and the table they produced is what every claim in this thread about destinations, basins and walls rests on.

What the second doubling bought, in order of value: the knowledge that the settling share converges rather than drifting, which is the difference between a bias with a size and a number that means nothing; a basin confirmed by a test it could have failed; a destination list whose on-ladder half is closed and whose off-ladder half is not; a steep-only list shown to be an upper bound by losing two members; and a wall shown never to have been located.

Five results from one afternoon of machine time is a good return, and four of the five are boundaries rather than discoveries. That is the ordinary shape of a round that re-samples something instead of measuring something new.

What a third doubling would buy

Eighty angles, another 1,280 runs, about three quarters of an hour. The predictions are worth writing down before anyone spends it.

The pooled settling share should fall by about a point, to somewhere near 29 per cent, and the increment should be smaller than this round’s 2.3. The basin count should grow to about 450, because it grows with the sampling and will go on doing so until the spacing is finer than a typical basin. One or two more destinations should appear, all of them off both ladders. The steep-only list should lose another member or two. And the wall should not settle down, because nothing about eighty angles makes a flat curve steeper.

If any of those comes out otherwise, the thing that came out otherwise is the interesting part. That is what a prediction written in advance is for, and it costs nothing to write.

The one number that did not move

The count of destinations sitting on the golden or Lucas sequence: five at nine angles, five at twenty, five at forty. Over a fourfold refinement of the sampling, not one on-ladder destination appeared that the first nine runs had missed.

That is the most stable quantity this table has produced, and it is worth more than the shares. It says the arrangements the whole subject is about are few, and that the rule falls into them from almost anywhere — while the off-ladder arrangements it also reaches are many and are found only from particular angles.

What is claimed

That the pooled settling share over the table is 40.6 per cent at nine starting angles, 32.3 at twenty and 30.0 at forty, falling by 8.3 points and then by 2.3, so the estimate is converging.

That the eleven angles added at twenty settle at 25.6 per cent and the twenty added at forty at 27.7, differing by 2.1 points against a standard error of 3.3 — so the bias belongs to the original nine rather than to the resolution.

And that the nine are therefore about 13.0 points high against ordinary angles and 10.6 against the whole table, where one doubling had extrapolated to fifteen.

How often a stem settles, at three samplings of the starting angle. The share of runs that reach a lattice, over the whole table of four falloff exponents by eight rises. The upper three bars are the pooled share at nine, twenty and forty starting angles; the lower three are the share for each group of angles on its own. The pooled figure falls 40.6 to 32.3 to 30.0 per cent, by 8.3 points and then 2.3, so it is converging. The eleven angles added at twenty and the twenty added at forty settle at 25.6 and 27.7 per cent, which differ by less than their own error.
Fig. 21 Three samplings of the starting angle, and what each doubling of them moved.

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. 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 widest basin, at twenty starting angles and at forty. The starting angles of one cell of the table, drawn as ticks, with the run of consecutive angles that all reach one destination filled. At twenty angles it holds 7 angles and spans 43.75 degrees; at forty it holds 14 and spans 47.5. Halving the spacing doubled the count and left the width where it was, which is what a real basin does and what a run of angles produced by the sampling does not. The other half of the same table goes the other way: 170 of 233 runs are a single angle, against 116 of 151. A basin that doubled A run of consecutive starting angles reaching one destination is a basin, and its width is a lower bound. Halving the spacing doubled the angles in the widest one and left its width alone, which is what a real basin does and a sampling artefact does not.

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.

BasinBiasClaim testingConvergenceHonest limitsInstrument settingMeasurement errorSample sizeSamplingSelection effectSettlingStarting angle