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

One rise per rung is a sample

Every census on this site takes one rise from each rung, because the question was always which pair. Any rule later scored on those rows inherits a variable that was never varied — and two of this collection's results turn out to be about the sampling as much as about the rule.
12 min read 24 figures State the claim, state the testCounted, not admired

Worth reading first: The survey this site cannot do · Counting the spirals · A head is a set of points.

Every census on this site is built the same way. Choose a set of lattices that spans the pairs and the branches worth covering, find a rise that produces each pair, grow the stem, do the experiment, record the row. It is a sensible design and it has answered the question it was designed for many times.

It also has a property nobody stated: within such a census, the counted pair and everything correlated with the rise move together. One rise per pair means one settled divergence per pair, one step ordering per pair, one front depth per pair. Any rule later scored across those rows inherits all of it.

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. 1 What a census cannot see: one rung, swept at a thousandth, with a row whose answer changes along it. A census takes one column of this and calls it the rung.

This essay is about what that costs, using two results from this collection where the cost has now been measured.

It is worth being clear at the outset that this is not a general complaint about sampling. Every experiment holds something fixed, and holding things fixed is what makes a comparison a comparison. The specific problem here is narrower and has a name: the quantity held fixed is not independent of the quantity varied. Pair and rise are chosen together, so a census that varies the pair across its rows is also varying the rise across them, in a pattern nobody selected and nobody recorded. Any reading sensitive to the rise is then being scored against a confound rather than against a control.

The mechanism

A rung is a range of rises over which a counter returns one pair. Inside it, the settled divergence slides, the two contact steps can change places, and the front deepens. None of that is visible to a counter, which is what makes it a rung.

The divergence slides along the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not.
Fig. 2 The divergence sliding across one rung, over a range where every stem returns the same pair.
The two contact steps change places inside the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not.
Fig. 3 And the two contact steps changing places inside the same range, which is the ordering an entire earlier reading was about.

So when a census picks one rise per rung, it is not picking a representative point on a plateau. It is picking one point from a range across which several quantities vary, and fixing all of them at once by that single choice.

The word plateau is doing the damage, and it is worth replacing. A rung is a plateau in the counted pair and in nothing else — the geometry underneath slides continuously through it, and the pair is a step function laid over a smooth one. Reading “rung” as “a set of equivalent stems” is the mistake, and the vocabulary invites it. The finest sweep this collection has run puts twenty-three rises inside one rung and finds the divergence moving through all of them without a single flat stretch.

How many offsets wreck, along the 5/8 rung. The count of offsets that never repair, at each rise on one rung. It runs from 1 at the coarse end to 5 at the fine end, while a counter returns 5 and 8 spirals at every one of them. The front — the run of recent organs at which a removal is felt at all — deepens as the rise falls, so there are simply more places a cut can land and fail to heal. That is the mechanism under the grid: the offsets that appear at the fine end are the ones beyond the smaller contact number, and those are the ones that keep the larger family.
Fig. 4 The third quantity the choice fixes: how many offsets are deep enough to wreck at all.

The consequence is precise. Any reading whose truth depends on one of those quantities will be scored, across the whole census, at one value of it per pair — and its score will be a joint statement about the reading and about which rises were chosen.

The size of the effect is not small. The 5/8 rung alone spans rises from eighteen thousandths to seven, over which the settled divergence moves by more than a degree, the step ordering reverses, and the number of wrecking offsets grows five-fold. A census samples one point from that. Two censuses built by different people, both entirely defensible, could pick 0.016 and 0.008 and disagree about a reading’s score without either being wrong about anything.

Which lag survives, at every lattice and every offset. A row for each of twelve lattices and a column for each offset an organ is removed at, with the lag whose hop survived written in the cell. 30 cells never repair. The lag left standing is one of the two contact families at 29 of them, and at 17 it is the second shortest step on the lattice rather than the shortest, which is what refuses the reading that a rule holds its nearest neighbours. The ringed cells are the five the offset rule gets wrong. Two lattices carry no cells at all because no single removal wrecks them.
Fig. 5 The census in question: one row per lattice, one rise per row, and thirty wrecked offsets scored as though they were thirty comparable tests.

Two results that moved

The shortest hop. Scored across the census, the reading that a wrecked stem keeps its shortest hop is right at twelve of twenty-nine — a refutation. Scored along one rung, where the pair is held and the ordering reverses, it is right at sixteen of thirty-one, which is a coin flip. The reading is dead either way; the number was reporting the sampling.

Which offsets give short hops, at a rise of 0.018. The two lowest points are at 5 and 8, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.
Fig. 6 The step ordering at one end of a rung, where the smaller family carries the shorter step.
Which offsets give short hops, at a rise of 0.008. The two lowest points are at 5 and 8, and those are the parastichy numbers. Offset 1 is high because a hop of one node is at least the rise, which is what makes a stem easier to count than a disc.
Fig. 7 And at the other end of the same rung, with the ordering reversed and the pair unchanged.

The offset rule. Scored across the census it is right at twenty-five of thirty, and its two clauses look like halves of one statement. That reading also had a single anomaly — one pair of runs agreeing on pair and offset and disagreeing on the answer — which is exactly what a confound produces when it is almost but not quite hidden. Sweeping the confounded variable turned the anomaly into a reproducible effect, which is the usual fate of a lone outlier in a census with a variable nobody moved. Swept along a rung they come apart: the first clause applies at every rise, the second has nothing to apply to until the front is deep enough, and one offset changes its answer between the coarse end of a rung and the fine one.

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. 8 The same rung without the marking, so the columns read as the sweep the census took one of.
A prediction and its opposite, scored on the same rows. The reading under test said the family whose member was removed is the one that breaks. Scored across every wrecked offset where the removed organ lies on exactly one contact chain — 9 of 30, the other 21 being silent because the organ lies on neither — it is right no times and its opposite is right nine. A chain that loses a member does not stop existing: the organs above the hole are still spaced at that lag and the rule that placed them is still minimising the same sum, while the other chain has lost the organ its members were positioned against.
Fig. 9 And the reading that replaced it, scored on rows selected by divisibility rather than by rise — which is why the sampling does not reach it.

What this is not

It is not a claim that the censuses were badly designed, and the distinction matters because the fix is different in the two cases.

A badly designed census would have chosen rises that produced the answer. These were chosen to produce the pairs, before any of the readings existed, and the rises attached were whatever the pair required. Nothing about that is careless, and the ordering matters: the census predates every reading scored on it, so there is no possibility of the rises having been selected, consciously or otherwise, to favour one. What is wrong is not the choosing but the reuse — a table built to answer one question, later asked a different one it was never arranged to answer.

The band that never heals is what two fixed edges leave over. Each row is one rung. A stem is grown to 400 organs, one organ is removed, and the stem is followed for 300 more. A filled mark is an offset the stem recovers from, with the number of organs it took printed above it; an open ring is an offset it never recovers from. At the 3/5 rung, a front of 5, every offset heals. At the 5/8 rung, a front of 8, the offsets 4, 5 never heal. At the 8/13 rung, a front of 13, the offsets 4, 6, 7, 8, 9 never heal. What is the same at every rung is the two edges: the first three offsets heal, and so do the last few of the front. What is left in the middle is the band, and it widens because the front does.
Fig. 10 The property the censuses were designed around: whether a stem is wreckable at all, which is a fact about where it sits.
The same rule, the same rise, two lattices, two fronts. How many organs back a single removal is still felt, on a stem seeded onto the golden lattice and on one seeded onto the Lucas lattice, at seven rises. Everything but the seed is identical at each rise — the rule, the spacing, the heights, the azimuth grid — and the two fronts differ at every one of them. Which branch has the wider front changes hands four times going down the range, so no function of the rise gives the column. The golden branch carries 5/8 at four of these rises, across a factor of two in the rise, and its front is eight at all four.
Fig. 11 And the coverage they were designed for: two branches, with fronts of different depth at the same rise.

The general form is that a sample assembled to vary one quantity holds others fixed by construction, and a rule fitted afterwards inherits the constancy as an untested assumption. The census is not wrong; it is answering a different question from the one later asked of it.

Every open question here needs under 28 specimens. The sample size at which each comparison reaches 80 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives.
Fig. 12 The version of this the collection already had, for claims about plants: how many independent cases a difference of a stated size needs.

What to do instead

Not build bigger censuses. A census twice the size, still at one rise per rung, has the same defect and twice the confidence in it.

Agreement between two windows happens only on a slow enough shoot. Five stems at each of four rates and five disturbances, each read through two overlapping windows of 250 internodes. A filled mark is agreement — both windows reported the same pair; a half mark is a disagreement; a small mark is one window reporting and one refusing; an open mark is silence. Agreement appears 0 times in 25, 1 times in 25, 14 times in 25, 14 times in 25 at 130, 250, 400, 700 nodes per rung, and the two rates it is almost absent from are the two at which a rung is no longer than the window.
Fig. 13 The shape a scored comparison should have: several stated readings, one table, and every score reported rather than the winner alone.

The fix is to notice which quantity a reading depends on and vary that on purpose. For a reading about step lengths, sweep the rung until the ordering reverses. For a reading about offsets, sweep until the front changes depth. For a reading about contact scales, sweep the rise until the contact numbers move.

There is a cheaper half-measure that is worth naming because it is often enough. A census does not have to become a sweep to stop being confounded: it only has to sample two rises per rung rather than one, at opposite ends. That doubles the cost, breaks the correlation between pair and rise, and turns any reading whose score differs between the two into a reading that has announced its own dependence. It would not have found the row that changes hands — that took twelve rises — but it would have found that something was wrong.

The deeper rule against the disturbance's memory, at three contact scales. How many of six seeds agree that the deeper rule passed more drift, swept across the correlation length of the disturbance, at three rises. The rule, the amplitude and the run length are identical in every panel; only the rise differs, and with it the contact numbers — 3 and 5, then 5 and 8, then 8 and 13. The shapes are not the same: 3/5 is crossing, 5/8 is no corner, 8/13 is crossing. A corner that sat at a fixed number of organs would look the same in all three, and it does not.
Fig. 14 The same discipline applied in a different thread: the one knob that moves the contact numbers while leaving the rule and the amplitude alone.
Where the deeper rule changes hands, and where it never does. One row per rise, with the contact numbers on the left and what the comparison does on the right. At the coarsest the deeper rule wins at both ends and loses in the middle; at the middle rise it wins throughout; at the finest it crosses once, from losing to winning. The contact scale runs over a factor of 2.6 across these three rises and the behaviour is not a translation of one curve — it is three different curves. That refutes a corner fixed at a short correlation, and it does not by itself establish one that tracks the contacts.
Fig. 15 And what it found — three rises, three shapes, where a fixed corner was expected.

That is more expensive per reading and much cheaper per conclusion, because a reading scored over its own variable either survives or does not, and either way the number means something.

The cost is real and should not be understated. A rung sweep at a thousandth is twelve settled stems plus a cut stem per offset per rise, each grown to settle again and compared against a control — where a census row is one settled stem and its cuts. Sweeping one rung costs roughly what a fifth of a census costs, and there are more rungs than the census has rows. That is why the answer is not “sweep everything” but “sweep the quantity the reading is about”, which is one rung per reading rather than every rung for every reading.

A window that fits inside a rung. Stems that climb the ladder at four rates, read over a window at the fine end. The condition is a ratio: the window has to be shorter than a rung. 250 internodes at 260 per rung is 0.96 rungs and agrees on 3 of 3; 400 internodes at 260 per rung is 1.54 rungs and agrees on 1 of 3; 250 internodes at 520 per rung is 0.48 rungs and agrees on 2 of 3; 400 internodes at 520 per rung is 0.77 rungs and agrees on 3 of 3. Read over the whole stem instead, every rate returns nothing — 0 of 3, 0 of 3 — because the quantity the comb is periodic in changes as the pattern climbs.
Fig. 16 The general form of reading anything inside one rung, where the pair is fixed and everything else is not.

How to tell whether a reading is exposed

There is a test, and it takes a minute rather than a sweep.

Write the reading down and ask what it would have to be given to be evaluated on a single stem. If the answer mentions only the counted pair, the offset, or which family is which, the reading is safe: those are all things a census row records and the rise cannot alter. If the answer mentions a length, a distance, a divergence, a depth, or anything measured in organs or degrees, the reading is exposed, because every one of those moves inside a rung.

By that test the shortest-hop reading was exposed from the day it was written — it is stated in step lengths — and the offset rule is exposed through its second clause, which is stated over a front. The reading about which family lost a member is safe, because divisibility of an offset by a counted number involves nothing the rise can touch.

The offsets where the removed organ belonged to one family. Each row is a stem that never repaired, with the family of the organ that was taken and the family that survived. An organ five places back on a stem counted at 5 and 8 spirals lies on the tip's five-chain, so the question can be asked there; an organ four places back lies on neither chain and it cannot. Of 30 wrecked offsets in the census, 9 remove a member of exactly one family and 21 remove a member of neither. On every one of the 9 the family that lost a member is the family left standing, which is the opposite of what the reading predicted.
Fig. 17 A reading the sampling cannot reach: its rows are selected by arithmetic on the pair, not by any quantity the rise moves.

The test is crude and it will occasionally flag something safe. It costs nothing and it would have flagged both of the results this essay is about.

The crude test says which readings could be exposed. There is a cheaper way to find out which ones actually are, and it needs no new stems.

Every row of every census here was grown at a stated rise, and a swept rung has stated ends. So each row has a number nobody has computed: where inside its own rung that rise sat, as a fraction — nought at the coarse end, one at the fine end. The 5/8 rows in the ablation census sit at 0.013 on a rung running from 0.018 to 0.007, a little under half way along. Nothing there is new information. It is arithmetic on numbers already recorded, and it turns a census that varies one quantity into a census that varies two.

With that column in hand, every reading already scored can be re-scored against it. If a reading’s failures cluster at one end of the rungs, it is exposed, and the column says in which direction. If its failures are scattered across the fraction, then the sampling is not what is wrong with it. Neither answer needs a single new stem, and the second is worth the more of the two, because it turns a caveat that currently applies to four censuses into one that applies to whichever of them the existing rows say it applies to.

The reason it was not available before is that a rung had no ends. A rung was a label attached to a pair, and a label has no inside; only sweeping one at a thousandth produced the two rises that bound it, and only then did where in the rung become a quantity at all. That is the ordinary way a confound becomes measurable — not by collecting more rows, but by finding the axis the rows already collected were spread along.

Two rungs have ends so far — the 5/8 rung swept at a thousandth, and the 3/5 rung at a tenth of that — so the column can be computed today for every row that fell inside either of them, and is a promise for the rest.

What is still scored the old way

Being specific is the point of writing this down, so: the ablation census, the two-organ table, the wreck destination list and the jugacy census are all one rise per rung, and every reading scored over any of them carries this caveat until it is swept.

A second cut moves the next organ, and does not move the boundary. Every pair of organs that can be taken out of a settled stem at a rise of 0.032, where the pattern is 3/5. A row is the offset of the nearer organ removed, counted back from the tip; a column is how many further places back the second one sits. The shade is how far the next organ ends up from where it would have been, from under a degree in the palest cells to a half-turn in the darkest. The run of shaded rows ends at 5, which is the larger parastichy number, and every row below it is blank across the whole width — the largest displacement anywhere past the front is 2.34°. So the nearer of the two organs decides whether the removal is felt at all, and the second one, wherever it is put, cannot make the pattern notice an organ it was not going to notice.
Fig. 18 One of them: every arrangement of two organs removed, at one rise.
The block is one of the two numbers, and not always the smaller. Every lattice a single organ was removed from, one row each: golden, rise 0.020 carrying 3/5; golden, rise 0.013 carrying 5/8; golden, rise 0.008 carrying 5/8; golden, rise 0.005 carrying 8/13; Lucas, rise 0.020 carrying 4/7; Lucas, rise 0.013 carrying 4/7. The two open ticks on each line are that lattice's own parastichy numbers; the filled dots are the blocks the stems that never recovered settled into, one per offset that failed. The claim this table was built to test is that the block is the smaller of the two, which held at the two rises it was first measured at. It does not hold here: 3/5 gives 5, 5/8 gives 5 and 8, 4/7 gives 4 and 7. What survives is weaker and still worth something — every filled dot but 1 sits on one of that row's own ticks, so the orbit carries a count of the lattice it was cut from, and which of the two it carries is decided by the offset rather than by the pattern.
Fig. 19 Another, in the quantity a wrecked stem is read by.
The block a wrecked stem settles into is the count it was cut from. A stem is cut in the middle of its front and followed for 300 organs. It does not come back to its lattice; what it does instead is repeat a fixed sequence of divergences exactly, to the resolution of the azimuth grid. Each row shows that sequence twice over, one bar per organ, drawn to the same scale. At the 5/8 rung the sequence is five angles long and advances -36.1° per block; At the 8/13 rung the sequence is eight angles long and advances 22.5° per block. Every block is the smaller number of the pair that was cut, so the second attractor carries the first one's count — and every precession is one part in twice the block, which makes each of these a two-jugate arrangement with 10 and 16 rows.
Fig. 20 And the destinations those tables feed, each measured at the rise its pair required.

The jugacy census deserves a particular mention, because the reading scored on it is about the arithmetic of the counted pair rather than about the geometry, and that is exactly the kind of reading this caveat does not reach. Whether a pair’s numbers share a factor is a property of the pair, and a census that fixes one rise per pair fixes nothing relevant to it. Sorting readings into those the sampling can touch and those it cannot is most of the work of applying any of this.

Two results are not affected and it is worth saying which, because a caveat that applies to everything is a caveat nobody applies. That the survivor is one of the two contact families is a statement about which lags exist, and the pair fixes those. That a wrecked stem keeps exactly one hop rigid is a statement about the motif it settles into, measured against its own control.

A cell's neighbours are its spiral families. Left: part of a 700-point golden, 137.508° head, with every contact between two cells drawn and coloured by the difference between the two nodes' placement indices. Right: the share each difference takes, across all 1459 contacts between interior cells. They are the parastichy numbers — 34, 55, 21, 89, 13, 8 — and the pair a person would count is 34 and 55. The six sides Euler forces are shared out among four of them, 5.64 edges per cell.
Fig. 21 Why the first of those is robust: the organs an organ touches are the members of its two contact families, at the two lags the pair names.
One wrecked stem, lag by lag — golden, rise 0.013, organ 4 back. How much each lattice hop moves from organ to organ in a stem that never repaired after a single removal, over its last 120 organs. The lag-one hop is the divergence itself and it swings by 65 degrees. The lag-5 hop swings by 0.11 degrees and sits 0.09 degrees from where the undisturbed stem put it, so that one family of the original lattice is still standing organ by organ. Its multiples inherit the same steadiness and nothing else comes within a factor of twenty. The block of angles this stem repeats has a period of 5, which is the surviving lag and not a coincidence.
Fig. 22 And how the second is measured: by lags, against a control sharing the stem’s history.

Where this leaves the collection

With a rule about its own method, which is the kind of result that is worth more than a figure and is harder to notice.

What the finer grid does to the rises already published. The two rises this site has argued from and the one it published as having no answer, each read at both azimuth grids, five stems apiece. A filled mark agrees with the position counter, a half mark contradicts it, an open mark is a refusal. At 0.013 and 0.005 the readings are identical at both grids, so nothing already written depends on the sample count. At 0.008 they are not: 384 azimuths gives 5/8, 5/8 and 1152 gives 8/13, 8/13, against a counter that says 5/8. That rise was chosen in the earlier work because the three shortest lattice offsets there are within a fifth of each other, and a stem with no answer answering differently on a different grid is the object behaving as it was said to.
Fig. 23 The neighbouring version of the same discipline, one level down in the instrument.
Every transition as the rise falls. The pair climbs 1/2 → 2/3 → 3/5 → 5/8 → 8/13. Consecutive transitions are 0.381, 0.382, 0.382 of the previous rise — 1/φ² is 0.3820.
Fig. 24 The ladder all of it sits on, and how much of each rung a census has actually visited.

Every future census here should record the rise beside the pair, and every reading scored over one should say which of the two it depends on. Neither is expensive. Both would have caught this years of essays ago, and the reason neither was done is that the question the censuses were built for genuinely did not need them.

The uncomfortable part is that this was findable at any point. Nothing in the sweep that exposed it needed a new instrument, a new library or a new idea — only the decision to move a parameter that every table already recorded and no reading had ever varied. What made it invisible was that the parameter was settled when the method was chosen rather than argued about anywhere in the results, and nobody reads a method twice. The same shape has already cost this collection a result in a different thread, where a window was mistaken for a neighbourhood for the same reason: a quantity was named once, early, and then quoted rather than measured.

Where this goes next

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. One rung, two answers The offset accounts for twenty-five wrecked stems of thirty and is refuted by a single pair of runs that differ in nothing but the rise. Sweep one rung at a thousandth and the refutation stops being an anomaly: the same offset on the same lattice keeps one family at the coarse end and the other at the fine one. The two contact steps change places inside the 5/8 rung. Measured at every rise on one rung, where a counter returns 5 and 8 spirals throughout. The settled divergence slides from 136.6406 to 137.8672 degrees. The ratio of the two contact steps falls to 1.0131 at a rise of 0.016 and the ordering changes hands at 0.015: above it the shorter step belongs to the 5-family and below it to the 8-family. Neither quantity is available to a counter, which is shown positions and reports a pair, and both of them move while that pair does not. The shortest hop was a coin flip The reading that a wrecked stem keeps its shortest hop was refuted at twelve of twenty-nine across the census. Re-scored along a single rung, where the counted pair is held and the step ordering reverses, it is right at sixteen of thirty-one — which is not a refutation but an absence of information. Every open question here needs under 34 specimens. The sample size at which each comparison reaches 90 per cent power at a 5 per cent false-positive rate, from the exact binomial rather than a normal approximation. The census question — do plants show consecutive Fibonacci pairs far more often than the geometry does — needs 4: 14.7% is the share of divergence angles giving a consecutive Fibonacci pair at a fine rise; 90% is what a grown history gives. The survey this site cannot do Four rounds of asking for a dataset, and it is still not here. What the work here can do instead is specify it — the fields, the sampling, the sizes, and which of this collection's claims each one would settle. Two of the four fields asked for turn out to be worth less than the asking implied, and one was never asked for at all.

What links here

Computed from the collection, not written here: the essays that point at this one.

Shares its objects with

Essays that name at least two of the same things, and that neither author linked.

  • A stem too fine to settle — both name counting blind, claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, rise, rung, underdetermination
  • One offset, two answers — both name claim testing, control, falsifiability, honest limits, lattice, measurement, negative result, parastichy pair, rise, rung, underdetermination
  • The organ that was nobody's neighbour — both name claim testing, control, falsifiability, honest limits, lattice, measurement, negative result, parastichy pair, rung, underdetermination
  • Two accounts of one number — both name counting blind, claim testing, honest limits, lattice, measurement, negative result, parastichy pair, rise, rung, underdetermination
  • The shallower front turns over — both name claim testing, honest limits, lattice, measurement, negative result, parastichy pair, rise, rung, sample size
  • Three organs and no mirror — both name counting blind, control, falsifiability, honest limits, lattice, measurement, parastichy pair, rise, rung

Named objects

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

Counting blindClaim testingControlFalsifiabilityHonest limitsLatticeMeasurementNegative resultParastichy pairRiseRungSample sizeSelectionSummary statisticUnderdetermination