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

The slide a counter holds constant

Inside one rung the settled divergence moves by more than a degree, monotonically, with no flat stretch anywhere — measured at a thousandth on one rung and at half a ten-thousandth on another. A rung is a plateau in one reported number laid over a geometry that never stops moving.
12 min read 27 figures Counted, not admiredState the claim, state the testOne parameter, one family

Worth reading first: A stem coarse enough to cut · Counting the spirals · A head is a set of points.

A rung is usually described as a plateau. The word is doing damage, and this essay replaces it with a measurement.

The damage is not in any single sentence. It is in what a plateau licenses without saying: that two stems on the same rung are the same kind of object, that a result measured on one transfers to the other, that “the 5/8 lattice” names a thing rather than a family. None of those is ever asserted outright, and all three are assumed by any census that takes one rise from each rung and treats the rows as comparable.

Across the 3/5 rung, swept at half a ten-thousandth — twenty-three rises, ten times finer than the ladder this collection normally draws — the settled divergence moves from 139.06° to 136.73°. It moves monotonically. There is no interval anywhere in the rung across which it is flat.

The 3/5 rung at a tenth of the ladder's step. Every rise of one rung, sampled ten times as finely as the ladder that found the locked band. All 23 are counted at 3 and 5 spirals and all 23 settle: the largest wander is 0.221 degrees, against the 0.5 degree threshold and against the 0.79 to 1.60 degrees the band on the coarse rung wobbles by. The divergence slides smoothly from 139.0625 to 136.7344 degrees with no rise stuck on a rational and none stuck on anything else. Whatever the band is, it is not something a coarser sampling was hiding here.
Fig. 1 The settled divergence across one rung at ten times the ladder’s resolution, sliding through the whole of it.

Across the 5/8 rung, swept at a thousandth, it moves from 136.55° to 137.87° — the same behaviour on a different rung, in the opposite direction, with the pair held throughout. Twelve rises there, twenty-three on the finer rung, and thirty-five settled stems between them: enough that the shape of each slide is read off a curve rather than inferred from its endpoints.

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 same measurement on the 5/8 rung, where the slide runs the other way.

So a rung is a plateau in exactly one thing: the pair a counter returns. Beneath it the geometry slides continuously, and every quantity built from the geometry slides with it.

The two sweeps together are the point rather than either alone. One rung swept at a thousandth could be a fluke of that rung; two rungs, at two resolutions, an order apart in step size, both sliding smoothly and neither showing a flat stretch, is a property of rungs. And the finer of the two was run for an entirely different reason — to look for a locked band — so its evidence about the slide is incidental rather than sought, which is the better kind.

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. 3 The ladder as usually drawn, where each rung is a step and the steps look like levels.

Why the direction differs

The two rungs slide opposite ways and that is not an error.

The settled divergence approaches the golden angle from alternating sides as the ladder is climbed, which is what the continued-fraction structure of the limit requires. A rung whose pair is a Fibonacci pair sits on one side; the next sits on the other. So a slide down within one rung and up within the next is the signature of converging on a limit by alternation rather than from one direction.

Continued fractions: why one number resists approximation. A large partial quotient means a very good rational approximation just ahead of it. The golden ratio's are all 1, the smallest they can be, all the way down.
Fig. 4 The arithmetic that produces the alternation: the convergents of the limit approach it from alternating sides.
How nearly each angle is a simple fraction of a turn. A dip means a rational approximation and a lattice with visible rows. The golden angle's floor is 0.008; the rational angle reaches exactly zero.
Fig. 5 And the landscape they approach through, where the rationals a divergence could stick to are crowded unevenly.

That is a satisfying check rather than a new result. It means the slides are not numerical drift or a settling artefact; they are the ladder doing what the arithmetic says it should, resolved finely enough to see within a single step.

It also gives the slide a direction that can be predicted rather than merely observed, which is the difference between a curiosity and a handle. If the alternation is the convergent structure, then the sign of the slide on any rung is known in advance from the pair, and a sweep that came back with the wrong sign would be evidence of something wrong with the stem rather than with the expectation. Neither sweep here does.

The size of it

A degree and a third across the 5/8 rung; two and a third across the 3/5 rung. Those are large numbers in this subject.

Rises that do not settle, at two resolutions. The count of rises whose divergence never settles, on the rung where the band was found and on the finer rung swept ten times as closely. The coarse rung has six of 19, all of them stuck on three eighths of a turn; the finer rung has none of 23. Sampling is not the explanation: if a band of the same kind sat inside the 3/5 rung it would need to be narrower than a ten-thousandth of rise to have been missed here.
Fig. 6 The two sweeps compared on the quantity that decides whether a rise is usable at all.

For comparison, the scatter of a settled stem’s own divergences — the wobble that decides whether it counts as settled — is a fifth of a degree at worst across the fine sweep. The slide within a rung is more than ten times the uncertainty of any single point on it.

Where each kind's lattice gives way. The largest amplitude at which every run still has a lattice, and the scatter it produces there. The amplitudes are incomparable — field 0.015 (fraction of the barrier), jostle 1 (degrees of azimuth), placement 0.8 (degrees of azimuth) — and the scatters agree to 19%. The boundary belongs to the pattern rather than to the disturbance: a lattice fails at about a degree and a half of scatter, and which of three mechanisms produced it does not move where.
Fig. 7 The tolerances this collection works inside, which the slide is an order larger than.

And for a different comparison: a real plant’s divergences scatter by about half a degree between organs. The slide across one rung is larger than the measurement error of the field measurement the whole subject rests on — which means the difference between two stems at opposite ends of one rung is, in principle, visible on a specimen. It is not a difference that lives below the noise; it is a difference nobody has looked for, because the pair says the two specimens are the same and nothing prompts a second look.

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. 8 What it would take to see a difference of that size in real specimens.
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 3 per cent error on each ring position leaves, so no ruler separates it from a flat disc.
Fig. 9 And which quantities survive a stated reading error on a real specimen at all.

What slides with it

Everything built from the geometry, which is most of what this collection measures.

The two contact steps. Their lengths are computed from the divergence and the rise, so both move, and their ordering reverses once inside the 5/8 rung.

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. 10 The ratio of the two steps across the same rises, crossing one partway down.

The front’s depth. The run of organs at which a removal is felt grows as the rise falls, so the number of offsets that can wreck a stem grows across a rung.

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. 11 The count of wrecking offsets across the same rung, growing five-fold.

And the answer to at least one experiment. At one offset the surviving family changes between the coarse end of the rung and the fine one.

That last item is the one that turns this from a vocabulary complaint into a result. If the slide moved only quantities nobody had drawn a conclusion from, it would be a footnote about description. It moves an outcome — a binary, reproducible outcome that an essay had reported as a property of a lattice — and that is what makes the interval worth insisting on.

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. 12 The grid where that shows: the pair constant along every row, one row’s answer not.

What does not slide

Two things hold across a rung, and they are the reason the count is worth taking at all.

Which lags exist. The two contact families are the two shortest steps, and which lags those are does not change inside a rung — only their lengths and their ordering do. So a wrecked stem’s menu of possible survivors is genuinely constant across the interval, which is the strongest result the ablation thread has and is untouched by any of this.

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. 13 The packing that fixes the menu: the organs an organ touches are the members of its two families.

And whether the arrangement is a lattice at all. Every rise on both rungs settles, counts consistently, and passes the same test. The interval is an interval of lattices, not a range that shades off into something else at its edges — which is what makes sweeping across one a controlled comparison rather than a walk between two regimes.

The spiral counts, band by band, in one head. The same flower gives 13/21, 21/34, 34/55, 55/89 at different radii. A caption saying "34 and 55 spirals" is a statement about one annulus.
Fig. 14 The check that a count belongs to the arrangement rather than to where it was taken.

Those two constants are what make the variable ones legible. A sweep in which everything moved would not be a sweep of anything; here the menu is fixed and the choice is not, and that separation is the whole reason the rung is the right unit to sweep along.

Three controls

Both sweeps stay on one rung. The pair is measured at every rise and the generators refuse to draw a sweep whose rises do not all return the same pair, so a sweep that wandered over a boundary would stop the build.

The two spiral families a counter finds between 0.43 and 0.67 of the radius. 21 spirals one way and 34 the other, found from the point positions alone — the counter is never told the divergence angle.
Fig. 15 The measurement that decides it, made on positions rather than on an assumed angle.

Every rise settles. The slide is a slide between settled values, not a trajectory through unsettled ones. The worst wander on the fine sweep is 0.221°, comfortably inside the threshold — and, more to the point, an order smaller than the slide itself. If the two were comparable the slide could be an artefact of where in each stem the average was taken; a factor of ten between them rules that out without needing a separate control.

Two runs of the same rule from unrelated starting angles. Both settle at 137.0°, within 0.5° of the golden angle, from seeds 166° apart.
Fig. 16 What settling looks like, which every rise in both sweeps does.
The 3/5 rung at a tenth of the ladder's step. Every rise of one rung, sampled ten times as finely as the ladder that found the locked band. All 23 are counted at 3 and 5 spirals and all 23 settle: the largest wander is 0.221 degrees, against the 0.5 degree threshold and against the 0.79 to 1.60 degrees the band on the coarse rung wobbles by. The divergence slides smoothly from 139.0625 to 136.7344 degrees with no rise stuck on a rational and none stuck on anything else. Whatever the band is, it is not something a coarser sampling was hiding here.
Fig. 17 And the wander at every rise of the fine sweep, against the threshold it has to fall below.

And the slide is not the grid. The azimuth grid the rule is evaluated over is finer than the slide by orders, so the divergences being reported are not quantised versions of a smooth curve. This is worth checking rather than assuming because a quantised slide would look identical at low resolution and would mean something entirely different — a staircase in the divergence would be a physical claim about preferred angles, which is the shape the locked band on the coarse rung actually has. Neither of these two rungs shows anything of the kind.

The 13/21 rung, at two azimuth grids. Five stems at each of three disturbances, all at a rise of 0.0019, read through a lag window of 45 from a seed of 40 nodes. A filled mark is a stem that returned 13/21, which is what the position counter finds in it; an open mark is a refusal. The only difference between the two rows is how many azimuths the placement rule samples when it takes its minimum: 384, which every run on this site has used since the earlier work, against 1152. At the coarse grid the rule locks onto its own sample points and the readout refuses 14 of 15 stems; at the fine one it reads all 15. The ceiling was a parameter of the program.
Fig. 18 The check that the grid is fine enough for the claim being made on it.
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. 19 And what a grid of a stated fineness can and cannot resolve, which is a separate limit again.

How wide a rung is, and why that matters more than it sounds

The 5/8 rung runs from eighteen thousandths of rise to seven — a span of eleven thousandths. The 3/5 rung runs from about thirty down to nineteen, a span of eleven again. Higher up the ladder the spans are wider in absolute terms and narrower relative to the rise; further down they compress quickly.

How the largest gap behaves as the head fills. The rational angle's gap grows by a factor of 3.8 over this range; the golden angle's stays within 1.30. This is the claim about 137.5° that survives measurement.
Fig. 20 The general form of that compression: what an arrangement looks like partway through filling is not a coarser version of what it looks like when full.

That has a practical consequence for every sweep this collection plans. A step fine enough to put a dozen samples inside the 5/8 rung puts six inside the next one down and three inside the one after. Two rungs finer than the sweeps here, a rung sweep stops being a sweep — before the settling ceiling is reached, which is the other limit and bites later.

The gaps close faster than the dips narrow. For each Fibonacci fraction, the distance to the nearest other rational with a denominator of 60 or less, and the half-width of its own dip at the smallest head that resolves it. The gaps fall from 0.763° at 3/8 to 0.0735° at 34/89; the dips stay between 0.0077° and 0.0155°. The dips never touch — the closest they come is a factor of 10 — so what stops the measurement is not the dips overlapping but the background between them ceasing to be flat. The clear offsets available fall from 72 to 53.
Fig. 21 Why the rungs narrow: the rationals the divergence must thread between crowd together as the denominators grow.

So the interval this essay insists on is not a fixed feature of the subject. It is widest exactly where this collection has done most of its work, and it shrinks towards the pairs real plants most often show — which is a reason to expect the sampling problem to matter less at the fine end and the settling problem to matter more.

That comparison points at something two other essays here both concluded was out of reach.

Where a stem sits inside its rung is the quantity almost every reading turns out to depend on, and it was described as unobservable without growing stems either side of a specimen until the pair changes — which can be done to a run and not to a plant. But the slide is itself a readout. The divergence moves monotonically across the rung, by a degree and a third here and two and a third on the rung above, and the divergence is the one quantity a specimen reports directly.

The arithmetic is friendlier than the half-degree scatter makes it sound, because a divergence is measured many times on one head. Scatter of about half a degree between successive organs, averaged over a hundred organs, gives a mean good to roughly a twentieth of a degree if the scatter is independent — which would cut a rung a degree and a third wide into something like twenty distinguishable positions. At a tenth of that resolution the reading would still separate the two ends of a rung comfortably.

The caveats are real and they are all of one kind. The estimate assumes the scatter averages down, and a systematic error — a tilted head, a consistent bias in how an angle is read off a photograph — does not average at all and would swamp it. Whether the scatter on a real head is independent between organs is a measurable thing nobody here has measured.

So the position within a rung is not recoverable from a single organ and may well be recoverable from a whole head, which is a considerably better situation than the one those essays describe. What it needs first is a rung swept finely enough to calibrate the conversion, which is what this one is.

That is the ordinary way a calibration comes about, and it is worth noticing when it happens: somebody measures one thing carefully for a reason of their own, and the measurement turns out to be the scale a different question was missing.

What to call a rung instead

Not a plateau. The accurate description is longer and it is worth using: a rung is the interval of rises over which a spiral count returns one pair. It is defined by the instrument, not by the arrangement, and it is a range of geometries rather than a set of equivalent ones.

The spiral counts four different divergence angles produce. Fibonacci counts come from one angle. The Lucas angle — which the same dynamical model reaches on a different branch — gives 47 and 76, and neither number is a Fibonacci number.
Fig. 22 The instrument’s own dependence on the angle, which is what makes a rung an interval rather than a point.

That phrasing costs nothing and it makes two mistakes harder to make. It makes “sample one rise per rung” look like the choice it is rather than the neutral act it seems, which is the sampling result of this round. And it stops “the 5/8 lattice” being spoken of as one object when it is a family of them.

A count of m and n pins the divergence to 221°/mn. Each dot is one reported pair, and its height is the total width of the divergence angles that could have produced it at some rise. 2/3 leaves 38.8° open; 34/55 leaves 0.118°. The line is 221°/mn, taken from the three highest pairs and drawn back through the rest.
Fig. 23 What the count is worth where the family behaves as one, which is most questions.
Every family but two is the sum of two others. Four heads and the contact families each actually has. An arc arrives at every family that is the sum of two smaller ones; the two with no arc arriving are the generators, and on every head they are the two smallest. whorled, 144°: 2, 3, 5 — golden, 137.508°: 8, 13, 21, 34, 55, 89 — Lucas, 99.502°: 11, 18, 29, 47, 76 — rational, 137.5°: 8, 13, 21, 34, 55, 89 — 137.0°: 8, 13, 21, 29, 50, 71, 92, 113. So once a pair is counted the rest is arithmetic, and a third counted family is a prediction rather than a second measurement.
Fig. 24 And the families a count is reporting, which are the same across a whole rung.

Where this leaves it

With a small, checkable, slightly deflating fact: the object this collection has been treating as a unit is an interval, and several of its results turned on where inside the interval they were measured.

None of the affected results is wrong in the sense of reporting a number that is not there. The shortest-hop refutation reached the right conclusion, the offset rule scores what it scores, and the census rows are all real measurements of real stems. What changes is what they are measurements of: not of a lattice, but of one geometry drawn from a range of them, chosen by a rise nobody recorded because nobody thought it varied.

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. 25 One of the tables built one rise per rung, in the quantity a wrecked stem is read by.
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. 26 And the coarsest version of the point: where a stem sits decides what it can do at all.

The deflation is worth having because the alternative is a vocabulary that quietly asserts something false. A plateau has no inside. A rung does, it is wide enough to hold a reversal and an outcome and a degree of divergence, and this round has spent most of its measurements there.

It is also, in the end, a compliment to the count. The reason a rung is an interval is that the count is robust — the same answer over a range of geometries — and robustness is what makes it worth taking on a plant with a hand lens. The measurement is sufficient for what it was invented for and insufficient for several of the questions this collection has since asked of it, and both halves of that are the same fact about what a topological reading is.

What the model settles on, against how fast the meristem grows. A broad golden branch, a transition, and then the two-whorl regime at exactly half a turn. 9 of 27 converged settings land within 4° of the golden angle; 15 land more than 20° away.
Fig. 27 The structure the whole ladder sits in, of which a rung is one interval.

Where this goes next

The 3/5 rung at a tenth of the ladder's step. Every rise of one rung, sampled ten times as finely as the ladder that found the locked band. All 23 are counted at 3 and 5 spirals and all 23 settle: the largest wander is 0.221 degrees, against the 0.5 degree threshold and against the 0.79 to 1.60 degrees the band on the coarse rung wobbles by. The divergence slides smoothly from 139.0625 to 136.7344 degrees with no rise stuck on a rational and none stuck on anything else. Whatever the band is, it is not something a coarser sampling was hiding here. The band was not the sampling Five rises in the middle of the coarse rung stick on three eighths of a turn, and the ladder that found them is swept at five thousandths — coarse enough that a band of the same kind could sit inside any finer rung unsampled. Swept at a tenth of that across a whole finer rung, nothing locks. 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 spiral families a counter finds between 0.68 and 0.92 of the radius. 34 spirals one way and 55 the other, found from the point positions alone — the counter is never told the divergence angle. What a count cannot decide A spiral count is the measurement this whole subject is built on, and it is deliberately blind to everything that varies inside a rung. Four results this collection now holds are results about that blindness rather than about the arrangements.

Shares its objects with

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

  • One offset, two answers — both name claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung
  • Two accounts of one number — both name counting blind, claim testing, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung
  • Half a turn, four at a time — both name counting blind, divergence angle, honest limits, lattice, measurement, negative result, parastichy pair, rational divergence, rung
  • The block is the count it was cut from — both name counting blind, divergence angle, honest limits, lattice, measurement, parastichy pair, the placement rule, rise, rung
  • The corner moves with the rise — both name control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung
  • The family that lost a member — both name claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rung

Named objects

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

Counting blindClaim testingControlDivergence angleHonest limitsLatticeMeasurementNegative resultParastichy pairThe placement ruleRational divergenceRiseRungScatter toleranceSummary statistic