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

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.
12 min read 23 figures State the claim, state the testCounted, not admired

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

Five rises in the middle of the 2/3 rung do not settle. They stick on exactly three eighths of a turn — 135.0000° to four decimal places — and wobble about it by 0.79° to 1.60°, while a counter shown their positions still returns the rung’s own pair and cannot tell them from a settled lattice.

The band was bounded there and not explained, and the obvious worry was recorded with it. The coarse ladder is swept at five thousandths of rise. That is coarse enough that a band of the same kind could sit inside any finer rung and never be sampled — in which case the 2/3 result would be a fact about the ladder’s step rather than about the 2/3 rung.

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 test: one rung swept at a tenth of the ladder’s step, with the threshold a rise has to fall below to count as settled drawn across it.

This essay runs that test. Twenty-three rises from 0.030 down to 0.019, in steps of five ten-thousandths, across the whole of the 3/5 rung. Nothing locks.

The 3/5 rung is the right one to test on and not merely the convenient one. It is the rung immediately finer than the coarse ladder’s range, so if stability at a low-denominator angle is something that fades as the front deepens, this is where the fading would be least complete. Testing on a much finer rung would risk finding nothing for a reason that has nothing to do with the band — the front there is deep enough that no low-denominator angle could plausibly be stable — and a negative from that rung would prove less.

What was swept

Every rise on the rung, at a tenth of the resolution that found the band. All twenty-three are counted at 3 and 5 spirals, which is what makes the sweep a sweep of one rung rather than a walk across two.

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. 2 The two sweeps side by side: rises that never settle, on the rung where the band was found and on the finer rung at ten times the sampling.

All twenty-three settle. The largest wander of the divergence over the last stretch of any of them is 0.221°, against a settling threshold that the coarse band exceeds by a factor of four to eight.

That margin is what makes the result readable rather than marginal. A negative that depended on the threshold — largest wander 0.6° against a band starting at 0.79° — would be a statement about where a line was drawn. A factor of three and a half between the worst settled rise here and the quietest locked rise there is not a line-drawing question, and it means the two populations do not overlap at all rather than merely separating on average. The same distinction between a threshold result and a separation is what decides whether a score is worth quoting.

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. 3 The settled divergence across the same rises, sliding from 139.06° to 136.73° with no rise stuck on anything.

The divergence slides smoothly through the rung. No rise sits on a rational; no rise sits on a value the neighbouring rises do not approach continuously; there is no interval of any width in which the behaviour changes kind.

The slide itself is worth noticing as a positive rather than only as the absence of a band. Across twenty-three rises the divergence moves by 2.3°, monotonically, while the counted pair never changes — which is the same phenomenon the ablation sweeps depend on measured on a different rung and at ten times the resolution. A rung is not a plateau in the geometry; it is a plateau only in what a counter reports about the geometry, and this sweep is the finest confirmation of that the collection has.

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. 4 What “stuck on a rational” would look like if it happened: the rationals a divergence could stick to, and how close together they are.

Why a tenth is the right factor

The band on the coarse rung is five consecutive rises wide at a step of five thousandths — an interval of about 0.025 in rise, which is a quarter of that rung’s whole width. A feature that occupied a quarter of the 3/5 rung would span about six of the twenty-three samples here.

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. 5 The ladder both sweeps sit on, with the coarse rung at one end and the rung swept here nearer the middle.

So the sweep is not merely finer than the one that found the band; it is fine enough that a band of the same relative width would be sampled six times over. For the band to have hidden here it would have to be more than an order narrower relative to its rung than the one on the 2/3 rung is, which is a different claim from the one the worry was about.

Stating the excluded width matters because “swept finer, found nothing” is the weakest form of a negative and the easiest to overstate. What is excluded here is a feature occupying more than about a twentieth of the rung. What is not excluded is a feature at a single rise between two samples, or one an order narrower than the coarse band. Neither of those is what the worry proposed, and both would be a different phenomenon from the one being tested for.

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. 6 What settling looks like when it works, which is what all twenty-three of these rises do.

That is the honest form of the negative: a band like the coarse one is not there. A far narrower band, or one at a rise between two samples, is not excluded by twenty-three points and could not be by any finite sweep.

What it does to the coarse result

It makes it a property of that rung rather than an artefact of the ladder, and that is a promotion rather than a demotion.

A negative result is worth what its margin is worth, and this one has a margin in two directions.

The first is how close anything came. The settling test bounds how far a stem’s divergences wander over its last stretch, and the coarse band’s members wander by 0.79° to 1.60°. The largest wander anywhere in this sweep is 0.221° — under a third of the smallest coarse-band value, at the worst of twenty-three rises. This is not a sweep that found everything comfortably inside a threshold with one or two rises pressed up against it. Nothing on this rung is near the condition being looked for.

The second is how large a band would have to be to hide here. The coarse band is five consecutive rises at a step of five thousandths, a stretch of rise 0.020 wide. The whole of the 3/5 rung is 0.011 wide. A band of that size does not fit inside this rung at all. What a proportionally narrower one would do to the sampling is the argument made above; what matters here is the shape it would leave — a run of neighbouring rises all wandering past the threshold together, which is the most conspicuous pattern this sweep is capable of producing and the last one anybody would miss.

So the sweep is not merely quiet. It is quiet in a place where the thing it was looking for would have been loud, which is the difference between a null and an absence of evidence, and the whole reason the sweep was run at a tenth of the step rather than at the same one.

What it cannot do is speak for rungs it never visited. The claim is about the 3/5 rung at ten times the coarse resolution, and the honest generalisation is narrow: whatever produces a locked band is not something every rung has and the ladder’s step was hiding.

Which rises are a lattice, from 0.05 to 0.09. How much the divergence wanders over the last sixty organs, at each rise up the coarse ladder. eight of the nine settle, scattering between 0.0000 and 0.3356 degrees. one do not: from 0.09 to 0.09 the divergence sticks on exactly 135.0000 degrees, which is three eighths of a turn, and wobbles about it by 1.60 to 1.60 degrees. A counter shown either kind returns the same pair, so the counts cannot tell them apart. The line is the threshold a rise has to pass before a cut is made on it, and it sits in the gap rather than among the measurements.
Fig. 7 The coarse rung where the band is, drawn over the rises that carry it.
What a two-organ cut does at each rise of the 2/3 rung. At every rise the coarse rung is a lattice on, all 36 arrangements of two organs removed, with the ones that never repair counted and split by where they end up. 22 of 324 cuts across the rung reverse the stem's handedness onto the mirror of the divergence they were cut from. 40 fall instead into a cycle whose mean is half a turn, which the lattice they came from has no number for. 4 rises give only the first, 4 give only the second, and at a rise of 0.075 both happen in the same table, which is what says the fate belongs to the cut and not to the rise.
Fig. 8 And what a cut does across the same rises, which is the thread the band interrupted.

A locked band that appeared at every resolution on every rung would have said something about the sweep. A locked band that appears where it was first found and not one rung finer says something about the 2/3 rung: that the rule prefers a particular rational there, at a place on the ladder where the arrangement has few enough organs in its front for a low-denominator angle to be a stable answer.

There is a reading of that which connects it to the rest of the collection. A front of three organs offers very few arrangements for the rule to choose among, and the number of arrangements a front can pass through is the same quantity that decides whether a cut stem can reach its own mirror. Both are statements about a front shallow enough that the rule’s options are countable. Whether they are the same fact is not established here and would need the wobble measured rather than bounded.

Everywhere a cut of one to five organs can send a 5/8 stem. Every settled divergence reached by any arrangement of up to five organs removed from a stem at the 5/8 rung, on one axis. There are six of them and no more. three are slips of the lattice the stem was cut from: each keeps the lag-5 family intact and sits a whole number of turns of it from the next, which is the ladder marked below the axis with rungs 72.0 degrees apart. The other three keep no lag at all and sit near a fraction of a turn, marked above: 175.0 degrees near 1 of 2 turns, 190.0 degrees near 1 of 2 turns, 235.0 degrees near 2 of 3 turns. A stem that is cut either slides along the ladder it was on or leaves it for a lattice with files in it.
Fig. 9 Where the rationals sit that a wrecked or unsettled stem can land on, which is the map the coarse band’s value comes off.

What it does not settle

Three things, and they are the questions the coarse essay left rather than new ones.

Why three eighths. Nothing here says why the rule prefers that rational over its neighbours. The finer rung has rationals of similar denominator available and approaches none of them — the 3/5 rung’s divergences pass within a degree of several eighths and ninths on their way through, and settle at none of them. So whatever selects three eighths on the coarse rung is not simply proximity, or the finer rung would show the same pull weakly rather than not at all.

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. 10 The crowding of the low-denominator rationals, which is where any answer to that would have to start.
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. 11 And the arithmetic that makes some angles harder to approximate than others.

Whether the wobble has a period. The coarse band wobbles by 0.79° to 1.60° about its rational, and nobody has asked whether that wobble is itself periodic or merely noisy. This sweep has no locked rises to ask it of. The question is not idle: a periodic wobble would make the band a cycle rather than a fixed point, and this collection has already found a wrecked stem settling into a cycle of four rather than onto an angle. If the band turned out to be the same object seen from a different side, the coarse rung would have one phenomenon instead of two.

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. 12 The kind of reading that would answer it: a sequence of divergences read as an orbit rather than as a mean.
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. 13 And the structure such an orbit would sit in.

And whether coarser rungs below 2/3 do the same. The ladder continues below the coarse rung, and no sweep of any resolution has gone there. If the reading is that low-denominator angles become stable when the front is shallow, the rungs below should show it more strongly rather than less — and a 1/2 rung, whose front is two organs, is where the reading would be most exposed. That sweep is cheap: coarse stems settle quickly and there are few rises to cover. It has not been run because nothing until now made it a question with an answer worth having, and the essay that established the coarse rung’s eligibility stopped at the boundary of what a cut could be made on rather than at the boundary of the ladder.

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. 14 The coarsest version of the point: where a stem sits on the ladder decides what it can do at all.

Four controls

One rung, checked rather than assumed. The sweep’s own generator refuses to draw a ladder whose rises do not all return the same pair, so a sweep that wandered across a rung boundary would stop the build rather than average two rungs together.

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. 15 The instrument’s resolution, which is a separate limit from the sweep’s step and is easy to confuse with it.

The settling test is the same one. A rise counts as settled by the criterion every other run on this site uses, at the same threshold, so the comparison between the two rungs is a comparison of like with like.

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. 16 The tolerances the rest of the collection works inside, which this sweep does not adjust.

The stems are the same construction. Same rule, same node count, same azimuth grid; only the rise differs across the sweep, and only the rung differs between the two sweeps. That last clause is the one worth pressing on, because a finer rung packs more organs into the same stem and could in principle be settling better for a reason that has nothing to do with rationals — simply more organs of settling per unit of rise. If that were the explanation the wander would fall smoothly across the rung as the rise falls, and it does not: the largest and smallest wanders here are not at the two ends.

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. 17 The check that the grid is fine enough for the claim being made on it.

And the negative is not a threshold artefact. The largest wander here is 0.221°; the coarse band’s smallest is 0.79°. The two do not come close enough for the answer to depend on where the threshold is put.

What the sequence sees that the scatter cannot. Each point is an ensemble at one amplitude, placed at the scatter it produces. A stem at three quarters of a degree of scatter has a lag-one correlation near zero if its noise arrived after the primordium was placed, and near 0.7 if it arrived before — and no measurement of the scatter can tell those apart. The separation closes above about a degree, because what the other two kinds preserve is the correlation of a lattice.
Fig. 18 And the general check that a scatter is the kind of scatter it is being read as.

Where this leaves it

The band is real, it is where it was found, and it is not a sampling artefact of the ladder. That is three statements and the sweep establishes the third, which was the one in doubt.

What a two-organ cut does at each rise of the 2/3 rung. At every rise the coarse rung is a lattice on, all 36 arrangements of two organs removed, with the ones that never repair counted and split by where they end up. 22 of 324 cuts across the rung reverse the stem's handedness onto the mirror of the divergence they were cut from. 40 fall instead into a cycle whose mean is half a turn, which the lattice they came from has no number for. 4 rises give only the first, 4 give only the second, and at a rise of 0.075 both happen in the same table, which is what says the fate belongs to the cut and not to the rise.
Fig. 19 The rung the band sits on, and the two fates a cut made on it can have.
A whorl and a spiral, from one lattice at two divergences. At 120° the nodes fall on 3 rows and the two counts share a factor: the elements of each turn are level with one another. At 137.51° they never are.
Fig. 20 And what a stem stuck on a low rational looks like beside one that is not.

What it also does is retire a worry that would otherwise have sat under every result on the coarse rung. The essays that cut stems there exclude the locked rises from their tables, and that exclusion is only defensible if the locking is a real property of those rises rather than a resolution effect that the same tables would show elsewhere at a finer step. It is real, and they are.

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. 21 One of the tables that exclusion feeds, in the quantity a wrecked stem is read by.
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. 22 And the form the caution takes if any of this is to be claimed about plants rather than runs.
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 And what a count is worth on an arrangement that has settled, which is the state all twenty-three of these rises reach and the coarse band’s five do not.

One more thing follows from the negative and is easy to miss. If the 3/5 rung has no locked band, then every rise on it is available to be cut — the whole rung is eligible, where the coarse rung loses five of its nineteen rises to the band before any ablation can be attempted. Finer rungs are therefore cheaper to sweep per usable rise as well as more informative, which is an argument for the finer sweeps this collection keeps deferring on cost.

A negative result that costs twenty-three settled stems and closes a standing doubt is a cheap one. The alternative was leaving the doubt open and hoping nobody asked.

It is also a template. Every sweep in this collection has a step, and every step is a claim that nothing interesting happens between two samples. That claim is almost never tested, because testing it means running the same sweep an order finer and reporting that nothing changed — which produces no new figure and no new result. This one produced both only because there was a specific feature to look for. The rung sweeps that found the survivor changing hands had the same shape and came out the other way, and the pair of them together is a better argument for sweeping finely than either alone.

Where this goes next

What a two-organ cut does at each rise of the 2/3 rung. At every rise the coarse rung is a lattice on, all 36 arrangements of two organs removed, with the ones that never repair counted and split by where they end up. 22 of 324 cuts across the rung reverse the stem's handedness onto the mirror of the divergence they were cut from. 40 fall instead into a cycle whose mean is half a turn, which the lattice they came from has no number for. 4 rises give only the first, 4 give only the second, and at a rise of 0.075 both happen in the same table, which is what says the fate belongs to the cut and not to the rise. Half a turn, four at a time At four of the nine coarse rises no wrecked cut reverses. What those stems do instead is stop settling: they repeat 171.09°, 269.53°, 189.14°, 90.23° without end, which adds to two whole turns over four organs. The mean is exactly half a turn and a counter finds four files where the lattice had three. Which rises are a lattice, from 0.04 to 0.13. How much the divergence wanders over the last sixty organs, at each rise up the coarse ladder. 13 of the 19 settle, scattering between 0.0000 and 0.3356 degrees. six do not: from 0.09 to 0.115 the divergence sticks on exactly 135.0000 degrees, which is three eighths of a turn, and wobbles about it by 0.79 to 1.60 degrees. A counter shown either kind returns the same pair, so the counts cannot tell them apart. The line is the threshold a rise has to pass before a cut is made on it, and it sits in the gap rather than among the measurements. A stem coarse enough to cut Below the 3/5 rung is a 2/3 rung, and it runs from a rise of 0.050 to 0.120. It is not a lattice across all of it: from 0.090 to 0.115 the divergence stops settling and sticks on exactly three eighths of a turn, wobbling by a degree and a half — while a counter goes on reporting 2/3 as though nothing had happened. 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.

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, the placement rule, rise, rung
  • One offset, two answers — both name claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung
  • The front deepens down a rung — 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
  • A stem on the other branch — both name counting blind, honest limits, lattice, measurement, metastability, parastichy pair, the placement rule, rise, rung
  • A survivor has to be a neighbour — both name counting blind, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, rung

Named objects

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

Counting blindClaim testingControlDivergence angleHonest limitsLatticeMeasurementMetastabilityNegative resultParastichy pairThe placement ruleRational divergenceRiseRungScatter tolerance