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

The panel with no corner

Sweeping the rise gave three shapes where one was expected, and the middle one is the informative panel: at the 5/8 contact scale the deeper rule wins at every correlation and there is no crossing to locate. That is either a fact about the lattice or a fact about the pair of exponents, and one measurement separates them.
13 min read 23 figures State the claim, state the testCounted, not admired

Worth reading first: A disturbance with a memory · Counting the spirals · A head is a set of points.

Sweeping the rise was supposed to separate two readings of the corner — the contact scale, or any memory at all — and it did, by refuting the second. What it also produced was three different shapes where a single shape sliding along an axis was expected, and the middle one is the least discussed and the most informative.

At the coarsest rise the deeper rule wins at both ends and loses across the middle. At the finest it loses at white noise and wins from about seven tenths upward. At the rise in between it wins essentially everywhere, and there is no crossing anywhere in the range.

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. 1 The three sweeps reduced to where the deeper rule changes hands, and the one where it never does.

A missing feature in the middle of a range is not a gap in a trend. It is either a fact about that lattice or a fact about the comparison being made on it, and this essay is about which.

The distinction is not academic, because the two readings say opposite things about whether the corner exists as an object. If the flatness is a fact about the lattice, then the corner is something a lattice either has or does not, and the thread has found a lattice without one — which is a property to explain. If it is a fact about the exponent pair, the corner is a feature of a comparison rather than of an arrangement, and the thread has been naming a relation as though it were a thing.

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. 2 The three panels without the crossings marked, so the middle one can be read as what it is.

What “no corner” means here

The comparison is between two rules with different falloff exponents, run on the same seeds with the same disturbance, and what is counted is how many seeds agree that the deeper one passed more drift. A corner is a correlation length at which that count crosses a majority.

Which rule passes more drift — exponent 1.5 against exponent 6The same disturbance is given to two placement rules, one with a falloff exponent of 1.5 and one of 6, and the bar counts how many of six seeds let more of it through under the deeper rule. Above the line means the deeper rule passed more. Reading left to right the disturbance is given a longer memory, from white noise to a correlation length of 99 organs. The two change hands: the shallower rule passes more up to a correlation length of 1.4 organs and the deeper one from 2.8 organs onward. The comparison is made seed by seed rather than between two averages, because the spread between seeds at one setting is larger than the difference being measured.63036white1/61.42/62.84/64.55/69.56/699.56/6seeds agreeing — up: the deeper rule passed morecorrelation length of the disturbance, in organsexponent 1.5 against 6 · 182/3 organs · 6 seedsgenerated from a stated rule, not drawn to look right
Fig. 3 One panel in its original form, seed by seed, which is the measurement the summaries are made from.

At the middle rise the count is a majority at every correlation tested, including white noise. There is nothing for a crossing to be between, and the margin is not marginal either: five of six seeds at the low end and six of six across most of the range.

Which rule passes more drift — exponent 3 against exponent 4. The same disturbance is given to two placement rules, one with a falloff exponent of 3 and one of 4, and the bar counts how many of six seeds let more of it through under the deeper rule. Above the line means the deeper rule passed more. Reading left to right the disturbance is given a longer memory, from white noise to a correlation length of 99 organs. The two change hands: the shallower rule passes more up to a correlation length of 9.5 organs and the deeper one from 2.8 organs onward. The comparison is made seed by seed rather than between two averages, because the spread between seeds at one setting is larger than the difference being measured.
Fig. 4 A different exponent pair at the same rise, which is where the alternative explanation starts.

That is a different situation from a corner at the edge of the range. A corner at zero would mean the deeper rule always wins in the region tested and might lose below it; here the deeper rule wins at zero, which is the region where the disturbance has no memory at all and where the original prediction said the shallow rule should do best.

It is worth restating that prediction, because the middle panel is its most complete failure. The reading under test was that a rule reading a deep neighbourhood should average away a fast disturbance and be defeated by a slow one, so the deep rule should win at white noise and lose at long correlations. The middle panel has the deep rule winning at both ends and everywhere in between: not the prediction reversed at a crossing, but the prediction failing at every point of the axis it was made over.

A memory manufactures nothing. The largest comb mean found in a kinematic lattice whose azimuth errors are an AR(1) process, against the coefficient of that process, over eight seeds at each point. The dashed line is where the rule's own stems sit, at 0.64; the shaded strip is three sampling bands. Every point is inside the strip — 0.031, 0.026, 0.022, 0.014, 0.015 at ρ = 0, 0.5, 0.7, 0.9, 0.97 — and the readout returns nothing on 40 runs out of 40. A correlated error is not a periodic one.
Fig. 5 Where a disturbance with a memory stops being distinguishable from one without, which is the axis’s low end.

Two readings, and the measurement that separates them

The lattice reading. Something about the 5/8 arrangement makes the deeper rule better at every correlation, so the comparison has no crossing on it. If that is right the middle panel is a fact about arrangements and belongs beside the other two as a data point.

Which offsets give short hops, at a rise of 0.013. 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 contact geometry at the middle rise, which is what the lattice reading would have to be about.
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. 7 And the packing that geometry describes, where an organ’s neighbours are the members of its two families.

The exponent reading. The panels compare one pair of exponents. If at the middle rise both of those rules happen to sit on the same side of whatever threshold matters, the comparison flattens without anything about the corner changing. On this reading the middle panel is a fact about the pair being compared, and a different pair would show a crossing on the same lattice.

Four ways to count the rule's neighbourhood, and one ordering. How many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by three different definitions. Counting the organs that carry half the profile gives 33, 12, 3, 2, 1. Counting the organs that carry nine tenths gives 182, 120, 30, 8, 3. Counting the organs that carry the weighted mean lag gives 68, 45, 21, 14, 11. They disagree about the size of the neighbourhood by a factor of 10 — the widest moves by 61 times across the sweep and the narrowest by 6.1 — and every one of them falls as the exponent rises. So no choice among them changes which rule is the deeper, and no redefinition can turn a result about the depth around.
Fig. 8 Why the exponent reading is plausible: what counts as the depth of a neighbourhood is itself a choice among summaries that disagree by a factor of ten.
A rule too long-ranged makes no pattern; every shorter one makes the same pattern. Each dot is 4 runs from a coarse start at one exponent, separated by 0.2° of placement noise. Below p ≈ 1.1 the divergence scatters by tens of degrees, which is what an arbitrary sequence gives. From p = 1.25 to p = 8 — a sixfold range — every run ends on 8/13 with the scatter between 0.75° and 1.06°.
Fig. 9 And how much the pattern itself moves with the exponent, which is the quantity the pair is chosen from.

The measurement that separates them is a third exponent at the middle rise. If the flatness survives — if every pair of exponents at that rise shows the deeper rule winning throughout — it is the lattice. If a different pair produces a crossing, it is the pair.

The exponents available are not unconstrained, which makes the test sharper than it sounds. The comparison requires the two neighbourhoods to differ by at least a factor of three, because pairs closer than that are a coin toss at every colour — a fact this thread established separately and which is asserted by the generator rather than assumed. So the third pair is drawn from a small set, and exhausting it is a bounded piece of work rather than an open-ended sweep.

Which rule passes more drift — exponent 2 against exponent 3. The same disturbance is given to two placement rules, one with a falloff exponent of 2 and one of 3, and the bar counts how many of six seeds let more of it through under the deeper rule. Above the line means the deeper rule passed more. Reading left to right the disturbance is given a longer memory, from white noise to a correlation length of 99 organs. The two change hands: the shallower rule passes more up to a correlation length of 2.8 organs and the deeper one from 4.5 organs onward. The comparison is made seed by seed rather than between two averages, because the spread between seeds at one setting is larger than the difference being measured.
Fig. 10 A third pair at one rise, which is the shape the deciding measurement would take.

That is one sweep of six correlations at six seeds. It is the cheapest outstanding measurement in this thread and it has not been run.

Why it was not run

Because the thread’s question was about the corner’s position, and a panel with no corner reads as a panel with no data. It took reducing all three panels to one line each before the absence became the interesting cell rather than the empty one.

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. 11 The same reduction with the crossings unmarked, where the middle row is a statement rather than a blank.

This is a recurring shape in this collection and it is worth naming. A sweep looking for where something happens treats the places it does not happen as background. The asymmetry is built into how a sweep is usually written: the code finds the crossing, returns its position, and returns null when there is none — and a null is easy to render as a blank cell and hard to render as a claim. The generator behind these panels prints the shape of each row rather than only the crossing, which is a small change that turned an empty cell into the subject of this essay. The census that looked for which family survives reports the offsets that heal for exactly this reason, and reports them because a table of only the wrecked offsets would be a picture of the selection.

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. 12 The same discipline in the ablation thread: whether a stem is wreckable at all is reported, not assumed.

What a flat panel would mean if it held

Suppose the third exponent comes back flat too, and the middle rise genuinely has no crossing at any pair. What follows is more interesting than the corner thread’s original question.

It would mean the deeper rule’s advantage is not a monotone function of anything — present at 3/5, absent at 5/8, present again at 8/13 — and a quantity that switches on and off along a ladder is not a scale. It would put the corner closer to a resonance than to a threshold: something that appears when the arrangement’s own numbers stand in some relation to the disturbance’s, and vanishes when they do not.

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. 13 The kind of quantity a resonance reading would have to be built on: how close the divergence sits to a rational.

That is a much stronger claim than anything the thread currently supports, and it would need the sweep run at every rung rather than at three rises. It is recorded here as what the flat panel would license if it survived, not as what it shows.

There is a third reading of the middle panel that neither of the two above covers, and it is the one that most weakens the case for treating its flatness as a finding.

Look at what the panel contains rather than at what it lacks. The deeper rule takes five of six seeds at white noise, then six of six at every correlation from a half to nine tenths, then four of six at the longest. It is not flat because the two rules are hard to tell apart there. It is flat because one of them is winning nearly every run, and a comparison that has run out of room at the top has nowhere to put a crossing.

That is a ceiling, and a ceiling is a property of the comparison rather than of the lattice. Six of six is the largest number the panel can print, so if the deeper rule’s advantage grew or shrank across those four correlations the panel would report the same six each time and no feature would appear. The middle rise is therefore not a lattice on which a corner was looked for and not found. It is a lattice on which the instrument was saturated, and the honest reading of a saturated cell is that the measurement was not made.

This has a cheaper test than the third exponent. Saturation is escaped by making the comparison harder rather than by adding a rule to it — raise the amplitude until the deeper rule stops taking every run, or narrow the gap between the two exponents so its advantage is smaller to begin with. Either brings the middle panel’s cells off the ceiling, and only then does asking whether it has a corner mean anything at all.

Until one of them is run, the middle panel supports a single sentence: at this rise and this amplitude, the deeper rule wins across the whole correlation range. Everything else anybody would like to draw from its flatness is waiting on the saturation being lifted.

What the middle panel already rules out

Whatever the explanation turns out to be, one thing is already excluded: the corner is not a property of the disturbance.

Three kinds of noise, matched at 0.75° of divergence scatter. The amplitudes differ — field 0.0056 (fraction of the barrier), jostle 0.15 (degrees of azimuth), placement 0.18 (degrees of azimuth) — and are in different units, so they cannot be compared directly. What can be compared is what they produce, and matched here they are within 27% of one another. Everything a finished pattern records about its noise is shared between the three.
Fig. 14 The disturbances themselves, at the same amplitude and different colours, identical across all three panels.
A lattice or a wreck, with nothing in between. Every run in the sweep, at both kinds of noise. The line at 6° is where a run stops being counted as a lattice, and nothing lands near it: the intact runs reach 1.97° and the destroyed ones start at 8.06°, a factor of 4.1 away.
Fig. 15 And the normalisation that keeps the amount of disturbance fixed while the colour changes.

The colours, the amplitudes, the run lengths and the seeds are identical in all three panels — the same six seed values driving the same two rules against the same six disturbances. If the corner were something the disturbance carried — a property of an autoregressive process rather than of the arrangement it is applied to — it would appear identically in all three, and it does not appear in one of them at all.

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. 16 The check that a disturbance is the colour it claims to be, made before any of the panels are read.
Two stems at 0.75° of scatter, one angle at a time. The divergence of each node, with the slow climb of the ladder removed. Both stems scatter by about 48.38° and a botanist would report them identically. The jostled one wanders in runs — its lag-one correlation is 0.70 — and the one with placement noise alternates about the mean at 0.21, because a displacement of one node enters two consecutive divergences with opposite signs.
Fig. 17 And the spread that any of this has to beat: two runs of one rule differing by more than two rules differ from each other.

That is a real narrowing. It leaves the corner as something produced by the interaction of a rule with an arrangement, which is where a mechanism would have to live, and rules out the tidiest alternative.

It also rules out the reading that was hardest to dislodge by argument. “Two organs is simply the shortest memory the run length can resolve” is a statement about the measurement apparatus, and no amount of reasoning about lattices touches it — only moving a parameter the apparatus does not depend on can. The rise is that parameter, which is why this sweep was the one that had to be run rather than another statistic computed over the sweeps already in hand.

Three cautions about the panel

Six seeds is few. A majority of six is four, and several cells sit at four or five. The claim rests on the pattern across a row rather than on any cell, and the row is monotone in the finest panel and flat in the middle one — but a seventh seed could move a cell. Six is not arbitrary: it is the number at which the seed-to-seed spread stops dominating the difference between two rules, measured rather than chosen. It is enough to make a row’s shape readable and not enough to make a single cell a result, which is exactly how the panels are read here.

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.
Fig. 18 The general form of the caution, at the power this collection usually asks for.

The correlation axis is coarse. Six values from zero to 0.99, unevenly spaced. A crossing between two of them would be located only to within the gap, and a crossing narrower than a gap could be missed entirely. The spacing is not uniform because the quantity that matters is the correlation length in organs rather than the coefficient, and length grows without bound as the coefficient approaches one — so equal steps in the coefficient would be wildly unequal steps in what the disturbance actually does. The axis is chosen to be roughly even in organs, which is the right choice for reading a crossing and the wrong one for claiming a crossing is absent.

The ratio follows the disturbance, not the rule. The ratio of the second comb to the main comb on stems grown by the placement rule and jostled by seven different disturbances, all at 0.25° of displacement per organ and all on the same rule. Independent errors and errors with a memory return 0.76–0.81, which is the value this site measured for the rule. A periodicity at the smaller parastichy number takes it down to 0.45; errors inherited from the contact neighbours take it up to 1.09, most of the way to the 1.24 a transported disturbance gives with no rule in it at all. So the quantity separates arrangements by how their errors are related, not by whether anything computed the positions.
Fig. 19 The colours the sweep runs over, which is the axis a missing crossing would have to hide in.

And the middle rise is one rise. It is a single point in a rung, chosen because it carries the 5/8 pair — which is precisely the sampling this round has learned to distrust. A second rise carrying the same pair would say whether the flatness belongs to the pair or to the rise, and it is the same cost as the third-exponent sweep. Running both would give a two-by-two — two exponent pairs at two rises on one rung — which is four sweeps and would settle the question in either direction rather than only in 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. 20 The rung that rise came from, where several quantities move while the pair does not.
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. 21 And the quantity most likely to matter here: the ordering of the two contact steps, which reverses inside the same rung.

That third caution is the one this round would not have thought to make a month ago, and it may be the whole answer. If the corner depends on which contact step is shorter, then a rung containing the reversal contains both behaviours, and sampling one rise from it gives whichever the rise happened to have.

Where this leaves it

The corner thread now has one refutation, one narrowing, and one cell whose meaning is undetermined between two readings that a single sweep would separate.

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. 22 The three panels once more, with the flat one now the outstanding question rather than the missing data.
The disturbance with the largest wander leaves none in the sequence. How much of a divergence sequence's variance survives being averaged over blocks, on kinematic lattices. The vertical quantity is B² times the variance of the block means divided by the variance of the sequence, which is one at every block size for independent errors — the arithmetic is normalised for a differenced stream, since a divergence is the difference of two organs' errors. A line that climbs is a sequence with power at frequencies below one per block. The disturbances that remember the last error climb to 32 at a block of 128. The ones inherited between touching organs do not climb at all — 1.06 and 0.83 at the same block — although their own deviates carry ×5 and ×49 an independent stream's variance in exactly this statistic. What they do instead is dig a hole: at block sizes of 8 and 13, which are the offsets they couple at, the statistic falls to 0.22 and 0.21.
Fig. 23 The quantity all of it is measured on: how much of a drift reaches the divergences.

Naming an undetermined cell and the measurement that would determine it is worth more than another statistic computed over the cells that are determined. The sweep is six correlations at six seeds at one rise with one new exponent, and it is the next thing this thread should do.

There is a temptation, with three panels and one of them flat, to report the two that agree and treat the third as noise. That would be the same error as reporting the offsets that wreck and not the ones that heal, and it is worth resisting for the same reason: the cells where nothing happens are cells, and a sweep that discards them is a sweep whose shape was decided before it ran.

Where this goes next

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. The corner moves with the rise The corner was either the contact scale or simply any memory at all, and nothing in the thread had ever varied the rise — the one knob that moves the contact numbers while leaving the rule, the amplitude and the run length alone. Swept over it, the comparison does not keep its shape. Which rule passes more drift — exponent 1.5 against exponent 6. The same disturbance is given to two placement rules, one with a falloff exponent of 1.5 and one of 6, and the bar counts how many of six seeds let more of it through under the deeper rule. Above the line means the deeper rule passed more. Reading left to right the disturbance is given a longer memory, from white noise to a correlation length of 99 organs. The two change hands: the shallower rule passes more up to a correlation length of 1.4 organs and the deeper one from 2.8 organs onward. The comparison is made seed by seed rather than between two averages, because the spread between seeds at one setting is larger than the difference being measured. The corner that does not move Read as degrees of drift getting through rather than as a ratio, and compared seed by seed, the deep and shallow rules change hands. The share that goes to the deeper rule climbs from twenty-three per cent under white noise to ninety-four at a correlation length of a hundred organs — and the crossing sits at two or three organs whether the two rules differ by a factor of four or sixty-one. Four ways to count the rule's neighbourhood, and one ordering. How many organs the placement rule is looking at, as its falloff exponent is swept from 1.5 to 6, by four different definitions. Counting the organs that carry half the profile gives 33, 12, 3, 2, 1. Counting the organs that carry nine tenths gives 182, 120, 30, 8, 3. Counting the organs that carry ninety-nine hundredths gives 284, 263, 149, 54, 9. Counting the organs that carry the weighted mean lag gives 68, 45, 21, 14, 11. They disagree about the size of the neighbourhood by a factor of 10 — the widest moves by 61 times across the sweep and the narrowest by 6.1 — and every one of them falls as the exponent rises. So no choice among them changes which rule is the deeper, and no redefinition can turn a result about the depth around. Four ways to count a neighbourhood How deep does the placement rule look? Counting the organs that carry nine tenths of its profile gives 182 down to 3 as the falloff steepens. Counting the ones that carry half gives 33 down to 1. The weighted mean lag gives 68 down to 11. The four disagree by an order of magnitude about the size and agree exactly about the order.

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 claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, underdetermination
  • One offset, two answers — both name claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, underdetermination
  • The shortest hop was a coin flip — both name claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, underdetermination
  • One rung, two answers — both name control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise, underdetermination
  • The band was not the sampling — both name claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, rise
  • The family that lost a member — both name claim testing, control, honest limits, lattice, measurement, negative result, parastichy pair, the placement rule, underdetermination

Named objects

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

Claim testingControlDriftHonest limitsThe range of the interactionLatticeMeasurementNegative resultNeighbourhoodNeighbourhood depthNoiseParastichy pairThe placement ruleRiseUnderdetermination