A population nobody chose

Every sentence in this collection of the form across the five patches is a sentence about five crease patterns somebody drew. They were drawn to be illustrative — one per tiling, at a turn angle that makes a good picture — and then they became the test set, because a test set is whatever was to hand when somebody needed one.

That is the failure the population question was raised about before, at the level of a single vertex, and again at the level of a whole pattern, where four defensible sources of examples turned out to agree on everything because all four were made of things a construction had produced. Both times the conclusion was the same: a number quoted without a population is a number about whatever was convenient.

The five patches are the same object one level further along, and this is what happens when they are replaced.

What the grid is, and why it is a grid #

The construction that draws these patches takes three parameters that change the object rather than its presentation: which tiling, how far each polygon turns, and how much of the available room the pleats take up.

The grid is every combination of four tilings, eight turn angles from 0.15 to 1.3 radians, and three pleat widths. Ninety-six patches, none refused by the construction, ranging from forty-nine panels to ninety-three. Every one is a genuine crease pattern: every vertex satisfies the four conditions, and the panels close.

The essential property is that the grid was declared before it was run and nothing was removed from it afterwards. That sounds like a small discipline and it is the whole of the method: a population that can be pruned is a population that will be pruned toward whatever the person pruning it expects, and the pruning will be invisible in the result. Eight turn angles were chosen because eight is enough to see a trend and the shallowest and steepest are the extremes the construction accepts; three pleat widths because the construction’s own default sits between two others.

None of that is a claim that the grid is the right population. There is no right population, which is exactly the point the earlier essay makes at length. It is a claim that this one was declared.

There is a second discipline in it that is easier to miss. The grid is swept over parameters of the construction, not over crease patterns. That distinction matters because there is no way to draw a crease pattern at random — a pattern is a graph with geometry and letters, and every sensible way of producing one is a construction with knobs on it. So a declared population here is always a declared construction plus a declared range, and the honest form of any claim from it names both.

That is why the ranges above are quoted in radians rather than described as “shallow to steep”. A range in words can be reread later to mean whatever the reader needs; a range in numbers cannot.

The nine the five could not contain #

Eighty-seven of the ninety-six have a consistent lettering, found in twenty-one to forty-seven steps. Nine have none at all, proved by exhausting the search rather than by failing to find one.

None of the five printed patches is anywhere near those nine. Every one of them is drawn at a turn of 0.35 radians and a pleat width of 0.62, and the nine are all at 0.15 or 0.25 with the two narrower widths. A collection measuring only the five would have gone on saying that every twist tessellation patch it had asked about has a consistent lettering, which is true, and would have had no reason to suspect the sentence had content.

The nine occupy a region rather than a threshold and the region’s boundary is a fact about sectors. What matters here is narrower: the region exists, it is eight per cent of the grid, and a hand-picked set of five examples chosen for looking good has a very poor chance of containing any of it — because the same aesthetic that makes a patch look good keeps it away from the shallow turns where the failure lives.

Nine of ninety-six is the wrong denominator #

The rate deserves the same treatment the essay gives the five patches, because it is a number about the grid’s shape rather than about the construction.

The nine failures are all at a turn of 0.15 or 0.25 and at the two narrower pleat widths. That is a block of four tilings by two angles by two widths — sixteen cells — and nine of the sixteen fail.

Outside that block, none of the remaining eighty fails.

So the honest reading is not nine and a half per cent of patches have no lettering. It is fifty-six per cent inside a sharply bounded corner of the parameter space and nothing anywhere else, and the grid’s 9.4 per cent is those two numbers averaged in whatever proportion the grid happens to sample them.

Which makes the rate arithmetic rather than a finding #

That decomposition says exactly what the rate is a fact about, and it is the grid’s design.

Sixteen of the ninety-six cells lie in the danger corner — a sixth of the grid — and nine sixteenths of those fail. Multiplying gives nine of ninety-six, so the headline rate is $\tfrac16 \times \tfrac{9}{16}$ and every factor in it is a choice somebody made about which angles and widths to include.

A grid running from 0.5 radians upward would sample none of the corner and report nought. A grid running from 0.1 to 0.4 would sample mostly the corner and report something near half. Both would be correct measurements of their own grids and neither would be a measurement of the construction.

What is a fact about the construction is the two numbers, not the average. Fifty-six per cent inside, zero outside, and a boundary that a sector crossing puts in a definite place — those survive any regridding, and the rate survives none of it.

That also sharpens the essay’s warning about a grid moving the choice rather than removing it. The choice is not merely written down here; it is separable, because the failures cluster tightly enough that the grid’s contribution and the construction’s can be pulled apart. A population whose failures were spread evenly would offer no such separation, and its rate would be irreducibly a fact about the sampling.

So the discipline the essay recommends has a second half. Declare the population — and then check whether the result decomposes, because a result that does is one whose scope can be stated and a result that does not is one whose scope is the population itself.

The tail belonged to one tiling #

The second thing the grid says is more uncomfortable, because it revises a measurement rather than adding one.

A great deal of work went into a heavy-tailed search cost: the same search on the same pattern costing eighty-six steps or fifteen thousand depending on the seed, with a restart curve fitted to the distribution and priced. All of it was measured on the rhombille patch, which is one of the five.

Over the ninety-six patches of the grid, not one has a tail. The coin costs twenty-five to fifty-six steps across the whole grid and a constant order costs twenty-one to forty-seven, which is a difference of no consequence at all. Twenty of the twenty-four rhombille patches have one.

So the phenomenon is real, it is confined to one tiling, and that tiling was in the five. Had the five been a different five — four uniform tilings and no rhombille — the heavy tail would never have been seen, the restart curve would never have been computed, and the coin that produced it would still be sitting unnoticed in the search — and the restart curve priced against it would never have been computed either.

The mechanism is worth stating because it explains why the tail was confined and the failure was not. Both belong to the shallow end of the construction, but they belong to it for different reasons: the failures are about which sector at a vertex is smallest, which is a property of every tiling whose sectors can reorder, while the tail is about how far a wrong letter propagates before it is contradicted, which is a property of the tiling whose side distances differ from vertex to vertex.

So one phenomenon is spread thinly across three tilings and the other is concentrated entirely in a fourth. A population that contained the first at all would probably contain some of the second, and a set of five patterns chosen for looking good contained the second and not the first — which is as clean an example as this collection has of a sample being unrepresentative in two directions at once.

Which is the luckier accident #

That is worth sitting with, because it cuts both ways and the collection’s habit is to report only one direction.

The convenient reading is that a hand-picked set nearly missed something. The inconvenient one is that the same hand-picked set caught something a declared grid over four tilings would have missed entirely — the grid above contains no rhombille at all, and on its own evidence the tail does not exist.

Neither set is better. The five contained one unusual member and gave it a fifth of the weight; the grid contains none and gives it none. What makes the pair informative is having both, and the correct statement about the tail is one neither could make alone: it belongs to the tiling whose vertices are not all alike, which is a structural description rather than a name.

The general form of that is a rule this collection has arrived at three times now from three directions. A measurement taken on a hand-picked set is not wrong; it is a correct measurement whose scope is unknown. What a declared population adds is not accuracy but scope — the ability to say which of several readings is about the family and which is about one member — and scope is the thing a reader cannot recover afterwards from a number.

Why the rhombille is not a fifth sample #

The grid covers four tilings and the rhombille is left out of it deliberately, for a reason that is structural rather than practical.

On the four, every vertex of the tiling is the same as every other, so the condition that gives each twist polygon its own side distance has nothing to propagate — every distance comes out equal and the construction could have assumed it. On the rhombille there are two kinds of vertex, the distances differ, and the propagation is the whole reason the pattern folds at all. The tiling the unit could not promise is where that distinction was established.

So the rhombille is a different object rather than a fifth member, and averaging it into a population would be the mistake the population question exists to prevent. It is measured separately and reported separately, and its three shallow patches — which also have no lettering — extend the region across it without being averaged into the rate.

The practical half is worth admitting too: a rhombille patch at a shallow turn takes six and a half minutes to draw, before any search begins, because the construction finds its own letters by a search of the same kind. A grid including it would cost hours rather than seconds, and a population whose cost decides its size is a population making the same compromise as one whose convenience decides its membership.

What the grid cannot say #

It cannot say anything about crease patterns in general. It is one construction, swept over its own parameters, and every member of it is a twist tessellation on a square sheet. A statement of the form eight per cent of crease patterns have no consistent lettering would be a wild extrapolation from ninety-six objects that are all nearly the same object.

Nor does a grid remove the choice — it moves it. Somebody picked the ranges, the number of steps, and which three parameters count. A grid from 0.5 to 1.3 radians would contain no failures at all and would support a confident and false conclusion. What a declared grid buys is not objectivity; it is that the choices are written down, so that a reader can see what was swept and a later measurement can sweep something else.

What it cost, which is the reason this is not routine #

A declared grid is more expensive than five examples in a way that decides how often anybody builds one, and the cost is worth quoting rather than waving at.

The ninety-six uniform patches take under two seconds to draw and about forty seconds to search to a verdict, which is nothing. The rhombille’s twenty-four take from a fifth of a second to six and a half minutes each, dominated entirely by the construction finding the pattern’s own letters — so a grid over five tilings rather than four costs roughly two hundred times as much as the four, for a fifth more members.

That ratio is the reason the grid stops where it does, and stating it is part of stating the population. A reader who knows the grid excludes the rhombille and does not know why might reasonably suspect the exclusion was chosen after seeing the results; a reader who knows the exclusion is structural and that including it would have cost hours can judge both halves.

Which theorem was checked, and how #

Every verdict in the grid is an exhaustion or a verified witness, and never a failure to find. The search distinguishes three outcomes and only two of them are used: a lettering, checked by writing it back onto the pattern and putting it past the four vertex conditions and a folded sheet rebuilt from scratch, or a completed search with nothing in it.

Every verdict is also checked under more than one order, because an exhaustion visits the whole tree and two orders that both finish must agree. That is asserted over the grid rather than assumed, and it is the check that would fail first if the search were quietly not exploring what it claims to.

The claim that no patch of the grid has a tail is asserted in the same way, as a requirement: every patch with a lettering is searched under the coin from several seeds, and the assertion fails if any of them spreads.

What the picture cannot show #

The rate. Nine of ninety-six is nine and a half per cent, and it is a fact about this grid rather than an estimate of anything — a grid with more shallow turns in it would produce a higher rate and one with fewer a lower, and neither number would be more true. There is no distribution over crease patterns here from which the grid is a sample, so there is nothing for a rate to be an estimate of.

Nor does the grid contain any pattern that a person would want to fold. The nine failures are all at shallow turns, and a shallow turn produces a patch whose twists barely turn — a picture nobody would print. The population is deliberately full of things nobody would choose, which is what makes it useful and also what makes it silent about the patterns anybody cares about.

The habit this leaves behind #

Three questions are now worth asking of any sentence in this collection that quotes a number across several patterns, and none of them is about the number.

What produced the members? If the answer is “somebody drew them”, the sentence is about that person’s taste, and the taste may be correlated with the property being measured — which is exactly what happened here, twice, in opposite directions.

Was the set declared before it was measured? A set assembled while looking at results is a set shaped by them, and the shaping leaves no trace in the numbers.

Which readings would survive a different set? This is the useful one and the hardest. A reading that every member agrees on is either a fact about the family or a fact about what the members have in common, and only a second population can separate those. Both times this collection has built one, the separation was the finding.

Where the ladder goes next #

The grid was built to answer a question about search orders and it answered a different one first, which is the ordinary way a declared population earns its cost. Where it goes next is into the family it excludes: the rhombille’s dial, and what makes one tiling of five unlike the others, which is the only place in this construction where a parameter changes the difficulty rather than the picture.