The patterns a checker is tested on

Every sentence in this subject that begins over some crease patterns is a statement about a population somebody chose, and four ways to draw a pattern is what happens when nobody says which: four reasonable constructions disagree about how far a pattern shrinks by a factor of three.

This site’s response was to name its populations and keep them. There are four — the printed shelf, a family of twists, a set of quadrilateral meshes, and a set of cut sheets — and every checker here is run over all of them. That makes a claim about typical instances into a measurement, and the measurement is only as good as what is in the population.

Asked a question none of them had been asked before, they answer badly.

The four answers #

Thirteen of thirty-three are known to have a flat folded state. Eleven certainly do not. Nine are past what an exhaustive ordering search will finish, and are reported as undecided rather than assigned to either side.

The mesh population is the sharpest case: two of six. Four quadrilateral meshes satisfying developability, Kawasaki, Maekawa and the big-little-big lemma at every vertex, placed consistently — and no ordering of their panels satisfies the non-crossing rules. They are patterns of something that cannot be made.

What the populations were for #

The four exist because almost every pattern fails is a statement that needs a denominator, and a denominator needs a construction somebody declared.

The printed shelf is the eight patterns this site prints at true scale for a reader to fold. It is a curated set and it is the population every figure about “the printed patterns” is measured over.

The twists are the twist family swept over polygon count and radius — every polygon admits a twist and three of them tile — so the sweep runs from a triangle to a seven-sided polygon at several sizes.

The quadrilateral meshes are the developable quad meshes the rigid-folding solver is exercised on: a Miura, a tapered version, a few others, each built so that its fold angles can be solved along a motion.

The cut sheets are patterns with a crease turned into a raw edge — what a cut buys, counted — and are the population every kirigami measurement runs over.

Each one was assembled for a question and each answers that question. What none of them was assembled for is the question below.

Two different ways of not folding #

The eleven split into two kinds and the split matters, because they are refused at different stages.

Five cannot be placed. Folding flat about a crease is reflection in that crease, so a walk over the panels places all of them — and every shared edge the walk did not use is a second route to a panel already placed. On five of the twist population the two routes disagree by nearly two sheet widths. There is no consistent flat placement of those panels at all, whatever their letters say, and the ordering question does not arise.

Six place and cannot be ordered. Their panels go down consistently; no arrangement of them in a pile satisfies the crease rule and the two non-crossing rules. Four of those are meshes and two are twists.

Both refusals are proofs rather than failures to find something, because every rule involved is necessary. What neither of them is, is new information about the patterns — the patterns have always been like this. What is new is that anything asked.

What this does and does not invalidate #

The temptation is to read this as “the measurements over these populations are wrong”. Mostly they are not, and being precise about which is the point.

A measurement of what the conditions do is unaffected. Which vertices are the random ones asks how a vertex’s sectors are drawn and what the conditions admit; a pattern that satisfies the conditions and does not fold is still a pattern that satisfies the conditions, and it belongs in that population by construction.

A measurement of what a folded object looks like is affected, and there is one: any statistic computed from a folded state — a footprint, a layer count, a shrink — is computed on a state that may not exist. Those statistics are still exactly what the placement produces, but describing them as properties of a folded object is a claim the population does not support.

And a measurement described as “over crease patterns” needs its population said out loud. That was already this site’s rule, and this is the rule with a number attached: typical over these four populations means “satisfies the vertex conditions”, not “folds”, and the two differ by twenty patterns in thirty-three.

Why the populations look like this #

Three of the four were built for a specific purpose and the purpose explains the numbers.

The printed shelf is a curated set. Its members are patterns a reader is expected to fold, so they were chosen to be foldable — and the four that fold are the four small enough to check. It is the only population selected on anything like the right criterion, and it was selected by hand rather than by a test.

The twists are a family swept over a parameter. Twelve twists at several polygon counts and several radii, and five of them do not place. A swept family contains whatever the sweep produces, and nothing in the sweep asked for a folded state.

The meshes were built to be solved, not folded. A quadrilateral mesh in this collection exists to be handed to the rigid-folding solver, which asks whether the fold angles are consistent along the motion — a different question with a different answer. Four of six having no flat ordering is not a defect in them; it is what happens when a set assembled for one question is used for another.

And the cut sheets do well — four of five — because a cut releases vertices from every condition and leaves the pattern with less to contradict.

What a population ought to be #

There is a version of this that is a criticism and a version that is not, and the difference is what the population is for.

A population used to exercise a checker should contain things the checker must refuse. A test set of only valid patterns tests half of a test; the site’s own habit is that an assertion which has never rejected anything proves nothing, and eleven refusals in thirty-three is a healthy proportion for that job.

A population used to estimate a frequency — how often does a pattern do X — must be built from the thing being estimated over. These four were not built that way and no frequency should be read off them.

So the finding is not that the populations are bad. It is that they are one kind and have occasionally been read as the other, and the number that separates the two readings did not exist until now.

The one population that was selected on the right thing #

The printed shelf is worth separating out, because it is the only one of the four whose membership rule had anything to do with folding, and its record is correspondingly different: four fold, four undecided, none refused.

The rule was that a reader should be able to fold it. That is not a computation — nobody could run one until now — and it was applied by judgement: these are the patterns of the subject, they are the ones the literature prints, they are the ones somebody has folded. Judgement did well. It selected eight patterns of which none is refused by either proof, against a swept family of twelve of which seven are.

And judgement was still wrong twice. Two of the eight went out with a lettering that has no folded state, and neither was caught by the rule “a reader should be able to fold it” because nobody in the loop had folded that particular lettering — the pattern was right, the letters were chosen afterwards by a routine, and the routine’s criterion was appearance.

So the shelf’s good record is a record about patterns and the two failures were about letterings, which is the same distinction the whole phase turns on. A curated set of foldable patterns is not a curated set of foldable crease patterns until somebody checks the letters.

Where undecided sits #

Nine of the thirty-three are neither. Their panels place consistently, their letters do not contradict themselves, and the search over orderings does not finish inside its budget.

That is not a small residue and it will not shrink. The orderings of n panels are n factorial, the search is refused past about a dozen and a half panels on these patterns, and the undecided nine are the large ones — the Miura, the Yoshimura, the waterbomb tessellation, the tapered corrugation, three twists and two fold-and-cut outlines. The patterns anybody actually folds are exactly the ones that cannot be decided, which is the standing shape of every counting argument here.

What can be said about them is what the cheap proofs say: none has a loop in the order its letters force, and all place consistently. That is two necessary conditions passed and no claim beyond it.

What it would take to build the right population #

If a frequency were wanted — how often does a crease pattern fold — the population would have to be built differently, and it is worth setting out how, because the difficulty is the finding.

The pattern would have to be drawn at random from a stated distribution over crease patterns, and there is no such distribution. A crease pattern is a planar graph with angles and letters, and every way of generating one makes choices — where the vertices go, what degree they have, whether the sectors are sampled or constructed — and four such choices already disagree by a factor of three about a simple statistic. Fixing the choices fixes the answer.

Then each draw would have to be decided, and nine of thirty-three here are not. A frequency computed over the decidable members would be a frequency over the small ones, and small patterns fold much more readily than large ones — the printed shelf’s four decidable members all fold and its four undecidable ones are its four largest.

So the honest position is that a rate is not available, that the obstruction is not effort, and that reporting one would require inventing both a distribution and a decision procedure the subject does not have. Naming a population and counting it is what is available, and this essay is that.

What can be said instead of a rate #

The refusal to quote a frequency is right and it is stronger than it needs to be. A rate cannot be estimated from these populations, because they are not samples of anything. But the counts do bound a quantity, and the bound is worth having because it is exactly what the undecided members cost.

Over this collection of patterns, some fold, some are proved not to, and the rest are unknown. So the share that folds is at least the proved-folding count over the total, and at most that count plus the undecided ones over the same total. On the numbers above that interval runs from a little under two fifths to a little under two thirds.

That is a genuine statement about these patterns and not about crease patterns in general, and it has the right shape: a lower bound established by exhibiting folded states, an upper bound established by exhibiting proofs of failure, and a gap that is the price of the search giving out. Nothing in it requires a distribution, an estimator or a sample.

The interval’s width is the finding restated. Almost a quarter of the collection is undecided, so the range of possible answers spans a quarter — which is a great deal for a question whose answer is a single number between nought and one, and it is why the essay declines to name one.

And which end of it to expect #

Partial identification is more useful when something can be said about where in the interval the truth sits, and one thing can.

The undecided members are the large ones — the Miura, the Yoshimura, the waterbomb tessellation, the tapered corrugation and the rest — and everything measured elsewhere in this collection says a large pattern is less likely to have a consistent order than a small one. The orderings grow factorially and the constraints grow with the pairs of panels sharing ground; on the patterns where both can be counted, the constraints have already reduced nine panels to a single ordering.

So the undecided ones are not a coin toss between the two ends. They are the members most likely to land at the refused end, which puts the truth nearer the lower bound than the upper — nearer two fifths than two thirds.

That is an argument rather than a measurement, and it should be read as one. What it does is convert the interval from something symmetric into something with a direction, which is the most that can be said without deciding the nine.

It also says what would narrow it, and the answer is not a better search. Deciding one large pattern would move the bound by a thirtieth; deciding all of them is the ordering problem at sizes nobody can reach. What would actually help is a cheap proof of failure that works at scale — a loop in the forced order, a placement disagreement, something that refuses without enumerating — because every such proof moves a pattern out of the gap and into the upper bound’s account without any search finishing.

Three things this does not say #

It does not say the eleven were mistakes. A pattern in a population is there to be measured, and being refused is a measurement. Nothing was drawn wrongly except the two that have since been repaired.

It does not give a rate. Thirty-one patterns from four hand-built families is not a sample of anything, and “a third of crease patterns do not fold” is a sentence this essay does not contain.

And it does not settle the eight. Undecided means undecided. Reporting them as folding because nothing refuted them would be exactly the error the whole rung is about, and it is the same error as reading a search’s silence as a negative.

The check this makes possible #

There is a use for the eleven that is worth more than the warning, and it is the reason the essay is not simply a correction.

A test set of only valid inputs tests half of a test. Every assertion in this collection is required to be able to refuse something; the site’s habit is that an assertion which has never rejected anything proves nothing, and each of this site’s own checks ends by feeding its machinery input it must throw on.

The ordering machinery arrived with two such inputs built by hand — a Miura of twenty-four panels, which it must refuse to enumerate, and a strip whose counts it must reproduce. Those are constructed cases. The populations supply thirty-three that nobody constructed for the purpose, and eleven of them exercise a refusal path.

Better still, they exercise two different refusal paths in known proportions: five that fail to place and six that place and fail to order. A single hand-built counterexample would have exercised one of those and left the other untested, which is the shape of most test suites and is why a population assembled for something else is worth having.

What a folder should take from it #

Ask what a test set was assembled for. A family swept over a parameter contains what the sweep produced; a curated shelf contains what somebody chose. Neither is a sample.

Satisfying the conditions is a membership rule, not a property. Every pattern in all four populations passes every condition in this subject, and twenty of thirty-three are not known to fold.

And a refusal in a test population is doing its job. The eleven that fail are the reason the checker can be trusted about the thirteen that do not.