A population that cannot fail

The patterns a checker is tested on are four populations of thirty-three crease patterns: the sheets this collection prints, twist tessellations at three turn angles each, developable quadrilateral meshes with no two vertices alike, and the fold-and-cut construction on seven outlines. Every member satisfies every condition at every vertex, which is what put it in the population.

Drawn by the same hand is the standing complaint about them: every one was produced by a construction written in the same place as the checkers, so the patterns are exactly the ones nobody expected to be difficult. Random segments on a square cross one another almost surely and the four constructed populations contain not one crossing between them.

There is a second version of that complaint, and it is about the letters rather than the lines.

The refusal with nothing to fire on #

A pattern’s letters can contradict themselves: each crease fixes which of its two panels lies above the other, and a chain of panels each of which must lie below the next is a proof that no folded state exists.

Run that over all thirty-three. It fires on none of them.

A nought there looks like a test with nothing to do, and that reading is available and wrong. The right reading is that the population cannot contain the failure, and the reason is in how the members were made.

Every construction here returns a lettering. The fold-and-cut construction letters its pattern from the skeleton; the mesh solver derives the letters from the fold angles; the printed patterns were lettered by somebody who folded the result and then wrote down what they had done. A lettering that arrives that way is a lettering that has already been tried, and a lettering that has been tried does not contradict itself.

So the test set has been assembled entirely out of patterns whose letters were chosen by something with an interest in their working.

The same patterns, relettered #

The repair is to reletter each member and ask again. Independent letterings come from propagating the vertex conditions to a fixed point and branching, wherever propagation stalls, on a crease with the two letters tried in an order a stream decides. Every draw passes every condition; the draws differ because the decisions do.

The printed shelf is between ninety and a hundred per cent: the Miura fifty-six of sixty, the tapered corrugation fifty-four, the Yoshimura and the waterbomb tessellation fifty-seven each. The meshes are all sixty of sixty except the fourteenth, which is forty-nine. The fold-and-cut outlines are sixty of sixty on all seven.

And the twists are where the population turns out to have been hiding something. A square twist at any angle is fifty-two of sixty; a hexagonal twist is ten of sixty, four of sixty, four of sixty at its three turn angles.

Seven per cent. A member of the standing test set on which nine of ten admissible letterings contradict themselves, and every checker that has ever been run over it saw the one lettering it arrived with.

What a hexagonal twist is doing in there #

The seven-per-cent member deserves a paragraph on its own, because it is the case that makes the argument concrete rather than methodological.

A hexagonal twist assembled unit by unit is twenty-five to thirty-three panels with forty-two to fifty-six creases, and twenty-four to thirty-two independent chains of panels for its letters to close. That is more chains than the Miura has and about as many as the Yoshimura, and it fails far more often than either — which is the pattern the twists show generally: a construction built out of rings joined by pleats offers the shortest chains there are, and the shortest chains are the ones that close.

So the member that turns out to be difficult is not an oddity. It is the member most like the patterns this collection has since found the failure in, and it was sitting in the test set the whole time with one lettering on it.

What the twists population is for #

The twists population is assembled deliberately in the way that looks wrong. Its members are built by assembling whole twist units on a square and running the outstanding pleats to the rim, which puts creases across other creases on every tiling but the square — and that is exactly why it is kept that way.

A population kept to exercise a checker has to contain patterns the checker must refuse. Cutting the same tessellations out of the plane instead removes every crossing, so a population built that way would be twelve patterns that all pass, which is a worse test set. Five of its twelve members are refused by the crossing sweep; five more have no folded state at all and cannot be relettered; seven can be asked.

That is a well-designed population for the failure it was designed for, and it is silent about this one by accident. Nothing in its design excluded contradictory letterings; the construction simply never produced any.

What “as drawn” means for the other checkers #

It is worth asking whether the same criticism applies to the rest of the ladder, and mostly it does not.

The crossing sweep reads the drawing and never the letters, so relettering changes nothing it sees. The vertex pass reads both, and every member passes it by construction — but a member that failed it would not be in the population, so its zero is a definition rather than a finding. The panel walk reads the drawing and the letters and computes where the paper goes; relettering does change what it sees, and it comes back clean on every redraw of every member here, which is a genuine result rather than an artefact.

Only the layer refusal has the shape this essay is about: a test that reads the letters globally, on a population whose letters were all supplied by something that had already solved the problem.

A test set that fires is not the same as a test set that is representative #

Two different objections are easy to run together here and they want different repairs.

The first is that the populations are unrepresentative: they contain no crossings because constructions do not draw them, no crowding because constructions space their creases, and no contradictory letterings because constructions letter their patterns. That is a statement about what a typical crease pattern is like, and the honest answer is that nobody knows, because there is no distribution over crease patterns that anybody has argued for.

The second is that the populations cannot exhibit a particular failure. That is much narrower and much more actionable: it is a claim about one test and one test set, it is checkable, and the repair is local. Reletter the members, or add members that were not lettered by a construction.

The second is what has happened here, and it has happened once before. The patterns a checker is tested on records the moment the crossing sweep was written and the population turned out to contain crossings that nothing had noticed. The same shape, one test earlier.

Why constructions letter well #

There is a mechanism behind “a construction that returns a lettering returns one that works”, and it is not that the constructions are careful.

Two of the four populations letter by derivation. The mesh solver computes fold angles and reads the letters off their signs, so its lettering is the one a physical folding produces and cannot be inconsistent — a motion that reaches the flat state has an order of the layers by having got there. The fold-and-cut construction letters from the skeleton in the same spirit: the perpendiculars and the arcs each have a role, and the roles decide the letters.

The other two letter by search. The printed patterns’ letters were found by enumeration or by a person; the twists’ by propagating the vertex conditions. And that is the case where the guarantee is weakest, because propagation returns whichever solution its branch order reaches first and has no opinion about layers at all. It is exactly where the guarantee failed: the square patch shipped with a lettering forcing a circle of twenty-eight panels for as long as nobody read its crease list that way.

So the sentence should be narrower than it first appears. A construction that derives its letters from a folding cannot produce a contradiction. A construction that searches for its letters can, and did.

The repair, and its limit #

The ladder above now includes the clipped tessellation patches, and with them the layer refusal fires — once.

Once, not five times, and the reason is worth following. Four of the five patches did find a clean lettering, because the builder checks and redraws until it does. So the population has been extended with patterns that were repaired on the way in, by the very test the population is meant to exercise.

That is not a mistake — a construction that knowingly shipped an unfoldable pattern would be worse — but it is a closed loop worth naming. A test set whose members are filtered by a test cannot measure that test.

The only clean escape is a population whose members are not produced by any construction of this collection’s, and the nearest thing available is sheets creased by folding them at random. Their letters come from the folding that made them rather than from a solver, and relettered they run from every draw consistent at eight panels to eleven of forty at forty-one.

The number that should be recorded instead #

There is a quantity the populations could carry that would have made all of this visible without anybody having to think of it, and it is cheap.

For each member: not does its lettering work, but what share of its letterings work. That is one number per pattern, obtained from sixty redraws, and it is a property of the pattern rather than of the construction that produced it. A member at a hundred per cent is a pattern nothing could go wrong with; a member at seven per cent is a pattern whose shipped lettering is one draw in fourteen.

Reported that way, the hexagonal twist would have stood out from the day it entered the population — not as a failure, since its own lettering is fine, but as the member on which every checker that reads letters has only ever seen an easy case.

What sixty redraws can carry #

The number this rung proposes recording is a good one and it is worth saying what it will and will not support, because a share reported to two figures invites comparisons it cannot settle.

At sixty draws, a member whose true share is around ninety per cent has a standard error of about 3.3 percentage points. The printed shelf’s four members come in at 93, 90, 95 and 95 — a spread of five points, which is a shade over one standard error.

So the shelf’s internal ordering is not established by this measurement. Whether the Miura or the waterbomb tessellation is the more fragile of the two is a question sixty redraws cannot answer, and reporting them as 93.3 and 95.0 makes them look separated when they are not.

Separating them properly is expensive. To distinguish two shares 1.7 points apart at that level, two standard errors of the difference, takes about sixteen hundred redraws each — twenty-six times the work, for an ordering nothing in this collection depends on.

Which is exactly the right resolution #

That is not an argument against the number. It is an argument for reading it as what it is.

The hexagonal twist comes in at seven per cent against the shelf’s ninety-odd. That gap is twenty-five standard errors, and it would have been unmistakable at ten redraws, let alone sixty. The measurement that matters here is an order of magnitude, and sixty draws resolve two of them comfortably.

So the right way to carry the number is as a band rather than a figure: everything above about eighty per cent is one class, everything under twenty is another, and nothing sixty draws produces should be read inside a class. The hexagonal twist changes class; the shelf does not move within its own.

That also sets the cost of adopting the proposal. Sixty redraws per member across thirty-eight patterns is a few thousand propagations, which is seconds — and the temptation to raise it to six hundred for a tidier number should be resisted, because six hundred still would not separate the shelf and would cost ten times as much to fail to.

A screening number that is honest about being a screen is worth more than a precise one nobody has the draws to justify, and this is one of the few places where the cheap version is not a compromise but the correct instrument.

The other populations this collection could have #

Three sources of crease patterns exist that are not constructions of this collection’s, and it is worth saying what each would and would not fix.

Published patterns. The field has thousands, and they would be a genuine outside sample — except that reproducing a designer’s crease pattern is not something this collection does, and the ones it may use are traditional bases and patterns published as mathematics, which is roughly the shelf it already has.

Random drawings. Segments placed at random on a square are a distribution somebody can state, and they are useless: they cross almost surely, so the crossing sweep refuses them before any other question arises, and the ones that survive are a strange conditioned sample rather than a natural one.

Sheets folded at random. These are the ones that work. A sheet creased by folding it and folding it again produces a pattern that is developable and flat-foldable because it was physically made, with vertices of every degree and no two alike, and letters that came from the folding rather than from a solver. Nine of them at three depths and three seeds are the closest this collection has to an outside opinion, and relettered they fail at rates between nothing and seventy-two per cent.

They are not in the four populations. Adding them is the obvious next thing and it is not free: a crumple has no name, no printable sheet and no argument attached, so a population of them is a population that can only ever be counted rather than read.

What the ladder’s other zeros mean #

Two of the five refusals report zero on these populations as well, and the same three-way question applies to each.

The vertex pass reports zero because passing it is what put a pattern in the population. That zero is a definition and it would be alarming if it were anything else.

The panel walk reports zero because every member places. That one is a genuine measurement: the members were not selected for closing, several of the twists population’s members famously do not close, and they are caught by the crossing sweep first and so recorded against that instead. So its zero means the cheaper test got there first, which is the third of the three readings and the only benign one.

Distinguishing them takes no work and is not usually done, which is the whole of the argument.

What to do with a zero #

The practical lesson is short and it generalises past this collection.

A test reporting zero on a population is reporting one of three things, and they are not distinguishable from the number alone: that the population has no instances of the fault, that the population cannot have instances of the fault, or that the test is broken. The first is good news, the second is a gap in the test set, and the third is a gap in the test.

Telling them apart costs one deliberate attempt to make the test fire. If it can be made to fire on a modified member, the test works and the population was the problem. If it cannot be made to fire at all, the test is the problem.

Here, sixty redraws of one member took it from zero to fifty-six of sixty. That is the check, and it took a minute — against the alternative, which is a test reporting zero for as long as anybody leaves it alone.