Refused at one lettering

Six developable quadrilateral meshes sit in this collection as its least regular population: nine panels each, no two vertices alike, every one satisfying every condition at every interior vertex. Four of the six have no ordering of their nine panels at all — every arrangement of the pile breaks one of the two non-crossing rules, so every one of those four must pass through itself, established by searching every ordering rather than by driving the mesh and watching for a collision.

That measurement was made at each mesh’s own labelling: the mountain-valley assignment its construction produced.

A mesh has more than one. Enumerating all of them changes two of the four verdicts.

The enumeration #

Each mesh has twelve creases, so four thousand and ninety-six labellings. Sixteen or thirty-two of those satisfy every condition at every interior vertex — the count varies with the mesh’s angles, because the smallest-sector lemma binds differently where the sectors are more or less equal.

For each admissible labelling: build the folded state, check the arcs the letters force for a circle, and where there is none, search every ordering of the nine panels for one that breaks neither non-crossing rule.

Mesh three: sixteen admissible labellings, all sixteen consistent in their arcs, none orderable. Mesh five: thirty-two admissible, thirty-two consistent, none orderable. Mesh eight: thirty-two admissible, thirty-two consistent, eight orderable. Mesh eleven: thirty-two, thirty-two, eight. Mesh fourteen: sixteen admissible, fourteen consistent — two of its labellings do close a circle — and fourteen orderable. Mesh nineteen: sixteen, sixteen, four orderable.

Meshes three, five, eight and nineteen are the four that were refused at their own labelling. Two of those four have labellings that fold.

What the earlier statement was #

It is worth being exact about what was claimed, because the claim was carefully worded and is still misleading.

What was reported is that four of six meshes have no ordering of their nine panels. That is true of the object measured — a mesh together with a labelling — and the essay was explicit that the letters of all four were consistent, which is a statement about a labelling and not about a mesh.

What a reader takes from it is that four of the meshes cannot be folded flat. That is true of two of them and false of two.

The gap is the same one this collection has now met several times: a construction’s labelling is one solution of a search nobody ran, and a verdict evaluated at it is a verdict about that solution. The mesh construction solves a closure condition to place the vertices and then propagates the letters; the letters that come out are whichever ones the propagation reached first, and nothing about them is preferred.

The two that fold at no labelling #

Meshes three and five are the stronger result, and they are the first patterns in this collection to carry it.

Every admissible labelling — sixteen for one, thirty-two for the other — was built, checked and searched. Not one produces an ordering of the nine panels that satisfies both non-crossing rules. So the statement this mesh cannot be folded flat is available for these two, quantified over every labelling the conditions permit, and established by exhaustion rather than by a single case.

That is a much stronger object than the earlier verdict, and it is worth noticing what made it affordable. Twelve creases is four thousand and ninety-six labellings, of which the vertex conditions leave thirty-two; nine panels is three hundred and sixty-two thousand eight hundred orderings, which a search over the constraints cuts to a few thousand nodes. The product is small because both factors were cut first, and neither cut is available on anything much larger.

The two that were rescued #

Meshes eight and nineteen are the correction, and the interesting part is how many labellings work.

Mesh eight has eight of its thirty-two admissible labellings orderable — a quarter of them. Mesh nineteen has four of sixteen, also a quarter. These are not knife-edge cases where one exotic labelling squeaks through; a quarter of the available labellings fold, and the construction happened to hand each mesh one of the other three-quarters.

That is worth dwelling on because it inverts the intuition the earlier result created. A mesh that fails at its own labelling is not thereby a difficult or marginal object. On these two the failure was a one-in-four accident of which labelling the propagation reached first.

What a labelling can and cannot move #

It is worth being precise about which parts of the problem a relabelling touches, because that is what decides whether a rescue is possible at all.

The positions of the folded panels do not move. A flat folded state places each panel by reflecting it across the creases on a path back to a fixed panel, and a reflection does not read the letter — so which panels overlap, and by how much, is settled by the geometry before any letter is written.

What a labelling decides is which of two overlapping panels must be above. Every crease contributes one such statement, and the two non-crossing rules turn the overlaps into constraints on the ordering: a panel may not lie between a crease’s own two panels where the crease’s image passes through it, and two folds wrapping the same edge may not interleave.

So a relabelling reshuffles the directions of a fixed set of constraints. On mesh eight, a quarter of the reshuffles admit an ordering and three quarters do not. On mesh three, none do — the overlaps are arranged so that no assignment of directions to them can be satisfied at once.

That is why the rescue is possible in principle and why it fails on two of the four. The geometry sets the board; the letters play on it.

Sixteen or thirty-two, and what the difference says #

The admissible counts are worth reading before the verdicts, because the two values that appear are not arbitrary and they report something geometric.

Each mesh has twelve creases and four interior vertices. Maekawa halves at every vertex; the smallest-sector lemma halves again wherever a sector is strictly smaller than both its neighbours. So the admissible count is $2^{12-4-b}$, where $b$ is the number of vertices at which the lemma actually bites.

That gives sixteen when $b = 4$ and thirty-two when $b = 3$, and those are exactly the two values observed. A mesh reporting thirty-two admissible labellings is a mesh with one vertex whose smallest sector is tied, and the count says so without anybody measuring an angle.

Three of the six are in that position. Which is a small thing to know and it is free: the census already computes the number, and the number carries a fact about the geometry that nothing else here reports.

A quarter, three times #

The orderable counts are more striking, and the striking part is that they repeat.

Mesh eight: eight of thirty-two. Mesh eleven: eight of thirty-two. Mesh nineteen: four of sixteen. Exactly a quarter, three times, on meshes with different admissible counts. Mesh fourteen is fourteen of sixteen, and meshes three and five are none of theirs.

So across six meshes the two non-crossing rules produce three outcomes and nothing in between: they remove everything, they remove exactly three quarters, or they remove nothing beyond what the arcs had already refused — mesh fourteen loses two labellings and both of them are the two its own letters contradict.

A quarter is what two more independent binary constraints would leave, which is a suggestive shape and not a proof of anything. What it does establish is that the quantity separating these meshes is discrete rather than graded. The essay’s open question — what distinguishes the pair that never folds from the pair that folds at a quarter — is therefore not asking for a threshold on some continuous measure of overlap. It is asking why a mesh lands in one of three buckets.

That is a much better-posed question than the one it replaces, and it changes what a larger population would have to report. Not a correlation between an overlap count and a share, but whether the buckets persist: whether twenty solved meshes also come in at nought, a quarter, and seven eighths, or whether six was too few to see a spread.

If the buckets persist, the constraint count is quantised and the reason will be structural. If they do not, three quarters was a coincidence of three meshes, and that is worth knowing too — it is the kind of regularity that is very easy to build a mechanism for and very embarrassing to have built for noise.

What separates the two pairs #

Nothing this collection can currently measure, and saying so plainly is better than offering a plausible mechanism.

All six meshes are generated the same way: pick a three-by-three grid of quadrilaterals, perturb the vertex positions, and solve for the lengths that make every interior vertex developable and flat-foldable. The seed is the only difference between them. Their panel counts are equal, their crease counts are equal, their vertex degrees are equal, and their independent chains number six each.

The two that fold at no labelling and the two that fold at a quarter of them differ in the geometry of their panels — how much they overlap when folded, and therefore how many of the non-crossing constraints bind — and this collection has no measurement of that quantity. What it has is the verdict.

There is a natural candidate. The two non-crossing rules are statements about overlaps, and a mesh whose folded panels overlap more has more constraints to satisfy; a count of overlapping pairs, or of the constraints they generate, would be the obvious thing to correlate against. Six meshes is not enough to correlate anything, which is the honest reason it has not been tried.

The one mesh where the arcs fire #

Mesh fourteen is the only member of the six whose letters ever contradict themselves, and it does so on two of its sixteen admissible labellings.

That matters because it is the control. The cheap test — read the arcs, look for a circle — fires on none of the six at their own labellings, and a test that never fires is indistinguishable from a test that has stopped working. Two firings out of ninety-six admissible labellings across the population is not many, and it is enough to establish that the instrument is live on these patterns rather than merely silent.

It also fits the earlier finding rather than contradicting it. The claim was that the linear proof and the exponential search answer different questions and neither contains the other, and mesh fourteen shows both halves at once: two of its labellings are refused by the arcs, fourteen are accepted by the arcs and also stack, and the two refusals are among the two the ordering search would also have refused.

What this changes about how a verdict is quoted #

One habit, and it applies to every negative result about a pattern in this collection.

A verdict about foldability is evaluated at a labelling. Where the labelling was chosen by a construction, the verdict is about the pair and should say so. Where the verdict is meant to be about the pattern, every admissible labelling has to be quantified over — which is affordable on twelve creases and not on a hundred and forty-two.

So there are now two grades of negative result here, and they should be named differently. Refused as drawn is what a single-labelling verdict establishes. Refused at every labelling is what an enumeration establishes, and only two patterns in this collection carry it.

The distinction is not pedantry. A designer handed this mesh does not fold would abandon it; handed this mesh does not fold at the letters it came with, and a quarter of its other letterings do, they would relabel it. Those are different pieces of advice and the earlier measurement supported only the first.

The same correction, three times over #

This is the third negative result in the collection to be weakened by asking about labellings rather than about patterns, and the three together make a pattern worth naming.

The first was a tessellation patch reported as having no consistent labelling on the evidence of two hundred draws, which turned out to have many once a search was pointed at it rather than a sampler. There the error was in the method: a sample cannot establish absence.

The second is subtler and is the one this rung shares. A verdict evaluated at the labelling of a pattern reads as a verdict about the pattern, and the definite article is doing work it cannot support — there is no such thing as the labelling of a crease pattern, only the one a construction stopped at.

The third is the direction that survives. Two meshes fold at no labelling, and that statement is quantified properly; a tessellation patch with no admissible labelling at any turn below a threshold is quantified properly too. Both were established by exhausting a space rather than by evaluating a case.

So the correction is not negative results here are unreliable. It is that a negative result has a scope, the scope is whatever was quantified over, and until this round of work the collection had been quantifying over one labelling and reporting about a pattern.

What a designer would do with this #

The practical reading is short and it changes what happens to a mesh that fails.

A mesh generated for a mechanism — a deployable panel array, a folding sheet with hinges — is generated by solving a closure condition, and its letters come out of the same propagation as everything else here. If the solved mesh then turns out to pass a panel through itself, the natural conclusion is that the geometry is wrong and the solve should be run again from a different seed.

What these six say is that a quarter of the time the geometry is fine and only the letters were unlucky. Relabelling is far cheaper than re-solving: sixteen or thirty-two candidates, each checkable in a few thousand nodes, against a numerical solve that takes seconds and produces a different mesh with different dimensions.

So the order of operations should be: solve, check, and if it fails, enumerate the labellings before touching the geometry. On two of the four failures here that would have recovered the mesh in hand.

That advice does not extend to the meshes that collide during their motion rather than at the end of it, where the geometry really is the problem — but it costs nothing to try first, and it is not currently what anybody does.

Why it was not caught earlier #

The earlier measurement was not careless, and the reason it stopped where it did is worth recording because it is the ordinary reason.

Searching the orderings of nine panels is expensive — seven to eleven thousand nodes per labelling on these meshes — and doing it once per mesh was already the most expensive thing in that essay. Doing it once per labelling multiplies by sixteen or thirty-two, which was not obviously worth it when there was no reason to think the answer would change.

What made it worth it was a different question entirely. The instrument built to ask whether a tessellation patch has any consistent labelling at all made enumerate the labellings into a routine operation, and once it is routine the mesh question is a few minutes rather than a decision. That is the usual way a measurement gets extended: not because somebody doubted the old one, but because a tool built for something else made the extension cheap.

Where the ladder goes next #

The immediate question is whether the two meshes that fold at no labelling do so for a reason that generalises. Both are three-by-three meshes with six independent chains; both refuse at every labelling; and nothing here says why. A larger population — twenty or thirty solved meshes rather than six — would say whether folds at no labelling is a common condition or a rarity, and the machinery is already built.

The second question is the one this rung cannot reach. All of this is about flat folded states: whether the nine panels can be stacked at the end of the motion. A mesh that folds rigidly drives its panels through one another somewhere in the middle of its motion on at least one of the six, and that is a separate condition evaluated at every angle rather than at one. Whether a relabelling can rescue a mesh from that refusal is not asked here, and the answer is probably no — the collision happens between panels that share no crease, and a letter does not move a panel.