The vertices nobody checks

Every crease pattern published here has been past four conditions at every vertex: the angles round the vertex must close, Kawasaki’s alternating sums must be equal, Maekawa’s counts must differ by two, and a strictly smallest sector must be flanked by creases of opposite letters. A pattern that fails any of them is refused rather than drawn, which is the arrangement that makes it reasonable to hand a reader a sheet and an instruction.

The sentence has a word in it that has been carrying more weight than it looks capable of. The conditions are checked at every interior vertex — every vertex with paper all the way round it. A vertex where a crease reaches the edge of the sheet has paper on one side only, and every one of the four conditions is silent there: the sectors do not close, so developability has nothing to compare; there is no cycle, so there is nothing to alternate round; the folded cross-section never comes back, so there is no turn for Maekawa to count.

Silence has been read as nothing to check. It is not. It is a different object, and the different object has a condition of its own.

The reduction, which is the whole rung #

Walk round an interior vertex and the sectors close: the last one meets the first, and every theorem in the subject is a statement about that ring. Walk round a boundary vertex and the walk stops. It begins at one edge of the paper, crosses the creases in order, and ends at the other edge — so the sectors are a line of segments with the creases as the points between them.

That is a strip. A one-dimensional crease pattern is a length of paper with creases at given points and a letter at each, and whether it folds flat is decided by three things kept separate here since the collection was young: where each segment lands, which is fixed by the crease positions alone; which layer sits above which, which the letters fix for each consecutive pair; and whether any two pieces of paper are asked to pass through one another.

Take the fan’s angular span as the strip’s length and each crease’s angle as a position along it. A sector of the paper becomes a segment; folding the fan flat sends each sector to an interval of the span exactly as folding a strip sends each segment to an interval of its length; and two sectors overlap in the folded fan precisely when the two segments overlap in the folded strip. The two questions are one question.

And the strip condition is not a necessary condition that happens to be checkable. In one dimension the stacking problem is solved rather than approximated, so the answer is exact — which is the reverse of the situation at an interior vertex, where four necessary conditions leave a gap that a whole essay is about.

How many of them there are #

The obvious defence is that this is a corner case. It is not a corner case; it is most of the vertices, which is a fact the edge of a folded model has already hinted at from the other side.

Counted over the eight patterns on the printable shelf: 105 vertices on the edge of the paper against 92 inside it. Twenty-seven of the boundary ones carry two creases or more, which is where the condition has something to say at all — a single crease reaching the edge folds whichever way it is lettered, and nothing is being decided.

The proportion is not an accident of what is on the shelf. Over four different ways of producing a crease pattern — the printed patterns, twist tessellations, quadrilateral meshes solved to fold, and fold-and-cut patterns — the share of vertices on the paper’s edge runs from 52 to 86 per cent. A crease pattern is mostly edge, and the reason is arithmetic rather than deep: creases run until something stops them, and what stops them is usually the paper.

What the condition says when it says anything #

At a fan of two creases spanning a straight angle, all four letterings fold. At three creases they mostly do. The condition bites where a sector is strictly smaller than both its neighbours — and the reader who has met the big-little-big lemma will recognise that immediately, because it is the same statement.

It is the same statement about a different object. The lemma at an interior vertex says a strictly smallest sector must be flanked by opposite letters, or the panel on one side has nowhere to go. On a strip the identical thing happens: a short segment between two long ones is a piece of paper that has to be crimped away, and if both of its creases turn the same way the layers collide.

So the constraint is not weaker at the edge of the paper. On the fan drawn above — sectors of 40°, 60°, 20° and 60° across a straight angle — exactly four of the eight letterings fold, and the four that do are the ones that crimp away the 20° sector.

How much the silence was worth #

The sweep gives enough to price the gap rather than describe it.

A fan of three creases has eight letterings. Where no sector is strictly smallest between two larger, all eight fold; where one is, four do. Fifty of the sixty-one swept positions are constrained, so across the sweep the share of letterings that fold is

$$\frac{50 \times 4 + 11 \times 8}{61 \times 8} = \frac{288}{488} = 59\%.$$

Two letterings in five are refused by a condition nothing was checking. Not a rare configuration inside a rare fan: the ordinary case.

The halving itself is exact and has a reason. A strictly-little sector forbids two of the four ways to letter the pair of creases bounding it — the two that match — and leaves two. So one little sector halves, and the count at the first round is $2^{d-L}$ for a fan of $d$ creases with $L$ little sectors. At the fan drawn above, $d = 3$ and $L = 1$: four of eight, which is what folding it gives.

Two little sectors can never be adjacent — neither can be strictly smaller than the other — so they never share a crease and their halvings are independent. That is why the arithmetic is a clean power of two rather than an inclusion-exclusion.

It is a first-round count and not the final one. Crimping away a little sector brings its neighbours together, and the merged sector may be strictly smaller than its neighbours, so a fan can be refused at the second round having survived the first. $2^{d-L}$ is therefore an upper bound, and the strip solver is what turns it into a verdict.

Which puts a number on the shelf #

The shelf carries twenty-seven boundary vertices with two creases or more. If each behaved like the swept three-crease family — which is a supposition and is flagged as one, since the fans on the shelf differ in degree and span — a lettering clearing every one of them by luck would arrive with probability $0.59^{27}$, or about one in a million and a half.

The published patterns all fold, checked one at a time. That number is not a claim that they should not; it is the size of the coincidence that a gate never firing on twenty-seven objects it does not examine is a gate that was never exercised on them.

The witness #

A count is an argument that something might have been missed. A witness is an argument that something was.

The pair is deliberately small and deliberately ordinary. The interior vertex is the standard flat-foldable degree-four one: sectors in supplementary pairs so Kawasaki holds by construction, with its letters found by enumeration rather than stated, which is how every pattern here is built. The sector lengths play no part in any of it, which an earlier rung established and which is why a fan can be described by angles alone. On the bottom edge sits a fan of three creases at 40°, 100° and 120°, and the two versions differ in whether that fan reads MMM or VMV.

The checker that gates every figure on this site passes both. Paper folds one of them.

That is not a large defect and it is a real one. Nothing already published is wrong — every one of the 105 boundary vertices on the shelf folds, checked one at a time — but the reason nothing is wrong is that nobody drew a pattern whose boundary fan happened to be constrained, and a gate that has never fired on a class of object it does not examine is a gate that has not been tested on it.

Which theorem was checked, and how #

Three separate things are asserted here and none of them is asserted by the same code.

The reduction is checked by the solver that knows nothing about vertices. The strip’s foldability is decided by the layer machinery written for one-dimensional patterns years before any of this, which has no notion of a fan, a sector or a sheet. It is handed a length, a list of positions and a list of letters.

The count is checked over the whole shelf, not over a chosen pattern: every vertex of every printed pattern is classified, and the totals reported are the totals.

The witness is checked in both directions. The bad pattern must be refused by the strip condition and accepted by the four conditions; the repaired pattern must be accepted by both. Either half alone would be a weaker claim: a pattern refused by everything is just a bad pattern, and a pattern accepted by everything proves nothing.

There is a refusal too, and it matters more than it looks. A fan is only a fan when the creases at the vertex all lie on one side of the paper’s edge. Creases on both sides mean the drawing has run off the sheet, and the reduction refuses that case rather than folding it — because a span computed from a vertex that is not a boundary vertex is a number about nothing.

What it costs to add the check #

Nothing, which is the least satisfying part of the finding and the part most worth stating.

The strip solver was written for a different question and has been in the collection since its second phase. Deciding a boundary fan is one call to it per vertex, over a list of positions no longer than the vertex’s degree, and the sizes involved are the sizes a folder meets: a fan of six creases is sixty-four letterings and a fan of ten is a thousand, both of which are exhausted faster than the drawing they belong to is laid out.

So the gap was never a matter of cost. It was a matter of nobody asking, and the reason nobody asked is that the four conditions are so completely the vocabulary of the subject that a vertex they do not apply to reads as a vertex with nothing to say about it. That is a fact about how a field’s language shapes what gets checked, and it is the kind of gap that stays open precisely because every individual step in it is correct.

Where the model stops #

One more necessary condition is not a decision procedure. A pattern all of whose vertices pass, boundary and interior alike, may still fail to fold: the global question is intractable and is nobody’s to solve here. What has changed is that a class of vertex which was being skipped is now examined, which moves the boundary of what is checked and not the boundary of what is decidable.

The condition is exact only at the vertex. In one dimension the layer problem is solved completely, so the verdict on a single fan is a decision rather than a filter. Two fans on the same sheet can each fold and still disagree about the order of the layers between them, which is the same gap the interior conditions leave arriving in a new place.

A corner is a fan too, and a fan of any span. Nothing above assumes a straight angle: a vertex at the corner of a square sheet has a fan of ninety degrees, a vertex on a fold-and-cut outline has whatever angle the outline gives it, and the reduction takes the span from the sheet rather than assuming one.

What the picture cannot show #

A fan drawn on the page shows sectors, and the thing being decided is a stack. Whether two sectors collide when the paper closes is a fact about layers lying over one another, and a flat drawing of the unfolded fan has no layers in it at all. The figures here show the input to the decision and never the decision itself, which is why the strip is drawn beside the fan: the strip’s folded state is the picture with the layers in it, and it belongs to a different figure.

Nor can any of these figures show that the reduction is exact. Exactness is a statement about every case, and a picture is one case. What the drawing can do is make the translation obvious enough that a reader can check it themselves on a sheet of paper in a minute, which is the whole reason the fan and the strip are side by side.

The idealisation, named #

The sheet has no thickness, so a sector may lie on another with nothing between them and a stack of any depth closes flat. The creases are lines with no width, so a fan of three creases at 40°, 100° and 120° has a sector of exactly twenty degrees rather than twenty degrees less two crease widths. And the paper does not stretch, which is what makes the angular span a conserved quantity and therefore what makes the strip’s length meaningful.

The third is the one this argument leans on hardest. If paper stretched, the fan’s sectors would not add to a fixed span, the map from fan to strip would not be an isometry, and the whole reduction would be an analogy rather than an identity.

The generalisation #

The useful statement is not about paper.

A theorem that is silent about a case is not a licence about that case. The four conditions do not fail at a boundary vertex; they are not defined there — a distinction the essay on what a checker cannot check makes about the sheet and which turns out to apply to a single vertex as well. A checker built from them inherits the silence and reports it as a pass, and the pass is indistinguishable from a verdict unless somebody goes and looks at what the domain of each condition actually is.

That failure has a shape worth carrying: the code was right, the theorems were right, the tests passed, and the gap was in the quantifier. At every interior vertex is a smaller statement than at every vertex, and the difference lived in one word in a function name for the whole life of the collection.

Who found it, and when #

The boundary vertex is not new mathematics and nothing above is a discovery. That a one-dimensional flat-folding problem is decidable in linear time has been known since Arkin, Bender, Demaine, Demaine, Mitchell, Sethia and Skiena studied simple foldability, and the reduction from a fan to a strip is the kind of observation that gets made in a seminar rather than published.

What is new here is the measurement: that the shelf carries more boundary vertices than interior ones, that fifty of sixty-one three-crease fans are constrained, and that a pattern exists which every check on this site passes and paper refuses. Measurements of that kind are statements about a population, and which population they are about is a question worth asking of every one of them. Those are facts about this collection and its machinery, and they were available for the asking from the day the layer solver was written.

Where the ladder goes next #

Two rungs are visible from here.

The first is the cut, which turns interior vertices into boundary ones on purpose, and which is where the arithmetic of released conditions gets its own essay — the vertex a cut makes has paper all the way round it and no ring to close, so it is a fan of a full turn rather than of a straight angle, and what a cut buys can be counted in conditions rather than described in words. The second is what happens when two boundary fans on the same sheet have to agree — the one-dimensional condition is exact at each of them and says nothing about the pair, which is where the local always stops and where a sheet stops being a list of vertices.