Crimp it away and ask again

Whether one vertex of a crease pattern folds flat is the first question this subject answers and the one it answers most completely. Two conditions at a point settles it: the sectors have to close to a full turn, the alternating groups have to be equal, the mountains and valleys have to differ by two, and the smallest sector has to be bounded by creases of opposite kind. Four tests, all of them cheap, all of them local. A checker built from them is what runs before every figure on this site is drawn.

That account is complete for a vertex whose sectors are all different sizes, and for nothing else.

The case everybody checks #

At a degree-four vertex drawn at no particular angles the four conditions are not merely necessary. They are sufficient, and the fact is easy to miss because nothing draws attention to it: a labelling that passes them folds, and a labelling that folds passes them, with no exceptions anywhere.

That is worth stating as a measurement rather than as a memory. Take a random developable vertex whose sectors satisfy Kawasaki by construction — two random compositions of a straight angle, interleaved — and put every one of its sixteen markings past two entirely separate machines. The first is the site’s own checker. The second knows nothing about theorems: it treats the paper round the vertex as a closed loop, folds that loop onto a line, and searches every ordering of the four sectors for one that breaks none of the three rules about which layer may lie where.

Over 373 four-crease vertices and 1,492 passing markings, the two machines agree every time. Nothing folds that fails a condition — which is the theorems doing their job — and, more surprisingly, nothing fails to fold that passes them all.

At six creases that stops. Of 3,208 markings that satisfy all four conditions across 329 vertices, 576 have no flat folded state, and they are not confined to strange geometry: 72 of the 329 vertices carry at least one. At eight creases the shortfall is 512 out of 1,440, on 32 of 58 vertices — better than a third of everything the conditions certify.

What the conditions cannot see #

The failure has a shape. Each of the four conditions reads something about the vertex as it lies on the paper: the sum of its angles, the alternating sums, the count of letters, the letters either side of the smallest sector. All four are properties of the drawing.

What decides flat-foldability is a property of the folding. When the smallest sector is folded away between its two neighbours — which is what the big-little-big lemma is about, and why that lemma is the only one of the four that is not about counting — the paper on either side comes together and makes a new vertex, of degree two less, whose sectors are not the sectors of the original.

That new vertex is not on the paper. Nothing about it can be read off the crease pattern, because it does not exist until part of the pattern has been folded. And the four conditions apply to it exactly as they apply to any vertex — including the big-little-big lemma, which can perfectly well be violated there by a pair of creases that satisfied it in the original.

Which is precisely what happens.

And at four creases, where two sectors tie #

The other place the conditions over-count is not at a higher degree at all. It is at a degree-four vertex whose two smallest sectors are equal — sectors a, a, 180° − a, 180° − a — which is the family where the lemma says nothing, and which is where the count of admitted markings jumps from four to eight.

Eight markings satisfy every condition there. Six of them fold. The two that do not are the ones in which the odd crease sits between the two largest sectors, and they fail for the reason above: with no strictly smallest sector there is no forced crimp, and neither of the moves that are available leads anywhere.

That is measured at 30°, 60°, 80°, 89° and at every other tied angle tried, and the answer is always eight and six. It corrects a number this site has published: the doubling at a tie is a doubling of what the conditions admit, and the number of objects a folder can reach at such a vertex is six.

The hypothesis is therefore not about degree. It is: a vertex whose sectors are all different sizes is decided by the four conditions, and a vertex with a tie anywhere in it need not be. Degree four with distinct sectors is the case everybody checks, and it is the only case in which the sufficiency has ever been safe.

The decision is a procedure, not a test #

So the answer at higher degree is a reduction. Find a sector strictly smaller than both its neighbours; check that the creases bounding it differ; fold it away; ask the same question of the smaller vertex. Repeat until two creases are left, which fold if and only if they are collinear and carry the same letter — a single straight crease through the point.

That is a different kind of answer from a condition. A condition can be evaluated at the vertex, in one pass, by anything that can measure an angle. A procedure has to be run, it produces intermediate objects that were not in the input, and its verdict on a vertex may depend on something that only becomes visible three steps in.

Against the exhaustive stacking search — a permutation search over the sectors that shares no line of code with the reduction — the two agree on 6,256 vertex-and-labelling pairs, at degrees four, six and eight, with no disagreement anywhere. The reduction is not an approximation to the answer; it is the answer, arrived at by a different route.

The word procedure is doing real work in that sentence, and it is worth separating from the ordinary sense in which any check is a procedure. The four conditions are a predicate: they take a vertex and return a verdict, and every intermediate value they compute is a number about the vertex handed in. The reduction is not. Its intermediate values are vertices, each of them a legitimate object with its own sectors and its own letters, none of them present in the input, and its verdict on the original is a verdict about the last of them.

The refusals are shallow, which is the practically useful part. At degree six, all 576 gap cases are refused after one crimp. At degree eight, 192 are refused after one and 320 after two. Nothing in the census needed a third.

The shortfall decomposes by crimp #

The three refusal counts look like a rising trend and they are better than that: they factor, and the factors say where the reduction’s verdicts actually come from.

At degree eight, 192 of the 1,440 certified markings are refused at the first crimp — 13.3% — and of the 1,248 that survive it, 320 are refused at the second, which is 25.6% of what reached that stage. Composing the two survivals gives $0.867 \times 0.744 = 0.645$, and $1 - 0.645 = 35.5\%$ against the measured 512 of 1,440. The shortfall is not one phenomenon; it is a hazard applied once per crimp.

That decomposition is what makes a comparison possible, and the comparison is the interesting part. A degree-eight vertex after one crimp is a degree-six vertex. If the crimped ones were like the random ones, its second-crimp refusal rate should match the degree-six census’s 576 of 3,208, which is 18.0%. It is 25.6% — half again as high.

Which says the intermediate vertices are a different population #

So the vertices the reduction manufactures are not drawn from the same distribution as the vertices anybody draws, and the crimp says why.

Folding a sector away merges three sectors into one of $\textit{prev} - \textit{here} + \textit{next}$, and since here is the minimum, the merged sector is at least as large as either neighbour. Every crimp therefore replaces two moderate sectors with one large one, and a vertex with one conspicuously large sector is a vertex whose remaining sectors are crowded — which is exactly the geometry in which a smallest sector is likely to find matching letters around it.

The reduction’s intermediate objects are systematically more lopsided than its inputs, and lopsided vertices refuse more often. That is a sharper statement of what the four conditions cannot see than “a vertex that is not on the paper”: the vertex that is not on the paper is also drawn from a worse population than the one that is, and the bias grows with every step.

And a number the next census can refute #

The composition gives a prediction rather than a trend line. A degree-ten vertex takes four crimps, and if the per-stage hazards continue anywhere near 13%, 26% and upward, its refusal rate must exceed one half — as does the cruder reading, since 0%, 18% and 36% at degrees four, six and eight extrapolate to 53%.

Two routes, one answer, and the census that would settle it is the one already written. Above degree eight, the four conditions certify more markings than they decide.

Where the reduction has nothing to work with #

The rule asks for a sector strictly smaller than both its neighbours, and the word doing the work is strictly — which is the same word that leaves the big-little-big lemma silent at exactly the vertices origami actually uses.

Halving produces equal angles by construction, so the standard bases have ties everywhere. At a vertex with ties there is no forced move at all: every sector that is no larger than both its neighbours is a candidate, only those whose bounding creases differ can be taken, and the reduction has to try them in turn.

That is a sharp division and an unexpected one. At generic angles the reduction is forced at every step and never searches. At the angles a folder actually produces, it searches at every step. The preliminary base’s four equal sectors branch on all sixteen of their markings; the census’s random degree-six vertices branch on none of theirs.

And ties do not always cost. A vertex whose sectors are all equal has no gap at all: at the Yoshimura pattern’s 60° six-fold vertex, thirty markings pass and thirty fold, and at the preliminary base’s eight 45° sectors, 112 pass and 112 fold. So a tie is not the villain either — what matters is whether the ties leave the reduction with a move that works, and at a completely equal vertex every candidate is available at every step.

The picture that holds across everything measured is this. Sectors all different: decided, at every degree tried. Sectors all equal: decided, at every degree tried. Sectors partly tied: not decided, and the shortfall runs from two markings at a four-crease vertex to twelve at the degree-six vertex a waterbomb repeats.

Which theorem was checked, and how #

Every number above comes from two computations that were required to agree and that were written to have nothing in common.

The conditions are computed by the site’s own flat-folding checker, unaltered: the same developability, Kawasaki, Maekawa and big-little-big code that every figure on this site is gated on. The vertex is built as a star of creases inside a square sheet and handed to the ordinary checker, so what is being reported is exactly what the site’s own machinery says.

The answer is computed by enumeration. The paper immediately round a vertex is a small circle, the creases cut it into arcs whose lengths are the sector angles, and folding flat maps that circle onto a line. So a vertex is the strip problem wrapped into a loop, with one extra crease joining the last sector back to the first, and the same three non-crossing rules apply: the letters decide which of a joined pair is on top, no sheet may pass through a fold, and two folds facing the same way may nest or stand clear but may not interleave. Every ordering of the sectors is tried.

The two are asked to agree on every vertex, at every degree, on every labelling — not on a summary. A gap in a total is a report; a gap at one labelling with two machines naming it is a fact.

There is one trap in that, and it is the kind that passes every test. The folded positions can be computed for any angles at all — the arithmetic does not object — and the three rules will happily have opinions about the result. A search then returns orderings of a figure whose two ends are in different places. So the loop is checked for closure before anything is enumerated, and an unclosed loop returns no stacking whatever its letters say. Closure is Kawasaki, arriving as a geometric fact rather than as a theorem to be quoted.

Where the model stops #

The idealisation is the usual one and it is worth naming: the paper has no thickness, so a stack of sectors is a stack of lines and two of them may sit arbitrarily close together. Real paper does not permit that, and at a vertex where a dozen layers converge the difference is visible to the hand long before it is visible to the arithmetic.

More importantly, this is still one vertex. Nothing here says anything about whether a pattern folds, and the distance between the two questions is the whole of local is not global: a reduction that decides every vertex of a sheet decides nothing about the sheet, because the vertices share creases and their reductions do not know about one another. Deciding the sheet is NP-hard and no procedure of this kind will settle it.

What the reduction does buy at the pattern level is a stronger filter. A checker built from the four conditions certifies markings that cannot be folded, so it over-counts — by nothing at degree four, by 18 per cent at degree six, and by a third at degree eight of everything it passes.

That matters here in a specific and slightly uncomfortable way, and it matters most for the patterns a folder actually makes, because those are the ones whose angles are halved and halved again until everything ties something. This site’s gate has been reading the four conditions since its first commit, and on every pattern it has ever passed with a vertex above degree four it has been reading a necessary condition and reporting a decision. Nothing shipped is wrong — the assignments the patterns carry were found by searches that ask for a folded state, not by taking the first labelling the conditions accepted — but the gate was doing less than it appeared to, and the amount less is now a number rather than a worry.

What the picture cannot show #

The figures draw the vertex flat, as a fan of creases, and every frame after the first draws something that is not on the paper. There is no honest way to show a crimped vertex as a folded object without drawing the fold, and drawing the fold would make the sequence a sequence of pictures of paper rather than a sequence of vertices — which is exactly the confusion the reduction exists to avoid. So each frame is a pattern, and the reader is asked to remember that only the first one is a crease pattern.

The stacking searches are not drawn at all. There is nothing to see: a legal ordering is a permutation, and forty thousand of them tested is a number rather than an image.

Who found it, and when #

The reduction is Thomas Hull’s, from his work on the combinatorics of single-vertex flat folds in the 1990s, and the ingredients are older: the crimp as a move is folklore, and the observation that the smallest sector must be crimped first is Jacques Justin’s lemma of 1986. Kawasaki’s condition and Maekawa’s count are both from 1979 and both carry names that are not the names of the people who proved them, which is the ordinary condition of this subject rather than an exception.

What is measured here is not the theorem. It is the size of the gap the theorem closes, on random vertices and on the patterns this site actually prints — and that measurement is available only to a site that runs both machines and compares them.

Where the ladder goes next #

Two things follow directly. The reduction produces, on a failure, a short object: the smaller vertex at which the crimp was unavailable, one or two steps in. That is a refutation somebody can check in a moment, and the complexity of a no is exactly the place to ask what happens to that when the question is asked of a whole sheet.

The other is the question this essay has carefully not answered. A marking that folds has a folded state; it may have several, and how many is a property nobody quotes. That is the next rung, and the answer separates degree four from everything above it for a second, unrelated reason.