The lettering that folds nowhere

Local is not global is the oldest warning in this subject and it is usually delivered with a constructed example: here is a pattern whose every vertex passes and which does not fold, drawn to make the point. How little the conditions decide sharpened it into a measurement of what the conditions entail — fix one crease and three others follow out of a hundred and fifty-eight.

Neither of those says how big the gap is. This does, and the way to measure it is to take a pattern small enough to enumerate, write down every mountain-and-valley labelling of it, and sieve twice: once by the conditions at every vertex, and once by whether the panels can be put in an order at all.

The second sieve takes 248 of the 256 away.

The second sieve #

The first sieve is the one this site has run since foundation: developability, Kawasaki, Maekawa and the big-little-big lemma, evaluated at every interior vertex. All four are conditions at a point, and a pattern passes when every point passes.

The second is the ordering of the panels, and it is a condition about the whole sheet. A crease says which of its two panels is higher; a panel may not sit between the two panels of a crease whose folded image runs through it; two creases in the same place may not interleave. Search the orderings of the panels for one that satisfies all of that, and either find one or exhaust the search.

The negative answer is a proof and the positive answer is not. Every one of those three rules is necessary — a crease where the paper turns the other way is not the crease that was drawn — so a search that exhausts the tree and returns nothing has shown there is no flat folded state. A search that returns an ordering has shown there is a candidate, which is all anything on this site ever claims, because deciding the general question is NP-hard.

That asymmetry is what makes the census worth running. The interesting number is the one the proofs produce.

Four patterns, counted exhaustively #

The preliminary base is the control and it comes out perfect: every lettering the conditions admit folds. That is not luck. The preliminary base has one interior vertex, and a single vertex’s flat-foldability is exactly what Kawasaki and Maekawa decide — there is no second vertex for a layer to be caught between, so the local conditions are the whole question and the second sieve has nothing to do.

The fold-and-cut triangle has one interior vertex too, and it loses twelve of thirty. The difference is that its creases run past each other in the folded plane: the straight skeleton’s arcs and the perpendiculars dropped from them land across one another, and a labelling can put a panel where it has to pass through a fold.

The square twist has four interior vertices and loses almost everything: eight of 256, three point one per cent.

The hexagon twist has six, and it cannot be counted whole — eighteen creases is 262,144 letterings and each survivor costs a search of its own. What can be counted is a slice, and a slice is a measurement rather than a sample as long as the slice is stated. Fix the six creases of the central hexagon to run as three of one letter and then three of the other, and enumerate the twelve pleat creases: sixty-four letterings satisfy every condition at every vertex, none of them forces a loop, and exactly one has a folded state.

The slice, and why it is not a sample #

The hexagon twist is where the honest procedure and the tempting one come apart, and it is worth being explicit about which was run.

The tempting procedure is to draw a few hundred letterings at random, count how many fold, and quote the fraction. That would be a sample, and a sample of a set this thin has an error bar wider than itself. Worse, the draws would not be uniform: they would come out of a search whose branch order decides them, which is a sampler over solutions and not a uniform one over them.

The procedure actually run fixes part of the lettering and enumerates the rest exhaustively. Six creases of the central ring are held at three of one letter followed by three of the other; the twelve pleat creases are enumerated whole, all 4,096 of them; sixty-four survive the vertex conditions, none of the sixty-four forces a loop, and one of them has an ordering. Every number in that sentence is a count and none of them is an estimate. What is not claimed is a figure for the whole pattern, because nothing here counted the whole pattern.

The slice is chosen for a stated reason and not for its answer. Three-and-three is the ring shape all eight of the square twist’s survivors have, so it is the shape the builder’s search tries first, and the reason it tries that shape first is that the same shape works on the triangle twist and the pentagon twist as well. Finding a survivor in it on the hexagon confirms the pattern rather than establishing it.

What this site had drawn #

The square twist printed here carried the lettering MMMM MVMV MVMV, and it has no folded state. Neither did the hexagon twist. Both had been on the printed shelf for years, at true scale, with a note recommending the square twist as the first one to fold.

The reason is not that nobody checked. It is that the thing checked was the first sieve, which both patterns pass, and the tie among the 256 survivors was then broken by something else. the twist builder searched specifically for a lettering whose central ring reads as one letter, on the stated grounds that a twist looks like a twist that way. the fold-and-cut builder took whichever candidate sat in the middle of its enumeration.

Both preferences are about appearance. Neither had any way to be wrong, because nothing downstream of them asked the question they were answering badly.

Thirty-two of the 256 have a uniform ring, and not one of them folds. The eight that do fold all have a ring that changes exactly twice: two mountains and two valleys, in adjacent pairs. The thirty-two with four changes — alternating letters all the way round — fold nowhere either.

So the criterion was not merely unrelated to foldability. It selected precisely the family that has none.

How fast sufficiency decays #

The census reports four survival rates and they can be read as a curve rather than as four facts.

The preliminary base keeps everything: a hundred and twelve of a hundred and twelve. The fold-and-cut triangle keeps eighteen of thirty, which is three fifths. The square twist keeps eight of two hundred and fifty-six, one part in thirty-two. The hexagon twist’s slice keeps one of sixty-four.

Take the fourth root of the square twist’s share and the sixth root of the hexagon’s — the geometric per-vertex figure — and both come out near a half: 0.42 and 0.53. So on the two multi-vertex patterns, each interior vertex costs roughly a factor of two in sufficiency, and the second sieve’s bite compounds with the vertex count rather than growing with it.

The preliminary base is the exception and it is exempt for a reason rather than by luck. Its eight creases all run from the single interior vertex out to the paper’s edge, so in the folded plane no crease’s image crosses another’s interior, and the rules that need crossings have nothing to fire on. A pattern whose creases never run past one another has a second sieve that does nothing at all — which is why one vertex is not the explanation and no crossings is.

The eight that fold have a shape #

The eight survivors of the square twist are described by their ring taking two letters in adjacent pairs, and counting the rings with that shape says how the eight are distributed.

A four-crease ring changing letter exactly twice, with the changes not adjacent to one another, is two arcs of two: MMVV and its three rotations. That is four ring letterings — swapping every letter maps the set to itself, so the complements add nothing new.

Four rings and eight foldings means two admissible pleat letterings for each ring, which is a much more informative statement than a bare count of eight. It says the ring is nearly the whole decision: choose which pair of the central square’s sides are mountains, and the twelve pleat creases are settled to within one remaining bit.

That also explains why the old criterion failed so completely rather than partially. It fixed the ring at zero changes, which is not one of the four; every one of the thirty-two letterings compatible with it was therefore outside the survivors before any pleat was considered. The choice that mattered was made first and made wrongly, and the two hundred and forty-eight refusals downstream of it were all consequences of a single decision about four creases.

Why a uniform ring cannot work #

Four of the thirty-two fail for a reason that can be read off the pattern without any search, and the reason is a loop.

Every crease settles which of its two panels is higher, so the letters give a directed graph on the panels with one arc per crease. A cycle in that graph is a proof of failure and it costs one pass over the crease list. The printed square twist has one, of length eight:

the first pleat is below the corner between it and the second, which is below the second pleat, which is below the next corner, which is below the third pleat … and round to the first.

Each corner region sits between two consecutive pleats — above one and below the other — and going round the four pleats the requirement closes on itself. No ordering of nine panels can satisfy it, and the sheet has nothing to do with paper: it is four inequalities in a ring.

The other twenty-eight fail further in, on the non-crossing rules rather than on the letters alone, and those need the search. But the cycle is the useful half, because it is the only test here that scales. Enumeration refuses past a few dozen panels and a cycle check does not, so on a Miura of twenty-four panels or a Yoshimura of sixty-five, a loop in the forced order is the only negative answer available at all.

What was changed, and what was not #

The pattern builders now choose the lettering by whether a folded state exists, and break the remaining tie by appearance rather than the other way round. The square twist goes out with two letters on its ring; the triangle and pentagon twists likewise; the hexagon twist too. The fold-and-cut triangle is chosen from the eighteen that fold instead of from the thirty that pass.

Nothing about the geometry changed. The sectors are the same sectors, fixed by Kawasaki as an identity rather than by design; the twist radius is the same; the printed sheet is the same size with the same creases in the same places. What changed is which of them are mountains.

And nothing about the earlier counting changed either. The 256 admitted letterings are still 256, they still fall into sixteen pieces no local change can cross, and the symmetries they keep are the symmetries they kept. Every one of those measurements was about the set the first sieve produces, and the set is unchanged. What was wrong was the single member of it that got printed.

The cost of the second sieve #

There is a reason the second sieve was not run for years, and it is not only that nobody thought of it.

The first sieve is arithmetic at a vertex: sum four angles, compare two alternating sums, count mountains against valleys, look at the smallest sector. It costs microseconds, and it is independent at each vertex, so a pattern with a thousand vertices costs a thousand times one vertex.

The second is a search over orderings of the panels. It costs a tree, the tree is factorial in the panels, and nothing about it is local — a panel at one corner of the sheet can be constrained by a crease at the other. On the square twist the search expands about nine thousand nodes; on the hexagon twist about 1.2 million; on a Miura of twenty-four panels it does not finish at any budget worth spending.

So the second sieve is available on small patterns and unavailable on large ones, and the printed shelf sits either side of that line. That is the honest position and it is the one the shelf now reports: three patterns with an ordering found, one with two, four undecided, and none with a proof of failure. Undecided is a real state and it is printed rather than rounded up to “verified”.

Three things this does not say #

It does not say the vertex conditions are worthless. They are necessary, they are cheap, and on a one-vertex pattern they are the whole answer — 112 of 112 on the preliminary base. What the census measures is how fast their sufficiency decays as vertices are added, and the answer is: at one vertex, not at all; at four, to three per cent; at six, to under one.

It does not say a pattern with a folded state folds. The search’s positive answers are candidates. The three rules it checks are necessary and this file makes no claim that they are sufficient, which is the same footing as everything else here.

And it does not extend to the patterns anybody actually folds. Four patterns were counted exhaustively and their creases number six, eight, twelve and eighteen. The Miura has thirty-eight and the Yoshimura eighty-six; two to the eighty-sixth letterings is not a set anything can sieve. What is available there is the cycle check, which is a proof when it fires and silence when it does not.

The count that did not change #

Worth stating once more, because it is the thing most likely to be misread from the table above: the middle column is not wrong.

Two hundred and fifty-six is the number of the square twist’s letterings that satisfy every condition at every vertex. Every essay on this site that quotes it quotes it correctly, every measurement made over that set is a measurement over the right set, and the conditions those letterings pass are exactly the conditions the subject states. Nothing about the first sieve has been revised.

What has changed is the interpretation attached to a single member of it. A lettering drawn from the 256 was being described as a crease pattern that folds, and that description was never earned by the arithmetic that produced it. The gap between “satisfies the conditions” and “folds” was in every essay in words; it was in no code until now.

What a folder should take from it #

A crease pattern with mountains and valleys marked on it is not a promise. The marks can satisfy every theorem in the subject and describe nothing that can be made out of paper, and the failure is not visible in the drawing — it is four inequalities that close on themselves, or a panel that would have to pass through a fold.

A twist whose ring reads as one letter is the specific thing to distrust. It is the version that gets drawn, it is the version that looks right, and on the square twist it is thirty-two letterings none of which folds.

And a pattern that has been checked has been checked against something. The useful question about any verified crease pattern is which sieve it went through. This site’s answer, for years, was the first one; it is now both, on every pattern small enough to ask.