Every move leaves the verdict

A folder holding a sheet that will not close has one obvious thing to try. Find the vertex that is fighting, push it through the other way, and see whether the model settles.

That gesture has an exact description on the crease pattern. Pushing a vertex through flips the letters on two creases meeting at that vertex — never one, because flipping one crease alone changes the mountain-valley count at both of its ends, and the count has to differ by two at every interior vertex. Two creases at a shared vertex is the smallest change that leaves the lettering admissible, and it is the only change a folder makes without unfolding the model first.

So the interesting question is what that move can reach, and the answer turns out to be almost nothing — twice over, in two different ways, and the second one is not the one this collection had already measured.

How rare a legal move is #

The first scarcity is already on the record and is worth restating because the second one is measured against it.

A patch of two hundred and eighty-two creases has seven hundred and fifty-six pairs of creases meeting at an interior vertex, so seven hundred and fifty-six candidate moves. Eight of them survive. The other seven hundred and forty-eight produce a lettering that fails a condition at one of the two vertices involved — usually the count, occasionally the smallest-sector lemma.

On the smallest patch the number is worse than eight. It is nought: two hundred and sixteen candidates, not one of which leaves an admissible lettering. That patch’s lettering is completely frozen, and a folder pushing at it would find every vertex refusing to go through.

That scarcity has a known cause. No move that survives the conditions ever changes a buried crease — a crease with an interior vertex at each end — and on the rhombille two hundred and twenty-two of the two hundred and eighty-two creases are buried. Almost every candidate move touches one, and touching one is fatal.

The second scarcity #

Take the eight moves that do survive, and ask what they do to the other question — whether the arcs the letters force close a circle.

The rhombille’s own lettering closes a circle of sixteen panels. All eight of its legal moves leave it closing a circle. A lettering found by search has no circle in it, and all eight of its legal moves leave it with none.

The same on every other patch. Three legal moves on the elongated, four on the hexagonal, four on the triangular; nineteen moves in all across two letterings of each of the five patches, and every one of the nineteen preserves the verdict.

This is not what the move was built to do. The move set was defined by one criterion — that the result still satisfies the four conditions at every vertex — and those conditions are statements about angles and counts at a point. The circle test is a statement about a closed chain of panels that may cross the whole sheet. Nothing in the definition of the move mentions it.

Why the move cannot cross #

The reason is short and it is the same reason a contradiction is never local.

A move flips two creases at one vertex, so it reverses exactly two arcs, and both of those arcs lie between panels that meet at that vertex. For a circle to appear or disappear, some closed chain of panels has to change from all-one-way to not, or the reverse — and a chain that runs through this vertex enters and leaves it, so it uses both flipped arcs or neither.

If it uses neither, nothing about it changed. If it uses both, they were reversed together, and reversing two consecutive arcs of a chain that was going round consistently leaves it not going round consistently only if the two arcs were consecutive in the chain — which is exactly the sub-chain round the vertex itself.

And that sub-chain is the four or six panels round one point, which Maekawa’s count has already closed: no admissible lettering sends them round in a circle, before or after any move. So the only chain a move could have flipped is the one chain that can never be flipped, and every other chain in the pattern is untouched.

The argument predicts the measurement, which is what makes the measurement worth having: nineteen for nineteen is not an accident of which patches were chosen.

Counting the candidates properly #

The two thousand nine hundred and sixty-four is worth unpacking, because a ratio of nineteen to that is the kind of number that invites a shrug and it should not.

A candidate is a pair of creases sharing an interior vertex. That is not the same as a pair of creases that happen to be adjacent in the drawing, and it is not the same as a pair sharing a boundary vertex either — a vertex on the rim of the sheet has no condition evaluated at it, so flipping two creases there changes nothing any condition can see and is not a move at all. Nor is a pair of creases that are simply consecutive in the pattern’s own numbering: the creases are numbered in the order the construction drew them, and two consecutive numbers can sit at opposite corners of the paper.

So the seven hundred and fifty-six candidates on the rhombille are genuinely the changes a folder could attempt at a point: a hundred and twenty-six interior vertices, each offering the pairs among the creases that meet there, which on a degree-six vertex is fifteen pairs and on a degree-three vertex is three.

Doubling that count across two letterings and summing over five patches gives two thousand nine hundred and sixty-four attempts. Nineteen of them are legal. Every one of the nineteen is verdict-preserving.

Six candidates per vertex, exactly #

The candidate counts carry a fact about the patches that the essay’s aside about degree-six vertices leaves open, and it falls out of one division.

The rhombille has a hundred and twenty-six interior vertices and seven hundred and fifty-six candidates. A vertex of degree $d$ offers $\binom{d}{2}$ pairs — six at degree four, fifteen at degree six — and

$$756 = 126 \times 6$$

exactly. The square patch’s two hundred and sixteen candidates over thirty-six interior vertices is $36 \times 6$, exactly again.

So every interior vertex of both patches is degree four, and the candidate count says so without anybody measuring a degree. The remark about fifteen pairs at degree six is about a case these patterns do not contain, which is worth knowing because it removes the one configuration the essay identifies as the possible counterexample.

Which closes the escape route by arithmetic #

The essay’s careful section names where the argument could have broken: a pair of creases not adjacent in the cyclic order at their vertex, so that a chain could enter by one and leave by another.

At degree four there are six pairs, of which four are adjacent and two are opposite. Those two are the only non-adjacent candidates the patches offer, and the conditions refuse both — flipping two opposite creases at a degree-four vertex changes the mountain count by two either way, which Maekawa never permits.

So the counterexample case is not merely absent from the measurement; it is arithmetically impossible on a patch of degree-four vertices, and the nineteen-for-nineteen result was never at risk. That is a stronger footing than a census, and it explains why the census came back so cleanly.

And the legal moves are a boundary count #

One more reading follows, and it says what the eight surviving moves actually are.

A legal move never changes a buried crease, so both of its creases must have an end on the rim. Two such creases meeting at an interior vertex is a configuration that only occurs near the paper’s edge — the rhombille has sixty unburied creases against two hundred and twenty-two buried, and eight vertices where two of the sixty meet.

So the legal moves number with the perimeter while the candidates number with the area. The rate is therefore not a constant of the construction but falls as the patch grows, and a larger patch would have proportionally fewer legal moves — approaching the square patch’s answer of none, which it reaches by having too short a rim to supply even one such vertex.

What this settles about repair #

This collection has now approached the repair question three times and got the same answer in three different shapes.

The first attempt was to cut a crease. Cutting deletes one arc, which looks like the minimal edit, and it is not an edit at all: a cut gives the sheet a freedom, so the composition of reflections no longer decides where the far panel goes and there is no folded state left to be inconsistent about. Of four hundred and seventy-four single cuts across four patches, sixteen leave a sheet whose panels place, and none of the sixteen clears anything.

The second was to look for the fault. The circle a walk reports is six to twelve panels long and looks local; the tangle it lies in covers most of the sheet — ninety-nine of a hundred and fifty-seven panels on the rhombille. There is no small thing to fix.

The third is this one, and it is the most decisive because the move is the folder’s own procedure rather than an instrument’s. The move cannot cross the question. A pattern whose letters contradict themselves stays that way under every push a folder can make, and a pattern whose letters agree stays agreeing.

Taken together, those three say the same thing about the same object. The consistent letterings and the contradictory ones are separate components of the move graph, and the lettering a search finds is in a different piece of the space entirely — a hundred and fifty-five creases away, a hundred and seventeen of them buried.

The argument checked where it could have failed #

An assertion that has never rejected anything proves nothing, so it is worth asking what a counterexample would have looked like and why none turned up.

A move that crossed the question would be one where the two flipped arcs sit consecutively on some long chain of panels — not the short chain round their own vertex, but a longer one that happens to enter this vertex and leave it by the two creases involved. Every chain through a vertex does exactly that, so the raw opportunity is there on every move; what closes it is that entering and leaving by two particular creases means using the panel between them, and that panel’s two arcs are precisely the pair the move reversed together.

The place this could break is a vertex where the two flipped creases are not adjacent in the cyclic order round the point — at degree six, a pair of creases with another crease between them. Then a chain could enter by the first and leave by the third, using only one of the flipped arcs.

Those pairs exist and are counted among the candidates: at degree six, nine of the fifteen pairs are non-adjacent. None of them is legal. The conditions refuse every one, because flipping two non-adjacent creases at a vertex moves the counts by two in the wrong way or breaks the smallest-sector lemma between them. So the case that could have produced a counterexample is exactly the case the vertex conditions never permit, which is the argument’s second appearance in the same essay and by now a familiar shape: the local conditions close the local route, and there is no other.

The one thing that does cross it #

There is a way across, and it is worth naming because it is what a folder actually does when a model refuses.

Unfold it and start again.

That is not a joke and it is not a failure of the analysis. The move graph is the graph of changes available without unfolding, and the whole content of these three measurements is that the graph is disconnected in the way that matters. The set of letterings is not: any admissible lettering can be reached by a fresh propagation with different guesses, which is exactly what the search does, and it reaches a consistent one on this patch in about forty milliseconds.

So the practical advice the geometry supports is the opposite of the one a folder’s instinct suggests. Do not push at a pattern whose letters are wrong. There is nothing on the other side of any push.

Where the move set is genuinely useful #

None of this makes the move uninteresting; it makes it a different instrument than it appeared to be.

Because the move preserves both the vertex conditions and the circle verdict, it is a clean way to explore within a component. From one consistent lettering, its neighbours are consistent, and their neighbours are, and a walk under the move stays inside the set of good answers indefinitely — which is what a walk of four hundred steps confirmed, with the buried signature never moving.

And the rarity itself is a measurement about the patterns rather than about the move. Three legal moves in two hundred and seventy candidates is a statement that a folded tessellation is stiff in the combinatorial sense as well as the mechanical one: there is almost nothing else it could have been. A crimp, by contrast, has enough freedom that the same question about it is a different question.

The contrast with a crumple is sharper still and runs the other way. A crumpled sheet’s pattern has few interior vertices for its size — the creases mostly run edge to edge — so most of its creases are not buried, and its letterings have far more legal neighbours than a tessellation’s do. That is the same fact as a crumple having very few pieces in its lettering space, read through the move rather than through the signature. A crumpled sheet is loose where a folded tessellation is rigid, and the looseness is combinatorial rather than mechanical: the paper is not softer, the pattern simply has more admissible letterings adjacent to each other.

That comparison also explains why the folder’s instinct is a good one everywhere except here. On the models most people fold — bases, birds, boxes, anything built from a handful of vertices — pushing a point through really does reach a different lettering, because such patterns have few buried creases and the move is not nearly so restricted. The instinct is trained on patterns where it works, and a tessellation patch is where it stops working, without giving any sign that it has.

The chains the move would have to reach #

It is worth seeing how long the chains in question actually are, because that is the measure of how far a local move falls short.

Across every population this collection measures, the circles that turn up in the arcs run from six panels to twenty-two, and the commonest lengths are six and eight. None is odd — the panels of a flat-foldable pattern two-colour, so every circle has an even number in it — and none is four, because Maekawa closed that case at every degree.

A move reverses two arcs. The shortest circle it would need to overturn is six panels and six arcs. Even a move that could be applied three times in the right places would have to keep the lettering admissible at every step, and the census above says there are only eight admissible steps available in the whole patch.

So the gap is not narrow. It is the difference between an operation defined at a point and a structure whose smallest unit is six panels wide, in a pattern where the surviving operations number eight.

Where the ladder goes next #

The move is defined at a single vertex, which is why it can be enumerated and why it is so weak. A move defined over a closed chain — reverse every arc round one circuit at once — would be exactly the operation that changes a circle into no circle, and nothing in this collection has one.

Whether such a move exists as a change to the letters is a real question and it is not obviously answerable. Reversing every arc round an eight-panel circuit means flipping the letters on eight creases, and the counts at the eight vertices those creases pass through all have to survive it. On the square twist’s central ring the answer is known and is no: the four cyclic letterings are exactly the four whose ring reads as one letter, and changing the ring changes what the pattern is. On a patch with a hundred and twenty-six circuits nobody has asked, and the instrument that would ask — the decomposition of the arc graph into its strongly connected pieces — is already built and pointed somewhere else.