The loop a vertex cannot close

Every crease of a flat-folded pattern says which of the two panels it joins lies above the other, and a circle in what those statements demand is a proof that the pattern has no folded state with the letters it carries. That proof is cheap and it is one-directional: it fires on some patterns that cannot fold and stays silent on others.

Which raises a question with an unusually clean answer. Where can a circle be?

A circle in the layer relation needs a circle in the panels — a closed chain of panels, each joined to the next by a crease. The panel graph of a crease pattern is not an arbitrary graph. Its shortest closed chains are the ones round a single interior vertex: a vertex of degree four has four panels between its four creases, and going round them once returns to the start.

So the smallest place a contradiction could sit is at one vertex, and that is the case worth enumerating first, because if it can happen there then every pattern in the collection is exposed to it and no amount of care in the drawing would help.

It cannot happen there. A degree-four vertex has sixteen labellings, of which eight satisfy developability, Kawasaki, Maekawa and the big-little-big lemma; a degree-six vertex has sixty-four, of which thirty do; a degree-eight vertex has two hundred and fifty-six, of which a hundred and twelve do. A hundred and fifty admissible labellings across the three degrees, and not one of them puts the panels in a circle.

Why it cannot, in one paragraph #

Going round the vertex, the panels alternate in orientation. The paper turns over at every crease, so if the first panel is face up the second is face down, the third face up, and so on — and because the degree of a flat-foldable vertex is even, that alternation closes consistently all the way round.

The arrow a crease contributes points from the lower panel to the upper one, and which of the two is which is the letter multiplied by the near panel’s orientation. Orientation alternates. So for every arrow round the chain to point the same way, the letter has to alternate too — mountain, valley, mountain, valley — because a term that flips at every step must be cancelled by another term that flips at every step.

A strict alternation has exactly as many mountains as valleys.

And Maekawa’s theorem says a flat-foldable vertex has two more of one than of the other. The difference is two, never zero. The one labelling that could have closed the shortest circle in the panel graph is the one labelling the subject’s oldest counting argument forbids.

The degree-four case, by hand #

At degree four the whole argument fits on a corner of the paper, and it is worth doing once rather than taking on trust.

Four creases meet, so four panels sit between them, and going round they are face up, face down, face up, face down. Call them one to four in order. For the arrows to run all the way round — one below two below three below four below one — each crease’s arrow has to point forwards along the chain, and the arrow’s direction is the letter times the near panel’s orientation.

Two of the four panels are face up and two are face down. So two of the four creases have their sign flipped by orientation and two do not, and the only way to end up with four arrows pointing the same way is to flip the letters on exactly the two that orientation did not flip. That is mountain, valley, mountain, valley round the point: two of each.

Maekawa’s count allows three and one, or one and three. It does not allow two and two, and the reason is not a convention: the cross-section of the folded paper at the vertex is a closed walk that turns by a right angle at each crease and has to come back to where it started, which balances only when the counts differ by two.

Three-and-one is one letter away from an alternation and one letter is enough: the chain of arrows reverses at the odd crease, and a chain with a reversal in it has a panel that everything points into and a panel that everything points out of, which is a perfectly orderable arrangement. The circle is not narrowly missed. It is missed by the whole of the theorem.

The theorem is doing it alone #

That last sentence is the one worth checking rather than believing, because a labelling refused by three conditions at once is a labelling about which nothing in particular has been shown. If Kawasaki also refused the alternation, then naming Maekawa would be arbitrary.

It does not. The three conditions are evaluated on the alternation directly, and two of them hold.

Kawasaki is a statement about the angles and does not read the letters at all, so it could not have refused a labelling. The big-little-big lemma reads both, and it asks something the alternation happens to satisfy: the two creases bounding the strictly smallest sector must carry different letters, which is exactly what an alternation guarantees. The lemma is, if anything, the condition an alternation is best at.

The census is also run at three different sets of sector angles — equal, and skewed two ways — so that it is a claim about the counting and not a claim about a symmetry. The sectors have to be laid out in equal consecutive pairs to keep Kawasaki while being skewed at all, which has a useful side effect: at equal sectors the big-little-big lemma has nothing to say, and at any skew there is a strictly smallest sector and the lemma is doing work. It refuses nothing either way. The numbers are identical at every skew.

Odd degrees, and the boundary #

Two cases are excluded here rather than handled, and both deserve a sentence.

A vertex of odd degree cannot fold flat at all — the sectors cannot alternate to zero round an odd number of them, so nothing meets at three and nothing meets at five. There is no admissible labelling to enumerate, and the census refuses such a vertex rather than reporting that none of its labellings closes a circle. A census over an empty set passes every claim made about it, which is the shape of a check that has quietly stopped checking, and the enumeration here counts its admissible labellings first for exactly that reason.

A vertex on the edge of the sheet has no chain of panels round it at all: the paper runs out, so the panels between its creases form a path rather than a ring, and a path has no circle in it. The vertices nobody checks are exempt from the vertex conditions for the same reason they are exempt from this — there is no closed thing there to be inconsistent about. On a tessellation patch, where most of the twist polygons touch the rim, that removes a great many candidate circles and the patch still fails; the circuits that matter are the ones in the middle.

Where a contradiction can be, then #

Not round one vertex. So a circle in the panel graph has to enclose more than one, which means the pattern needs a closed circuit of creases — a chain of creases returning to where it started, going round something other than a single point.

That is a strong condition and a lot of patterns do not meet it. It is also exactly the structural difference between the patterns in this collection that fail this test and the ones that do not.

How much room a pattern has for such a chain is a number, and it is one the drawing already fixes. The independent closed chains in a panel graph are its edges less its nodes plus its pieces, and on twelve of the thirteen patterns measured here that count comes out equal to the number of interior vertices — which is Euler’s relation and not a coincidence, and the one exception is a patch six of whose vertices have had the paper clipped away on one side. So the chains that generate all the others are the vertex chains, and Maekawa has closed every one of them.

What is not closed is a combination. Two vertex chains sharing a panel boundary combine into a longer chain enclosing both, and there is nothing local standing in its way. The number of such combinations grows as two to the power of the count, so a pattern with three interior vertices has a handful of candidate chains and one with a hundred and twenty-six has more than anything has a name for.

That is the difference between the two ends of this collection’s shelf. A fold-and-cut pattern has one to three interior vertices, so one to three generating chains; every one of the seven outlines it folds is consistent in all forty draws. A tessellation patch has thirty-six to a hundred and twenty-six.

A four-panel circle, and where it came from #

The strongest evidence for a claim of this shape is a case where the theorem was watched doing the work, in something that was already lying around rather than something built to demonstrate it.

The waterbomb tessellation has a family of repeating lettering rules — a nine-bit rule deciding, cell by cell, which way each crease of the unit goes. There are five hundred and twelve of them, and thirty-two survive on a patch containing every kind of vertex.

Run the layer test over all five hundred and twelve and a hundred and twenty of them force a circle. Every one of those hundred and twenty circles is four panels long — the shortest a circle can be — and every one goes round a single interior vertex of degree four.

Which looks, on first reading, exactly like a counterexample. It is not. None of the hundred and twenty is among the thirty-two that pass, and every one of them fails Maekawa at the very vertex the circle goes round. On all four hundred and eighty refused rules the refusal is Maekawa’s alone: Kawasaki holds at every failing vertex and so does the big-little-big lemma.

So every place in this collection where a labelling puts four panels in a circle is a place where the count has been broken, and there are a hundred and twenty of them. That is the argument above running in reverse, observed rather than derived. Break Maekawa and the shortest circle becomes available immediately; keep it and the shortest circle is closed at every degree. The census comes out identically on a three-by-three patch and a four-by-four, so it is a fact about the rule rather than about the size.

What this changes about where to look #

It changes the question from is this pattern faulty to which of its letterings are, and those are different searches with different answers.

A vertex condition is a property of the pattern near a point, and a pattern that fails one has a place to be repaired. A circle is a property of the pattern’s global structure together with a choice of letters, and there is no point to repair — the circuit is not a mistake, it is the twist. The square twist is not defective for having a central polygon; the polygon is what a twist is. What can be wrong is the letters put round it, and those are chosen.

That is why the interesting number about a tessellation patch is not whether it folds but how many of its letterings do. On a square patch of forty-nine panels it is twenty-six in two hundred. On a rhombille of a hundred and fifty-seven it is none in two hundred, and the patch’s own construction hands it one of the failures.

What the vertex conditions were never asked #

There is a habit of speech worth correcting here, because this collection has used it too. The four conditions at a vertex are routinely described as the local conditions on flat-foldability, with the implication that they are everything a neighbourhood can say. They are not. They are everything a neighbourhood can say about angles and counts, and the layer relation is a fifth thing a neighbourhood could in principle have been asked about — whether the panels round this point can be stacked at all.

The answer turns out to be that they always can, which is why nobody ever wrote the condition down. A vertex is never the problem. But always yes is a result rather than an absence, and it is a result that depends on Maekawa: at a vertex where the count is broken, the local layer question has a genuine no, and eight of the sixty-four waterbomb rules are it.

How little the conditions decide measured the same conditions from the other side — how much they entail when one crease is fixed, which is almost nothing — and the two findings point in opposite directions about the same four statements. They constrain a pattern’s letters hardly at all, and they nonetheless close off the entire class of local layer contradictions. Weak in one respect and complete in another, which is the usual shape of a necessary condition that has survived two hundred years.

The shape of the argument #

Three things about this are worth separating, because they are true to different degrees.

The enumeration is a fact: a hundred and fifty admissible labellings of a single vertex, none of which closes a circle, checked at three sets of angles. Nothing about it is sampled.

The mechanism is an argument — orientation alternates, so the arrows can only agree if the letters alternate, so Maekawa is what stands in the way — and it is the kind of argument a reader can check with a pencil at a degree-four vertex in about a minute. It also predicts the enumeration’s result at every degree, including degrees not enumerated here, which is what an argument is for.

The consequence — that a contradiction requires a closed circuit of creases — follows from the mechanism and is what the rest of these measurements are organised around. It is not a claim that circuits cause contradictions. Any tiling makes a twist and every one of those twists has a circuit; most letterings of most of them are perfectly consistent. A circuit is where a contradiction can be, and that is a much weaker statement than where one is.

It also explains a result that had no explanation when it was measured. The lettering that folds nowhere found that every labelling of a square twist whose central ring reads as one letter passes every vertex condition and has no folded state — established by enumerating the orderings of nine panels and finding none. The circle is why, and it is the ring itself: eight panels round the central square, proved in one pass instead of seven and a half thousand nodes. A result that needed a search now needs a sentence.

What it does settle is the thing that looked most puzzling about the sampled numbers. The printed patterns are consistent in ninety per cent of their draws and the patches almost never, and the difference is not that the patches are larger — the Yoshimura folds to more panels than the square patch and is consistent in ninety-five per cent of its draws. The difference is how many independent closed chains of panels each of them has — fifteen on the Miura, twenty-two on the Yoshimura, thirty-six on the smallest patch and a hundred and twenty-six on the largest — and that count is the number of interior vertices rather than the number of panels.