One marking, many objects

A crease pattern with its assignment written on it is treated, in practice and in print, as a complete specification. Hand it to a folder, hand it to a solver, hand it to a machine: the creases say where, the letters say which way, and what comes out is the model.

More than one way to lie flat already denies that for a strip of paper, where the legal stackings can be counted exactly and the count is routinely more than one. This asks the same question of a vertex, which is a different object with a different answer — and the answer divides the subject at exactly the place the local theorems divide it, for a completely unrelated reason.

What a stacking is, and why counting it is a separate question #

The paper immediately around 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. Every sector lands somewhere on that line; each one sits at some height in the stack; and the whole question of which model has been folded is the question of what those heights are.

The letters do not settle the heights. What a letter settles is one relation: of the two sectors a crease joins, which is above the other. That is one constraint per crease. The heights are a permutation of the sectors, and a permutation of n things has far more freedom than n constraints remove — so the arithmetic leaves room for several answers and the only question is whether the two non-crossing rules take that room away.

At four creases they take all of it #

Take a degree-four vertex, any angles, any marking that folds. The stacking is unique. Not usually unique, not unique for the vertices anybody draws: unique, over every four-crease vertex the census contains.

The claim is worth stating carefully, because “unique” is a strong word and the census is finite. What was tested is 373 vertices drawn from a seeded stream of Kawasaki-satisfying angles, every one of their sixteen markings, with the stackings enumerated exhaustively rather than sampled. Every marking that folded, folded one way. That is not a proof and it is not offered as one; it is a measurement with no exception in it, and the argument below says why an exception would be surprising.

The reason is short. At degree four, one sector is folded between its neighbours and the pair rule at each of its two creases fixes both of that sector’s relations; the remaining two sectors are joined to each other round the back of the vertex, and their relation is fixed by their own crease. Four sectors, four creases, four relations — and four relations on four things, if they are consistent at all, leave exactly one order.

That is a coincidence of counting rather than a principle, and the moment the degree rises the counting stops working out. At degree six there are six sectors and six creases, so the counting still balances — and it balances wrongly, because two of the six relations can now be implied by others rather than adding anything. Constraints on a permutation do not compose by addition, and the place that becomes visible is exactly where two creases land in the same place in the folded image, which is where the taco-taco rule has an opinion and the letters do not.

At six, and at eight #

At the degree-six vertex a waterbomb tessellation repeats — sectors of 90°, 45°, 45°, 90°, 45°, 45° — eighteen of the sixty-four markings fold. Twelve of those name exactly one object. The other six name six each.

Six is a large number for an object with six pieces in it. It means that a marking which a checker calls flat-foldable, which a solver returns as the answer, and which a diagram would show as a single step, is in fact a choice among six, and that five of those choices are wrong for whatever the pattern was drawn for. There is nothing on the sheet to distinguish them.

At the eight-crease vertex in the middle of a preliminary base, with all sectors 45°, every one of the 112 foldable markings is permitted by Maekawa alone and every one of them folds. Sixteen of them name one object. Sixty-four name three. Thirty-two name four.

That last is worth sitting with. The preliminary base is the first thing most folders learn, its central vertex is the most-folded vertex in the subject, and a marked crease pattern of it does not say which of four objects to make. The count is not evenly spread, either. The sixteen markings that name one object are the ones whose letters happen to force every relation; the ninety-six that do not divide into two families of three and four, and which family a marking falls into depends on how the mountains are distributed round the vertex rather than on how many there are. Maekawa fixes the number of mountains at three or five; it says nothing about their arrangement, and the arrangement is what the multiplicity reads.

What settles it in practice is the sequence — a folder collapses the base by bringing corners together in an order, and the order chooses the stacking. Nothing on the pattern records that order, and nothing has to: the pattern is the specification of a set of objects, and the tradition has quietly been carrying the extra information in the diagrams.

The count is a property of the angles #

The obvious hypothesis is that it is a property of the degree — more creases, more room, more answers. It is not.

Two degree-six vertices, the same crease count, and the largest multiplicity differs by a factor of three. What separates them is the sizes of the sectors, and specifically how many of them are small enough to be folded away between their neighbours without meeting anything. A vertex with two large sectors and four small ones has room in the middle of its stack; a vertex with six equal ones does not.

The random census makes the same point from the other side. A degree-six vertex drawn at no particular angles has every marking naming exactly one object — the multiplicity is a phenomenon of the symmetric vertices, which is to say of the vertices a folder actually produces.

The two phenomena are opposite in exactly the same place. At a vertex with generic angles, the conditions over-count the foldable markings and every foldable marking names one object. At a vertex with ties, the conditions are exact and the markings name several objects each. Ties buy decidability and spend uniqueness.

There is a practical reading of that. A tessellation drawn at generic angles is a pattern a solver can decide and a folder can fold exactly one way; a tessellation drawn at the angles a grid produces is one whose every vertex offers a choice. What the second buys is that it can be folded wrong without any condition noticing, which is a familiar experience at the paper and has not previously had a number attached to it.

Which theorem was checked, and how #

The stackings are enumerated, not derived. Every ordering of the sectors is generated and each is put past the three rules: the assignment rule at each crease, the taco-tortilla rule at each crease against every sector that crosses it, and the taco-taco rule at each pair of creases whose folded images coincide. Nothing is pruned, so the count is the count.

Two independent things had to hold for the numbers to be trusted. The first is that the set of markings with at least one stacking must agree, vertex by vertex, with the crimp reduction — a recursion on angles that never builds a stacking at all. It does, on 6,256 vertex-and-marking pairs. The second is the same closure trap: a vertex whose alternating sums are unequal produces folded positions that the rules will cheerfully evaluate, and orderings will be found, of an object whose two ends are in different places. Closure is checked first.

Where the model stops #

The count is a count of combinatorially distinct stackings, and two stackings can be combinatorially distinct and physically almost identical: swapping two layers that never touch changes the permutation and not the shape of the object. The census does not distinguish those, deliberately, because the alternative is to decide what “almost identical” means and there is no principled place to draw it.

Zero thickness is doing work here too, and more than usual. A stack of eight sectors at 45° is eight sheets converging on a point, and real paper at a crease has a radius that grows with the number of layers below it. Some of the four objects a preliminary base’s centre admits are much harder to make than others, and nothing in this account says which.

There is a subtler limit as well, and it is about what “the object” means. Two of the preliminary base’s four stackings differ by exchanging a pair of sectors that are separated by two other layers; a folder producing one rather than the other has made a model that behaves differently under a subsequent fold, because the layers a later crease has to go through are not the same. So the multiplicity is not idle even where the finished object looks identical: it is a difference that shows up one step later, which is the ordinary way an origami sequence goes wrong.

And this is still one vertex. A pattern’s folded states are not the product of its vertices’ folded states — the vertices share creases, and a choice at one is a choice at its neighbour. The product is an upper bound and a poor one.

What the picture cannot show #

The stacking figures draw the folded loop as a set of bars at heights, which is the honest picture of the combinatorics and not a picture of the paper. A real folded vertex is a cone of sectors converging on a point, seen edge on it is a stack, and the bars are that stack with the thickness taken out. Nothing in the figure shows what a reader would see looking down at the model, because from above the four objects are indistinguishable — which is the inverse problem’s subject and is exactly why the count matters.

Who found it, and when #

That a flat-foldable pattern has a set of folded states rather than one is old and is usually credited to Jacques Justin’s 1986 analysis of the layer conditions, which is where the non-crossing rules are first written down as rules rather than observed as difficulties. The counting is not classical: it is what a computer does with those rules, and the small numbers here have been obtained many times by many people without being collected anywhere, because a count of the folded states of a named vertex is a fact about that vertex and not a theorem.

What is worth saying about the credit is that the multiplicity is exactly the part of the subject that the diagram tradition solved by not writing it down. A folding sequence carries the order in which flaps are brought together, and that order picks the stacking; publishing the crease pattern instead — which is what made the patterns checkable at all — dropped the information without anybody having to decide to.

The observation this essay adds is the division at four. Not that it is unique at four — that is a two-line argument once it is looked for — but that the vertices origami actually uses are precisely the ones where it fails, and that the same tie which makes the local conditions exact is what makes the object ambiguous.

One more consequence is worth naming because it changes what a count of patterns means. Enumerations in this subject are almost always enumerations of markings: how many assignments of a pattern fold, how many patterns of a given size exist, how many ways a strip of stamps can be folded. The last of those is a count of folded states and the first two are counts of specifications, and the ratio between them is exactly the multiplicity measured here. So two censuses that look like the same census — one counting labellings, one counting objects — differ by a factor that is three or four at a single vertex and compounds across a sheet.

The exchange rate between the two censuses #

The closing observation — that a census of markings and a census of objects differ by a factor — has an exact value at each vertex measured, and it is worth computing because it is the number that converts one kind of published count into the other.

The factor is the mean multiplicity: total objects divided by total foldable markings. At the preliminary base’s centre, sixteen markings name one object each, sixty-four name three and thirty-two name four, so the objects number $16 + 192 + 128 = 336$ against 112 markings. Exactly three. At the waterbomb’s degree-six vertex, twelve markings name one and six name six, giving forty-eight objects against eighteen markings: two and two thirds. At the Yoshimura’s, twelve name one and eighteen name two: forty-eight against thirty, which is one and three fifths. At every degree-four vertex it is one.

So a published count of what a vertex admits is a count of specifications, and multiplying by that factor gives the count of things a folder could make. The preliminary base’s centre admits a hundred and twelve markings and three hundred and thirty-six objects, and the second number has never been written down anywhere because nobody was counting objects.

The ordering of the four factors is the essay’s own finding restated as one column. One, one and three fifths, two and two thirds, three — rising with how much room the sector sizes leave in the middle of the stack, and not with the crease count, which is six for the two middle entries and four and eight for the outer ones.

What it does across a sheet #

Compounding is where the honesty has to be careful, because the arithmetic that suggests itself is an upper bound and a bad one.

A waterbomb tessellation patch of nine degree-six vertices, marked so that every one of them is in the six-way class, would offer at most six to the ninth — a little over ten million objects for one crease pattern with one assignment written on it. That number is not the answer and is not close to it. The vertices share creases, so a choice made at one constrains its neighbour, and the layers of one vertex’s stack are the layers of the next; the true count is smaller by however much that sharing removes, and nothing here computes it.

What the bound does establish is the shape of the problem. The multiplicity is not a rounding error that washes out over a sheet — it is a factor per vertex, and a factor per vertex compounds even when it compounds badly. A pattern with a hundred vertices, each offering a choice between two, has a number of folded states that no diagram of the pattern distinguishes and no condition in the subject constrains.

And it is exactly the patterns anybody folds that have it. The factor is one at a generic vertex and rises with ties, so a tessellation drawn on a grid — which is every tessellation in the tradition — carries a multiplicity at every vertex it has, and a mesh solved at no particular angles carries none at all.

Where the ladder goes next #

If a marked pattern does not name an object, the natural question is what does — and the natural candidate is the object itself. The shadow does not name the pattern showed that an outline and a layer count are not enough to recover a crease pattern. The next rung hands the observer the stacking as well, which is everything a folded object physically is, and asks whether that closes the gap.