The order does not name it either

A photograph of a folded model carries an outline and a thickness at every point of it, and that is the whole of what it carries. It is not enough: of the 233 folded profiles that four creases on a twelfths grid can reach, 71 are produced by more than one genuinely different crease pattern.

The natural reading of that result is that the photograph is the problem. A photograph flattens; it loses the inside of the model; a better record — one that could see between the layers — would do better. This asks how much better, by handing the observer the layer order as well.

Two things the object could be asked #

Before the census, it is worth being clear about which question is being answered, because there are two and the subject runs them together.

The first is whether a folded object determines its crease pattern. The second is whether a folded object determines its marked crease pattern — the creases together with which way each one turns. They are not the same question, and the second is easier: the letters are visible in the object, because a fold that turns one way exposes a different pair of layers from one that turns the other.

This essay asks the first, which is the one a reader of a photograph cares about. The creases are what a folder needs; the letters follow from them and from the stacking. So two patterns count as the same only when the creases are in the same places, and a pair that differs only in its letters is not an ambiguity at all.

What the richer observation is #

An observer holding the folded object, allowed to take it apart layer by layer without unfolding it, learns more than a photograph. At every point of the folded outline there is not merely a number of layers but a stack, and at every fold along the edge of the object there is a specific pair of layers being joined.

So the observation is: the widths of the bands, the number of layers over each, and for every fold, which two heights in the stack it connects. That is everything the object physically is. Nothing about the paper’s history survives in it, but nothing about the paper’s present is missing from it either.

It is worth being precise about what is not in the observation, because the list is short and each item is one somebody might expect to help. The order in which the folds were made is not there — the object does not record its own construction. The sequence of intermediate shapes is not there. Which end of the strip was held is not there, and would not help, because the object can be turned round. What is there is the finished geometry and its combinatorics, exhaustively.

Two symmetries have to be divided out or the answer is nonsense. The object can be turned end for end, and it can be turned over — reversing the heights. Neither changes the thing; both change the description. Every key below is taken as the smallest of the four descriptions those two operations produce.

The census, and what it says #

The census is the same one, run again with the richer key.

At four creases on twelfths, 71 profiles are ambiguous. The layer order settles 69 of them. It does not settle two.

At three creases on sixteenths, 47 profiles are ambiguous, and the order settles 12. Thirty-five survive.

At three creases on twelfths, 16 are ambiguous and 6 are settled; ten survive.

The direction of that is backwards from the obvious expectation. More creases should mean a more complicated object, more ways to be confused, a harder inverse problem. Instead the four-crease case is almost completely resolved by the order and the three-crease case is barely touched.

The reason is that the order’s information content grows with the number of folds and the ambiguity’s does not. A three-crease object has three folds, so the order contributes three pairs of heights out of four possible layers — a handful of bits. A four-crease object has four folds among five layers, which is enough to separate almost anything that the profile alone had confused. The ambiguity, meanwhile, is caused by coincidences of position, and those do not become rarer as the strip gains a crease.

A further reading is available and is worth resisting. It is tempting to say that the order settles four-crease objects and does not settle three-crease ones, and to conclude that the inverse problem gets easier with size. The census does not support that: what it shows is that the fraction settled rises, while the number surviving falls from thirty-five to two but does not reach zero. Two survivors at four creases on twelfths is two crease patterns among a few hundred that no observation of the object can separate, and there is no reason visible here for that number to be zero at five creases or at fifty.

The pairs that survive #

What remains is not a near miss. The two patterns fold to the same object.

Nothing about the folded object distinguishes them. Not the outline, not the thicknesses, not the stacking, not which layers are joined where. An observer with the object in hand, permitted to measure anything about it that does not involve unfolding it, has no question left to ask.

There is a second way to say what has happened, and it is the one worth carrying away. Adding the order does not make the observation richer in kind. It makes it richer in quantity: the profile is a list of bands, the order is a list of pairs, and both are finite lists of small integers read off the object. The survivors are the cases where two patterns produce the same finite list, and no amount of further reading of the same object produces a longer one. Everything an observer can learn without unfolding has been used.

The pattern in the survivors is that the two crease sets are related by moving a block of the strip: the creases sit at different places, but the lengths of the pieces between them, and the order in which those pieces are travelled, work out the same. The strip is longer at one end and shorter at the other by exactly enough.

Which theorem was checked, and how #

Every profile in the census comes from one routine that records breakpoints and layer counts exactly rather than by sampling, so two profiles are compared as strings and never as pictures. Every stacking comes from an exhaustive permutation search put past the three non-crossing rules. The census over four creases on twelfths reproduces the earlier essay’s numbers exactly — 233 profiles, 71 ambiguous — which is the check that the machinery has not drifted since it was written.

The order key is computed twice for every survivor: once inside the census, and once again from the crease set and the stacking, and the two must be the same string. That is a small check and it catches the mistake that would otherwise make the whole result meaningless, which is a canonical form that depends on the order the members were visited in.

One thing the survivors are not is symmetric copies of one another. A strip read from the other end is the same strip, and the census divides that out before it counts anything; a pair that survives is two patterns that are different after every symmetry has been used up. That is what makes the residue interesting rather than an artefact of bookkeeping — the obvious way to get a false ambiguity in a census like this is to forget one of the symmetries, and the obvious way to get a false resolution is to invent one.

The grid pushes the other way #

The census varies two things and the essay reads only one of them. Holding the grid at twelfths and moving from three creases to four takes the settled share from six in sixteen to sixty-nine in seventy-one, which is the surprise reported above. Holding the crease count at three and moving from twelfths to sixteenths takes it from six in sixteen to twelve in forty-seven — from 38 per cent down to 26.

So a finer grid behaves exactly as intuition says it should. More available positions means more crease sets, more chances for two of them to coincide in what they fold to, and a smaller share that the order rescues. The ambiguity nearly triples, from sixteen profiles to forty-seven, while the strip is no more complicated than it was.

Both axes are moving in the full census and only one of them is counter-intuitive. Stating them together is what makes the mechanism legible: the crease count buys resolving power and the grid buys coincidences, and which of the two wins on a given family is a matter of arithmetic rather than of intuition.

The arithmetic of the two axes #

That arithmetic can be done roughly, and roughly is enough to show the two effects have different sizes.

What the order can say is bounded by how many distinct orders there are. A strip with $k$ creases has $k+1$ layers and, by the count this collection reproduces from the stamp-folding sequence, sixteen foldings at three creases and fifty at four. So the order carries about four bits of separation at three creases and about five and a half at four — it gains roughly a bit and a half per crease, and it goes on gaining.

What has to be separated is the crease sets. On twelfths there are a hundred and sixty-five ways to place three creases and three hundred and thirty to place four, so the population doubles: one bit. On sixteenths with three creases there are four hundred and fifty-five, which is nearly three times the twelfths figure — one and a half bits — and the order gains nothing at all, because the crease count has not changed.

That is the whole asymmetry. Adding a crease adds about a bit and a half of order against a bit of population, so it gains. Refining the grid adds population and no order, so it loses. The essay’s surprise is that the first effect exists at all; the arithmetic says it exists and is small, and that a census run at a fine enough grid would show the four-crease case looking as unresolved as the three-crease one does now.

Which also says the two survivors at four creases on twelfths are not a residue that is on its way to zero. They are what a bit and a half against a bit leaves over at one particular size, and a marking that names several objects has a counterpart here that no finer observation of the object removes — because the observation has already used everything the object holds, and past a dozen layers even enumerating the orderings gives out.

Where the model stops #

This is one dimension. A strip is the tractable case precisely because overlaps on a line form a chain, and the two-dimensional inverse problem is not this problem with more indices in it. What carries over is the shape of the conclusion — that the object under-determines the pattern — and what does not carry over is any number.

The grid is doing work too. Creases are placed on twelfths or sixteenths, which is what makes an exhaustive census possible and is also what manufactures coincidences: two patterns agree because two grid multiples agree, and a continuum of crease positions would make exact agreement a measure-zero event. That is the right objection and it has a short answer — a folder works on a grid, and the patterns anybody actually makes have their creases at rational positions for exactly the reasons box pleating exists.

Zero thickness matters here more than usual. The observation includes which layers a fold joins, and in real paper a fold joining layers three and four in a stack of six is physically different from one joining one and two — the outer fold is looser, the inner one is tighter, and a careful hand can tell. That difference is information the model throws away and paper does not.

What the picture cannot show #

The figures draw the folded object as bars at heights, which is the combinatorics rather than the paper. A photograph of the two objects in the pair figure would be one photograph: they are the same object, and there is nothing to see twice. That is the finding, and it makes the figure awkward — the only honest way to show that two things are identical is to draw both and let the reader check.

The crease patterns above the objects are the only place the two differ, which is why they are drawn at all.

The generalisation #

The statement that survives all of this is not about strips and not about grids. It is that the map from crease patterns to folded objects is not injective, and that enriching the description of the target does not make it injective — it only shrinks the fibres.

That is a different claim from the one the subject usually makes about the inverse problem, which is that recovering a pattern is hard. Hardness is a statement about effort and admits the answer “with enough effort, yes”. This is a statement about information, and it admits no such answer: where two patterns share an object, no procedure, however expensive, separates them from the object alone.

What would separate them is anything the object does not contain — the folding sequence, a partly-folded intermediate, a measurement of how tightly each fold is set, the designer’s own account. And a process record has its own trouble — an exact construction is not an accurate one — so what is kept is not free of doubt either; it is merely doubt of a kind that more measurement can reduce. Every one of those is a record of the process, and the moral is that in this subject the process is not recoverable from the product and has to be kept.

Who found it, and when #

The one-dimensional folding problem, and the recognition that overlaps on a line form a chain, goes back to the stamp-folding literature of the 1960s and to the algorithmic treatments of the 1990s. The inverse question — what a folded object records about its pattern — has no standard name and no standard reference, which is part of why the counts are worth making: the subject has a great deal of machinery for going from a pattern to an object and almost none for asking what the map forgets.

The distinction being drawn is an old one in other fields, where it goes by identifiability: a model is identifiable when different parameters produce different observations. This is that question about crease patterns, and the answer is that they are not identifiable from their folded states, at either resolution of observation tested.

There is one practical consequence, and it belongs to the part of the subject that reads photographs for a living. A crease pattern recovered from a picture of a finished model — which is how a great deal of reverse-engineering in this field is done — is a hypothesis whose competitors have not been excluded, and the number of competitors is not small. At the sizes tested here it is one competitor in five for a small object even with the inside of the model available, and the tradition’s usual defence, that a skilled reader can tell, is a claim about the reader rather than about the object. It is the same trouble a published count of foldings has when it is counting records, read from the other end: there, one object was counted several times; here, several patterns arrive at one object and no count of the object can tell them apart.

Where the ladder goes next #

Two continuations sit immediately behind this. The first is the two-dimensional version, which needs a layer-ordering solver this collection does not have and is a body of work rather than one essay. The second is narrower and reachable: the survivors have a structure — a block of the strip moved from one end to the other — and whether that structure is the whole of the residue is a question the census can be made to answer rather than illustrate.