The shadow does not name the pattern

The crease pattern is the object on this site — that is the whole of the reason the figures are patterns rather than photographs — and the argument for it has always been about completeness: a pattern is checkable and reproducible and a photograph is neither.

There is a sharper argument available, and it is a measurement rather than a preference. A photograph of a folded model does not merely record less than the pattern does. It records too little to name it.

What a photograph of a folded thing carries #

Be precise about the evidence, because the argument turns entirely on what is being claimed.

A photograph of a folded model, taken with every advantage — perfect lighting, edge-on, with the layers countable — carries two things. It carries the outline: the region of the plane the folded paper occupies. And it carries the layer count: how many thicknesses of paper lie over each point of that region, which a careful photographer can get from the edge or from transmitted light.

Call the two together the profile. It is more than a silhouette and less than a folded state: it says where the paper is and how deep, and says nothing whatever about which piece of the sheet is which, or which piece is on top.

The question is what the profile determines. A folder’s intuition says: everything, more or less. Unfold the thing and the creases are right there.

That intuition is about unfolding, which is a physical operation on the actual paper and which does indeed recover the pattern. It is not about the photograph.

The census #

The cleanest place to settle it is one dimension, where the objects are small enough to enumerate completely.

Take a strip of paper of unit length. Put k creases on it at positions that are multiples of one twelfth. Mark each crease a mountain or a valley. Fold the strip flat if it will fold flat, and record the profile. Do that for every position and every marking, and group the results by profile.

Four creases on twelfths gives 233 distinct profiles. Of those, 71 — nearly a third — are produced by more than one crease pattern. One profile is produced by four different patterns. Patterns are counted up to reading the strip from the other end, because a strip read backwards is the same strip and counting mirror images would make the answer trivially yes.

The proportion is not an artefact of the grid. Three creases on twelfths gives 69 profiles of which 16 are ambiguous; three creases on sixteenths gives 184 of which 47; four creases on tenths gives 94 of which 24. Somewhere between a fifth and a third of the folded objects a short strip can make are objects that more than one pattern makes.

What is being conflated, and what is not #

Two other kinds of multiplicity live nearby and this is neither of them, so it is worth saying which is which.

A marked crease pattern does not name a folded object: the legal stackings can be counted, the count is routinely more than one, and two stackings of the same pattern give the same outline and the same layer counts and different objects. That is the forward direction being many-to-one, and it is a fact about layer order.

What is measured above is the backward direction being many-to-one: several patterns, one profile. The creases are in different places on the strip. The distances between them are different. Unfold any two of them and they are visibly not the same piece of paper.

The third neighbour is the count of stampings — the number of ways a strip of n stamps folds — which counts orderings of a fixed pattern and not patterns at all.

There is one more distinction that matters, and it is the one a reader is most likely to slide past. Two patterns sharing a profile is not the same as two patterns being equivalent. The patterns in the census fold to the same object and are different objects themselves: they use different amounts of crease, they put their layers together in different orders, and folded from real paper they behave differently under a thumb. What they share is exactly the thing a camera can see.

Why it happens #

The mechanism is not mysterious and it is worth having, because it says which patterns collide.

Folding a strip flat is a walk: start at one end, travel along the paper, and reverse direction at every crease. The folded position of a point is where that walk has got to. So the profile is determined by the sequence of segment lengths and by nothing else about the creases — and the same multiset of lengths can be laid out in different orders along the strip.

Not every permutation folds, which is why the ambiguity is a minority rather than the rule: reordering the segments usually produces a strip that will not fold flat at all, or one that folds to a different profile because the reversals land differently. But enough of them survive, and the survivors are the census’s answer.

There is a second source, and it is the one that makes the point about photographs sharpest: two patterns that differ only in their letters — the same creases, mountains and valleys exchanged — can fold to the same profile. The preliminary and waterbomb bases are the standing example in two dimensions, and the strips have their own smaller instances.

It is worth counting what survives, because the ratio is the interesting number rather than the fact. Of the roughly six hundred crease-and-letter combinations that fold flat at four creases on twelfths, the census sorts them into 233 profiles — so on average a profile has between two and three patterns behind it, and the reason the ambiguous count is only 71 is that most of those patterns are mirror images of one another. Take the mirrors out and the collisions are between genuinely different objects, and there are still 71 of them.

The letters never enter the profile at all #

The section above names two sources of collision — permuted segment lengths, and patterns differing only in their letters — and the second is not a source. It is a certainty, and saying so sharpens the whole census.

Fold a strip flat and travel along the paper. At every crease the paper turns through a straight angle, so the direction of travel reverses — and it reverses whether the crease is a mountain or a valley, because a mountain and a valley are the same turn seen from opposite sides. So the folded position of every point on the strip is the alternating sum of the segment lengths up to it, and no letter appears anywhere in that computation.

The same is true in two dimensions for the same reason: a panel’s place in the folded plane is found by reflecting across one crease after another, and a reflection does not ask which way the crease folds. The folded image of a sheet is determined by its creases alone; the assignment decides only which layer is on top of which.

So two patterns with identical creases and different letters do not merely sometimes share a profile. They always do, necessarily, and there is nothing to measure.

Which makes the census a census of positions #

That changes what the numbers count. The profile is a function of the crease positions, so the ambiguity the census reports is entirely about where the creases are and not at all about how they are marked.

The two squares folded in opposite orders are the cleanest case. They have the same cross of creases and differ only in which line changes its letter at the crossing — so by the argument above they were guaranteed to produce the same outline and the same layer counts before anything was folded. The demonstration is worth doing with paper because it makes the point vivid, and the geometry never had a choice about it.

Which leaves the interesting half unaltered and better located. The seventy-one ambiguous profiles are seventy-one cases where different sets of crease positions produce the same alternating partial sums — the same multiset of turning points, with the same multiplicities, arrived at by segments laid out in a different order along the strip. That is a genuine coincidence between two drawings, it is what the permutation mechanism describes, and it is the whole of what the census found.

It also says what a photograph cannot possibly give even in principle. A camera sees the folded image; the folded image is a function of the creases; so a photograph carries information about crease positions and, strictly, none at all about the assignment. Everything a reader thinks they can infer about mountains and valleys from a picture is inferred from shading, from how the layers catch the light, and from the small departures a real sheet makes from the ideal one — which is to say from the ways the paper is not the model.

In two dimensions, with a square #

The census is one-dimensional because that is where exhaustion is affordable. The phenomenon is not.

Fold a square in half left-to-right, then in half top-to-bottom. Now fold a fresh square in half top-to-bottom, then left-to-right. Both give a quarter-sized square, four layers thick everywhere, and both leave the same cross of creases on the paper. What differs is the letters: in the first the horizontal crease is a single fold and the vertical one changes assignment across it, and in the second it is the other way about.

This is worth doing with real paper, and it takes about twenty seconds. Two squares, four folds in total, and two objects that a photograph cannot tell apart even though the paper says plainly which is which the moment either is opened. The pattern of creases is the same cross in both; what differs is that in one the vertical line is a single unbroken fold and the horizontal line changes its letter where it crosses, and in the other the roles are exchanged.

That the two are both flat-foldable is Maekawa’s three-to-one split being satisfiable two ways at a four-crease vertex whose sectors are all right angles — which is the degenerate case the big-little-big lemma declines to speak about, arriving here for a completely different reason.

The two objects are indistinguishable by outline and by layer count. They are distinguishable by unfolding, and by nothing else that a camera can do.

What this does to the argument for patterns #

The case for publishing a crease pattern instead of a folded model has usually been made on grounds of reproducibility: a pattern can be folded again and a photograph cannot.

The census supplies a stronger version. A photograph is not merely hard to work from; it is, for a measurable fraction of objects, consistent with several patterns, and there is no amount of care in the photography that fixes it, because the ambiguity is in the object rather than in the picture.

There is a pleasant symmetry with the layer-order multiplicity in the other direction. One pattern makes several objects; several patterns make one object. Neither map is a function in the direction anybody would want, and the pattern is the only one of the three descriptions — pattern, marked pattern, folded object — from which the others can be computed.

That has a practical edge. Attributing a design from a photograph of the finished model is a common activity in this subject and is sometimes contentious, and the census says plainly that it cannot always work — not because the photograph is bad but because two designers can arrive at the same object from different patterns. The evidence a record actually supplies is a question this site has had to ask before, in a field where the answer was about dates rather than about geometry, and the shape of the answer is the same: a record establishes what it establishes and no amount of confidence adds to it.

What the folder loses and what the folder keeps #

None of this makes a photograph worthless, and it is worth being clear about which half of the practice it touches.

A photograph is excellent evidence of existence. It shows that a piece of paper was persuaded into that shape, which is a real claim about the world and is often the claim anybody wanted. It is good evidence about proportions, about how many layers a designer was willing to put through a fold, and about how a model was shaped after the base was made — the part of a design that no theorem on this site reaches and that the crease pattern is silent about.

What it is not is evidence about the pattern, and the census says how badly. Roughly a third of the time, at these sizes, it is consistent with an alternative that nobody would guess by looking.

The subject’s own history is instructive here. The move from publishing sequences to publishing patterns was made for reasons of practicality — a pattern is shorter than a sequence for a complicated model — and it happened to be a move to the only representation of the three that determines the other two. That was not the reason it was made and it is the reason it lasted.

What would be enough #

Since the profile is not enough, it is worth asking what would be.

The crease pattern with its letters determines the folded object entirely, except for the stacking. The stacking on top of that determines it completely. Going the other way, the folded object with its stacking — which is to say, knowing not only how many layers lie over each point but which piece of the sheet each layer is — does determine the pattern, because the pieces can be traced back and the creases are where they join.

So the missing information is the identity of the layers, and a photograph gives none of it. A cross-section gives some. Taking the model apart gives all of it, and taking the model apart is unfolding.

That is a tidy resolution and it makes the practical advice unambiguous: a photograph is evidence about a model, and the model is evidence about the pattern only when it can be opened. Which is why a collection that wants its arguments checkable publishes patterns, and why one that publishes photographs is publishing something else.

Where the model stops #

The census is one-dimensional and the grid is coarse. Strips with four creases on twelfths are small objects. What the proportion of ambiguous profiles does as the strip gets longer is not measured here, and there is no reason to expect it to be constant.

The profile is a generous model of a photograph. A real photograph gives the outline well and the layer count badly, and gives nothing at all about layers hidden behind other layers. So the true evidence is weaker than the profile, and every ambiguity found here survives in the real case while others are added.

Unfolding is not affected. Physical unfolding recovers the pattern exactly, and nothing above suggests otherwise. The claim is about the folded object as a record.

Nothing here is about how hard the search is. Finding which patterns share a profile was done by exhaustion because exhaustion is affordable at these sizes; how one would do it at a useful size, and what that would cost, is a question about algorithms and belongs elsewhere.

Where the ladder goes next #

The measurement above counts profiles that coincide exactly. The more useful question for anybody actually looking at a photograph is how near two patterns can be in profile while being far apart in pattern — a distance rather than an equality, which would say how much a slightly better photograph is worth.

The other direction is the extra evidence. A folded object seen from the side reveals its layer order as well as its layer count, and layer order is a great deal more information than a count. Whether outline, count and order together determine the pattern is a question the census above is one term short of answering, and the answer would say exactly what a folder gains by being allowed to pick the model up.