The file records no verdict

A file has no paper: the field’s interchange format records vertices, the edges between them and a letter for each edge, and nothing in it forbids two edges from crossing. A pattern that could not exist as a sheet is a valid file, and the boundary between a drawing and a sheet turns out to be a convention rather than part of the object.

That is the format failing to notice something about the drawing. There is a second and larger thing it fails to notice, and it is about the letters.

Two hundred and forty-eight valid files describing nothing #

The square twist has four thousand and ninety-six mountain-valley labellings. Two hundred and fifty-six of them satisfy every condition at every vertex. Eight have a folded state.

Every one of the two hundred and fifty-six makes a valid file. The coordinates are the same; the edges are the same; only the letters differ, and a letter is a letter. There is no field a validator could consult to tell the eight from the other two hundred and forty-eight.

Ninety-seven per cent of the files that pass every check the format and the subject can apply describe an object nobody can hold.

It is worth being precise about what “valid” is doing in that sentence, because a format has several kinds of validity and only the weakest is at issue.

A file is well-formed if its arrays line up: every edge names two vertices that exist, every letter comes from the alphabet, every coordinate is a number. All two hundred and fifty-six pass that.

A file describes a flat-foldable drawing if every interior vertex satisfies the conditions the subject has. All two hundred and fifty-six pass that too, which is what made them the two hundred and fifty-six.

A file describes a folded object if some ordering of its panels breaks none of the non-crossing rules. Eight pass that, and no format anywhere checks it, because checking it is the hard problem.

The same gap, in the other direction #

There is a mirror of this that is worth stating so the criticism is not one-sided.

A file can be more correct than it looks as well as less. A pattern exported without letters at all — every edge marked unassigned — is a legitimate file describing a shape, and whether that shape has a foldable labelling is a question the file does not raise and cannot answer wrongly. How many assignments fold is the count of what such a file leaves open, and on a small pattern it is a large number.

So the format supports two honest positions: here is a shape, and here is a shape with letters. What it does not support is the third, which is here is a shape with letters that has been checked, and that is the position most files are read as taking.

Where the verdict would go #

The format is not naive about this. It has a field — faceOrders — for exactly the missing information: which panel lies above which, pair by pair, in the folded state.

Nothing fills it in. Not this collection, which computes orderings and does not export them; not the tools that produce most of the files in circulation, which draw crease patterns rather than folded states; and not the people exchanging patterns, for whom the file is a picture to be folded from rather than a record of a fold.

The half no notation records is the measurement of how much is left out, and it is most of the object on any pattern past a few panels. This is the consequence for what a file means: with the field empty, a file is a statement about a drawing and its letters, and it makes no claim at all about foldability.

Two things a file could have said and does not #

There are two verdicts a format could carry and it carries neither.

That the letters agree with themselves. This is cheap — one pass over the crease list — and it would eliminate four of the square twist’s two hundred and forty-eight bad files. On a tessellation patch it would eliminate most of them: on a rhombille of a hundred and fifty-seven panels, no lettering found in two hundred draws passes it.

That an ordering exists. This is the real verdict and it is expensive: an exhaustive search over orderings, which refuses at about twenty panels. But the verification is cheap — hand over an ordering and checking it costs one pass — which is precisely why a format with a field for it would be so much stronger than one without.

That asymmetry is the ordinary one between finding an answer and checking one, and it is what a certificate field exists to exploit. A file with its face ordering filled in is a file whose foldability anybody can verify in linear time. A file without it is a file whose foldability nobody can verify at all without redoing the work.

What the format does refuse #

It is worth being fair about what the format does check, because it is not nothing and the refusals it makes are exactly the ones a drawing-level format should.

Vertices must be listed before they are referenced. Edges must name two vertices. Letters must come from the format’s own small alphabet — mountain, valley, boundary, flat, unassigned — and this collection has found the alphabet binding in one place: it writes a cut as a sixth kind of assignment, meaning paper separated along an edge with paper on both sides, and the format has no letter for it.

So a pattern carrying a cut is refused rather than exported. Rewriting it as a boundary edge would produce a valid file describing a different pattern under the same name, which is the one thing an interchange format exists to prevent.

That refusal is the model for the thing this essay is about. The format is careful about the alphabet and silent about the verdict, and being careful about the alphabet is what makes the silence look like rigour.

The reader is the validator #

There is a version of this that has always been true and that the file makes worse.

A published crease pattern on paper has the same defect: it is a drawing with letters, and whether it folds is left to the reader. What has changed is who is expected to catch a mistake. A diagram in a book went through an editor and a test folder; a file goes from an editor to a repository, and the first person to discover that it does not fold is whoever spends an evening on it.

The reader decides the junction is the same shift seen at the level of the drawing: a person reading a crease pattern decides at each place two lines meet whether they meet or pass, and nothing in the file says. This is that shift at the level of the letters, and it is larger, because a junction is a local ambiguity a reader resolves correctly almost always and a layer order is a global one nobody can resolve by looking.

What it does to a claim of authorship #

There is a consequence for the history of the field, and it is why this belongs here rather than among the checkers.

A crease pattern file is increasingly how a design is published — publishing the pattern instead of the sequence is the change that made a pattern the unit of exchange — and a file is therefore what a claim of authorship attaches to. What is being claimed is a folded object; what is being transmitted is a drawing with letters on it.

Those come apart in both directions. A file can describe a pattern nobody can fold, in which case the claim is empty. And two files with different letters on the same lines are two different objects, so a pattern published with the wrong letters is not the design that was made — a difference no reader can detect from the file, and one that produces exactly the experience of a pattern that will not go together.

A record is not a proof is the standing argument about historical claims, and this is the same argument about a contemporary one. A file is a record of a drawing. It is evidence that somebody drew it, and no evidence at all that it folds.

What a format cannot be blamed for #

Two of the three gaps are not the format’s fault and it is worth separating them, because a format is a place people put blame.

The hard verdict — whether an ordering exists — is NP-hard to decide. No format can require a field nobody can compute, and requiring it would mean no large pattern could be exported at all. Leaving it optional is the only available design.

The letters themselves are genuinely part of the drawing rather than the verdict. A pattern published without letters is a legitimate thing to publish — it is a shape, and the letters are the reader’s problem — and several published patterns are exactly that. The format’s unassigned letter exists for it.

What is left, and what is the format’s own, is that a file with the verdict field empty is indistinguishable from a file with it deliberately omitted, which is indistinguishable from a file whose author never had a folded state to record. Three quite different situations, one representation.

What would be enough #

Very little, which is the frustrating part.

Filling in the face ordering would do it, for any pattern small enough for anybody to have computed one — and once computed it is a fixed list of pairs, written once, verified by anybody in linear time thereafter. Nothing about the format needs to change.

For patterns too large to compute an ordering for, the cheaper verdict is still available and still worth recording: that the letters have been checked against one another and found consistent. It is a weaker claim and it is checkable in one pass, so it costs nothing to produce and nothing to verify.

Neither is done. The reason is not technical and it is worth naming plainly: the people producing files are producing them from crease pattern editors, which know about lines and letters and have no notion of a folded state at all. The field for the verdict exists because somebody who understood the problem put it there, and it is empty because the tools that write the files were never asked to answer it.

The one place a file could be checked cheaply #

There is a third verdict between the two above, and it is the one a file format could most usefully require, because it is cheap to produce, cheap to check, and rules out a great deal.

It is the crossing sweep: does the drawing contain two edges that pass through one another with no vertex recorded there? A file has no paper is the account of why the format permits it, and the sweep that catches it is quadratic in the edges — thirty-nine thousand comparisons on the largest pattern here, which is a fraction of a second.

Requiring it would cost nothing and would refuse a class of file that describes no sheet at all. It is a stronger candidate for a validator than either of the layer verdicts, because it never has to be left empty: every file can be checked and the answer is always available.

That it is not required is the same silence as the rest. A format that checks its alphabet and not its geometry is a format that has decided its job is syntax, and the field has treated the output as though it were semantics.

What a certificate would have cost the field #

It is worth asking why nobody has done this, since the obstacle is not technical.

A face ordering for a pattern of n panels is a list of pairs — which of each overlapping pair is above — and for the patterns anybody exchanges that is a few hundred entries at most. Producing it requires a folded state, which means either running a search or reading it off a physical fold, and the second is available to anybody who has folded the model.

That is the part worth noticing. A designer who has folded their own model has the certificate in their hands, and recording it is a matter of writing down which flap is above which. No computation is needed and no hard problem is solved; the ordering exists because the paper is in it.

So the missing field is not missing for want of an algorithm. It is missing because the tools that write files draw crease patterns, and the person with the answer is holding paper rather than sitting at the tool.

How big the missing field would be #

The certificate has a size and it is worth computing, because the number turns out to explain the emptiness better than the difficulty does.

Only panels that overlap need ordering, so the field is not every pair of panels. Every point of the folded footprint carries a stack of some depth, and the pairs to be ordered there number about half the square of that depth. Summing over the footprint and dividing by a panel’s area gives a count of roughly

$$\frac{n \times d}{2}$$

for $n$ panels at mean depth $d$ — and mean depth is the reciprocal of how far the pattern shrinks, which this site measures for every printed pattern.

Put the numbers in. The square twist has nine panels at a mean depth of 4.1, so its ordering is about eighteen pairs — smaller than the file’s own edge list. The waterbomb tessellation, fifty-two panels at depth thirty-one, needs about eight hundred. The Yoshimura, sixty-five panels at depth sixty, needs about two thousand.

Which says why the field stays empty #

That is a second reason, independent of the hard problem, and it points the same way.

The certificate’s size grows with how well the pattern packs. A pattern that folds to a sixtieth of its area has sixty layers over every point and therefore sixty-choose-two orderings to record there. The patterns worth exchanging are the ones that pack well, and those are exactly the ones whose certificates are thirty times the size of the file they would be attached to.

So the three verdicts sort themselves by cost in a way the essay’s argument has been assuming rather than showing. The crossing sweep is free and always available. The letters’ self-consistency is one pass and always available. The face ordering is expensive to find, large to write, and largest precisely where it would be most valuable.

None of that changes the conclusion — an eighteen-pair certificate for the square twist is not being written either, and the reason there is genuinely that nobody asked the tool for it. What it changes is the expectation. A format field that scales with the object’s depth will be empty on the objects that matter, and a field that scales with the object’s size might not be. The self-consistency verdict is one bit, and one bit is a field that could be filled in everywhere.

What this collection does #

It writes files, refuses to write one for a pattern carrying a cut, and does not fill in the face ordering — which puts it in exactly the position described above and is worth stating rather than leaving implicit.

The orderings are computed. They exist for the four printed patterns small enough to search, and they are used to draw the pile from the bottom up. They are not exported, and there is no good reason for that beyond nobody having done it.

What can be said in mitigation is that the four patterns whose orderings exist are the four for which the question is easiest, and the ones a reader would most like certified — the Yoshimura, the Miura, the waterbomb tessellation — are the ones where the search does not finish. A format’s certificate field is most valuable exactly where nobody can fill it in, which is a real limitation of the idea and not merely of the practice.