The drawing does not say what is glued

The premise this collection is built on is that the pattern is the object: a crease pattern is not a picture of a model but the model itself, written down, complete and checkable.

It is not complete, and the missing part has been invisible because it had only one value.

Four sheets, one drawing #

Take a rectangle of a repeating pattern. Leave it as it is, or declare its left edge to be its right, or its top to be its bottom, or both.

Four objects. One drawing.

Their vertex counts are identical, because the rectangle’s edges are placed to miss every vertex. Their crease angles are identical. Every condition this subject checks at a point holds or fails identically on all four.

Their free letters, their panel counts, their Euler numbers, their search costs and — on some drawings — their verdicts are not.

The case where the verdict moves #

Counts moving is one thing. A drawing that folds on one sheet and not on another is another, and it happens.

An $n$-by-$n$ cell of the plain grid folds flat, cut out of the plane, at every size. Glued into a torus it folds when $n$ is even and has no flat folded state at all when $n$ is odd, because a loop that cannot be shrunk crosses $n$ creases and crossing a crease exchanges which face of the paper is up.

So the answer to does this pattern fold depends on a fact that is not in the pattern.

What a drawing does record #

It is worth being exact, because the drawing records a great deal and the gap is narrow.

A crease pattern records where the creases are: their endpoints, therefore their lengths and angles, therefore every sector at every vertex. It records which are creases and which are boundary. It usually records an assignment of mountain and valley, and sometimes a layer order.

From those, everything local follows. Developability, Kawasaki, Maekawa and the big-little-big lemma are all computed from the drawing and nothing else, which is why a checker reading a pattern can certify it — for what a local check certifies.

What is not recorded is which boundary points are the same point. There is no field for it, in any format, and the reason is that on a disc of paper the answer is none of them and a constant is not recorded.

A drawing on a table #

The clearest way to feel the gap is with paper, and it takes two rectangles.

Print or rule two identical rectangles of grid, three squares by three, and crease every line on both. They are the same drawing; put one on top of the other and every crease matches.

Now roll the first into a tube and tape its two edges. Leave the second flat.

Press each. The flat one folds — a three-by-three grid folds, in many ways. The tube does not: three creases run along it, three is odd, and pushing it flat forces a fourth crease to appear.

Two objects, one drawing, different answers. And the difference is a piece of tape, which is not a mark on the paper and is not a crease, and which no drawing of the pattern would show.

That is the whole essay in one experiment, and it is worth doing because the drawing under-determines the object is an abstract sentence and this one folds and that one does not is not.

What the tape is, exactly #

Worth a paragraph, because the tape is doing something specific and it is easy to describe it wrongly.

The tape is not a crease. It does not fold; the paper on either side of it is continuous through it, exactly as the paper is continuous through the middle of a panel.

The tape is not a boundary either. Before it went on there were two edges; after it there is one line of ordinary paper.

What the tape does is remove boundary. The sheet had four edges and now has two, and the two edges it lost were the ones the tape joined.

So the operation is a subtraction from the sheet’s boundary rather than an addition to its drawing, which is why no addition to the drawing could record it. A drawing records what is on the paper, and this is a fact about where the paper stops.

The formats, and why they have no field #

The absence is not an oversight in any particular format and it is worth saying why.

A crease pattern format records vertices, edges, assignments, and sometimes faces and layer orders. Every one of those is a feature of the paper’s interior. The boundary appears only as a special kind of edge — an edge that is not a crease — and there is nowhere in that scheme to say that two such edges are one.

Saying it would need a new kind of record: a relation between boundary edges, with an orientation, held outside the list of edges rather than inside it. That is a structural addition rather than a new attribute, which is why no format has one.

And the reason no format has one is that nobody has needed it. Every object the formats were designed for is a disc of paper, and for a disc the relation is empty.

How much the ambiguity is worth #

The size of the gap is measurable and it is worth quoting, since under-determined is a qualitative word.

Free letters differ by up to a fifth between the four sheets of one rectangle: forty against thirty-two on a two-period square twist cell.

Panels differ by more: twenty-five against sixteen on the same cell.

Search costs differ by a factor of twenty-six at three periods on that family, and by three orders of magnitude at four.

Verdicts differ outright on the grid at every odd size, on the Miura at every odd size in one direction, and on the Yoshimura at two sizes in three.

So the ambiguity is not a technicality. A number quoted for a drawing without a sheet is a number that could be wrong by a factor of a thousand, and a verdict quoted without one could be the opposite of the truth.

The shape of the omission #

This is not the first time the drawing has turned out to under-determine something, and the family resemblance is useful.

The vertex the list does not have is about two creases crossing: a drawing showing an X may mean one vertex of degree four or two creases passing over each other, and the file has no field for that either.

Two creases that cross and the crease the drawing cannot show are the same shape again.

Each is a case where the picture is compatible with two objects and the format records the picture. The gluing is the largest instance, because it changes the sheet rather than a local feature, and everything global follows the sheet.

What a complete description would need #

Short, and worth writing down since the fix is not conceptual.

A drawing: the crease pattern as recorded now.

An identification: a list of pairs of boundary edges, each with a direction, saying which points of one are which points of the other. For a rectangle glued into a torus that is two pairs; for a cylinder one; for a disc none.

That is the whole of it. Two facts rather than one, and the second is empty for every sheet anybody normally folds — which is exactly why it was never written.

No file format has a field for it, and the absence is a separate matter from the absence in this collection’s own thinking.

Every figure here is a rectangle #

A consequence for how this collection’s own pictures should be read.

A glued sheet cannot be drawn. It has no boundary to put at the edge of the picture and no distinguished place to cut it open, so every figure of one is a rectangle with a note saying which edges are identified.

The rings marking where creases cross the rectangle’s edges are the closest a static picture gets to saying so. They are a notation rather than a feature: on the object there is nothing there at all, and a reader looking at the picture is being asked to supply the identification themselves.

That is the same act of imagination any tiling figure asks for — the reader supplies the continuation — and it is slightly more consequential here, because in a tiling figure the continuation is decoration around a specimen and here it is the object.

So a figure of a glued cell is a figure of a rectangle plus a sentence, and the sentence is in the caption. There is no better arrangement available and it is worth knowing that it is the arrangement.

What can be printed #

The paper rule in this collection is that a pattern which folds gets printed at a stated size, and a glued sheet raises the obvious question.

What gets printed is the rectangle, with an instruction: cut it out, crease it, and tape one pair of edges together.

For a cylinder that produces the object exactly. For a torus it does not and cannot, because a flat rectangle joined at both pairs of edges is a doughnut and paper will not make one without stretching.

So of the four sheets, two are printable and foldable, one is printable with an instruction that produces a physically impossible object, and the flat one is what everybody has always printed.

That asymmetry is worth carrying: the cylinder is a real object that this collection could not previously describe, and the torus is a description that no printing can realise. Both are needed for the comparison and only one is paper.

Two other things a drawing does not say #

Since the essay is about under-determination, two further gaps are worth listing so that the gluing does not look unique.

The drawing does not say which patch it is. A rectangle of a tessellation is a specimen, and reading a property off it and attributing the property to the tessellation is the standing hazard of patches. The rectangle’s size and position are choices the drawing does not record.

The drawing does not say what folded state. A pattern with an assignment may have many folded states, differing in the layer order, and the layer order is usually not recorded either. That gap is old, well known, and the subject of a good deal of this collection.

The gluing sits between them in kind. Like the first, it is a fact about which object the drawing is a drawing of. Like the second, it changes the answer to whether the thing folds.

Which claims are affected #

The honest audit, since the essay is a criticism of a premise the collection holds.

Local claims are unaffected. Anything computed at a vertex is computed from the drawing and the sheet does not enter. That is most of what this collection says.

Claims about patches are unaffected and are about patches. A cut rectangle costs one node of search per panel is exactly what was measured.

Claims about tessellations are the ones to watch. A tessellation has no boundary; a patch of one has four sides; and a claim of the form this tessellation costs so much measured on a patch is a claim about a sheet the tessellation is not.

The premise, and why it was worth holding #

It would be easy to read the essay as an argument against the collection’s own premise, and it is not.

The pattern is the object did a great deal of work and continues to. It is what makes a crease pattern checkable rather than illustrative, what makes a figure evidence rather than decoration, and what lets a reader with a printer reproduce every claim here exactly.

The alternative — that a pattern is a picture of a model, and the model is the real thing — is what the subject looked like before it was mathematical, and it is why folding sequences were the unit of publication for a century.

So the premise is right. What it needed was the observation that a pattern is the object given a sheet, and that the sheet has been constant for so long that it stopped looking like an input.

That is the ordinary fate of a constant. It stops being written, then stops being thought about, then stops being noticed when it changes.

An analogy that carries #

There is a familiar version of this in another subject and borrowing it makes the situation less peculiar.

A graph drawn on paper is a set of vertices and edges. Drawn on a torus it is the same set of vertices and edges, and it may be embeddable there when it is not embeddable in the plane — which is a fact about the surface rather than about the graph, and every account of graph embedding states the surface as a parameter.

Nobody says this graph is planar without meaning in the plane, because the surface is written into the word.

Folding has no equivalent word. This pattern folds flat has no surface in it, and there has been no need for one, so the sentence looks complete.

The correction is to acquire the habit that the other subject already has: name the surface. Four words, and the sentence becomes true or false rather than under-determined.

Where to look for the next one #

If a constant that stopped being written is the shape of the problem, it is worth asking what else in this collection is constant.

The paper has no thickness. Varied deliberately and often, and the collection has a whole anchor about it.

The paper does not stretch. Varied occasionally.

The paper is a disc. Varied here, for the first time.

The folding is flat. Varied, in the rigid-folding work.

The paper is a single connected piece. Not varied. Two sheets joined at a point, or a sheet with a piece hanging off, are objects nobody here has built.

The creases are straight. Varied, in the curved-crease work.

The sheet is finite. Not varied, and the torus is the closest anything has come.

Two of those seven have never been moved, and the one that has just been moved produced four essays’ worth of consequences. That is not a prediction about the other two, and it is a reason to look.

The correction, in practice #

Four words per claim: name the sheet.

A rectangle of this pattern, cut out of the plane, costs one node per panel. A cell of it glued into a torus refuses at odd sizes. A tube of it flattens when an even number of creases run along it.

Each of those is a claim somebody can act on. This pattern folds is not, and it was the form every claim here took until there were four sheets to choose between.

What the premise survives as #

The pattern is the object is still the right premise and it needs one clause.

The pattern is the object on a stated sheet. A crease pattern together with a sheet is a complete description of a folding problem: everything else follows, the checks are exhaustive, and the reader can fold it.

A crease pattern alone is a complete description of a folding problem on a disc, which is the case it has always been used for and is not the only case there is.

That is a smaller correction than it sounded like three paragraphs ago, and it is the correct size. The premise was right and it was missing an argument, and the argument was invisible while it had one value.