The sixth thing that is not true

Four things that are not true named the idealisations every theorem here rests on: the paper has no thickness, it does not stretch, a crease is a line, and the memory is perfect. A fifth turned up later — the paper has no grain — and it was invisible for a different reason from the others: not because it is small, but because no theorem in the subject mentions a direction, so nothing in the apparatus could have noticed.

There is a sixth, it is more basic than any of them, and it is invisible for a third reason again.

The paper is a disc.

Why it is different from the other five #

The five are all about the material. Thickness, stretch, crease width, memory and grain are properties of what the sheet is made of, they are measurable in a laboratory, and each of them is a small deviation from an ideal that the theorems assume exactly.

The sixth is not about the material at all. It is about the shape of the sheet — not its outline, which the subject varies constantly, but which of its boundary points are the same point.

And it is not an approximation. A sheet is a disc or it is not, exactly, and every theorem quoted here holds exactly on a disc and can fail outright on anything else.

What changes when it is dropped #

Three of the subject’s statements change, and none of the four vertex conditions is among them.

The two-colouring becomes a condition. On a disc it is implied: every closed path bounds, so the parity is forced by the vertices inside it. On a sheet with a loop that cannot be shrunk it is an independent requirement, and a grid glued at an odd size fails it.

The closure condition changes its right-hand side. Composing the reflections round a loop has to give the gluing map rather than the identity, and on a disc the gluing map is the identity, which is why nobody writes it.

Mountain and valley stop being globally defined. On a sheet with one side there is no consistent choice, and Maekawa’s condition survives because it reads a difference while the letters it counts do not.

What does not change #

The four local conditions, exactly and completely.

Developability, Kawasaki, Maekawa and the big-little-big lemma each read the creases meeting at one interior point of the paper. Every point of every sheet here is an ordinary point of paper, the angles are what they were, and the conditions hold or fail identically on all four sheets.

So the sixth idealisation is unusually clean in what it touches. Everything local is exact; everything global has a hypothesis.

Why nobody named it #

The five material idealisations were named because they are violated by every real sheet. Any piece of paper has thickness, stretches, has creases with width, forgets, and has a grain. A theorem assuming otherwise is visibly assuming something, and the interesting engineering is in the gap.

The sixth is satisfied by every real sheet anybody folds. Paper comes flat, in discs and rectangles, and it stays a disc unless somebody deliberately joins two of its edges.

So there was no counter-example in the room, and a hypothesis with no counter-example does not get written. That is the ordinary reason a hypothesis goes unstated and it is a shape this collection has now met four times.

What the violation costs #

The other five idealisations have a cost: how thick the paper is, how much it stretches, how wide a crease is. The gap is a number, and engineering lives in it.

The sixth has no such number. A sheet is a disc or it is not, and a sheet that is not is a different object with different theorems rather than the same object with an error term.

That is the sharpest way to see why it belongs on the list and does not resemble the rest of it. The five are approximations and the sixth is a hypothesis, and both are things the theorems assume without saying.

A hypothesis, not an idealisation #

The essay has been calling it the sixth idealisation and the word is not quite right, which is worth being honest about.

An idealisation is a simplification: the real thing is nearly the ideal thing, and the error is small and quantifiable. Zero thickness is an idealisation of paper about a tenth of a millimetre thick.

A hypothesis is a condition: the theorem applies to objects satisfying it and says nothing about the others. There is no error term, because there is no approximation.

The sixth is a hypothesis. A tube is not nearly a disc; it is a different sheet, and the theorems do not degrade gracefully as one approaches it, because there is no way to approach it.

The reason for putting it on the list anyway is that it occupies the same position in the subject’s reasoning as the five: it is assumed silently, it is true of everything anybody looks at, and dropping it changes what is true. Whether the word for that is idealisation or hypothesis matters less than that it belongs beside them.

Why the vertex conditions are untouched #

It is worth being explicit about this, since the essay claims a clean split and a clean split is the kind of claim that hides exceptions.

Each of the four local conditions is a statement about the creases meeting at one interior point of the sheet. Developability is about the sectors summing to a full turn; Kawasaki about their alternating sum; Maekawa about the letters; the big-little-big lemma about the smallest sector.

An identification of a sheet’s boundary edges changes none of that, and it changes none of it by construction: the rectangle’s edges are placed to miss every vertex, so a crease crossing an edge does so at an ordinary interior point of itself, and every vertex is exactly where it was at exactly the same angles.

That is checked rather than assumed. Every figure here reporting counts on the four sheets asserts that the interior vertex count is the same on all of them, and refuses to draw when it is not — which has fired once, on a cell whose corner had a crease running through it, and that was a defect in the identification rather than a counter-example to the claim.

So the split is: local conditions are about points, points are unchanged, and everything that changes is about paths.

What a sheet with one side does to the list #

The Möbius band deserves its own line, because it violates something the other three sheets do not.

A cylinder and a torus are orientable: the paper has two sides, a walker returns the way they went, and mountain and valley are consistently definable.

A Möbius band is not. There is no consistent choice of side, so the letters are not globally defined, and the parity condition inverts rather than merely existing.

So the sixth assumption really contains two: the paper is a disc, and — weaker — the paper has two sides. Dropping the first while keeping the second gives cylinders and tori, where everything survives with a hypothesis attached. Dropping both gives a sheet where the subject’s basic vocabulary loses a term.

Nobody has ever needed the distinction, because both are satisfied by every sheet in the world of folding. Having a sheet that satisfies the second and not the first, and one that satisfies neither, is what made the two separable.

The one that might be seventh #

Having found a sixth by asking what has always been constant, it is worth asking what else is.

The sheet is connected. Two pieces of paper joined at a point, or a sheet with a piece hanging off, are objects nothing here has built. Every theorem assumes one connected piece.

The sheet is finite. Every object here is bounded, and the torus is the closest anything comes to an infinite one. What a theorem says about an infinite sheet is not obviously the limit of what it says about large finite ones — the enumeration of stackings, for one, needs a least element and an infinite stack has none.

The sheet is two-dimensional. Trivially true and not worth a line, except that thick-panel origami is precisely the study of what happens when it is not, and that subject has its own literature.

Two of those three are genuine candidates. Neither has been built, and the reason for listing them is the reason this essay exists: the hypotheses that go unstated are exactly the ones nothing available violates, and finding them means building something that does.

Where it is violated in practice #

Almost nowhere in folding and almost everywhere in engineering.

Nobody folding a model works on anything but a disc. Nobody teaching the subject uses anything else. Every diagram, every pattern, every book.

Every folded structure that gets manufactured is a tube or a shell — a boom, a stent, a bellows, a packed antenna — and each of those is a sheet joined to itself. So the violation is confined to exactly the applications the subject exists to serve, which is an uncomfortable place for an unstated hypothesis to have been sitting.

How the other five were found #

The provenance of the list is worth recalling, because the sixth was found a different way and the difference is instructive.

The first four were found by folding. Anybody who folds a complex model discovers thickness — the layers pile up and the model will not close. Anybody who wet-folds discovers stretch. Anybody who unfolds a model discovers memory. The material announces itself.

The fifth was found by measuring: a census of crease directions against the sheet’s grain across every pattern here, which turned up an alignment worth up to forty-two per cent of a pattern’s crease length and a ceiling no placement beats. Nothing announced it; somebody had to look.

The sixth was found by building an object that violates it, for an unrelated reason. Gluing a rectangle’s edges was done to ask what a boundary costs a search, and having the object made a whole family of questions askable that had not been askable before.

Three routes: the material announces it, a measurement finds it, or somebody builds the counter-example. The third is the only one available for a hypothesis that nothing in the world violates, and it is the slowest, because it requires wanting the object for some other reason first.

What this does to the theorems #

An accounting, since the essay claims a hypothesis was missing from a body of results.

Nothing is retracted. Every theorem quoted in this collection is true of discs, was proved about discs, and applies to every sheet anybody folds.

Three statements gain a clause. The two-colouring, the closure condition, and the definability of mountain and valley — each is stated in a form that is exactly right on a disc and needs a hypothesis elsewhere.

Two pieces of machinery were wrong and are repaired: the consistency test that assumed a cycle is a contradiction, and the closure comparison against the identity. Both were correct on every object they had been run on.

One piece is still wrong and is recorded: the stacking enumeration, which starts from a bottom layer.

So the cost of the missing hypothesis was two repairs and one outstanding item, on a body of work that has been running for a long time. That is a low price and it is low because the hypothesis was true of everything.

The list, updated #

Zero thickness. Material, approximate, violated by every sheet.

No stretch. Material, approximate, violated slightly and deliberately in wet-folding.

A crease is a line. Material, approximate, violated by a measurable radius.

Perfect memory. Material, approximate, violated over time.

No grain. Material, exact for the theorems and violated by every sheet of paper, which no theorem can see because none mentions a direction.

The paper is a disc. Not material, not approximate, satisfied by every sheet anybody folds and violated by every folded object anybody builds.

The sixth is the odd one out on every axis, which is a reasonable explanation for why it took longest to name.

Reading the earlier essays now #

A reader coming to the four and the fifth after this one may wonder whether they need amending, and they do not.

Four things that are not true is about the material, exactly, and everything in it stands. Its four idealisations are approximations with costs, and the costs are where the engineering is.

The fifth thing that is not true is about the grain, and its argument — that no theorem here mentions a direction, so nothing in the apparatus could notice — is itself the observation that a hypothesis can be invisible because the vocabulary has no word for its subject.

That is very nearly this essay’s argument, arriving one idealisation earlier. The grain went unnamed because no theorem mentions directions; the sheet’s shape went unnamed because no theorem mentions sheets. In both cases the missing thing is missing from the language rather than from anybody’s attention.

Which suggests a way of looking for the seventh: not by asking what is approximated, but by asking what the theorems never mention. They never mention connectivity, and they never mention the sheet being finite, and both of those are on the list above.

The engineering in the gap #

Each of the five material idealisations has engineering in the gap between the ideal and the real: thickness accommodation, wet-folding, crease allowances, choosing a paper that holds a fold, laying a pattern with the grain.

The sixth has engineering too, and it is a different kind.

There is no gap to work in, since the sheet is a disc or it is not. What there is instead is a check: given a folded object that closes on itself, count the creases a loop round it crosses and require the count to be even. That is arithmetic, it is exact, and it is the whole of the engineering the sixth idealisation calls for.

For a designer of folded tubes that is a genuinely useful thing to have, and it is cheaper than any of the other five. Thickness accommodation is a redesign; grain alignment is a placement decision with a measured payoff; the parity is an addition performed before anything is cut.

So the sixth idealisation is the odd one out here as well: the only one whose violation is checkable in advance and free.

The one-sentence version #

Five idealisations are about what the paper is made of, and each is a small deviation with a cost.

The sixth is about what shape the paper is, it is exact rather than approximate, it is satisfied by every sheet anybody folds and violated by every folded object anybody builds — and it went unnamed for the ordinary reason, which is that nothing in the room had ever violated it.

What naming it buys #

One clause, added where it is needed.

A pattern folds flat becomes a pattern folds flat on a disc, and the general statement — that the signs multiply to one round every closed path — is what covers the rest.

Composing the reflections round a loop gives the identity becomes gives the sheet’s own identification map, and the disc’s version is the case where that map is the identity.

Every crease is a mountain or a valley becomes on an orientable sheet, and the difference of the two counts survives where the letters do not.

Three clauses, each of them saying nothing on a disc, and each of them the difference between a theorem and a theorem about the objects people build.