The fifth thing that is not true

Four things that are not true names the idealisations every theorem on this site rests on: zero thickness, no stretch, creases that are lines, and perfect memory. Each of the four has had an essay. The crease has a radius took the third, wet-folding the second, the sheet remembers the fourth, and the first runs through the whole rigid-folding ladder.

There is a fifth, and the reason it has never been named is instructive. The other four are assumptions that are stated and then relied on. This one is not stated anywhere, because the theorems do not need it — they are silent about the thing it is about, and silence is easy to mistake for generality.

Paper has a grain, and no theorem in this subject mentions a direction.

What the silence costs #

Developability, Kawasaki, Maekawa and the big-little-big lemma are all invariant under turning the sheet. So is flat-foldability, so is rigid-foldability, so is every count and every census this site has published. Turn a crease pattern through a right angle and every statement about it is unchanged.

Turn a sheet of paper through a right angle and it is a different object to fold. Its fibres lie mostly along one direction — the direction it travelled through the machine that made it — and a crease along that direction is a different physical thing from a crease across it. It folds more sharply, it resists less, and it cracks less.

The consequence is precise and is worth stating as a gap rather than as a complaint. The subject can tell a folder everything about a pattern except which way round to put it on the paper, and the choice is not small.

How much is at stake #

The census measures the share of a pattern’s crease length lying within 22.5° of the grain — a band that splits the quarter turn evenly into three named parts, stated rather than tuned.

On the tapered corrugation it runs from 42.5 per cent at the best placement to none at all at the worst. On the Yoshimura it runs from 28.6 per cent to none. On the hexagon twist, 30 per cent to none. On the waterbomb tessellation, 21 per cent to none. On the Miura, 54.2 to 45.8 — the smallest swing in the census, because the Miura’s creases run in three directions rather than two and turning it exchanges rather than removes.

And on two patterns the swing is exactly zero. The preliminary base has its creases at 0°, 45°, 90° and 135°, which is a set closed under a quarter turn, so every placement is the same placement. The square twist is the same story with two families at right angles.

Eight placements, and only two of them #

A square sheet has eight symmetries: four rotations and four reflections. A grained square has two — the identity and a half turn — because everything else moves the grain.

So a pattern that could be laid on plain paper eight indistinguishable ways can be laid on grained paper in four genuinely different ones. And turning the sheet over is not one of them: the grain runs through the paper, so flipping it leaves the grain where it was and changes only which face is up.

Four placements, of which two are distinguishable by this census — because a direction taken modulo a half turn is unchanged by a half turn of the sheet. That is not a limitation of the measurement; it is a fact about what a grain is, and it means a folder’s real choice is between two options rather than four.

A folder is not restricted to quarter turns either. A pattern can be laid on the sheet at any angle it fits at, and the sweep figures show what happens in between — which is where the argument’s shape is visible, and where the best placement usually is not at a multiple of ninety degrees.

Every pattern has the same average #

There is an invariant under all this that the census’s spread of numbers hides, and it settles what the census is actually measuring.

Take any single crease and turn the sheet through a half turn. Its direction sweeps the whole range, so the fraction of that sweep during which it lies within $\beta$ of the grain is $2\beta/180°$ — and that is true of every crease, whatever its direction and whatever the pattern.

Summing over the pattern, the share averaged across all placements is $2\beta/180°$ for every pattern in the census: 25% at the essay’s 22.5° band, 16.7% at the narrower 15° one. Not approximately, and not on average across patterns — exactly, for each of them.

The two zero-swing patterns confirm it directly. The preliminary base’s creases sit at 0°, 45°, 90° and 135°, spaced exactly a band-width apart, so at every placement exactly one family of the four lies within 22.5° of the grain. Its share is 25% always, which is the invariant with no swing left in it.

So the census measures sensitivity, not quality #

That reframes the whole table. A pattern with a large swing is not a pattern that can do better; it is a pattern whose result depends more on the folder’s choice. The total available is the same everywhere.

It also says the reported worsts are not worsts. The Miura’s 45.8% cannot be its minimum over all placements, because a minimum of 45.8 and a maximum of 54.2 cannot average to 25 — so the Miura has placements, away from the quarter turns, at which its share falls far below anything in the table.

The lever a folder has is placement and only placement, and the essay’s real finding is which patterns hand that lever a long arm: the tapered corrugation, from a quarter of its length above the average to none of it, and the preliminary base, which offers no choice at all.

The ceiling #

There is a bound on how well any placement can do, and it needs no census at all.

Two crease families at an angle φ cannot both lie along the grain. Turning the sheet to favour one moves the other away by the same amount, so the worse of the two is left at least φ / 2 away, whatever the folder does.

On the Yoshimura φ is 60°, so some family is always at least 30° off the grain — which is most of the way to being across it. On the Miura φ is 69.9° and the bound is 35°. On the preliminary base and the waterbomb tessellation φ is a right angle and the bound is 45°, which is the worst case: two perpendicular families are as far from agreeing as two families can be, and no placement helps either.

That bound is computed from the pattern’s own directions with no measurement in it, and it is the reason the census’s best-placement figures are what they are rather than being close to one.

The two patterns where the question does not arise #

The preliminary base and the square twist have a swing of exactly zero, and the reason is worth separating from the rest because it is a design property rather than an accident.

Both have crease directions closed under a quarter turn. The preliminary base’s are 0°, 45°, 90° and 135°; the square twist’s are 0° and 90°. Turning either pattern by a right angle maps its set of directions to itself, so the census cannot tell the two placements apart, so the folder’s choice is genuinely free.

That is a property a designer could aim at. A pattern whose directions are closed under a quarter turn is one that cannot be laid on the sheet badly — and the price is that no placement is good either, because closure under a quarter turn means the families are spread over the whole half turn and some of them are always across the grain.

A pattern is either indifferent to the grain or exposed to it, and it cannot be aligned to it. That is the honest summary of the census: the best any of these eight does is 54 per cent of its crease length near the grain, and half of that is the Miura’s third family arriving by accident.

Which theorem was checked, and how #

Four things, and the first is a decision rather than a check.

The sheet’s own edges are left out. Every pattern here is drawn on a square, two of whose sides run along any grain at all, so counting the boundary would report every pattern as partly aligned for a reason that has nothing to do with folding. The census is over creases only, and the check requires it.

A crease is a line and not an arrow: a crease drawn from the other end lies the same distance from the grain, so directions are taken modulo a half turn. Without that, a pattern’s census would depend on which way its generator happened to emit each edge.

Turning the sheet over changes nothing. The grain runs through the paper rather than along one of its faces, so flipping the sheet is not one of the placements — and the check mirrors a pattern and requires the census to be unchanged, which it is to the last bit a double holds.

And the ceiling has to be capable of being zero. A pattern whose creases all run one way is not caught between anything, and the bound returns zero for it — which is what makes the bound on the others a statement about them.

What a folder already does about it #

The gap this essay is about is not one folders live with unknowingly. The practice has a well-established answer and it is worth setting beside the census, because it shows what the geometry is missing.

A bookbinder cuts stock so that the grain runs the way the spine will fold. A folder working on a large model chooses which way to cut the sheet from the roll and notices, without measuring anything, that the wet-folded curves come out better one way round. An origami paper sold in squares has been cut from a machine-made web and its grain is parallel to one pair of edges, so the choice is available and is not usually written on the packet.

What the practice does not have is a number, and what it does not have a number for is the comparison between patterns. A folder knows that this model went better this way round; nobody has said that the tapered corrugation puts 42 per cent of its folding with the grain in one placement and none of it in the other, or that the Yoshimura’s ceiling is thirty degrees.

That division of labour is the same one the record-and-proof distinction is about: an accumulated practice that is reliable and unquantified, beside a theory that is exact and silent.

Where the model stops #

The census also says nothing about cutting. A sheet is cut from a larger one, and which way the grain runs in the piece is decided at that moment; a folder handed a square has already had the choice made for them, and the census’s advice reaches only somebody who cuts their own.

Nothing here is a material property. The grain enters as a direction and as nothing else; no modulus, no fibre orientation distribution, no bending stiffness and no tear resistance appears anywhere, and every number above survives with the physics of paper deleted. How much easier a with-grain crease actually is belongs to a subject about materials, and this essay does not borrow from it.

That is a deliberate narrowing and it costs the essay its punchline. The census says how much crease length a placement puts near the grain; it does not say what that is worth, because saying so would require a quantity this site does not compute and would not check.

The census is also over the site’s own eight printed patterns, which are mathematical objects rather than designs. A designed model has creases at many more angles — the fold-and-cut triangle, with six directions among six creases, is the closest thing here to that case — and its census would be flatter, because a pattern with creases everywhere has no placement that favours many of them.

What the picture cannot show #

None of these figures shows paper. A grain is a statistical fact about fibres a few tenths of a millimetre long, and what is drawn is a horizontal dashed line standing for a direction — which is the right abstraction for the argument and is not a picture of the thing.

Nothing shows the difference between a with-grain crease and an across-grain one, because that difference is what the essay declines to compute. A reader who has folded a sheet the wrong way round knows what it looks like and no figure here will remind them.

The generalisation #

The useful form of this is about what an idealisation is. The four in the earlier essay are simplifications: the paper does have thickness and the theory says it does not. The fifth is different in kind — the theory does not say the paper is isotropic, it simply never mentions direction, and a reader supplies the assumption without noticing because the alternative never came up.

That is a more dangerous shape of idealisation than an explicit one. An explicit assumption can be checked, relaxed and priced; an assumption that lives in what a theory declines to mention cannot be, because there is no sentence to point at.

The general test is the one this essay applies: take a symmetry the theory has and ask whether the material has it. Every theorem here is invariant under turning the sheet, so the theory has a rotational symmetry; paper does not; and every place where a theory is more symmetric than its subject is a place where it has stopped saying something.

Applied to the rest of this site, that test finds one more immediately. The theorems are invariant under turning the sheet over as well, and paper — printed on one side, coloured on one side, sized on one side — usually is not. That one is at least visible, because the two-colouring makes it a design variable rather than an oversight.

Who found it, and when #

Grain is not a discovery. Every paper mill has known about it since paper was made by machine, every bookbinder works with it, and every account of folding aimed at practitioners mentions it — usually as advice about which way to cut a sheet rather than as a property with consequences for a pattern.

What is not written down anywhere is the arithmetic: that a pattern’s crease directions determine how much a placement can buy, that the answer is a third to a half on the patterns this site prints, and that two families at an angle set a ceiling nothing can beat. It is a small computation and the reason nobody appears to have done it is that the two communities who would care — the theorists and the folders — are separated by exactly this gap. The theorists have no direction in their objects and the folders have no census.

The test applied to the rest of the site #

If the general test is take a symmetry the theory has and ask whether the material has it, it is worth running down the list once rather than leaving it as a slogan.

Rotation. The theory has it; paper does not, because of the grain. That is this essay.

Reflection through the sheet. The theory has it; paper often does not, because one side is coloured or coated — and this one is at least visible, since it is the basis of a design technique.

Scaling. The theory has it — no flat-folding condition mentions a length — and paper does not, because thickness does not scale with the pattern. That is the whole content of the thickness ladder, and it is the one place where the site has already followed the test all the way through.

Translation. The theory has it; a sheet does not, because it has edges, and what the boundary does to the conditions is a rung of its own.

Four symmetries, four failures, and three of them already have essays. The test is not subtle and its value is that it produces the list rather than requiring somebody to notice each item separately.

Where the ladder goes next #

The immediate continuation is the one the model stops short of: what a with-grain crease is worth, which needs a measurement of paper rather than of patterns. The geometric half is done and the number it would multiply is a single ratio.

The other direction is a design question with a clean statement. Given a target crease pattern and a grained sheet, the placement is a one-parameter choice and the census is a function of it; but a designer choosing the pattern has more freedom than that, and a pattern whose families are closed under a quarter turn — like the preliminary base’s — is one for which the placement question does not arise. Whether that closure is worth designing towards is a trade nobody has priced.