Half the recipe is decoration

Look up how to fold a Miura and the letters arrive as a pair of instructions. Something like: the horizontal zigzag lines alternate mountain, valley, mountain, valley up the sheet; the vertical lines change from mountain to valley every time they cross one of them. The second half is always flagged as the difficult part and it is always the part a beginner gets wrong.

The first half rules out nothing. It is not wrong; it is empty — a description of the picture the writer was looking at, promoted to an instruction.

The enumeration #

A repeating rule for a grid corrugation is six bits: a letter for the even rows, a letter for the odd rows, and a letter for each of the four combinations of column parity and row parity. Sixty-four rules in all, and every one of them can be built and put past all four conditions at every interior vertex.

Sixteen fold. They are exactly the rules whose column creases carry different letters above and below a row — the second instruction, stated as a fact about bits — and among those sixteen the row letters take all four forms: both rows valley, both mountain, valley then mountain, mountain then valley.

Four row patterns times four column patterns is sixteen, and there is nothing else in the list. The two halves of the recipe are not two conditions. They are one condition and a free choice.

What the four row patterns actually produce #

It is worth being concrete about what changes when the row letters change, because the pattern folds identically is a strong claim and it deserves unpacking.

The four permitted row patterns give four different-looking drawings. With both rows valley, every zigzag line on the sheet is a valley and the columns carry the alternation alone. With both mountain, the same in reverse. With the rows alternating, the drawing has the striped look most pictures of a Miura have.

Folded, they are the same corrugation seen differently. Turning the sheet over swaps every letter, which maps one of the four onto another; shifting the pattern by one row swaps the two row letters, which maps a third onto a fourth. So the four are two objects, each seen from both sides — and the two objects are the same corrugation with its zigzag phase shifted by a cell.

That is why no condition distinguishes them: they are related by operations that leave a crease pattern the same crease pattern. What a recipe should say is that the rows are free; what a recipe usually says is which of the four the writer’s own diagram happened to be.

Why the empty half gets taught #

The mechanism is worth naming because it is not carelessness, and it produces this kind of instruction reliably.

A teaching text needs a picture. A picture of a Miura has to show some letters on its rows — there is no way to draw a crease pattern with the row letters left open — and whichever letters the picture shows get read off and written into the prose. The writer is describing what is in front of them, accurately.

What turns a description into an instruction is the sentence that surrounds it. The rows alternate is a true description of the drawing and a false claim about the pattern, and nothing in the drawing marks the difference.

This is the same mechanism that produces a wrong reason attached to a right conclusion, and it has the same tell: a statement about a pattern that could only have been arrived at by looking at one example of it.

What a learner loses #

An empty instruction is not harmless, and the harm is specific.

A learner folding from a diagram checks their work against the picture. If they have written the rows the other way round — which is one of the four permitted forms, and a natural thing to do if the sheet was turned over at some point — the pattern in front of them will not match the picture, and the recipe tells them they have made a mistake.

They then “correct” a perfectly good pattern. And if they correct it by changing the rows without changing the columns, they will produce something that still folds; if they correct it by changing a column crease to match, they will produce something that does not.

So the empty instruction manufactures a class of false alarms, and the repair for a false alarm is a real error. That is a worse outcome than saying nothing about the rows at all.

The claim is that the sixteen form a rectangle #

“One condition and a free choice” is a strong statement about a set of sixty-four objects, and the arithmetic says exactly what would have to be measured for it to be false.

Six bits give sixty-four rules. The surviving condition is one demand per column class — that the class’s letter differ above and below a row — and there are two column classes, so it kills one bit each. Sixty-four becomes sixteen, which is $2^{6-2}$, and the fraction is a quarter.

But the count alone does not establish the claim. Sixteen survivors could have been any sixteen of the sixty-four; what makes the rows free is that the survivors factor as $4 \times 4$ — every one of the four row patterns appearing with every one of the four column patterns, with no cell missing. The set is a rectangle, and a rectangle is what independence looks like when it is written down.

That is the thing the enumeration checks and a recipe cannot. Had the two halves interacted at all — one row pattern admitting only three column patterns, say — the total might still have come out near sixteen while the free-choice claim was false, and no amount of looking at drawings would have said so.

And it makes the waterbomb’s silence measurable #

The same reading turns the pattern nobody writes a rule for into an open arithmetic question rather than an absence.

Nine bits give five hundred and twelve rules and thirty-two fold, which is $2^{9-4}$: four bits killed against the Miura’s two, and a survival fraction of one in sixteen rather than one in four.

What nobody has asked is whether those thirty-two also form a rectangle. If they do, a waterbomb recipe is four forced clauses and five free bits, statable in one sentence exactly as the Miura’s is. If they do not, the pattern genuinely resists a rule and the teaching-by-object approach is the right one rather than merely the traditional one — and that is a distinction the enumeration already run could settle without folding anything.

The reason usually given for the half that matters #

The second instruction is real, and the explanation attached to it is usually wrong in the way this collection has met before.

The reason given is normally something like the alternation is what lets the pattern collapse in both directions at once, which is a statement about the folded behaviour and reads as though the letters were chosen to produce it. They were not chosen at all; they are forced, and what forces them is a counting argument at a single point.

At an interior vertex of a Miura, four creases meet. Two are the halves of one zigzag row running straight through, so they are collinear and a repeating rule gives them the same letter. The other two are the halves of a column crease, one above the vertex and one below, and they are in different rows.

The counting theorem requires three of one letter and one of the other at every flat-foldable vertex. With the two row-halves already equal, that leaves exactly one possibility: the two column-halves must differ. Not because of anything about collapsing, and not as a choice — the count leaves no alternative.

The auxetic behaviour everybody quotes is downstream of that, not upstream. It is what the pattern does once its letters are what the count allows.

The one thing the row letters do change #

To be fair to the recipe, the row letters are not entirely without effect, and it is worth saying what the effect is so that free is not overstated.

Two of the four row patterns put every zigzag on the sheet the same way — all valley or all mountain — and the other two alternate. Folded, all four give the same corrugation; unfolded and looked at, they do not, and a folder scoring a sheet works from the unfolded drawing.

An all-valley set of rows is easier to score in one pass: every horizontal line gets the same treatment, and the hand does not have to keep track. An alternating set is what most diagrams show and what most folders will have learnt, so it is the one that will look right.

Neither of those is a folding condition, and neither belongs in a statement about which patterns fold. They belong in a note about how to work, which is a different kind of instruction and could usefully be labelled as one: any row pattern folds; this one is easiest to score.

The corrected recipe #

One sentence covers it: a column crease changes letter every time it crosses a row.

That is the whole rule. The rows may be written any way at all, the pattern folds identically, and a learner who is told this and nothing else can neither be wrong about the rows nor made anxious by a picture that disagrees with their sheet.

It is also, unusually, a rule with a reason short enough to teach beside it: two creases at a vertex are one straight line and carry one letter, the count wants three and one, so the other two must differ. That is three clauses and it is complete.

The subject does not often get an opportunity like this. Most of what a folder needs to know about letters is not teachable at all, because the questions are global and their answers are searches. The Miura’s rule is one of the few that is local, exact and one sentence long, and it has been taught with a redundant half and a wrong reason attached.

The same shape, in a pattern nobody writes a rule for #

It is worth looking at what happens where no recipe exists, because the contrast says something about why recipes are written the way they are.

The waterbomb tessellation has five hundred and twelve repeating rules of which thirty-two fold, and no teaching text states any of them. It is taught as an object rather than a rule: fold a waterbomb base, then another, then join them. The letters come out right because the base’s letters are learnt by hand and repeated.

That is a completely different pedagogy and it has a different failure mode. It does not produce false alarms about rows, because nobody is comparing a rule against a picture; it produces a folder who can make the pattern and cannot say why it works, and who has no way to check a pattern they have not been shown.

Between the two, the Miura’s rule is much the better arrangement even with its empty half. A rule that can be stated can be corrected; an object learnt by hand cannot.

The instruction that would have been worth stating #

There is a third thing a recipe could have said and none of them does, and it is the one a folder most needs.

The Miura’s difficulty in the hand is not the letters. It is the order — which creases to set first, how far to collapse before moving on — and that is a question about the layers rather than about the assignment. A folder who has the letters right and the order wrong gets a stubborn half-collapsed object; a folder who has them the other way round finds out immediately.

Nothing about layer order has ever been teachable as a rule, which is exactly why recipes fill their space with the part that can be stated. Two instructions about letters, one of which is empty, is what a text writes when the thing it would like to say has no short form.

That is a more sympathetic account of the empty half than carelessness, and it fits the evidence better. The instruction is not there because somebody was sloppy; it is there because a recipe with one line in it looks incomplete, and the free bits are the only material available to lengthen it with.

How much of the subject’s advice is like this #

The honest answer is that nobody has checked, and this collection is in no position to be superior about it — the Miura’s rule is repeated in these essays too, in the two-instruction form, in more than one place.

What can be said is that the conditions for producing an empty instruction are common. Any pattern with a symmetry that the conditions do not see will have letters that are free, and any picture of it will show one choice of those letters, and any description read off that picture will present the choice as a requirement.

The tapered leaf corrugation is the immediate second case. Its rules are the Miura’s rules, exactly — sixteen of sixty-four, the same sixteen by number — because the taper is in the column widths and no vertex condition reads a width. So any recipe for lettering a leaf corrugation has the same empty half, and inherits it from the same source.

What the collection itself said #

It is worth quoting this collection against itself, because the two-instruction form appears in these essays too, and appeared there first.

The account of the Miura given here has always said that the vertical creases change assignment every time they cross a row, and has always called that alternation the whole trick. That much is exactly right, and it is right for the reason given above.

Beside it sits a description of the rows, written as a rule about the row’s parity and presented as part of the same construction. Nothing marked it as free. Anyone reading the account came away with two rules, because two rules were what was written.

Nothing about the printed pattern changes. A sheet has to be drawn with some letters and this one keeps the letters it has; what changes is that the freedom is now stated beside the constraint instead of being silently absent, which is the whole of the correction and the whole of what a recipe needed.

That is the smallest possible instance of a general problem in this subject. A construction records a choice and a condition in the same notation, and everything downstream reads both as conditions. The letters a construction produces are one solution among many, and the only way to tell which parts of them were forced is to sweep the alternatives — which is what a rule sweep does for a family with six bits, and what nothing does for a pattern with two hundred and eighty-two creases.

What would settle it generally #

A rule sweep is the instrument, and it is cheap: build the pattern under every rule in the family, check every vertex, and read off which bits the survivors agree about and which they do not.

Applied to a family, it produces exactly the distinction this essay is about. A bit the survivors all agree on is a condition; a bit that takes both values among the survivors is a choice; and a recipe should state the first and not the second.

Four families have now been swept here. In two of them — the Miura and the leaf — two of the six bits are free. In the Yoshimura, twenty-six of sixty-four rules survive and the freedom is larger still. In the waterbomb, thirty-two of five hundred and twelve, and no recipe exists to be checked against.

Where the ladder goes next #

The obvious continuation is the survey nobody has done: take the standard accounts of the half-dozen patterns that get taught as rules — the Miura, the Yoshimura, the accordion, the waterbomb where anyone attempts it — and check each stated instruction against its family’s sweep. The instrument exists and each sweep is seconds.

The more interesting question is about the pictures rather than the prose. If a pattern’s free bits are what a picture accidentally fixes, then a good picture of a corrugation would show them as free — two rows drawn in a way that says either, or two versions side by side. No diagramming convention in this subject has a way to say that a letter is unconstrained, and the notation’s whole vocabulary is mountain, valley, boundary and cut. A fifth mark, meaning either, would be a small addition with a large amount of received advice behind it.