Sixty-four rules, sixteen fold

The Miura fold is the most reproduced crease pattern in this subject, and the way its letters are usually explained is a recipe. Draw the zigzag rows, draw the columns, make each row’s creases one letter, and make the columns change letter every time they cross a row. That last part is always flagged as the trick, and it is the part beginners get wrong.

A recipe is not a measurement. This collection has taken one corrugation’s letters apart before — the waterbomb tessellation has five hundred and twelve repeating rules and thirty-two of them fold — and the same question about the Miura had never been asked, though it is the pattern everything else here is compared against.

There are sixty-four rules. Sixteen fold.

What a repeating rule is #

A tessellation is meant to repeat, so the letterings worth considering are the ones that repeat with it. That is a much smaller set than all letterings and it is the set a folder actually works in: nobody memorises two hundred and eighty-two letters, they memorise a rule and apply it.

A grid corrugation has two kinds of crease and three parity classes between them. A row crease can take its letter from the parity of its row — two bits, one for the even rows and one for the odd. A column crease sits between two rows and in some column, so it can take its letter from the parity of its column and the parity of the row below it — four bits.

Six bits. Sixty-four rules. The family contains the constant rules, all-mountain and all-valley, which is deliberate: they are what somebody who has not met the pattern would draw, and it turns out they fail in a way that is worth seeing.

The sixteen, written out #

The result is not a scatter of survivors. It is a product.

All sixteen surviving rules make a column crease carry different letters above and below a row. Not one of the forty-eight rules that fail does; not one of the sixteen that pass fails to. The column alternation is exactly the condition, stated as a fact about the bits rather than as advice.

And the row letters are free. All four ways of writing them — both rows valley, both mountain, valley then mountain, mountain then valley — appear among the sixteen, and each of them appears with all four ways of alternating the columns. Four times four is sixteen and there is nothing else.

That is a stronger statement than the recipe, and a simpler one. Rows one way, columns the other is two instructions of which one is empty. What has to be learnt is: a column crease changes letter every time it crosses a row. The rows can be anything.

Why the rows do not matter #

The reason is visible at a single vertex, and it is worth doing by hand because it explains the whole table.

An interior vertex of a Miura has four creases. Two of them are the halves of one zigzag row running through the point, and they are collinear, so they lie opposite each other. The other two are the halves of a column crease, one going up and one going down, and they lie opposite each other too.

Under a repeating rule the two row-halves are in the same row, so they carry the same letter — call it r. The two column-halves are in the same column but in the rows above and below, so they carry v above and w below, and those are separate bits.

The counting theorem requires three of one letter and one of the other. With the two row-halves already equal, that forces the two column-halves to differ: v and w cannot both equal r (that is four of one) and cannot both differ from r (that is two and two). One of them must match r and the other must not.

So the condition at every vertex is exactly v ≠ w, whatever r is. The row letters cancel out of the requirement, which is why they are free, and the column alternation is the requirement, which is why it is forced.

Every refusal is one theorem’s #

The forty-eight rules that fail all fail the same way, and the collection has now seen this often enough that it is starting to look like a fact about corrugations rather than a coincidence.

At every vertex where a failing rule fails, the count is what refuses it. The angle condition holds — a Miura’s vertex has two collinear creases and two making equal angles above and below, so the alternating sums are a hundred and eighty degrees exactly whatever the letters are. The smallest-sector lemma holds too, and for a reason worth noticing: it only speaks when there is a strictly smallest sector, and a Miura vertex has its sectors in equal pairs, so at the standard proportions it has nothing to say anywhere in the pattern.

That leaves one theorem carrying the entire family, which is exactly what the waterbomb’s five hundred and twelve rules reported: four hundred and eighty failures, every one the count’s alone. Two families, two rule spaces of different sizes, the same verdict.

The forty-eight, in three groups #

The failures are as structured as the survivors, and counting how badly each one fails makes the structure visible without any argument.

The pattern swept has fifteen interior vertices. The forty-eight failing rules break the count at six vertices, at nine, or at all fifteen — and there are exactly sixteen rules in each group. Nothing fails at one vertex, or at two, or at fourteen.

The reason is the same v ≠ w condition applied class by class. A column crease’s letter depends on the parity of its column, so the pattern has two classes of column, and each class either alternates down the rows or does not. If both classes alternate, every vertex is satisfied and the rule is one of the sixteen. If only the class in the odd columns fails, the vertices in those columns fail — three columns by three rows, nine of them. If only the even class fails, that is two columns by three rows, six. If both fail, all fifteen.

Four cases, sixteen rules each, and the sixteen in each case are the four row patterns times the four ways of choosing which letter each class starts on. The whole sixty-four is a product of four independent binary choices, only one of which the folding cares about.

That is why the sweep is worth running rather than reasoning about. The reasoning above is two paragraphs and could have been written without folding anything; what makes it trustworthy is that a machine built the pattern under all sixty-four rules, checked every vertex of each, and returned a table with three sixteens in it.

Why a rule rather than a lettering #

There is a question of principle underneath the choice of what to sweep, and it deserves a paragraph because the answer decides what the sixty-four are for.

A crease pattern’s letterings are the objects the conditions are about, and a repeating rule is not one of them — it is a description of one, and a very restrictive description. Sweeping the rules therefore answers a question about descriptions rather than about the pattern: which of the compact instructions somebody could write down produce a foldable object.

That is the right question for a corrugation, and only for a corrugation. A tessellation exists to be repeated, a folder scores it by applying a rule across a sheet, and a lettering that is not expressible as a rule is one nobody would ever produce by hand however foldable it is. So the sixty-four are the realistic candidates, and the sixteen are the realistic answers.

For a pattern that does not repeat — a base, a fold-and-cut outline, a crumple — the question is empty, because there is no parity class to hang a rule on. Those patterns are lettered by a propagation whose answer is one of many, and a sweep over their descriptions would be a sweep over a set with one member.

The tapered leaf gives the same sixteen #

The same sweep run on the tapered corrugation — the fold a plicate leaf packs into, broad in the middle and narrow at both ends — produces an identical table. Sixteen rules survive, they are the same sixteen by number, the forty-eight failures fail the count alone, and thirty-eight of them close a circle.

That identity is not a surprise once it is stated, and it is worth stating because it is the second half of a finding this collection made earlier from a completely different direction.

The leaf’s taper is in its column widths, and the column widths never enter any vertex condition: at an interior vertex the two row creases are collinear and the two column creases leave at an angle set by the row heights alone. So a corrugation may be tapered along its fold lines and may not be tapered across them, and the widths are free — every one of them, independently.

If the widths are invisible to the conditions, they are invisible to a sweep over the conditions. The leaf and the Miura are, as far as their letters are concerned, the same object. That was derived; this measures it.

What the sweep does not settle #

Three limits, and the third is the one that matters most.

The sweep is over repeating rules, which are a vanishing fraction of the letterings a Miura admits. A six-by-four Miura has thirty-eight creases and its admissible letterings number in the millions; sixty-four of them repeat. So sixteen of sixty-four is not a share of anything, and a rule that fails is not a lettering that fails — the same pattern lettered irregularly may well fold, and most of its irregular letterings do.

The sweep is at one size. The waterbomb’s rules taught this lesson expensively: fifty-six of its five hundred and twelve rules pass on a two-by-two patch and only thirty-two on anything larger, because a two-by-two grid has one corner vertex inside it and a three-by-three has four. The conditions did not get stricter; the patch stopped hiding them. The Miura sweep is run at six columns by four rows, which has fifteen interior vertices of one kind, and there is only one kind — so there is nothing for a small patch to hide, and the count does not move with the size.

And the sweep asks about the vertex conditions, not about folding. A rule that passes all four conditions at all fifteen vertices has passed a necessary test, and passing every local condition is not folding. What the sixteen have additionally been checked for is that their arcs close no circle, which they do not — but the non-crossing rules are not in evidence here at all.

Two to the bits, less the vertex kinds #

Two families have now had their rules counted, and the two counts are one formula.

The condition at a vertex — that the two column-halves differ — is a single equation over the two-element field: two bits summing to one. A repeating rule makes every vertex of a given kind identical, so a family with $k$ kinds of vertex imposes $k$ such equations on its $b$ bits, and the survivors number

$$2^{\,b-k}$$

The Miura has six bits and two kinds of vertex — one per row parity — and sixteen survive. The waterbomb has nine bits and four kinds, and thirty-two survive. Both exact.

So the rule table is a linear code, and its size is fixed by counting vertex kinds rather than by folding anything. That is why the survivors come out as a clean product rather than a scatter: a coset of a linear subspace is a product of independent choices, which is exactly the four-by-four the essay reads off the sixteen.

The family that breaks it #

The Yoshimura has six bits and twenty-six survivors, and twenty-six is not a power of two.

The reason is its vertices. At degree four the counting theorem is exactly a parity condition — three and one are the odd counts, and odd is a sum modulo two. At degree six it is not: parity admits zero, two, four and six mountains, while the theorem wants two or four, so the two extreme cases have to be excluded by hand and the exclusion is not linear.

A non-power-of-two rule count is therefore a signature of degree-six vertices, and a power of two is what a family of degree-four vertices gives. The three grid families in this collection are all sixteen or thirty-two; the one with six creases at a point is twenty-six.

A prediction the sweep does not make #

There is one more thing the sixteen almost certainly are, and it is worth recording as a prediction rather than a result because the measurement is cheap and has not been run.

The waterbomb’s thirty-two surviving rules were folded and compared panel by panel, and they turned out to be one object — thirty-two markings of a single folded state, differing by turning the sheet over and by shifting the starting cell.

The Miura’s sixteen have the same shape: four row patterns times four column starts, with the row letters known to produce identical folds. The prediction is that all sixteen fold to one object, and the check is the same one the waterbomb had — build each, place its panels, and compare centroids and areas.

If it holds, sixteen rules is a count of descriptions and the Miura has exactly one repeating lettering, which is a considerably tidier sentence than the recipe.

The recipe, corrected #

It is worth writing down what a folder should actually be told, since the usual form of it is half wrong and the correct form is shorter.

Every column crease changes letter where it crosses a row. That is the whole rule. It is sixteen rules only in the sense that the rows and the starting letters can be chosen freely, and choosing them differently produces the same fold turned over or shifted by a cell.

The reason the recipe usually includes an instruction about the rows is that a picture of a Miura has to show some row letters, and whichever ones the picture shows get copied into the description. The picture in this collection shows valley then mountain. A picture showing both rows as valleys would be equally correct, would look noticeably different, and would fold to the same object.

What the sweep says about the sampler #

The rule sweep is exhaustive over a small set, and this collection’s other statements about corrugations come from a sampler over a very large one. The two answer different questions and it is worth checking they do not contradict each other.

The sampler reports that a Miura’s irregular letterings are consistent about two-thirds of the time at thirty-five independent chains of panels, and that the share falls as the pattern grows. The sweep reports that sixteen of sixty-four repeating letterings satisfy every vertex condition, and that all sixteen are consistent.

Both are true and neither implies the other. The repeating letterings are extraordinarily special members of the admissible set — a set with millions in it — and there is no reason a share measured over the whole set should match a share measured over sixty-four hand-picked members. What is notable is the direction: every repeating rule that passes the vertex conditions also passes the circle test, which is a hundred per cent against the sampler’s two-thirds.

That is not a coincidence and it is not a puzzle. A repeating rule is a lettering with the pattern’s own symmetry, and a symmetric lettering’s arcs are symmetric too; a circle in them would have to be a circle the symmetry maps to another circle, and on a corrugation the only circuits available round a single vertex are the ones the count has already closed. The sampler’s failures are irregular letterings closing irregular circuits, which no rule can produce.

Where the ladder goes next #

The forty-eight failures are not all alike, and the difference between them is the next rung. Thirty-eight of them do something stronger than failing the count: their letters send four panels round in a circle, which is a proof that no arrangement of the layers exists rather than a vertex that will not close. Ten do not.

Which ten, and why, turns out to be readable off three of the six bits without folding anything — and the closed form agrees with the arcs on all sixty-four rules of both grid families.

And there is a family here that behaves differently. The Yoshimura’s sixty-four rules produce twenty-six survivors and not one of its thirty-eight failures closes a circle, which is a fact about the degree of its vertices rather than about its letters. Where a repeating rule can close a circle is the rung that separates the two cases.