Two sentences for the Yoshimura

A recipe needs degree four proves a negative. A lettering instruction of the kind taught for the Miura fold is a list of same-and-differ clauses — these two creases carry the same letter, these two differ, an odd number of these four are mountains — and a set of rules picked out by such clauses is closed under adding any three of them bit by bit. The Miura’s sixteen folding rules are closed that way, and two clauses pick them out exactly. The Yoshimura’s twenty-six and the waterbomb’s thirty-two are not, so no list of such clauses describes either family, however cleverly it is chosen.

The reason was one vertex. Where three pairs of opposite creases are tied to one letter, the counting theorem asks that the three letters be not all alike, and that is not a parity of anything. So the negative result came with a positive question attached. If a recipe for these patterns has to contain a sentence of that other kind, how short can it be?

Short. The Yoshimura needs two sentences.

The rules, and the two kinds of sentence #

A repeating rule for the Yoshimura pattern gives a letter to each of six classes of crease. The courses — the horizontal creases — take a letter by whether their row is even or odd, which is two classes. The zigzags take a letter by which band between two courses they are in, even or odd, and by which way they lean, which is four classes. Six letters, sixty-four rules, and a recipe needs degree four found that twenty-six of them fold.

A recipe may now say two kinds of thing. A parity says that an even number, or an odd number, of some chosen classes are mountains; “these two differ” is the parity of a pair. A prohibition says that some chosen classes are not all alike — not all mountain and not all valley. The first kind is what the Miura’s recipe is made of. The second is the kind the six-crease vertex needs.

The recipe is then checked the only way that means anything: every one of the sixty-four rules is put through its clauses, and the rules it allows are required to be exactly the twenty-six that fold, no more and no fewer.

One parity, and what it lets through #

Among the Yoshimura’s folding rules one parity holds and only one: an even number of the four zigzag classes are mountains. Every folding rule obeys it, and it is the only same-and-differ statement they all share.

On its own it allows thirty-two rules, and the second figure draws them with the folding ones filled. Six obey the parity and do not fold. They are the rules in which all four zigzag classes carry one letter and a course carries that letter too: all four zigzags valley with an even-row course valley, or an odd-row course valley, or both; and the same three with every letter reversed.

The second sentence #

That description of the six is already the missing clause, stated backwards. Forbid it: no course carries the letter all four zigzag classes share. When the zigzags are mixed there is nothing to forbid, and when all four agree, both courses must take the other letter.

Written as clauses the checker can run, the prohibition is two statements, one for the even-row courses and one for the odd-row courses, and each is a ‘not all alike’ about a course and the zigzags. The third figure runs the whole recipe in order. The parity takes sixty-four rules to thirty-two. The prohibition about even-row courses refuses four of those, leaving twenty-eight. The one about odd-row courses refuses two more, leaving twenty-six — and the twenty-six left are exactly the twenty-six that fold.

Each prohibition in the figure names a course and only three of the four zigzag classes, which looks like a different sentence from the one in words. It is the same sentence. Once an even number of the four zigzag classes are mountains, three of them agreeing forces the fourth to agree as well — three mountains need a fourth to make an even count, three valleys need a fourth valley — so “the course and these three zigzags are not all alike” and “the course does not carry the letter all four zigzags share” refuse exactly the same rules. The search that found the recipe prefers the shorter set of letters; a teacher would prefer the sentence.

So the whole instruction for lettering a Yoshimura by rows and bands is:

Letter the zigzags so that an even number of their four classes are mountains. Never give a course the letter all four zigzags share.

Two sentences, one of them a parity and one a prohibition, and they allow the twenty-six rules that fold and nothing else.

What the two sentences say on a sheet #

A recipe is easiest to trust by drawing what it produces. The fourth figure letters the Yoshimura by rule 3: every course mountain, every zigzag valley. The zigzags are all alike, so the parity holds — none of the four is a mountain — and the courses carry the other letter, so the prohibition is kept. Every vertex satisfies every condition.

That is the lettering the pattern is usually drawn with, or its mirror — the kind of familiar drawing taught with a wrong reason checks against the conditions it is taught with: one letter for all the horizontals and the other for all the diagonals. The recipe says it is one of twenty-six, and the most uniform of them: one letter for each kind of crease. It is not the safest of them — its zigzags all agree, so the second sentence is the one holding it, and recolouring either set of courses would break it.

The fifth figure breaks the second sentence and keeps the first. Rule 0 letters everything valley. The zigzags are all alike, so the parity holds, but the courses carry the zigzags’ letter, which is exactly what is forbidden — and the count fails at all twenty-two interior vertices of the patch.

The sixth figure lays out the six rules the parity lets through and the prohibitions refuse, shaded by which prohibition catches each. The two all-alike rules, all valley and all mountain, are caught by both; each of the other four is caught by exactly one. Neither prohibition could be dropped.

One prohibition for each kind of vertex #

The count of prohibitions is not arbitrary, and the reason for it connects the recipe back to the vertex.

The Yoshimura patch has twenty-two interior vertices, all of degree six, and they come in two kinds. Twelve sit on odd-row courses and ten on even-row courses. At each, the two halves of the course are tied to one letter by the rule, and the four zigzags meeting there are one from each zigzag class. So the counting theorem at a vertex on an even-row course asks for four of one letter and two of the other among the even-row course, twice, and all four zigzag classes once each — and if the zigzags are mixed that is automatic, while if they agree it holds exactly when the course differs from them. That is the prohibition for even-row courses, arriving from the vertex. The odd-row vertices give the other.

So the Yoshimura’s recipe is the vertices’ own conditions, one prohibition per kind of vertex, with the parity they share pulled out in front. Stated as a recipe, the pattern’s two kinds of vertex are two sentences’ worth of instruction, and the parity is what they have in common.

The waterbomb needs fewer prohibitions than it has vertices #

The waterbomb tessellation is where the count of prohibitions and the count of vertex kinds come apart, and the difference is the more interesting result.

Its rules have nine bits: one saying which rows of horizontal creases are mountains, and eight giving a letter to each half of each diagonal in the light cells and in the dark cells of a chequerboard. Thirty-two of its five hundred and twelve rules fold. Its patch has two kinds of four-crease vertex, at the centres of the cells, and four kinds of six-crease vertex, at the grid corners, depending on which cells meet there and which row they are on.

Stated vertex by vertex, all four kinds of six-crease vertex are needed: remove any one of their conditions and some rule that does not fold gets through. But the shortest recipe found needs only two prohibitions. Three independent parities come first. The four-crease vertices at the centres of the light and the dark cells account for two of the three, the corners impose the third together, and the figure states them as three relations among half-diagonals that between them say all of that — taking 512 rules to 256, 128 and 64. Then two prohibitions, each about just three half-diagonals, take 64 to 48 and 48 to 32.

The two prohibitions refuse sixteen rules each and share none. Neither is the condition at any one corner: each mixes the two halves of one diagonal in a light cell with one half of the other diagonal in a dark cell. What happens is that the parities have already tied the corners’ letters together so tightly that four different kinds of corner can go wrong in only two different ways. The waterbomb’s recipe is shorter than its list of vertices, because what the parities say about one corner they also say about another.

Why the horizontals drop out #

The collapse can be followed at a single corner, and doing so explains both why the waterbomb gets away with two prohibitions and why the Yoshimura does not.

At a corner of the waterbomb grid, six creases meet. Two are the halves of a horizontal line, tied to one letter. The other four are half-diagonals, one from each of the four cells around the corner: the two cells diagonally opposite each other share a colour on the chequerboard and contribute the two halves of a rising diagonal, and the other two contribute the two halves of a falling one. The counting theorem wants four of one letter and two of the other. If the four half-diagonals are mixed evenly, two mountains and two valleys, the horizontal pair makes it four and two whichever letter it carries. If the four half-diagonals all agree, the horizontal pair must carry the other letter — which is where the rule that breaks the count found every failing waterbomb rule already failing.

Now the difference between the two patterns. The waterbomb’s rules give the horizontals alternating letters, row by row, whatever else they say — mountains on even rows and valleys on odd, or the reverse. So an arrangement of corners that repeats from row to row meets horizontals of both letters. If the four half-diagonals at that arrangement of corner all agree, some row’s horizontal carries their letter, and the count fails there. The prohibition therefore does not need to mention the horizontals at all: these four half-diagonals are not all alike, full stop. The even-row corner and the odd-row corner of the same arrangement, which are two kinds of vertex, become one sentence. And since the parities already fix how many mountains those four half-diagonals have, three of them agreeing is enough to decide, which is why each prohibition in the figure names three.

The Yoshimura’s courses do not alternate. Its rules give even-row courses and odd-row courses independent letters, so both can carry the letter opposite to the zigzags — which is exactly what rule 3 does. A prohibition that ignored the courses would forbid rule 3, and rule 3 folds. So each kind of course needs its own sentence, and the Yoshimura’s two kinds of vertex stay two prohibitions.

A prohibition shrinks when some letter it would have to mention is forced to take both values. The waterbomb’s horizontals are forced to; the Yoshimura’s courses are not.

That is the answer to the question a recipe needs degree four left for the waterbomb. Thirty-two rules, one object explains why the pattern is taught by folding a base and repeating it rather than by lettering a sheet, and the recipe does not overturn that. It says what the alternative would cost: five sentences, three of them parities about four half-diagonals and two of them prohibitions about three, against the Miura’s two sentences and the Yoshimura’s two. A teacher could state that, and a folder could follow it with a pencil.

An engineer has seen the first half before #

The same-and-differ half of a recipe has a name outside paper folding, and it is worth knowing because it says what the second half is not.

In coding theory, a set of bit strings picked out by parity checks is a linear code, or a coset of one, and the checks are written as the rows of a parity-check matrix: each row names some bits and says their sum is even. The Miura’s sixteen folding rules are exactly such a set, and its recipe — a column crease changes letter at every row, once for each class of column — is its parity-check matrix read aloud. Richard Hamming built the first error-correcting codes this way in 1950, and the property that makes them easy to work with is the one a recipe needs degree four used as its test: add three members bit by bit and the result is a member.

The Yoshimura’s twenty-six and the waterbomb’s thirty-two are nonlinear codes: sets of bit strings no list of parity checks describes. What the recipes above find is that these two particular nonlinear codes are very nearly linear. Each is a coset of a linear code with a few ‘not all alike’ conditions cut out of it — two slices of a thirty-two-rule set for the Yoshimura, two slices of a sixty-four-rule set for the waterbomb. A nonlinear code in general can be as far from linear as it likes, and these are about as close as a nonlinear one can be.

The connection also says why the prohibitions are the part a notation struggles with. A parity check is local to its bits and can be verified by counting, and a vertex that expresses only parities is never the material the gadgets that make folding hard are built from. A prohibition is a statement about a pattern of agreement across several classes that may be far apart on the sheet, and it is the kind of condition coding theory handles by listing exceptions rather than by an equation.

What the recipes do not show #

They are recipes for repeating rules. A sheet lettered by hand need not repeat, and a recipe for every lettering of a Yoshimura patch — rather than for the sixty-four that repeat by rows and bands — is a different and much larger object. What these recipes establish is that the simplest letterings of both degree-six families have a short description once a prohibition is allowed, which the negative result had left open.

Shortest means shortest in this search. The recipes were found by taking every parity all the folding rules obey and then trying every combination of up to three minimal prohibitions for the fewest that refuse everything left over. A recipe in some other language — one allowing statements like “exactly two of these”, or conditional sentences — might be shorter still, and nothing here rules that out. Within the two kinds of sentence, two prohibitions is the fewest for both families.

Nothing here measures a learner. A prohibition about letters far apart on the sheet is plausibly harder to apply than a parity, and “never give a course the zigzags’ shared letter” is plausibly easier than a prohibition about three half-diagonals in two different cells. The recipes say how much there is to state, not how hard it is to follow.

And the order the clauses are applied in is a presentation choice. The parities come first because they do the most refusing; the rules the whole recipe allows do not depend on the order.

How the recipes were checked #

Every rule of each family is built as a crease pattern and put past all four conditions at every interior vertex, so the twenty-six and the thirty-two are found rather than assumed.

The counting condition at each kind of vertex is read off the rules directly. Each bit is flipped once and the creases round each vertex that change letter are recorded, so a vertex’s letters become fixed offsets plus sums of bits, and those conditions over every kind of vertex are required to allow exactly the folding rules on their own. They do, for the Miura, the Yoshimura and the waterbomb alike.

Each recipe is required to allow exactly the rules that fold, each clause to refuse something the clauses before it let through, and each prohibition to refuse some rule the others do not. A recipe with an idle clause would not draw.

And the two-sentence recipe is checked on the drawn patterns themselves: on every rule the pattern figure draws, the recipe’s verdict is required to agree with the verdict of the vertex checks.

Still open: whether the recipe survives a larger patch #

The waterbomb once kept fifty-six of its rules on a two-by-two patch and thirty-two from three by three on, because a small patch hides vertices at which a rule would fail. The recipes here are computed on the patches the family sweeps use — six by five for the Yoshimura, four by four for the waterbomb — and the vertex kinds they find are the kinds those patches contain. Whether a larger patch or a different boundary adds a kind of vertex, and with it a clause, is the question to ask before calling either recipe final, and the answer turns on whether every kind of interior vertex of the infinite pattern already appears in the patch.

The more interesting open question runs the other way. The waterbomb’s four kinds of corner collapse to two prohibitions because its parities tie them together, and the collapse was found rather than predicted. Which patterns’ vertex conditions collapse like that, and by how much, is a question about how the rule’s bits are shared between vertices, and a family designed so that its prohibitions collapse all the way to one would be a degree-six pattern that teaches almost as easily as a Miura.

The habit worth carrying is about instructions generally. Before deciding that something can only be taught by example, try writing its rules in the two kinds of sentence — what must add up, and what must not all agree. The first kind is cheap and composes; the second kind is where the real instructions are, and there are often far fewer of them than the thing being taught has parts.