A recipe needs degree four

Half the recipe is decoration sweeps the Miura’s sixty-four repeating rules and finds its familiar two-part lettering instruction has an empty half. Sixteen rules fold, the row letters take all four possible forms among them, and the one real instruction is that a column crease changes letter every time it crosses a row. The survivors factor as four row patterns times four column patterns — a rectangle, which is what independence looks like written down.

Then it asks the question this essay answers. The waterbomb tessellation has five hundred and twelve rules of which thirty-two fold, and nobody teaches it as a rule. If its thirty-two also form a rectangle, a waterbomb recipe is a handful of forced clauses and some free letters, statable in one sentence; if they do not, the pattern resists a rule and teaching it as an object is the right choice rather than merely the traditional one.

They do not, and neither do the Yoshimura’s. The reason is decided at one vertex, and it is its degree.

What a recipe of that kind can say #

A repeating rule gives every crease in a family of creases the same letter, mountain or valley, so a rule is a short string of bits. The Miura’s rules are six bits, the waterbomb’s nine.

A teaching instruction of the Miura’s kind is made of clauses about pairs of letters: these two are the same, these two differ. Written as bits, “the same” says the two bits add to zero and “differ” says they add to one, counting modulo two. A clause may mention more than two letters — an odd number of these four are mountains — and it is still the same kind of clause, a statement that a sum of bits has a stated parity.

A set of rules picked out by clauses of that kind has a property that can be tested without knowing the clauses. Take any three rules in the set and add them bit by bit; the result is in the set too. Each clause is a parity, and three rules satisfying a parity add to a rule satisfying it, because adding three copies of a stated parity gives the same parity back. A set closed under that operation is what linear algebra calls affine, and its size is always a power of two.

So there is a test for whether a family has a same-and-differ recipe, and it is purely a test on its surviving rules: are they affine? If they are, the clauses exist and can be found. If they are not, no set of same-and-differ clauses picks them out, however cleverly chosen.

The Miura and the leaf pass #

The Miura’s sixteen survivors are affine. Every sum of three of them is among them, and exactly two independent clauses hold on all sixteen: for each of the two classes of column crease, the letters above and below a row differ. Two clauses on six bits allow sixteen rules, and the sixteen they allow are the sixteen that fold. The recipe is exact.

The tapered leaf gives the same answer with the same numbers, as the leaf’s rules are the Miura’s found: its taper is in widths no vertex condition reads.

That is the formal version of the corrected Miura recipe. “A column crease changes letter every time it crosses a row” is two differ clauses, one per column class; the row letters appear in no clause at all, which is what being free means; and the sixteen folding rules are an affine set of size two to the fourth.

Twenty-six cannot be a recipe #

The Yoshimura pattern fails the test before it is run. Twenty-six of its sixty-four rules fold, and twenty-six is not a power of two, so no affine set has that size and no recipe of same-and-differ clauses picks the twenty-six out.

Running the test anyway shows how far short it falls. Only one clause holds on all twenty-six survivors — an even number of mountains among four particular letters — and one clause on six bits allows thirty-two rules. Six of the thirty-two do not fold.

The six are informative. They are exactly the rules in which the four letters of the clause are all alike, all mountain or all valley, and the two remaining letters do not compensate. Any correct Yoshimura recipe therefore has to contain an instruction of a different kind: something like these four are not all the same, which is not a parity of anything.

Thirty-two is a power of two and still not a recipe #

The waterbomb tessellation is the interesting case, because its count passes the first test. Thirty-two is two to the fifth, and five free bits with four forced clauses would be exactly the rectangle half the recipe is decoration hoped for.

The survivors are not affine. There are three folding rules whose bitwise sum does not fold. And the clauses that do hold tell the whole story: three independent parities hold on all thirty-two survivors, three clauses on nine bits allow sixty-four rules, and exactly half of those sixty-four fold.

So the waterbomb’s thirty-two is a power of two by coincidence rather than by structure. Its survivors are half of an affine set, not an affine set, and the half is carved out by the same kind of instruction the Yoshimura needed: letters that must not all agree.

The vertex decides #

Both failures come from the same place, and it is a single vertex.

A repeating rule gives a vertex’s opposite course-halves the same letter, as where a rule can close a loop established for every family here. So at an interior vertex some pairs of creases are tied: they are forced to carry one letter by the rule itself. What the counting theorem — why the difference is two — then asks of the remaining letters depends on how many pairs are tied and how many creases the vertex has.

At a degree-four vertex with one opposite pair tied, the letters are a, a, b, c. The theorem wants three of one letter and one of the other. Of the eight ways to choose a, b and c, four satisfy it, and they are exactly the four in which b and c differ. That is a parity: the other two differ. It is the Miura’s column clause, arriving at one vertex.

At a degree-six vertex with three opposite pairs tied, the letters are a, a, b, b, c, c. The theorem wants four of one letter and two of the other, so the number of mountains among a, b and c has to be one or two — not zero, not three. Six of the eight patterns satisfy it: every pattern except all mountain and all valley. Six is not a power of two. No parity picks out six patterns of eight, and the instruction the vertex gives is the three are not all alike.

That is the whole mechanism. A family whose vertices are all of degree four gets only parity clauses from its vertices, and the conjunction of parity clauses is affine, so its survivors are affine and a recipe exists. A family with degree-six vertices gets not-all-alike clauses, their conjunction is not affine, and no recipe of the Miura’s kind exists — whatever the count of survivors happens to be.

Two kinds of constraint, and a famous boundary between them #

The distinction between “differ” and “not all alike” is not a peculiarity of paper. It is one of the best-known boundaries in the study of logical constraints.

A problem made of parity constraints — these bits add to an even number — can be solved by the Gaussian elimination taught for simultaneous equations, whatever its size. A problem made of not-all-equal constraints on triples — these three bits are not all the same — is one of the standard hard problems: deciding whether a large instance has a solution is NP-complete. Thomas Schaefer’s dichotomy theorem of 1978 classified exactly which kinds of constraint on Boolean variables give tractable problems, and parity constraints are on the tractable side while not-all-equal triples are not.

The Miura’s vertices are parity constraints and the waterbomb’s are not-all-equal constraints. A repeating rule has so few bits that nothing about hardness applies to the families here — every rule can simply be tried — but the vocabulary of the teaching instruction is decided by exactly the same line. A recipe is a small system of equations when every vertex has degree four, and it is a small system of not-all-equal clauses once a vertex has degree six.

That also says something about the gadgets that make flat-folding hard in general. Hardness proofs for crease patterns build their difficulty out of vertices that act like logical clauses, and the kind of clause a vertex can act as is set by its degree. Degree four gives parities, and parities alone are never hard.

Why these two are taught as objects #

The result reads directly onto how the patterns are taught.

The waterbomb is taught as an object: fold a waterbomb base, fold another, join them. The letters come out right because the base’s letters are learnt by hand and repeated. Half the recipe is decoration suspected that this might be the right choice rather than merely the traditional one, and it is: an instruction list for the waterbomb’s letters has to contain a clause of the form not all of these alike, and a clause of that form is not something a diagram’s letters express.

The Yoshimura is in the same position for the same reason. Its degree-six vertices mean that any correct list of instructions about its letters contains a prohibition rather than a parity, and prohibitions are the kind of instruction learners find hardest to apply — they say what not to do at a point, in terms of three letters that may be far apart on the sheet.

The patterns that are taught as recipes are the ones whose vertices have degree four. That is not a historical accident about which patterns got diagrams. It is what the counting theorem allows a recipe to say.

What the clauses look like on a sheet #

A recipe is easiest to judge by drawing what it produces, and the Miura’s case makes the contrast with the degree-six families concrete.

The Miura’s rule numbered 22 writes its rows valley then mountain and alternates every column crease across every row. It satisfies both of the family’s differ clauses, so every interior vertex has its column halves on opposite letters, and every vertex satisfies every condition. The rule is one of sixteen the two clauses allow, and all sixteen fold.

A waterbomb recipe could not be checked this way crease by crease. At a degree-six vertex the tied letters are three, and whether they satisfy the count depends on all three together: two mountains and a valley pass, and so do two valleys and a mountain, and three of a kind fail. Looking at any pair of them says nothing. The census in the rule that breaks the count reads its failures the same way: every one of the hundred and twenty waterbomb rules that sends four panels round a circle had already broken the count at the vertex the circle goes round — a failure of the whole set of letters at one vertex, which no pair of them reveals.

That is also the shape of the misconception taught with a wrong reason records about the preliminary base, whose symmetry suggests four mountains and four valleys. A symmetry suggests a clause about pairs of creases; the counting theorem asks about the whole set at once. And six creases and the same straight line marks where the difference stops: a search letters the degree-six family exactly as cheaply, crease for crease, as a grid, so what the degree changes is what a recipe can say and not what a search has to do.

A fifth mark would not be enough #

Half the recipe is decoration proposed a small notational repair: since the Miura’s row letters are free, a diagram could mark them with a fifth symbol meaning either. For the Miura that works exactly, because its freedom is freedom of individual creases — two free bits, each belonging to a class of row crease.

For the degree-six families it does not. Their freedom is not a set of free creases; it is a relation among creases. At a waterbomb vertex no single tied letter is free, and no single one is forced — any one of the three can be mountain or valley, provided the other two are not both the same as it. A mark on a crease cannot say that. The notation would need a mark that spans three creases and means not all alike, which is a different kind of symbol from any the subject has, and which the history of the dashed line suggests would be hard to introduce.

So the survey the earlier essay called for has a sharp answer for two of its four patterns, and the answer is negative. A recipe can mark a free crease; it cannot mark a free relation.

What the sweep cannot show #

The sweeps are over repeating rules, which are a tiny corner of all letterings. A Miura patch has thousands of creases and far more letterings than sixty-four, and a recipe for the general lettering — as opposed to the repeating one — is a different question. What the sweep establishes is that even the simplest, most symmetric letterings of the degree-six families have no same-and-differ description, which makes it unlikely that the general ones do.

It also cannot show what a learner experiences. A not-all-alike clause is harder to state than a differ clause, and it is plausible that it is harder to apply; nothing here measures that. The claim is about what an instruction list can say, not about how well it is followed.

And the degree argument is about vertices at which opposite creases are tied. A family whose rules tied different pairs, or tied none, would give different local clauses, and a degree-six vertex with no ties has sixty-four patterns of which fifteen satisfy the count — also not a power of two, but for a different reason.

The rules the sweep assumes #

A repeating rule assigns letters by row and column parity, or by the analogous classes of the family, and nothing else. The bits are the rule; the pattern is drawn from them; the conditions are checked at every interior vertex.

The counting theorem is the condition that refuses. In all four families every refused rule is refused by it, with the angle condition and the smallest-sector lemma holding at every failing vertex, so the grammatical form of the counting theorem’s clauses is the form of the whole refusal.

And a recipe means same-and-differ clauses, including parity statements about several letters. A recipe allowed to say not all of these alike would describe every family here, at the cost of being a different and harder kind of instruction.

How the families were sorted #

Every rule of every family is built and checked at every interior vertex, so the survivors are found rather than predicted.

Affinity is tested by closure: every triple of survivors is added bit by bit and the sum looked up. The Miura’s and the leaf’s sixteen are closed; the Yoshimura’s twenty-six and the waterbomb’s thirty-two are not, and a triple witnessing it is found for each.

The clauses are found by trying every parity: every subset of a family’s letters is tested for having the same parity on all its survivors, which gives the relations and the size of the set they allow.

The single-vertex count is checked against the families. At degree four with one pair tied the theorem is required to keep four of eight patterns forming a parity, at degree six with three pairs tied six of eight forming none, and the families whose vertices are all of degree four are required to be exactly the affine ones.

Still open: a recipe for the Yoshimura in the language it needs #

The negative result leaves a positive question. If a Yoshimura recipe has to contain prohibitions, what is the shortest recipe that does?

One clause holds on its twenty-six survivors, and six rules satisfy the clause and fail to fold; those six are the ones in which a particular set of letters is all alike. So a correct recipe is the parity clause plus a prohibition, and whether that is two sentences or several depends on how the six fall, which the sweep already has. Writing it down would give the first statable lettering rule for a degree-six family, and it would be a rule of a kind no diagram currently expresses.

For the waterbomb the same question is larger — three parities and a prohibition carving sixty-four into thirty-two — and its answer would say whether the pattern taught as an object could be taught as a rule at all, at the price of one new kind of instruction.

The habit worth carrying is a test for any claim that something can be taught as a list of rules. Ask what grammatical form the rules need. If every constraint is a parity, a short list exists and can be found by elimination; the moment a constraint forbids agreement among three things, the list needs a different kind of sentence, and that is where teaching by example begins to earn its place.