Half the waterbomb's rules have no stack

Two sentences for the Yoshimura wrote a recipe for lettering a Yoshimura pattern and another for the waterbomb tessellation, and it ended by asking whether they would survive a larger patch. The question it had in mind was about vertices: a small patch can hide a kind of vertex at which a rule would fail, and the waterbomb’s count once fell from fifty-six to thirty-two for exactly that reason. The answer to that question is short. A repeating rule’s letters depend on row parity, band parity and lean, every kind of interior vertex has appeared by three cells by three, and neither count changes on any larger patch drawn below.

So the recipes survive, and that turns out to be the wrong thing to have asked. Every count in that line of essays, from half the recipe is decoration on, said a rule “folds” when it passed Kawasaki’s condition, Maekawa’s count and the smallest-sector lemma at every vertex. A unit that folds is not a tessellation said in its last section that this was not the same as having a folded state, and left it there. Thirty-two rules, one object compared the panels’ positions, which do not need an order of layers. A teacher handing a folder a recipe is promising something stronger than the recipes were checked for: that following it gives a sheet that can actually be folded.

Asking every rule for a stack #

The question can now be put directly. For each repeating rule that passes every vertex, search for a folded state: an order of the layers that keeps every crease’s letter, lets no panel pass through a crease and no two creases interleave, and closes no cycle. The search stops at the first stacking it finds, and when it fails it has exhausted the possibilities, so its “no” is a proof about that patch.

Sixteen of the waterbomb’s thirty-two rules have no folded state, on every patch from three cells by three to four by four, with up to fifty-two panels. Four of the Yoshimura’s twenty-six have none, from three by three up to six by five. The Miura loses nothing. The rules these two families usually get drawn with are among the survivors: the waterbomb’s rule 46 and the Yoshimura’s rule 3, every course mountain and every zigzag valley, both stack. So no printed pattern was wrong. What was wrong is the count, and with it the sentence that “the twenty-six left are exactly the twenty-six that fold”, which should have read “that pass their vertices”.

The search was checked two ways. On the Yoshimura’s twenty-one panel patch a second search that lists every folded state rather than stopping at the first finds none for those four rules and between 41 and 9,790 for each of the others, so the two agree on all twenty-six. The waterbomb’s refusals are small enough to check by hand, and that is the next section. A larger patch is a separate matter: a stacking on four by four is no proof of one on forty by forty. Nothing here proves the sixteen stack for ever, and what has been shown is that the patch can be taken to fifty-two panels without losing any.

Neither of the usual explanations covers these refusals. No rule here sends panels round a forced circle, the shape the rule that breaks the count caught, since the count holds at every vertex. None has the ring of flaps that a mirror image was refused for. Something else is doing it.

Four panels and two creases #

Take away panels from a refused waterbomb patch for as long as what is left stays refused. On every one of the sixteen rules the process stops at the same size, four panels: a chain of two triangles and two squares, joined by three creases.

The refusal takes five lines. Call the triangles $A$ and $B$, sharing a horizontal crease, and the squares $D$ and $C$, joined to $A$ and $B$ across two half-diagonals that are mirror images of one another in that horizontal.

• The three creases’ letters say $B$ lies above $A$, $D$ lies above $A$, and $C$ lies below $B$.
• Folded, the squares $C$ and $D$ reach across the crease line between $A$ and $B$, so neither may lie between those two: a panel cannot pass through a crease.
• $D$ is above $A$ and not between $A$ and $B$, so $D$ is above $B$.
• $C$ is below $B$ and not between them, so $C$ is below $A$. The whole order is forced: $C, A, B, D$.
• The creases $A$–$D$ and $B$–$C$ land on the same folded line, so one pair of panels must nest inside the other. Here $A$–$D$ spans the second to the fourth layer and $B$–$C$ the first to the third. They interleave, and the paper would have to pass through itself.

Both prohibitions in that argument are old ones, known in the literature on layer ordering as the tortilla and taco conditions, and neither is a condition at a vertex. The two half-diagonals are at opposite ends of a horizontal crease and the four panels have no vertex they all share. So a recipe checked at vertices has no way of hearing this argument, and every patch large enough to hold a mirror pair across a horizontal can hold it.

If the two half-diagonals carry different letters, the argument changes at its first line and the order is no longer forced into a crossing. So the proof says what the rule has to do: a half-diagonal and its reflection in a horizontal crease must carry different letters.

There is a reason it happens across a horizontal and never across anything else. A reflection in a crease is what folding does, so two half-diagonals that are mirror images in a horizontal crease land on one line whenever that crease is folded. The waterbomb has straight horizontal creases running the whole width of the sheet and no vertical ones, so the horizontals are the only lines along which whole rows of panels are laid back onto one another. The argument above is what that stacking asks of its neighbours, and a pattern with no long straight creases has nothing to bring two distant half-diagonals together in the first place.

The recipe that was proved impossible #

A recipe needs degree four proved a negative. A set of rules picked out by same-and-differ clauses must be closed under adding any three of them bit by bit; the waterbomb’s thirty-two are not closed; so no such recipe exists, and the later recipe needed two prohibitions, sentences of the form these are not all alike. The negative was correct about those thirty-two.

The sixteen with a stack are a different set, and they are closed.

The waterbomb’s recipe has no prohibition in it once the layers count. Five parities pick out the sixteen exactly, and a same-and-differ recipe is precisely what the earlier essay’s test had ruled out. The prohibitions were there to refuse corners where four half-diagonals all agreed. A rule whose mirror pairs differ can never have four half-diagonals agreeing round a corner, so a sentence about the layers has made both prohibitions redundant, and a stronger requirement has produced a shorter recipe.

The earlier recipe is worth seeing beside it, because its last two lines are what has gone.

The two recipes share their first three lines and differ in what they do after sixty-four rules remain. The vertex recipe forbids agreement among three half-diagonals, twice. The layer recipe requires disagreement between two half-diagonals, twice, and in doing so refuses everything the prohibitions refused and sixteen more. A parity cuts exactly half of what it is applied to and these two prohibitions cut a quarter and then a third, which is why two parities reach sixteen where two prohibitions stopped at thirty-two.

The two new clauses read as a pair of same-and-differ statements between half-diagonals of neighbouring cells, and the geometry says what they are. In both, the two half-diagonals meet at a corner of the grid and are each other’s reflection in the horizontal running through it. The figure names two, and the parities supply the other two. Written out in full it is a single instruction, and the machine check confirms it as stated: among the rules passing the vertices, those with a stack are exactly those in which every half-diagonal differs from its reflection in the horizontal it meets.

So the instruction for lettering a waterbomb sheet that can really be folded has three vertex parities, about the centres of the cells and the corners, and one sentence about reflections. For comparison, the vertex-level recipe was three parities and two prohibitions. A teacher gets something shorter and stronger at once, and it has an obvious picture: look across any horizontal and the diagonals meeting it are chevrons, one mountain and one valley. The usual drawing has this, and anyone who has folded a waterbomb from its base has seen it without being told. The even-row horizontals remain free in all of it, as they always were, since the sixteen come in pairs differing only in that one letter.

The Yoshimura’s refusals are not small #

The same question to the Yoshimura gets a different answer in every respect.

Its refused rules are 24, 27, 36 and 39, and they have a simple description. The four zigzag classes are split into two mountains and two valleys in one particular arrangement, so the even-band right and odd-band left zigzags carry one letter and the other two the other, and both courses carry the same letter. Give the two courses different letters and the same zigzags stack. So the Yoshimura’s rule is an exception inside a family of rules, which is the kind of thing a parity cannot state: twenty-two is not a power of two, and the set is not closed. Its recipe gets longer, from two prohibitions to four.

It also has no small proof. Taking panels away from a refused Yoshimura patch stops at sixteen panels of the twenty-one, on all four rules, not at four. That process finds a minimal set rather than a smallest one, so a smaller refusal may exist elsewhere in the patch, but what it finds is nothing like the waterbomb’s. On the three-by-three patch the refusal needs most of the sheet, and on two by two, where the same letters are drawn on ten panels, the rule stacks without difficulty. A folder cannot be handed the Yoshimura’s rule as a five-line argument about two creases; the argument that exists is an exhaustive search.

That is the distinction this measurement has found between the two families, and it is about teaching rather than about geometry. A refusal that fits on four panels becomes a sentence a teacher can say, and a recipe gets shorter by it. A refusal that needs sixteen panels stays an exception, and the recipe grows by it.

What this does to the earlier sentences #

The earlier essays are unaffected in what they computed and need rereading in what they said. The twenty-six and the thirty-two are correct counts of rules passing every vertex. Thirty-two rules, one object is correct that all thirty-two give the same panels in the same places, but its object now has sixteen labels that can be folded and sixteen that are only positions. Thirty-two rules, thirty-two pieces counts letterings that pass the vertices, and its pieces are not all pieces of folded states. The practical reading of taught with a wrong reason gets one more example, since “a tessellation is verified because its unit is” fails here in the same way it failed there, one level up. And the first thing about layers said there had never been a rule about layer order simple enough to teach. On this one family, at the level of repeating rules, there now is.

The history in it, and what it does not show #

The conditions are not new. Bern and Hayes used layer-order conditions of this kind in their 1996 proof that deciding flat foldability is NP-hard, and “taco” and “tortilla” are the usual informal names for them. A prohibition of this shape is standard knowledge among people who compute layer orders. What is measured here is its consequence for one particular kind of instruction, the repeating lettering rule, and no claim is made that nobody noticed it on a waterbomb sheet. Folders plainly did, since they fold the right ones.

A rule is a repeating lettering, and a sheet lettered by hand need not repeat. The sixteen refused rules are refused as rules. Whether a non-repeating lettering of the same patch gets its letters from one and another in different places is a separate and much larger question. The rule’s own bits are what the recipe describes.

The search is exhaustive on each patch and silent beyond it. The 2 by 2 waterbomb shows why that matters in one direction: it keeps forty rules, of which twenty-four later go, because the patch holds too few vertices and mirror pairs. Nothing has been found in the other direction, a rule stacking on every small patch and failing on a large one, and nothing rules it out. The rim is part of every patch tested, and a sheet whose border is finished differently might behave differently: the ring-of-flaps refusals were rim effects, and these are not, since they sit inside the sheet, but the survivors’ stacks do use the freedom a rim gives. A folded state here has no thickness and no motion: having a stacking does not mean the sheet can reach it by a folding sequence, which one fold at a time treats as its own question.

And the figures are of letters and orders rather than of folded sheets. The strip’s section is a diagram of layer order, with its panels drawn as lines. A reader wanting the waterbomb itself needs the pattern figures of the essays above.

A stronger question with a shorter answer #

The unexpected result is the direction of the change. Adding a requirement ordinarily makes a description longer, since there is more to state. Here the condition with more in it has a recipe of a simpler kind, because the extra requirement happens to fall along a line the structure of the rule already respects: each new clause is a statement about two letters, the kind of statement a rule made of bits can absorb without any exception. Counting vertices had produced a set that was slightly the wrong shape, and asking for the real thing removed the corners that made it awkward. It is the same reason half the recipe was decoration on the Miura: the true constraint on a repeating rule can be tidier than the conditions anyone tested first.

Still open: why the Yoshimura needs so much of its sheet #

The Yoshimura’s refusal has only half an account. The heights at which the Yoshimura folds flat finds these four rules forcing nearly the whole sheet onto one rising chain, which says why no small piece refuses; it does not yet turn into a proof. A smallest refused set rather than a minimal one would be the place to start, by exhausting subsets instead of deleting greedily, and whether its size grows with the patch would settle whether the failure is really global.

The other families are waiting. Eight creases close the circle too already asked its grid for stackings and found that all 144 of its rules passing the vertices do stack; the tapered leaf has not been asked. A census across every repeating family, with the question of which ones come out closed under adding three, would say whether the waterbomb’s linear survivors are typical or lucky.

The habit worth carrying is about what a recipe was checked against. A recipe is exactly as strong as the test it was fitted to, and a correct recipe for a weaker test will be taught as though it were a recipe for the real thing. Before calling an instruction complete, ask whether following it produces the object or only passes the checks written down so far.