No height to swap

A folded strip is a stack of segments along a line, so a stacking of it is a permutation and the smallest change anybody can make to one is a swap of two heights. That is what nothing slides past anything counted: over evenly creased strips of four to seven segments and every marking of each, 672 stackings, of which 560 have no legal swap available at all and a hundred and twelve do.

A folded sheet has no heights to swap. Its panels lie over one another in a region, two panels are in each other’s way only where they share ground, and the order between two panels that never meet is not a fact about the object at all. What replaces the permutation is a set of pairwise statements, and what replaces the swap is a statement about an overlap.

The counts that come out are not the strip’s counts made larger. They are the opposite shape.

What the sheet asks that the line does not #

On a strip, the rule for which of two neighbours is higher reads the crease’s letter and the direction of travel, and the two non-crossing rules are stated between intervals. All three survive into two dimensions with one word changed, and the change is the whole difference.

The crease still decides its own pair. A valley brings the far panel up and over, a mountain takes it under, and which of those a reader sees depends on whether the near panel has itself been turned over on the way. On a strip the near panel’s orientation is the direction of travel along the line. On a sheet it is whether the panel’s motion reverses orientation, which is the same quantity read off a plane isometry instead of off an interval.

Taco-tortilla becomes a statement about a segment lying inside a polygon, and the polygons are placed by reflection rather than by any angle being computed. A crease’s folded image is a line segment in the folded plane; a panel whose interior that segment runs through cannot sit between the crease’s own two panels, because the crease is where the paper turns and there is no gap in it.

Taco-taco becomes a statement about two segments lying along one line. Two creases whose folded images overlap are two U-turns in the same place, and they may nest or stand clear but may not interleave.

There is a third rule in the usual statement of this — tortilla-tortilla — and it is not checked here, because it cannot fail. It exists in the literature to catch a contradiction between pairwise variables that no single rule notices, and this measurement works with total orders rather than with pairwise variables. Given a total order in which neither panel of one crease lies between the panels of another, the two pairs are already nested or clear. Transitivity is free when the thing being enumerated is a permutation.

Two routes to one number #

A strip is a sheet whose creases are parallel. So the same paper can be handed to both instruments, and if the two disagree, one of them is wrong.

Six is what fits on a page. The check that runs on every rebuild of these pages takes eighty: five sets of crease positions, every mountain-and-valley string over each, up to five creases and six segments, and the two routes agree on the ordering count and on the state count in all eighty.

They did not agree at first, and the disagreement is worth recording because it is the kind that produces a plausible answer rather than a crash. A crease at the very end of a folded strip has an image lying along the edge of the panel it terminates, and whether a point on that edge counts as inside the polygon depends on which way a crossing count rounds. Counting it as inside makes the fold at the end of the strip appear to run through the panel it is the edge of, which invents a taco-tortilla rule that is not there and removes stackings that exist. The sheet reported one where the line reported two. The fix is to require a point to be strictly inside — a stated distance off every edge — and the eighty cases then agree exactly.

What a printed pattern actually has #

Four of the eight patterns this site prints have few enough panels for the search to finish. Here is everything it returns.

One. Not one in the sense of “the first one found” — the search is exhaustive over all orderings of the panels, and on the preliminary base, the square twist and the hexagon twist there is exactly one ordering that satisfies every rule. The fold-and-cut triangle has two, and they are two genuinely different folded objects rather than one object counted twice.

That is not what a strip does. Evenly creased strips of four to seven segments admit 672 stackings between them, and on such a strip an ordering and a folded object are the same thing, because every segment lands on every other. A five-segment strip alone has fifty. A sheet with nine panels has one.

Why the answer collapses #

The reason is countable and it is in the third column of that table.

A strip of n segments has n − 1 creases, and the number of non-crossing rules it generates grows roughly like n. A sheet’s panels grow with the area of the paper, and the pairs of panels that share ground grow with the square of that — 21, 28, 36, 66 across the four patterns above, against 7, 8, 9 and 13 panels. The rules written between those pairs grow with them: 15, 12, 48, 96.

So the two quantities that decide the answer move in opposite directions as the paper gets bigger. The orderings grow factorially and the constraints grow quadratically — and the constraints win, because each one removes a constant fraction of what is left rather than a constant number of orderings. Nine panels is 362,880 orderings and forty-eight rules, and the forty-eight rules leave one.

The counting argument predicts the wrong sign #

The explanation offered for the collapse — orderings growing factorially, constraints growing with the square of the panel count, and the constraints winning — is the natural one, and the growth figure contradicts it.

Read that figure again. A Miura patch of six panels returns three orderings and three states; nine panels, six and six; twelve panels, eleven and eleven. Every ordering the search finds is a different folded object, and the number of them rises with the size of the patch rather than falling to one. If constraints really outgrew orderings over this range, twelve panels would be more determined than nine, not less.

These counts are not the ones this essay first printed. It reported one state at nine panels and five at twelve, from an overlap test that missed the pairs of panels a folded Miura slides along one another — one choice with eleven answers draws the blind spot and repairs it. The printed patterns in the table above lose no pair to it and are unchanged. The Miura’s corrected counts make the contradiction sharper, not weaker.

What the constraints do not reach #

The earlier version of this section blamed a falling share of overlapping pairs, and with every pair counted the share does not fall over this range: a two-by-two Miura has all six of its pairs sharing ground, a three-by-two all fifteen, a three-by-three all thirty-six, a four-by-three all sixty-six and a four-by-four all hundred and twenty. Only the printed six-by-four, at 228 of 276, has pairs that miss.

So the rules are written between every pair, their count really does grow with the square of the panels, and the states still rise. What the count of rules cannot see is where the rules leave room. A Miura collapses each row into a zigzag, and the two end columns of a zigzag of three fold onto the same side of the middle one — the arrangement of a letter fold, whose two flaps are not joined to each other by any crease and so are ordered by no rule. One sheet down found that same arrangement behind every pattern with more than one folded state. A Miura patch has such a pair of end columns in every row, and a longer patch has more of them; the constraints multiply everywhere else and leave those places free.

Which repairs the explanation rather than removing it. The mechanism is right: what decides the answer is constraints against orderings. What the counting version leaves out is that the constraints are not spread evenly over the pairs — they are dense where panels are joined through creases and absent between flaps that are only stacked — and that is a property of the shape the pattern folds into rather than of its panel count.

A pattern that collapses toward a point stays determined however large it gets, because every flap it has is wrapped round a neighbour. The Miura, which is the one everybody builds, has flaps that are only stacked, and its freedom grows with it. The four decided printed patterns are all of the first kind.

That also says which way the refused patterns are likely to go. The Miura at twenty-four panels, the waterbomb tessellation at fifty-two and the Yoshimura at sixty-five are refused, and the first of those is a pattern whose freedom has been growing at every size measured. Nothing here decides them — but the reasoning that suggested they would be as determined as the small printed patterns is the reasoning this section has just withdrawn.

Nothing to rearrange #

The move a folder makes is the reason the strip count mattered: pushing on a pleat swaps two layers, and the question was how often the swap is legal. Asked of a sheet, the move has to be restricted before it means anything — swapping two panels that never lie over one another changes nothing a reader could see, so it is bookkeeping rather than rearrangement, which is exactly the distinction one dimension needed too.

Restricted properly, the move is: take two panels that are neighbours in height and share ground, exchange them, and ask whether the result still satisfies every rule.

Across the four decided patterns there are thirty-nine such swaps available, and not one of them is legal. The Miura patches add twenty-one folded states and a hundred and eighty-seven swaps, and two of those are legal, both on the three-by-three — the only rearrangements a single exchange of neighbours can make on any sheet measured.

On the strips, 3,740 swaps were tried and a hundred and twelve were legal — about one in thirty-three. On the sheets, two hundred and twenty-six were tried and two were legal, about one in a hundred and thirteen, and none at all on the printed patterns. A folded sheet is much harder to rearrange than a folded strip, and on the patterns this site prints it cannot be rearranged at all.

Part of that is arithmetic rather than physics. A pattern with exactly one ordering has no legal move by definition, because a legal move would produce a second one. The informative statement is the uniqueness, and the move census is how a folder finds out about it: the sheet will not be argued with, and the reason is that there is nothing to argue about.

The two cases with more than one state are the interesting ones, because they could have come out the other way. The fold-and-cut triangle has two states; the Miura patch of four columns by three rows has eleven. In both, every state is isolated — the several states exist, and no sequence of single swaps connects any two of them. They are reachable only by taking the sheet apart.

Which rule is doing the work #

The rules split into two kinds, and which kind holds a pattern down is not readable off its drawing.

The preliminary base generates no taco-tortilla rule whatever — twelve taco-taco rules and nothing else — because every one of its creases runs from the single interior vertex out to the edge of the paper, and in the folded state those images land on panel boundaries rather than inside panels. The square twist generates thirty-six of the first kind and twelve of the second. The Miura generates a hundred and forty-four and eighty-four.

The Yoshimura is the extreme case and it is worth pausing on: sixty-five panels, two thousand and fifty-five pairs sharing ground, one thousand one hundred and eighty-seven taco-taco rules, and zero of the other kind. A pattern that folds into a long thin object stacks its creases on top of each other and never lays one across the middle of a panel.

Where the search stops, and why it is a node count #

Four of the eight printed patterns are refused: the Miura at twenty-four panels, the tapered corrugation at twenty-eight, the waterbomb tessellation at fifty-two, the Yoshimura at sixty-five.

The limit was a panel count first and the panel count was wrong in both directions. Thirteen panels is 6.2 billion orderings and the hexagon twist settles in a fraction of a second, because the crease rule fixes one relation per crease before the search starts and the tree is mostly pruned before it is grown. Twenty-four panels does not settle at any budget. What separates them is how much the letters and the overlaps constrain the order, and that is not readable off the panel count — so the budget is spent in the currency the search uses, which is nodes of its own tree, and the answer past it is unknown rather than an estimate.

Unknown is a real answer here and it is said rather than smoothed over. Nothing in this essay claims to decide whether a Miura fold has one folded state or a thousand. What it claims is that four printed patterns have their orderings counted exhaustively, and that the counts are one, one, one and two.

What this does not settle #

Everything above is about orderings that satisfy the two non-crossing rules and the crease rule. Those rules are necessary — a crease where the paper turns the other way is not the crease that was drawn, and a panel cannot pass through a fold — and necessity is what makes the negative answers proofs. When the search exhausts the tree and returns nothing, the pattern has no flat folded state, full stop.

The positive answers are not proofs in the same way. An ordering that satisfies every rule here is a candidate, in exactly the sense every other assertion on this site means it: deciding flat-foldability of a general pattern is NP-hard, and a checker that returned a certificate would be a different and much larger claim. What has been shown is that the candidates are unique, not that the unique candidate is realised.

The other limit is the one the strip did not have. A strip’s segments all lie along one line, so every pair either overlaps or is separated by a crease. A sheet’s panels can miss each other entirely, and the pairs that miss are invisible to the whole apparatus. The preliminary base has twenty-eight pairs of panels and all twenty-eight share ground, and so do all sixty-six pairs of the Miura patch of four by three; the printed six-by-four Miura is the first on which pairs miss, 228 of its 276 sharing ground. A long enough sheet has panels stacked on nothing in common, and the ordering question is only ever about the part that is.

What a folder should take from it #

A sheet is not a strip with more segments in it. The one-dimensional case is where the whole problem is tractable and where every one of these rules can be seen whole, and it is the only case in which a folded object has a spare parameter to move. Everything printed here that can be checked has one folded state, and the hand that tries to re-stack it is not being clumsy.

The order is decided by the drawing, not by the folding. A pattern with one ordering was going to fold that way whoever folded it, and the “which way was it tucked” question that a strip poses has no analogue on any of these sheets. That is the same shape of answer the letters gave when the question was which of them a hand can still change.

And the arithmetic runs the wrong way for anybody hoping to enumerate. The panels grow with the area, the pairs with its square, and the orderings with the factorial of the panels. Twelve panels is the largest patch of the commonest tessellation in the subject that can be counted at all, and the tessellation itself is normally drawn with hundreds.