The crumple keeps its options

A crumpled sheet reads as the most jammed object in the subject. It has more creases per unit of paper than anything anybody designs, they were put there by force rather than by intention, and the sheet has already written itself a lettering and is enclosed in it — sixteen to forty-four decisions it cannot revisit.

That is the story about its letters. The story about its layers runs the other way.

The comparison #

Every printed pattern on this site whose panels can be ordered has exactly one folded state. The preliminary base, the square twist and the hexagon twist have one each; the fold-and-cut triangle has two. Across all four there are thirty-nine ways of swapping two panels that are neighbours in height and share ground, and not one of the thirty-nine is legal.

Run the same search over sheets folded at random and the numbers are different in kind.

One crumple of four folds admits nine distinct folded states. Nothing designed on this site admits more than two.

The share of legal swaps is the sharper reading, because it is what a hand does. On the printed patterns it is nought in thirty-nine. On four-fold crumples it is six in three hundred and thirty-five, about one in fifty-six — and a legal swap is a rearrangement the sheet will accept without being unfolded.

Why this is not the opposite of the letter result #

The two findings look contradictory and are not, and the reason is worth being precise about because it is the whole shape of the subject.

The decision a crumple has taken counts buried creases — creases with an interior vertex at each end — and reports that a crumple has many, so its letters cannot be changed by any local move. That is a statement about which mountains and valleys the sheet has.

This counts orderings of the panels given the letters. It is a statement about which pile those mountains and valleys permit.

A sheet can be completely settled about the first and undecided about the second. The letters say which way the paper turns at each crease; they do not say, when three pieces of paper end up over the same square millimetre, which is on top. On a designed pattern the two non-crossing rules take that last freedom away. On a crumple they usually do not, and the reason is geometric: a crumple’s panels are irregular, so fewer of them overlap.

A four-fold crumple averages nine panels, and the panels are triangles and quadrilaterals of wildly different sizes scattered over a small footprint. A square twist’s nine panels are the same nine panels every time and every one of them lies over all eight of the others. Fewer overlaps means fewer rules; fewer rules means more orderings survive.

What a legal swap looks like #

A swap is legal when two panels that are neighbours in height and lie over one another can exchange places and every rule still holds. On paper it is the thing a hand does without thinking: push a corner, and one flap slides to the other side of its neighbour.

On the printed patterns that never works, and the reason is arithmetic rather than stiffness — a pattern with one ordering has no legal swap by definition, because a legal swap would produce a second one. On a crumple it works about one time in fifty-six at four folds, and the crumples where it works are the ones with the most panels not lying over one another.

So the sheet that gives under the thumb is the untidy one. That matches what anybody who has flattened a receipt knows and it is not what the crease count suggests: the crumple has more creases, more vertices and more buried letters than a square twist, and more freedom than one too.

The population is the unit, not the sheet #

A crumple is a random object and reporting one of them is reporting a coincidence, so every number above is a population figure with the population stated: twenty-four sheets at each fold count, seeded so that the same twenty-four come back on every rebuild of these pages.

That matters because the spread is enormous. At four folds the sheets run from eight panels to twelve, from nine non-crossing rules to forty, and from one folded state to nine. Quoting the one with nine would be quoting the best case; quoting the median would hide it. The table gives the mean, the maximum and the count with more than one state, which between them say what the distribution looks like without pretending to fit it.

The same discipline is what four ways to draw a pattern was about: a sentence beginning over some crease patterns is a statement about a construction nobody declared. The construction here is stated — fold the sheet in half about a random line, k times, keeping every crease — and no claim is made about crumples produced any other way.

Refusals, counted #

Five of the twenty-four four-fold crumples could not be searched at all, and eighteen of the twenty-four five-fold ones. Their panel counts run past what an exhaustive ordering search will finish, and the search says so instead of returning what it found before it gave up.

That matters here more than anywhere else in these essays, because the refused sheets are the large ones — and large means more panels, which usually means more states. The averages in the table are therefore biased low, and the bias grows down the column: the five-fold row is an average over the six simplest sheets of twenty-four, which is why it comes out below the four-fold row rather than above it.

The honest reading of that row is not “five folds gives fewer states than four”. It is “the five-fold sheets that can be searched give 2.33, and the eighteen that cannot are the ones that would have raised it”.

The one that had nine #

The four-fold crumple with nine states is worth looking at on its own, because it is the extreme of the population and it shows what the extreme is made of.

Twelve panels. Fifty of its sixty-six possible pairs share ground, so sixteen pairs never meet. Forty non-crossing rules between the fifty, which is four fifths of a rule per overlapping pair — on the square twist the same ratio is forty-eight rules over thirty-six pairs, more than one each. Thirty orderings satisfy all forty rules, and those thirty collapse to nine distinct folded states once the pairs that never meet are discounted.

All nine are isolated: none of its ninety-three available swaps is legal. So this sheet is ambiguous about its pile without being loose — nine different piles its own pattern permits, and no route from any of them to another that does not go through unfolding.

And the freedom is in two panels rather than spread across twelve. Sorted, the panels lie over 3, 3, 9, 9, 9, 9, 9, 9, 9, 9, 11 and 11 of the others: ten of them are stacked on nearly everything and two are stacked on almost nothing. Those two are what the sixteen absent pairs are made of, and they are where the nine states come from — the heap in the middle is as decided as any designed pattern, and the pair that landed clear of it is not. The same effect at the rim of a designed sheet is worth two panels of degree; here it is worth a factor of nine, because a crumple is mostly edge.

The arithmetic of the loose pair #

The degree list of that twelve-panel sheet says more than the essay reads off it, and the extra is exact rather than suggestive.

The panels lie over 3, 3, 9, 9, 9, 9, 9, 9, 9, 9, 11 and 11 of the others. Those sum to a hundred, which is fifty pairs, which is what was reported — so the list is consistent with the count and can be trusted for the rest of the arithmetic.

Sixteen pairs are absent, out of the sixty-six a twelve-panel sheet has. Each of the two low-degree panels misses eight others, since eleven others exist and it meets three. Eight and eight is sixteen, exactly the number absent — so the two loose panels do not miss one another. If they did, that pair would have been counted twice in the sum and the absent total would be fifteen.

That is a small deduction with a real consequence. The freedom in this sheet is not two independent panels lying clear of the heap in different places. It is a loose pair, sitting together, overlapping each other and almost nothing else — one region of the sheet that landed off to one side while the other ten panels piled up.

Nine is three by three #

The state count invites the same treatment. Nine is a square, and there are two loose panels, which suggests each of them has three positions available and the two choices are independent.

That reading fits what the degrees say. A panel overlapping three others has its height constrained relative to those three and to nothing else; among the heights the heap leaves, three slots is a plausible count for such a panel, and two panels constrained only by the heap and by each other would multiply their choices. Three times three is nine, and the thirty orderings that satisfy the rules collapse to those nine once the pairs that never meet are treated as not distinguishing anything.

It is a reading rather than a derivation, and the essay should say so: nothing above computes which three slots, and a coincidence between a square number and a pair of loose panels is the kind of thing that is right about four times in five. What makes it worth stating is that it is checkable — count the admissible heights of each loose panel separately, and see whether the product is the state count.

Which relocates the finding #

If it holds, the crumple’s looseness is not a property spread through the sheet. It is a property of the panels that missed the pile, and the pile itself is as decided as any designed pattern — ten panels each lying over nine or eleven of the others, with no room in them at all.

That changes what the comparison with the printed shelf is about. A designed pattern collapses everything onto one footprint, so it has no panels that miss; a crumple scatters, so it has a few. The states come from the scatter, they are located rather than distributed, and the number of them should be predictable from the degree list alone — which is a much cheaper thing to compute than an ordering search, and is the measurement this population is now set up to make.

The check the population exists to make #

There is a claim buried in every number above and it needs stating separately, because if it failed the whole essay would be about the machinery rather than about paper.

A crumple has a folded state by construction. It was made by folding a sheet; the folded sheet is sitting there. So a search that failed to find an ordering for one would not have discovered anything about crumpling — it would have found a bug.

Every crumple measured here returns at least one ordering. That is the strongest single check the ordering machinery gets anywhere on this site, because it is the one case where the answer is known in advance and known for a reason that has nothing to do with the code: the object exists.

The printed patterns cannot make that check. They are verified against the vertex conditions, which is exactly what the second sieve was written to go beyond, so a wrong answer on one of them would look like a finding. A wrong answer on a crumple looks like what it is.

Where the layers actually are #

The layer counts add a second reason a crumple is loose, and it is the one visible in the hand.

A crumpled sheet’s footprint has a few deep places and a great deal of shallow. The printed patterns are the opposite: the preliminary base, the square twist and the fold-and-cut triangle each have a point where every panel of the sheet lies over it, and the Yoshimura is sixty layers deep almost everywhere.

Deep means constrained. Where twelve panels overlap, every pair of them generates rules, and the rules multiply while the orderings do not. Where two panels overlap there is one rule and two orderings. A crumple spends most of its area in the second condition.

Against a designed pattern of the same size #

The cleanest comparison is at equal panel counts, because panel count is what the search cost depends on and it is the obvious confounder.

A four-fold crumple averages 8.9 panels. The preliminary base has eight and the square twist nine. So the three are the same size, and they are not the same object at all: the two designed patterns have twenty-eight and thirty-six pairs of panels sharing ground — every pair, in both cases — while the crumples average eighteen to fifty pairs out of the thirty-six to sixty-six available.

The designed patterns are complete: every panel over every other. That is what a pattern that collapses toward a point does, and the printed shelf is full of them because a pattern that collapses to a point is a pattern somebody designed to collapse.

A crumple collapses toward nothing in particular. Its footprint is a blotch, its panels are scattered over it, and the pairs that never meet are the pairs that leave the ordering room.

What this does not say #

It does not say a crumple is easy to fold flat. The states counted here are orderings of the panels of a pattern that did fold, and finding them is a search. Nothing about a crumple is easier than anything else; what it has is more answers, not cheaper ones.

It does not extend past a dozen panels. Six folds is twenty-one panels and every one of those is refused. A real crumple — the one in a pocket — has hundreds, and this essay says nothing whatever about it. What the populations show is a trend over the sizes that can be counted, and the trend is up.

And it says nothing about which state the sheet is in. The paper picked one when it was crumpled, and the pattern does not record which. That is the same gap the folded object has everywhere: a crease pattern is not a folded object, and a folded object with nine states is nine objects sharing a drawing.

One more comparison closes the argument, and it is the one a reader can make with their hands. Take a sheet, crumple it, flatten it out and press it down: the layers shift as it is pressed, and pressing it again shifts them differently. Take a square twist, collapse it, and press: nothing moves, and pressing harder does not help. The measurements above are that difference, counted — nought of thirty-nine against six of three hundred and thirty-five — and the reason is not stiffness or friction or the state of the paper. It is that one of the two patterns admits more than one arrangement of its panels and the other does not.

What a folder should take from it #

The tidy sheet is the stiff one. A designed crease pattern arrives with its pile already decided, and the reason it will not be argued with is not that it is well made — it is that its panels all lie over one another, and overlapping panels are what the rules are written between.

A crumple gives because it is irregular, not because it is loose. Its creases are as settled as anything on the shelf. What it has spare is the order.

And the number to watch is overlaps, not creases. Two sheets with the same number of creases can have very different amounts of room in them, and what separates them is how much of each panel lands on another one. The creases decide the letters; the overlaps decide the pile.