The decision a crumple has taken

A crease pattern drawn at random satisfies the flat-folding condition at essentially none of its vertices. A sheet crumpled at random satisfies it at every single one, on every seed, at every size — and the reason is a tautology that is very easy to miss: the pattern is the record of a folding that happened, so of course it folds.

That is a statement about the pattern. It leaves untouched a statement about the sheet, and the sheet is the thing in the reader’s hand.

The folding also wrote a lettering. Every crease it made is a mountain or a valley, decided by which way the paper went, and the resulting string is one of the many the pattern admits. Being in one is not the same as being able to leave it, and the machinery for asking which letterings a pattern’s own conditions let a hand walk to is now sitting there ready to be pointed at a pattern nobody designed.

What grows, and how fast #

Crumple a square by taking hold of the whole stack and folding it about a random line, over and over, and count what comes out.

Two sheets at each depth, from two streams. The pairs diverge widely — a fold that lands near an edge marks little and one through the middle of the stack marks a great deal — and the trend through them is unambiguous.

A sheet folded twice has no buried creases at all. Everything it has made runs out to a raw edge, and its letterings are one connected set: whatever it did could be undone by pushing on it.

A sheet folded seven times has forty. Two to the fortieth is about a million million, and the sheet is in one of them.

Not one change touches them #

The count is a prediction from the graph. The check is a census of the changes the sheet actually admits.

Take the crumple’s own lettering. For every pair of creases that meet at an interior vertex, flip both and ask whether the pattern still satisfies every condition. On a six-fold crumple from one stream there are seventy-two such pairs and nine of them survive; from another, seventy-eight pairs and twelve survive; from a third, a hundred and forty-four pairs and five survive.

Not one of the twenty-six surviving changes alters a buried crease. Not on any of the three sheets.

And a walk confirms it over a longer baseline: from the sheet’s own lettering, take a surviving change at random, then another from wherever that landed, sixty times. The walk visits between sixteen and fifty-four distinct letterings, and the letters on the buried creases at the end are the letters they had at the start, on every sheet.

It is not stuck, it is enclosed #

The distinction that makes this a finding rather than a triviality is that the sheet has changes available to it.

Nine surviving moves is nine things a hand could genuinely do: push here and the mountain becomes a valley, push there and another pair swaps. The paper is not rigid and it is not saturated. What it cannot do is get anywhere — every route out of the piece it is in passes through a lettering that fails a condition at a vertex, which is to say through an arrangement of paper that will not lie flat.

That is a physical statement and it is testable at the table. Take a crumpled sheet, flatten it, and try to persuade one region to sit the other way. It goes, locally. Then try to persuade the whole thing into a different crumple without opening it out. It does not, and the reason is not that the paper is tired.

Why a crumple has so many buried creases #

A designed pattern’s creases are drawn to do something, and most of them do it by running from one place to another — which usually means from an edge to somewhere, or from somewhere to an edge. A crumple’s are not drawn at all.

Each fold of a crumple is a straight line across the whole stack, so it marks every layer it crosses, and the marks land wherever the previous folds put the paper. After a few folds the sheet is a dense arrangement of segments crossing each other in the interior, and a segment that crosses two others has an interior vertex at each end by construction. Every facet is a layer counts the same density from the other side.

The ratio bears that out. On the printed shelf the buried share runs from zero to 58 per cent of the creases; on a six-fold crumple it is 50 to 67 per cent, and on a seven-fold one 61 to 68. A crumple is more thoroughly committed than anything on the shelf, at a fraction of the crease count.

The count that a hand could check #

The prediction is a number read off a drawing, and a reader with a flattened crumple can check it without any of the machinery.

Flatten the sheet. Take a crease — one line segment between two crossings, not a whole fold line. Follow it to each of its two ends. If both ends are places where other creases cross, that crease is buried and its letter is settled. If either end is on the edge of the paper, it is not.

Count the buried ones. The number of mutually unreachable arrangements the sheet’s letters could have taken is two to that count, and the sheet is in one of them.

On a sheet folded three or four times the count is small enough to do in a minute, and the answer will be somewhere between zero and a dozen. On a sheet folded eight times it is not, and that is the point: the quantity is trivially computable and astronomically large, which is the combination that makes it worth computing rather than estimating.

The count without tracing anything #

The procedure offered for a reader with a flattened sheet is to follow every crease to both of its ends, and on a sheet folded more than four or five times that is a great deal of tracing. There is an identity that removes it.

Every interior vertex of a crumple has degree four — its creases are straight segments crossing, and two straight lines cross in twos. So counting the crease-ends that arrive at interior vertices gives $4V$. Each buried crease contributes two of those ends and each crease with one interior end contributes one, so

$$2B + H = 4V,$$

writing $B$ for the buried creases and $H$ for the half-buried ones. And every crease is buried, half-buried, or has neither end interior, so $B + H + N = C$. Eliminate $H$ between the two and

$$B = 4V - C + N.$$

On a crumple $N$ is nought or one — a crease with neither end at a crossing is a fold that marked a layer nothing else had touched, which happens at most once or twice on a sheet with any depth to it. So to within one:

$$B \approx 4V - C.$$

Which the table confirms exactly #

Put the table’s own numbers in. Twelve interior vertices and thirty-two creases gives forty-eight less thirty-two, which is sixteen — the measured figure. Nine and twenty-four gives twelve, measured twelve. Ten and twenty-seven gives thirteen, measured thirteen. Five and sixteen gives four, measured four. Three and ten gives two, measured two.

The two deepest sheets are one out — twenty-four vertices and fifty-eight creases predicts thirty-eight against a measured thirty-nine, and twenty-seven and sixty-five predicts forty-three against forty-four — which is exactly the $N = 1$ case, one crease on each of those sheets running between two rim points without crossing anything.

So a reader with a flattened crumple does not have to trace a single crease. Count the crossings, count the segments, and the buried creases are four times the first less the second. Both counts are things a person can do by eye on a lightly crumpled sheet and a scanner could do on any of them.

That also makes the piece count directly readable: two to the power of four times the crossings less the segments. Twelve crossings and thirty-two segments is two to the sixteenth, which is sixty-five thousand five hundred and thirty-six arrangements a sheet folded six times has chosen among and cannot revisit.

The sheet that folded itself is the only certified one #

There is a reason a crumple is a useful object to point this instrument at, and it is not that crumples are interesting in themselves.

Every other pattern on this site is a candidate. Its letterings satisfy the conditions at every vertex, deciding the global question is out of reach, and the essays say so on every page. A crumple is different: its own lettering is a folding, because the folding happened, and the paper is the certificate.

So a crumpled sheet is this repository’s only source of patterns that are certified flat-foldable rather than merely admissible — which is what makes it the control that the site’s random-pattern measurements are compared against. And it turns out the certificate comes with a constraint attached: the certified lettering is enclosed, so the certificate does not transfer to any of the pattern’s other letterings.

That is a slightly melancholy conclusion for anybody hoping to bootstrap from it. A crumple hands over one guaranteed folding of a pattern and no route from that folding to any other, which is exactly the situation the subject is in generally.

What this says about the sheet in the hand #

Three things, and the second is the one that changes how the object is described.

A crumple is a decision procedure that has already run. The letters were not chosen by anyone and they were not arbitrary either: each one was determined by which side of the stack the paper was on when the fold was made, and the sequence of folds is a sequence of decisions that the flattened sheet records without recording their order — which is the half no notation records arriving in a pattern nobody wrote down.

“Disordered” is the wrong word for it. A crumpled sheet is usually described as random, and its crease pattern genuinely is — the angles and positions have no structure anybody imposed. The lettering is not random in the sense that matters: it is one point of a space with a million million pieces, it satisfies every condition, and it cannot be moved.

And the sheet remembers in two ways. The paper remembers a crease physically — the fibres are damaged and the fold reopens where it was — and that is a material fact this site has measured. This is a second and completely independent memory: the sheet’s letters cannot be changed even on a perfectly elastic, perfectly ideal sheet with no material memory at all, because the obstruction is combinatorial.

The lettering the crumple could not have written #

There is a converse worth stating because it bounds the result.

The sheet is in one piece of the space. The other pieces are perfectly good letterings of the same pattern — they satisfy every condition at every vertex — and each of them is a folded state some other crumple could have arrived at with the same creases in the same places.

So the pattern does not determine the folding. It admits a vast number of them, the sheet realised one, and the realised one is not distinguishable from the others by any local test. Which is the inverse problem in a new guise: a flattened crumple hands over the creases and the letters, and there is no way to read back the sequence of folds that produced them.

Two seeds that behave completely differently #

The pairs in the table diverge by a factor of two and it would be easy to read that as noise. It is not, and what it is says something about crumpling.

At six folds one stream produced thirty-two creases and sixteen buried; another produced fifty-eight and thirty-nine. Same number of folds, same procedure, and one sheet is nearly twice the other on both counts.

What differs is where the folds landed. A fold through the middle of a tall stack marks every layer and produces a dense patch of crossings; a fold near an edge of the stack marks a few layers and produces little. The random line is drawn over the bounding box of the current stack, so a sheet that has folded itself small early gets more out of each subsequent fold than one that has not.

That means the number of decisions a crumple has taken is not a function of how many times it was folded. It is a function of the folding, and two sheets crumpled the same number of times can be committed to very different degrees. Which is a familiar shape of statement in this subject — the same is true of the layer count and of the shrink — and it is the reason every table here carries two sheets rather than an average.

What the crumple is not asked #

Two things are held back deliberately.

Whether the pattern folds globally. Every condition here is local, and the global question is NP-hard; the crumple’s own lettering demonstrably folds because the paper did it, and the other letterings in the space are candidates rather than objects. So the piece count is a count of pieces of a candidate set — except for the piece the sheet is in, which contains at least one genuine folding.

And whether the crumple’s lettering is typical. It is a fair question, it was the first one asked here, and the answer is not interesting: the crumple’s mountain-and-valley balance sits comfortably inside the distribution its pattern admits, at every depth measured. Whatever is special about the sheet’s own lettering, it is not the count of mountains.

Against the sheet that was folded on purpose #

The comparison worth making is with a designed pattern of the same size, because it separates what the crumpling did from what the density did.

A six-fold crumple has thirty-two creases, twelve interior vertices and sixteen buried creases. A square twist has twelve creases, four interior vertices and four buried. Scale the twist up to the crumple’s crease count by taking a hexagon twist — eighteen creases, six vertices, six buried — and the ratio is stable at a third.

The crumple’s ratio is a half at six folds and two-thirds at seven. So a crumple is not merely bigger; it is denser in interior vertices per crease, and the buried count follows the vertices rather than the creases.

The mechanism is the one above and it is worth putting in design terms. A designer draws creases that go somewhere, and going somewhere usually means reaching an edge — a pleat runs across the sheet, a bisector runs out from a vertex. A crumple’s folds are chords of whatever shape the stack happens to be, and a chord of a folded stack lands in the interior of the flattened sheet far more often than at its rim.

The one thing a crumple can be talked out of #

The result has an obvious exception and it is worth naming so that a reader does not test it against the wrong thing.

Creases that reach the edge of the paper are negotiable. A six-fold crumple has thirty-two creases of which sixteen are buried, so sixteen run out to a raw edge, and those letters can be changed by hand — which is why the move census finds anything at all.

That is also why a lightly crumpled sheet feels different from a heavily crumpled one. At two folds nothing is buried and the sheet will take any lettering its creases permit. At seven, two-thirds of the letters are settled, and what is left to push on is a rim — the freest paper on any sheet.