The letters a crumple was given

The creases a sheet gives itself are what happens when a square is folded along a random line, then along another, then another: at each step the whole stack is folded, so every layer takes a crease, and unfolding the result leaves a pattern nobody designed. It is developable and flat-foldable at every interior vertex — necessarily, since it was made by folding — and its vertices come in every degree with no two alike.

It also arrives with a lettering. Each crease is a mountain or a valley depending on which way that fold went and how many layers were under it, and the labelling is not a choice: it is a record of what happened.

That labelling has a property worth isolating.

It cannot be wrong #

A pattern’s letters can contradict themselves — each crease says which of the two panels it joins lies above the other, and a chain of panels each of which must lie below the next is a proof that no folded state exists.

A crumple’s letters never do. Fifteen sheets folded at three depths from three starting streams, and every one is consistent as it stands.

The reason is not subtle and it is worth stating exactly. A contradiction in the letters is a proof that no ordering of the panels exists. A crumple has an ordering of its panels: the one the paper was in when the folding stopped. So a contradiction would be a proof that the thing on the table cannot exist, and the machinery is not wrong about that.

The guarantee is therefore not a property of the drawing, of the vertex degrees, or of anything a checker reads. It is a property of the pattern’s provenance: this lettering came from a physical stack, and a physical stack is an ordering.

It is worth checking that the machinery could have said otherwise. A test that can only pass is not a test, and this one is asserted rather than reported: the measurement refuses to draw the figure if any crumple’s own letters force a chain, so a change to the folding simulator that produced an impossible stack would fail on the spot rather than quietly reporting a hundred per cent.

The same creases, lettered again #

Which raises the obvious question. Keep the lines and change what is written on them: how many of the other admissible labellings agree with themselves?

Independent labellings come from propagating the vertex conditions to a fixed point and branching where propagation stalls. Every draw passes every condition at every vertex; the draws differ because the decisions do.

At eight panels, forty of forty. At twelve, forty of forty. At twenty-one, thirty-five. At thirty-five, twenty-five. At forty-one, eleven of forty.

So a crumple is not a pattern that is somehow easy to letter. It is a pattern that has been handed the answer.

What separates the two facts #

There are two things a crease pattern can be said to have and they are routinely run together.

A pattern has a drawing: coordinates, segments, angles. That is what determines whether it is developable, whether Kawasaki holds, where the panels land when it is folded, and how many independent closed chains its panels form. All of that is settled before anyone writes a letter.

A pattern has a labelling, and the labelling is what decides whether it folds. Two sheets with identical drawings and different labellings are one perfectly foldable object and one with no folded state at all, and nothing about the picture distinguishes them.

That distinction is easy to lose because most of this collection’s patterns come with their letters attached and nobody has occasion to separate them. The crumple is where the separation is cleanest, because its letters have an unusually strong warrant and its drawing has none at all — the drawing is whatever the random folds produced.

What a redraw is not #

One thing the redraws are emphatically not: alternative ways the sheet could have been folded.

A folding is a sequence, and the letters it produces are constrained by which layers were where at each step. The relabellings here are constrained only by the four conditions at each vertex, which know nothing about sequences. So most of them do not correspond to any folding at all, and the ones that agree with themselves may still be unreachable by any order of simple folds.

The fold a machine can make is the standing account of that gap: a folded state can exist and be unreachable by any sequence of folds through all the layers. So the shares here bound something from above. Of the labellings that pass every vertex condition, some fraction have consistent letters; of those, some fraction have a folded state; of those, some smaller fraction can actually be folded by hand. The crumple’s own labelling is in the innermost of those sets by construction, and it is one point.

Why the share falls #

The crumples fall along the same axis everything else here does: the number of independent closed chains their panels form, which is edges less nodes plus pieces on the panel graph and equals the count of interior vertices.

A crumple at twenty-three to twenty-nine chains is twenty-eight to sixty-eight per cent consistent. A Miura at twenty chains is eighty-one per cent and a Yoshimura at twenty-two is ninety-five. So an irregular pattern does worse than a regular one with the same room, and the crumple is the most irregular pattern this collection has.

The decision a crumple has taken is the account of how much a crumple’s own state settles: each fold in the sequence removes freedom, and by the end almost nothing about the sheet is undetermined. The share above is the complement of that — how much of what remains admissible would have worked.

The penalty, per chain #

The comparison between a crumple and a Miura is made at different chain counts — twenty-nine against twenty — so it mixes two effects: having more chains to fail in, and failing more often in each. Dividing them apart takes one step and it makes the irregularity penalty a number.

If the chains were independent, a pattern with $c$ of them would be consistent with probability $p^{c}$ for some per-chain rate $p$. Take the $c$-th root of each measured share and the rate falls out.

A crumple at twenty-nine chains is consistent in eleven draws of forty, which gives a per-chain rate of 0.957. One at thirteen chains is consistent in twenty-five of forty, giving 0.965. The two agree to within a per cent, which is what a per-chain rate is supposed to do across different sizes.

Now the designed patterns. A Miura at twenty chains is eighty-one per cent consistent, for a rate of 0.990. A Yoshimura at twenty-two chains is ninety-five per cent, for 0.998.

A crumple’s chain fails about four times as often as a Miura’s — four and a bit per cent against one — and about twenty times as often as a Yoshimura’s.

Which separates the two things the raw shares confuse #

That is a better statement of the irregularity penalty than the raw comparison, because the raw comparison would show a difference even if the two patterns’ chains behaved identically.

Read as raw shares, a crumple at twenty-eight per cent and a Miura at eighty-one look like a threefold gap. Most of that gap is the extra chains: twenty-nine against twenty, at any rate below one, costs a substantial factor on its own. The residue — the part that is genuinely about the chains being worse rather than more numerous — is the four-to-one in the rate.

It also says the trend and the seed-to-seed spread really are one effect, as the essay claims. The three seeds differ in chain count and their rates come out within a per cent of each other; the shares differ because the exponents do. A quantity that is constant across seeds and across depths, with the variation living entirely in the exponent, is a quantity that has been correctly separated from the thing it was tangled with.

Where the redraws go wrong #

The contradictions the redraws produce are short. Across every family measured here the commonest lengths are six panels and eight, and the crumples contribute heavily to the six-panel row.

Six panels is two adjacent interior vertices — the smallest combination of chains there is, since a chain round a single vertex can never be closed. So a crumple fails at its adjacent pairs, and a crumple has a great many of them: its folds cross one another repeatedly, and every crossing is a vertex sitting close to several others.

That is the mechanism behind the irregularity penalty. A Miura’s adjacent pairs are all the same pair repeated, and if that pair does not close in one place it does not close anywhere. A crumple’s adjacent pairs are all different, so each one is an independent draw against the same small probability, and enough independent draws will find the bad case.

What it says about deep folding #

There is a limit worth pointing at, and it is the one thing here that a person folding paper would notice.

A sheet folded eight or ten times is not producing a crease pattern that anybody could have lettered by hand: forty-one panels with twenty-nine chains, at a share of twenty-eight per cent, means that a person copying the pattern and guessing the letters — even guessing well enough to satisfy every vertex — would produce something unfoldable seven times in ten.

That is the practical content of publishing the pattern rather than the sequence. A crease pattern is a complete description of a folded object only if its letters come with it, and for anything past a handful of folds the letters are the hard part — not because they are hard to write down, but because most of the ones that pass every published condition do not work.

The three seeds, and why they differ #

The three curves on the first figure are three starting streams for the random folds, and they do not lie on top of one another — at seven folds they are twenty-seven, eleven and twenty-one of forty. That spread is larger than the sampling error and it is worth saying what it is.

A crumple’s shape is decided by where its folds happened to go. One stream produces folds that pile up near a corner and another spreads them; the first gives more vertices in less paper, the second fewer and better separated. At seven folds the three sheets have thirty-five, forty-one and thirty-nine panels and twenty-three, twenty-nine and twenty-seven chains, and the ordering of their shares follows the ordering of their chain counts exactly.

So the spread is the same effect as the trend, seen sideways: it is not that some crumples are luckier, it is that a random folding produces sheets of different complexity from the same number of folds.

What the machine that made them is #

One detail about the simulator matters for reading any of this, and it is the difference between two ways of folding at random.

The one used here folds the whole stack: at each step every layer under the fold line takes a crease, which is what happens when a person folds a piece of paper in half and then in half again. A second machine folds only part of the stack, which is what happens when paper is crumpled in a fist rather than folded.

The two produce different patterns from the same number of steps, and the boundary between them is measured elsewhere. Everything here is the first machine, so the guarantee — that the letters came from a folding and cannot contradict themselves — is exactly as strong as the simulation is faithful, and it is faithful to folding rather than to crumpling.

What a folding knows that a drawing does not #

Three things follow, in ascending order of how much they generalise.

A crumple is a certificate. Its lettering is a witness that the pattern folds, obtained the only way a witness of that kind can be obtained, which is by folding. Nothing this collection computes has that status: every other lettering here was found by a search or a rule and is a candidate until something checks it.

A construction that derives its letters from a folding cannot produce a contradiction, and one that searches for them can. The mesh solver derives from fold angles and never fails; the tessellation builder propagates conditions and failed for several rounds of work without anybody noticing. That is a real distinction between kinds of construction and it is not usually drawn.

And a population of physically-made patterns is the only outside opinion available. Every test set here was drawn by the same hand as the checkers, and the crumples are the nearest thing to an exception — their vertices were not chosen, their degrees were not chosen, and their letters came from paper rather than from a solver.

An ordering is a certificate and a lettering is not #

There is a distinction here that is worth making sharply, because it explains why the crumple’s guarantee is so much stronger than anything computed.

A lettering is a claim: these creases go this way. It can be checked against the vertex conditions, and it can be checked against the layer relation, and passing both leaves it a candidate — the non-crossing rules may still refuse every ordering of the panels, and finding out costs a search that does not finish past about twenty panels.

An ordering is a certificate: this panel is above that one, all the way through the pile. Given one, every claim about the pattern can be verified in a single pass — each crease checked against its pair, each overlap checked against the rules — and no search is needed. The asymmetry is the ordinary one between finding an answer and checking one.

A crumple hands over an ordering. Not as a list of panel heights, but physically: the sheet is in a stack, and the stack is the certificate. Reading it off is a matter of looking rather than of computing, and it is available for a pattern of any size, which nothing else here is.

That is why the sentence a crumple’s letters cannot contradict themselves is a statement about certificates rather than about paper being clever. It would be equally true of any pattern anybody had actually folded, and this collection’s printed shelf is exactly that — which is why the shelf is consistent as drawn too.

The one thing the crumple cannot settle #

It cannot say whether the typical crease pattern behaves like this, because a crumple is not a typical crease pattern either. It is the output of one particular procedure — fold the whole stack along a random line, repeatedly — and that procedure has its own biases: every crease runs edge to edge of whatever it was folding, the creases accumulate near where the paper was already thick, and the facets that result are layers rather than a free arrangement.

There is one more caution, about the model rather than the paper. This whole account treats a crumple as a flat-foldable crease pattern with an exact ordering, and real crumpled paper is neither: its creases have a radius, its facets are not flat, and it is not folded flat at all but into a bundle with air in it. The simulator here folds an idealised sheet along exact lines, and it is that idealisation the guarantee applies to. Four things that are not true is the standing list.

What it does settle, and settles better than anything else available, is the separation this essay is about. The drawing and the letters are two objects, the guarantee travels with the second, and a pattern arriving with a guarantee has usually been folded rather than computed.