Origami & Folding
what one sheet can do
Curves and material

The crease that stops in the middle

A sheet folded flat at random writes a crease pattern that satisfies every condition in the subject, everywhere. Leave one layer behind on each fold — one layer out of a dozen — and it stops writing crease patterns at all: the creases stop in the middle of the paper, and a crease with a loose end is a thing no flat folded sheet can have.

Assumes The creases a sheet gives itself.

19 min read 8 figures Paper is not idealThe pattern is the object

A sheet that has been folded flat is flat folded. That tautology carries a real measurement behind it: fold a square nine times at random, through the whole stack each time, and open it out, and the crease pattern left behind satisfies developability and Kawasaki at every interior vertex — while a sheet with the same number of creases drawn on it at random satisfies them essentially nowhere.

The condition that makes it work is hiding in the phrase through the whole stack.

Folded at random, and drawn at randomLeft, the creases a square is left with after eight folds along randomly chosen lines, unfolded. Right, the same number of creases drawn on an uncreased square at random. The two patterns are equally disorderly and their vertices are nothing alike: every vertex of the folded sheet satisfies the flat-folding condition and almost none of the drawn one does.folded 9 times, then unfolded39 interior vertices, all of degree 439 of 39 satisfy Kawasakithe folding is the reason, not the drawing54 creases drawn at random726 interior vertices, all of degree 40 of 726 satisfy Kawasakisame count, same sheet, nothing folded
Fig. 1 The pattern a sheet gives itself, next to the same number of lines drawn at random. The one on the left came from folding and passes every condition; the one on the right was drawn and fails at almost every vertex.

The machine that folds part of the stack

The subject already distinguishes machines by what they are allowed to fold. A machine that must take the whole stack is weaker in some ways and stronger in others than one that may pick out some layers, and the comparison between them is a small lattice of models with a clean answer.

The crumple is the same distinction applied to a random folder rather than a purposeful one. On each fold, take the top some of the layers and turn them over the line; leave the rest lying where they are. Everything else is identical — the same random lines, the same square, the same number of folds.

What each machine can reachFor three one-dimensional crease patterns, the number of mountain-and-valley assignments that fold flat at all, and the number each kind of folding machine can actually reach. Every bar is a separate search: the top one over stackings, the next three over sequences of folds, the last over rewritings of the segment lengths.evenly spaced — 3 creasesany flat folding8 of 8some-layers8 of 8all-layers8 of 8one-layer2 of 8crimping only0 of 8one short segment — 3 creasesany flat folding4 of 8some-layers4 of 8all-layers0 of 8one-layer2 of 8crimping only0 of 8a machine that takes fewer layers is weaker, not more patientthe paper is joined, so what it declines to hold it also cannot move
Fig. 2 The machine models this subject already separates, and what each can reach. Which layers a machine may take is a real distinction with real consequences, and this essay is about a consequence nobody had asked for.

What comes back is not a crease pattern.

Creases with loose ends

A fold through the whole stack leaves a crease that runs from one side of whatever it crossed to the other. A fold through part of the stack leaves a crease that runs only as far as the part that moved — and the part that moved has a boundary inside the sheet, so the crease stops there.

Take the record without tidying it up and count. Eight random folds through half the stack leave twelve creases with a loose end inside the paper. The same eight folds through the whole stack leave none.

One layer left behind, and it stops being a crease patternLeft, the record of a folding that took part of the stack each time, with the interior vertices of odd degree marked. Right, the same folds through the whole stack, which has none. A crease that stops inside the paper makes a vertex an odd number of creases meet at, and no flat folded sheet has one.50% of the stack, 9 foldsthe whole stack, 9 folds13 interior vertices of odd degree13 creases with a loose end before planarisingrefused by the first condition it is put pastno odd vertex anywhereno loose end anywhereand every vertex satisfies Kawasaki
Fig. 3 The record of a folding that took half the stack each time, beside the same folds through all of it. The marked vertices on the left are the ones where an odd number of creases meets — the loose ends, after the pattern has been tidied so that every crossing is a vertex.
Folded at random, and drawn at randomLeft, the creases a square is left with after eight folds along randomly chosen lines, unfolded. Right, the same number of creases drawn on an uncreased square at random. The two patterns are equally disorderly and their vertices are nothing alike: every vertex of the folded sheet satisfies the flat-folding condition and almost none of the drawn one does.folded 8 times, then unfolded33 interior vertices, all of degree 433 of 33 satisfy Kawasakithe folding is the reason, not the drawing45 creases drawn at random485 interior vertices, all of degree 40 of 485 satisfy Kawasakisame count, same sheet, nothing folded
Fig. 4 Creases with loose ends, drawn beside the folding that made them. Every crease here came from an all-layers fold and every one runs edge to edge; the ones that stop in the middle are what a partial fold would have written instead.

A crease with a loose end is not a small irregularity. It is an interior vertex of degree one, and a vertex of degree one cannot satisfy Maekawa’s demand that the two counts differ by two — one crease cannot be split into two counts differing by anything. It is the same impossibility as three creases meeting at a point, at the smallest possible degree.

What the tidying up does to it

There is a subtlety here that is more interesting than the loose end itself, and it took a planariser to find.

Before anything is tidied, the partial fold’s crease genuinely dangles. But a crease that stops in the middle of a sheet usually stops on another crease — because the part of the stack that moved is bounded by earlier folds — and once the pattern is planarised, so that every crossing becomes a vertex, the dangling end is no longer a vertex of degree one. It is a vertex of degree three.

Which is a fine outcome for the argument, because degree three is the impossibility this subject established first and understands best. The partial fold’s record fails not by producing something exotic but by producing the oldest forbidden thing there is.

The boundary is at every layer

The obvious guess is that this is a matter of degree — that folding most of the stack gives mostly a crease pattern, and the trouble accumulates as more layers are left behind. It is not.

Fold nine times, leaving out one tenth of the stack — which for most of the run means leaving one layer behind out of nine or ten. Across five seeds the record has 13, 4, 14, 4 and 8 odd-degree interior vertices. Across the same five seeds with the whole stack taken every time, it has 0, 0, 0, 0 and 0.

Folded at random, and drawn at randomLeft, the creases a square is left with after eight folds along randomly chosen lines, unfolded. Right, the same number of creases drawn on an uncreased square at random. The two patterns are equally disorderly and their vertices are nothing alike: every vertex of the folded sheet satisfies the flat-folding condition and almost none of the drawn one does.folded 10 times, then unfolded64 interior vertices, all of degree 464 of 64 satisfy Kawasakithe folding is the reason, not the drawing82 creases drawn at random2019 interior vertices, all of degree 40 of 2019 satisfy Kawasakisame count, same sheet, nothing folded
Fig. 5 The boundary is at every layer, on a deeper sheet: ten folds, and the creases whose ends stop in the middle of the paper marked. A fold that takes part of the stack leaves one of these at every layer it did not take.
One layer left behind, and it stops being a crease patternLeft, the record of a folding that took part of the stack each time, with the interior vertices of odd degree marked. Right, the same folds through the whole stack, which has none. A crease that stops inside the paper makes a vertex an odd number of creases meet at, and no flat folded sheet has one.90% of the stack, 9 foldsthe whole stack, 9 folds14 interior vertices of odd degree14 creases with a loose end before planarisingrefused by the first condition it is put pastno odd vertex anywhereno loose end anywhereand every vertex satisfies Kawasaki
Fig. 6 Nine folds that left a tenth of the stack behind, beside the same nine through the whole of it. The difference between the two pictures is one layer, and it is the difference between a crease pattern and a record of something that happened.

One layer. The boundary is not somewhere in the middle of the range; it is at the very top of it, and everything below the top behaves the same way. Leaving three quarters of the stack behind gives between three and eighteen odd vertices, and leaving a tenth behind gives between four and fourteen — the same order of magnitude, with no trend that survives the seed-to-seed variation.

That is the sharp form of the tautology this ladder began with. A sheet that has been folded flat is flat folded — and “folded flat” has to mean the whole of it. A stack with one layer left standing was never flat.

There is one more comparison worth making, because it separates two things that could be confused. A partial fold is not a sloppy fold. Sloppiness would put the creases in slightly the wrong places, which is a perturbation and has a size; the site has measured what that costs and the answer is that a pattern jittered by a fortieth of a cell sits less than a degree from folding. A partial fold puts every crease exactly where the geometry says, and produces an object that is not in the space of crease patterns at all.

One layer left behind, and it stops being a crease patternLeft, the record of a folding that took part of the stack each time, with the interior vertices of odd degree marked. Right, the same folds through the whole stack, which has none. A crease that stops inside the paper makes a vertex an odd number of creases meet at, and no flat folded sheet has one.34% of the stack, 6 foldsthe whole stack, 6 folds2 interior vertices of odd degree2 creases with a loose end before planarisingrefused by the first condition it is put pastno odd vertex anywhereno loose end anywhereand every vertex satisfies Kawasaki
Fig. 7 A third of the stack, six folds. Fewer creases than the all-layers version — because the moving part is smaller and crosses less — and more of them wrong.

What the checker did, and what it should have done

The machinery here refuses the partial fold’s record, and the refusal comes out of the first condition it tests: an odd number of creases meets at a vertex, so Kawasaki’s two alternating sums have an odd number of sectors to alternate over and the test is not even well posed.

That is exactly right and it is worth saying why, because there was an obvious alternative that would have been wrong. A checker could have repaired the record — extended the dangling crease to the sheet’s edge, say, or deleted it — and returned a pattern. It would have been a pattern of something nobody folded.

The subject has been caught by that shape of error before, from the other side: a search whose budget ran out once reported “no assignment exists” for a pattern that had demonstrably been folded, because exhaustion looked like a result. Repairing an input looks like helpfulness and is the same failure in the opposite direction — an answer about an object that was not the one asked about.

What it means for a machine

The subject’s machine models were built to answer a question about reachability: which folded states a machine of a given kind can arrive at. The measurement here answers a different question about the same machines, and it is one nobody had asked.

A machine that folds all the layers is the only one whose history is legible. Whatever it did, the sheet it hands back records it as a crease pattern, and every condition in the subject applies to that pattern and holds. A machine that folds some of the layers is more capable — the models are ordered, and the one that may choose reaches more — and the sheets it hands back are not patterns.

There is a moral in that for anybody instrumenting a folding process. If the record is going to be checked against the theorems — and the whole value of a crease pattern is that it can be — then the process has to take every layer, or the record has to be treated as something else and checked some other way.

Why the all-layers fold is the special one

It is worth having the reason rather than the measurement, because the reason is short.

At an interior vertex of a flat folded sheet, the paper goes round the vertex once and comes back to itself, crossing every crease that meets there. Crossing a crease turns the sheet over. So the number of crossings has to be even, which is the parity behind the two-colouring of the panels, and it is a statement about a walk that closes.

A fold through the whole stack preserves that everywhere: every layer turns together, so at any point of the sheet the walk is unchanged or reversed, and either way it still closes. A fold through part of the stack turns some layers and not others, and at the boundary of the moved part the walk no longer closes — the sheet on one side has been turned and the sheet on the other has not.

Two creases the all-layers machine cannot foldA strip of three segments that folds flat, drawn above the four ways an all-layers machine could begin. Each opening move leaves the remaining crease covered by paper with no crease in it, and a machine that must fold every layer cannot fold through solid paper. All four are shown because four is all there are.segments 0.30 · 0.25 · 0.45, assignment MVMVit folds flat — a stacking exists and the layer-ordering rules find itfold at 0.30, left side overthe other crease now sits under paperthat has no crease therefold at 0.30, right side overthe other crease now sits under paperthat has no crease therefold at 0.55, left side overthe other crease now sits under paperthat has no crease therefold at 0.55, right side overthe other crease now sits under paperthat has no crease there
Fig. 8 The one-dimensional witness for the same distinction: a state a machine taking all the layers can reach and one taking some layers cannot, or the reverse. Which layers move is not a detail of how a fold is performed; it is what the fold is.

The count that does not grow

One more measurement is worth having because it disposes of a hypothesis a reader is likely to form.

The obvious guess is that the trouble accumulates: more folds, more layers left behind, more odd vertices. It does not, in any way that survives the noise. Nine folds leaving a tenth of the stack behind give between four and fourteen odd vertices across five seeds; nine folds leaving two thirds behind give between five and thirteen. Fewer creases are written when less of the stack moves — a partial fold crosses less paper — so the fraction of vertices that are wrong rises while the count does not.

What that says is that the failure is not a dose. It is a property each fold has or does not have, and a folding is spoiled by whichever of its folds had it. A single partial fold in a sequence of otherwise complete ones is enough to leave a record that no checker will accept.

The property each fold has or does not have

The measurement says the trouble is not a dose: a folding is spoiled by whichever of its folds was partial, and leaving one layer behind is as bad as leaving eight. That is the right reading and it leaves the property itself unnamed. It can be named, and naming it weakens the all-layers rule in a way that is worth having.

Follow one fold. The crease it writes on the flat sheet is the fold line pulled back through every layer that moved. On a layer that moved, that crease runs from one side of the layer to the other — and the sides of a layer are either the raw edge of the paper or an earlier crease. On a layer that did not move, no crease is written at all.

So a loose end appears exactly where the fold line passes from paper that moved to paper that did not, at a point that is not on the sheet’s own edge. Nothing else can produce one.

Which gives the condition, and it is not all layers. It is: at every point of the fold line, take all the layers there or none of them. A layer the fold line never crosses can be left lying where it is, at no cost, because the fold writes nothing on it and there is no crease to leave dangling.

That is a genuine weakening rather than a restatement, because in a folded stack a great many layers do not extend across the whole footprint. A sheet folded eight times has layers of every size, and a fold line drawn across it typically crosses some of them and misses others entirely. A machine obeying the weaker rule may leave every missed layer alone and still hand back a crease pattern.

Why taking the top fraction almost never satisfies it

The measurement’s generator takes the top share of the stack, and that is what makes the failure so reliable — reliable enough that leaving a tenth behind fails on every seed.

The top layers of a folded stack are the small ones. They are the paper that has been turned most often, so their boundaries lie deep inside the footprint rather than on the sheet’s raw edge, and a line drawn at random across the footprint crosses those boundaries. Taking the top nine tenths therefore cuts the stack at the edge of a small layer somewhere along almost any line, and each such cut is a loose end.

The one-layer case is the extreme of that and it explains the empty column. Leaving out a single layer is leaving out one region of the footprint, and a random line either misses that region entirely — in which case nothing happens and the fold is harmless — or crosses it, in which case it enters and leaves, and two loose ends appear at once. The counts of four to fourteen odd vertices over nine folds are what that coin-flip looks like when it is tossed nine times against regions that cover most of the sheet.

So the finding survives with a sharper edge on it. A folding whose record is a crease pattern is not one that took every layer; it is one whose every fold took every layer it went through. The all-layers machine satisfies that automatically and is therefore the only machine guaranteed to be legible — but the guarantee comes from a condition on lines and layers rather than from taking everything, and a machine that checked the condition could be strictly more capable at no cost to its record.

Why the loose end is not the diagnosis

There is a subtlety in how the failure is reported, and it is worth getting right because the obvious description is slightly wrong.

The obvious description is: the record has a crease with a loose end, and a loose end is a degree-one interior vertex, and Maekawa cannot be satisfied at one. That is true of the raw record and it is not what the checker sees. By the time the pattern has been tidied — every crossing made into a vertex, which is what any pattern needs before it can be checked at all — the loose end has usually landed on another crease, and what it makes is a vertex of degree three.

So the fault is reported as odd degree rather than as a loose end, and that is the better diagnosis: it is the same impossibility three creases meeting at a point established, and it is checked by a condition the subject already had rather than by a special case written for this. A record that fails an old test is more convincing than one that fails a new one.

Where the model stops

The simulation is exact and the paper is not. Every fold here is a perfect reflection of a set of polygons about a line, with no thickness, no radius at the crease and no slip between layers. Real paper has a crease radius and a real partial fold would leave a smeared boundary rather than a point.

Random folds are not what a crumpling sheet does. This is a machine model, not a model of crumpling: a sheet crushed in the hand does not choose lines at random through its whole stack, and nothing here claims it does.

The share of the stack is measured from the top. Taking the top fraction is one way to fold part of a stack and there are others — alternate layers, a middle band — and the counts above would differ. What would not differ is that any of them leaves a boundary inside the sheet.

Odd vertices are counted, not classified. The record’s vertices are refused because their degree is odd; whether the even vertices in a partial fold’s record satisfy the angle conditions is not measured here, and there is no reason to expect them to.

One consequence of that is worth stating for anybody who folds by hand. Nobody folds by hand through part of a stack on purpose — but everybody does it by accident, when a layer slips out from under the thumb on a thick fold. The result is not a slightly worse fold; it is a sheet whose record has a crease that goes nowhere. The visible symptom is a short scored line that stops abruptly in open paper, and it is worth recognising, because it means the fold has to be redone rather than eased.

Where the ladder goes next

The natural continuation is the repair question. Given a partial fold’s record, what is the nearest thing to it that is a crease pattern — and does the answer look like anything a folder would recognise? That is an approximation problem with a real object at the end of it, and the machinery to state it exists.

The other direction is the crumple this ladder keeps not doing: the one with a material in it. Everything measured on this site about crumpling has been geometry — counts of creases, facets and layers — and the reason a real sheet crumples the way it does is about where energy goes, which is somebody else’s subject and is named here rather than borrowed.

Where this goes next

folded 8 times, then unfolded33 interior vertices, all of degree 433 of 33 satisfy Kawasakithe folding is the reason, not the drawing45 creases drawn at random485 interior vertices, all of degree 40 of 485 satisfy Kawasakisame count, same sheet, nothing folded The creases a sheet gives itself 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. interior vertices, by number of creases meeting therenone3odd594evennone5odd326evennone7odd18even8 patterns, 92 interior vertices, and not one of them with an odd number of creases Nothing meets at three Every interior vertex of a flat-foldable pattern carries an even number of creases and at least four. So a crease cannot stop in the middle of the sheet, three creases cannot meet anywhere, and every crease pattern in the subject ends up looking the same way — all crossings and no stars. creases at 0.40, 0.50 — assignment MVflat1 layerno sequence of all-layers folds finishes this pattern — the search exhausted 5 states The fold a machine can make A theorem that says a folded state exists says nothing about getting there. A machine that folds every layer at once is stopped by a strip with two creases in it — one that folds flat perfectly well, and that a pair of hands folds in about four seconds.

Named alongside this one

Essays reaching for the same objects. Nobody chose these; they are what the concept index makes visible.

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Every essay whose body links to this one.

The objects this essay names

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The all-layers simple foldCrease patternCrumplingIdealisationKawasaki's theoremThe machine modelParityVertex degree