Origami & Folding
what one sheet can do
Curves and material

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.

Assumes Almost every pattern fails.

18 min read 8 figures Paper is not idealFlat is rare

Every crease pattern elsewhere on this site was designed. A twist, a Miura, a molecule, a circle packing turned into creases — somebody or something chose where the lines went, and the choosing was constrained by the theorems the pattern then satisfies.

There is another way to get a crease pattern, and everybody has produced one: crumple a sheet of paper, then flatten it out and look at what is on it.

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. 1 Left, 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 are equally disorderly to look at and their vertices are nothing alike.

The obvious prediction is that such a pattern is hopeless. Kawasaki’s condition is one equation per interior vertex and a drawing satisfies an equation with probability zero; this site has measured that and the failure is total. So a crumpled sheet, which looks like the most arbitrary crease pattern anybody could produce, ought to fail everywhere.

It fails nowhere. Not approximately, not usually — nowhere, at every vertex, on every seed.

Two random processes, and only one of them is random about lines

The reason is a tautology, and the tautology is worth stating slowly because it is what the prediction missed.

A sheet that has been folded flat is flat folded. Every vertex it carries is a vertex of an object lying in a plane, so every local condition that a flat folded state must satisfy is satisfied there — not because the folds were chosen well, but because there is nothing else for them to be.

So random is doing two completely different jobs in the two experiments. Drawing lines at random puts randomness in the crease pattern directly, and the pattern is then asked whether it folds. Folding at random puts the randomness in the process, and the pattern that comes out is whatever a folded sheet happens to have on it. The first is a sample from the space of drawings. The second is a sample from the space of foldings, and every point of that space folds.

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 6 times, then unfolded24 interior vertices, all of degree 424 of 24 satisfy Kawasakithe folding is the reason, not the drawing34 creases drawn at random341 interior vertices, all of degree 40 of 341 satisfy Kawasakisame count, same sheet, nothing folded
Fig. 2 Two random processes, and only one of them is random in the way that matters. A drawn vertex satisfies the conditions essentially never; this sheet’s vertices were not drawn, and every one of them satisfies everything.

The rule, and what it is a model of

The simulation is deliberately plain. Start with a square. Pick a line at random, fold every layer that crosses it, and repeat. Unfold at the end and the creases left behind are the pattern.

That move — turn everything on one side of a line through 180° — is the all-layers simple fold, which this site already decides for strips as a question about machines. Here it is run in two dimensions on a stack of polygons, because a crumple is a plane object and a strip cannot be crumpled.

Each layer carries the piece of the sheet it is, and the isometry that has brought it to where it now lies. So the crease a fold leaves is found by intersecting the fold line with the layer where it currently is, and carrying that intersection back through the isometry — which is why a crease made on the fourth fold can land anywhere at all on the sheet, including across an earlier one, exactly as it does on paper.

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 6 times, then unfolded11 interior vertices, all of degree 411 of 11 satisfy Kawasakithe folding is the reason, not the drawing
a sheet folded at random — sheet 150×150 mm — 15 mountain, 17 valley, 1124.71 mm of crease
Fig. 3 A sheet after six folds, alone. Thirteen interior vertices, every one of degree four, every one satisfying every condition. The pattern is drawn from the creases the simulation recorded, planarised so that crossings become vertices, and then checked as though somebody else had drawn it.

It is a model of a crumple and not a crumple. A real crumpled sheet is a three-dimensional object with curved ridges and conical points, and its creases are not made one at a time by a person choosing lines. What this shares with it is the thing being claimed: the creases were produced by folding, and the claim is about what that does to a crease pattern. Paper is not the ideal sheet in several ways that matter here and are named at the end.

Every vertex, and the numbers behind that word

Five seeds, six folds each, forty-eight interior vertices between them: forty-eight satisfy Kawasaki. At eight folds, at ten, at twelve — the same. The worst alternating-sum residual anywhere in the sweep is at the level of arithmetic noise.

The drawn control is measured the same way with the same routines: at sixteen lines it has 359 interior vertices and none of them satisfies Kawasaki; at twenty-four lines, 797 vertices and none. Not one, ever, at any size.

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. 4 Every vertex, and the numbers behind that word: ten folds, eighty-two creases, and every interior vertex satisfying every condition. Nothing checked them as they were made; the making is what guarantees them.

The pattern is not nearly foldable — it is exactly foldable

One more distinction deserves stating, because “satisfies Kawasaki” can mean two rather different things and only the strong one is true here.

A pattern can satisfy a condition within a tolerance, which is what a measurement of a physical object gives and what a fitted construction gives. The crumple’s vertices do not satisfy Kawasaki within a tolerance; they satisfy it the way a reflected angle equals the angle it was reflected from. Each fold is an isometry, the sectors on the two sides of a crease are images of one another, and the alternating sum of the four sectors at a crossing is a difference of quantities that are equal by construction.

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. 5 The pattern is not nearly foldable — it is exactly foldable, and here is a deeper one saying so: eight folds, with the sheet beside the pattern it wrote. Every vertex of it closes exactly, because a fold made it.

That is why the claim survives at every fold count without degrading. A construction that satisfied a condition approximately would drift as the folds accumulated and the residual would grow with depth; there is nothing to drift.

The letters were not searched for either

The stronger half of the claim is about the mountain-valley assignment, and it needs care, because it would be easy to cheat here without noticing.

Kawasaki is a condition on angles alone, so it can be checked without deciding anything. Maekawa and the big-little-big lemma need letters. The tempting move is to search for letters that work and report success — which would be a weaker claim, since a search finding something is a fact about the search.

The letters are not searched for. Every layer in the simulation is either showing its front face upward or its back, which is the sign of the isometry that brought it there, and the whole stack turns the same way when it folds. So the letter each fold writes on each layer is decided by that sign, and it is written down as the fold happens. Those letters — the folding’s own, not a solver’s — pass Maekawa and the big-little-big lemma at every vertex of every sheet in the sweep.

A vertex that folds flatFour creases at one point, with the sectors between them measured and both flat-folding conditions evaluated. Kawasaki constrains the angles and Maekawa constrains the assignment; a vertex needs both, and they are independent of one another.VMMM60°90°120°90°Kawasaki60° + 120° = 180°90° + 90° = 180°both 180° — satisfiedMaekawa3 mountains, 1 valleysdifference 2exactly 2 — satisfiedangles sum to 360°which is what a flat sheet requiresmountainvalley
Fig. 6 The four conditions at a single vertex, on a pattern built to satisfy them. A crumpled sheet’s vertices pass all four with the letters the folding gave them, and nothing anywhere in the simulation consults a condition.

That distinction earned itself. An earlier version of this work did use a solver, and on some sheets the solver reported that no assignment existed — for patterns that had demonstrably been folded flat, since folding them is how they were made. The solver was wrong: its search budget was shared across the whole recursion, so a long search exhausted the budget and reported the exhaustion as a contradiction. A negative answer from a search looks exactly like a result, which is a lesson this site has now learned twice.

Even degrees, which nothing asked for

There is a second witness, and it is independent of the first.

Every interior vertex of every crumpled sheet in the sweep has degree four. Not merely an even degree — four, throughout. Nothing in the simulation enforces parity, checks a degree, or knows that an interior vertex of a flat-foldable pattern must have an even number of creases and at least four. It falls out.

The drawn control has degree four everywhere as well, which is the point: degree is not what distinguishes them. Straight lines drawn across a square cross in pairs and make degree-four vertices as reliably as folds do. The two populations of vertex are identical in their combinatorics and completely different in their angles, and only the second is visible to a theorem.

Why the drawn control fails, exactly

The control fails at every vertex and the essay reports that as a measurement. It has a reason, it is two lines long, and having it makes the contrast sharp rather than merely large.

Two straight lines crossing at a point cut the neighbourhood into four sectors, and opposite sectors are equal: the sizes read aa, 180°−a180° - a, aa, 180°−a180° - a round the point. Kawasaki asks the alternating sum to vanish, which here is 2a−(360°−2a)2a - (360° - 2a), and that is nought only when aa is a right angle.

So a vertex made by two drawn lines satisfies Kawasaki if and only if the lines cross squarely, and lines drawn at random cross squarely with probability zero. The control’s 359 vertices and its 797 do not fail for a statistical reason that might occasionally relent. They fail because every one of them is two straight lines crossing at an angle that is not ninety degrees, and there is no other kind of vertex a drawing of straight lines can produce.

And why the crumple’s vertices are not that

Which raises the question the contrast now demands. If two straight creases crossing can only pass at a right angle, and a crumple’s vertices are crossings, how do all of them pass?

The answer is that a crumple’s creases are not straight lines in the flat sheet. A fold line is straight in the folded stack, and the crease it writes on a layer is that line carried back through whatever isometry brought the layer there — so where a later fold crosses an earlier crease, the two halves of its preimage are related by a reflection in that earlier crease. Unfold the sheet and the later crease has a bend in it, and the bend sits exactly at the earlier crease.

Now redo the arithmetic at such a vertex. One crease runs straight through, so the sectors on either side of it pair up to a straight angle twice over. The other crease arrives at some angle and leaves at another. Writing the four sectors in order, Kawasaki’s demand comes out as: the angle the bent crease leaves at must equal the angle it arrived at, measured on the other side of the straight one.

That is exactly what reflection means. The condition Kawasaki imposes at a crumple’s vertex is the later crease reflects in the earlier one, and reflecting in the earlier one is what folding along it did.

So the tautology at the top of this essay has a local form as well as a global one. The sheet’s vertices pass not merely because the object is flat folded, but because the specific geometric relation Kawasaki demands at a four-crease vertex — a bend that mirrors — is the relation a fold physically performs. A drawing has no reason to produce it and never does; a fold cannot do anything else.

What the crease pattern still does not settle

None of this says the sheet folds flat as a whole. Every condition used is local, deciding the global question is NP-hard, and the fact that these particular patterns did fold flat is known from the construction rather than from any check. What has been established is the converse-shaped statement: a pattern produced by folding satisfies the local conditions everywhere, which no amount of drawing achieves.

The simulation folds cleanly and paper does not. A real fold has a radius, the sheet stretches slightly at every crease, and a crumpled ball is held in three dimensions by ridges that are curved rather than straight. The crease has a radius and that radius is where the model and the material part company.

Every fold here takes the whole stack. A person crumpling paper does not fold every layer at once; they crush a region, and different parts of the sheet do different things. The all-layers model is the strongest of the three machine models and the one furthest from a hand, and a model that folded only some layers would produce a different population of pattern. Whether it would produce one with the same property is a question the same machinery could answer and this essay does not.

A physical crumple is not a sequence of chosen folds. It is a buckling process in which many creases form at once under load, and what a sheet does under load is structural-engineering-statics.com’s subject. Nothing here is a claim about force, energy or why a crumpled ball springs back; every number above is an angle, a length, a count or an area.

The refusal the model has to survive

A simulator that quietly did nothing would pass every claim above. No creases, no vertices, no failures — a perfect score over an empty set, which is the shape of result this site distrusts most.

So the check is run against the two cases it must not survive. One fold leaves a crease and no interior vertex at all, and a census of it must come back empty rather than perfect: there is nothing for a condition to hold at, and reporting that as a pass would be reporting the absence of a test as the result of one. Eight folds must leave a pattern with vertices in it, so that the claim being made has something to be about — thirty-three of them on the seed the figures use.

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 4 times, then unfolded4 interior vertices, all of degree 44 of 4 satisfy Kawasakithe folding is the reason, not the drawing
a sheet folded at random — sheet 150×150 mm — 3 mountain, 9 valley, 571.83 mm of crease
Fig. 7 Four folds, which is about the smallest number that produces a crossing at all. The pattern has four interior vertices; below this the sheet has creases and no vertices, and the census has nothing to report.

Between those two the claim is a claim. Above about ten folds the patterns get dense enough that the interesting quantity stops being whether they pass — they always do — and starts being how the counts grow, which is the next rung.

Why the prediction was wrong, and what kind of wrong it was

The prediction that a crumpled sheet fails everywhere is not a careless one. It follows from a correct result — flat-foldable patterns are of measure zero among drawn ones — applied to the wrong population.

That is a specific and recognisable mistake: taking a genericity statement about one probability distribution and applying it to a sample from another. The set of flat-foldable patterns is tiny inside the set of all patterns and it is everything inside the set of patterns produced by folding. Both facts are about the same set; they differ in what they are a fraction of.

The reason it is worth an essay rather than a footnote is that the intuition it corrects is load-bearing. “Flat-foldability is rare” is one of the more quotable results in the subject, and it gets used to explain things it does not explain — including, occasionally, why crumpled paper is such a mess. Crumpled paper is not a mess in this sense at all. It is a flat-folded object that has been opened out, and every theorem in the subject applies to it exactly.

The pattern, and where its panels landEvery panel of the pattern drawn at the place folding puts it, at the same scale as the pattern itself. The outlines are left in so the layers can be counted; which panel lies above which is a separate question this construction does not answer.the patternthe panels, foldedsheet 12.000footprint 1.966 · 6.11 layers on average · 12 at the deepest1.966 × 6.11 = 12.007, which is the sheet
Fig. 8 Where a designed pattern’s layers land, for contrast. A crumple’s stack has the same kind of structure and much more of it — the layer count runs into the hundreds after a dozen folds — and it is a stack of the same kind of object.

Who has looked at this, and how

Crumpling has a substantial physics literature and almost none of it is about crease patterns. The interest there is in energy: how the energy of a crumpled sheet scales with how hard it is squeezed, how it concentrates on ridges, why a ball of paper is mostly air. Those are questions about a three-dimensional object under load and they are answered with elasticity, not with Kawasaki.

The geometric side has been studied mostly in the other direction — from a crumple’s ridges and conical points toward the surfaces they belong to — and the crease pattern left behind, considered as a crease pattern, is not much looked at. That is understandable. It arrives with no design intent, it repeats nothing, and it teaches nothing about how to fold anything.

What it does carry is a clean statement about the subject’s central condition, available nowhere else: a population of vertices that satisfies it universally and was produced by no one’s care. Everything else on this site satisfies Kawasaki because a generator was written to; a crumple satisfies it because it was folded, which is the only other way there is.

Where the ladder goes next

The natural continuation is what more folding does. Each fold adds creases, and the counts turn out to be tied together exactly: the facets of the flattened sheet and the layers of the folded stack are the same number, always, on every seed — which makes the depth of a crumpled stack readable off its flat pattern without folding anything.

The other direction is the gap between this model and the material. A crumple’s real creases are not sharp, its ridges are curved, and the same region of sheet is worked repeatedly until the fibres give. Each of those is a way the ideal sheet is wrong, and each of them is measurable against a pattern that would otherwise be exactly right.

Where this goes next

-3-2.5-2-1.5-1-8-6-4-20tolerance (log₁₀ radians)fraction inside it (log₁₀)1 vertex · slope 1.002 vertices · slope 2.013 vertices · slope 3.0140,000 random vertices, none of them constructed to fold and none of them folding Almost every pattern fails Kawasaki's condition is one equation for each interior vertex, and a drawing satisfies an equation with probability zero. Every pattern on this site folds because it was constructed to, and the fraction that would fold by accident can be measured. no thicknesslayers add up; a 64-grid model is millimetres thick at the coreno stretchpaper stretches a little, which is why wet-folding works at allcreases are linesa crease has a radius; sharp folds tear and soft ones springperfect memorypaper relaxes, so a model opens slightly the moment it is put downthe theorems are exact statements about a sheet nobody has ever folded Four things that are not true Zero thickness, no stretch, creases that are lines, and perfect memory. Every theorem on this site rests on all four, every one of them is false, and the interesting engineering is exactly where each fails. 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.

What this makes readable

Essays that name this one as a prerequisite.

Named alongside this one

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

What links here

The 8 essays that link to this one and share the most of its objects, of 14 that link here.

The objects this essay names

Each one links to every other essay that touches it.

AssignmentGenericityIdealisationKawasaki's theoremSimple foldVertex degree